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[Anton Bolyachkin](https://orcid.org/0000-0003-0420-1806), Ekaterina Dengina, [Nikita Kulesh](https://orcid.org/0000-0001-7046-2671), [Xin Tang](https://orcid.org/0000-0001-6762-6145), [Hossein Sepehri-Amin](https://orcid.org/0000-0002-7856-7897), [Tadakatsu Ohkubo](https://orcid.org/0000-0003-3548-1951), [Kazuhiro Hono](https://orcid.org/0000-0001-7367-0193)

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[Tomography-based digital twin of Nd-Fe-B permanent magnets](https://mdr.nims.go.jp/datasets/abf10a43-880c-4713-88f1-785c040ea7f4)

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Tomography-based digital twin of Nd-Fe-B permanent magnetsnpj | computational materials ArticlePublished in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Scienceshttps://doi.org/10.1038/s41524-024-01218-5Tomography-baseddigital twinofNd-Fe-Bpermanent magnetsCheck for updatesAnton Bolyachkin 1,2 , Ekaterina Dengina1,3, Nikita Kulesh 1, Xin Tang1,2, Hossein Sepehri-Amin 1,3,Tadakatsu Ohkubo 1 & Kazuhiro Hono1,3Many functional materials have been designed at the multiscale level. To properly simulate theirphysical properties, large and sophisticated computer models that can replicate microstructuralfeatures with nanometer-scale accuracy are required. This is the case for permanent magnets, whichexhibit a long-standing problem of a significant offset between the simulated and experimentalcoercivities. To overcome this problem and resolve the Brown paradox, we propose an approach toconstruct large-scale finite element models based on the tomographic data from scanning electronmicroscopy. Our approach reconstructs a polycrystalline microstructure with actual shape, size, andpacking of the grains as well as the individual regions of thin intergranular phase separated by triplejunctions. Such amicromagneticmodel can reproduce the experimental coercivity of ultrafine-grainedNd-Fe-B magnets along with its mechanism according to the angular dependence of coercivity.Furthermore, a remarkable role of thin triple junctions as nucleation centers for magnetization reversalis revealed. The developed digital twins of Nd-Fe-B permanent magnets can assist their optimizationtoward the ultimate coercivity, while the proposed tomography-based approach can be applied to awide range of polycrystalline materials.Permanent magnets are essential functional components of motors andgenerators in modern electric/hybrid vehicles, robotic systems, wind tur-bines, and other applications aimed at achieving carbon neutrality1. In thishigh-tech sector, Nd-Fe-B magnets are dominant owing to their superiorperformance at room temperature. After the decades of research since theNd2Fe14B compound was discovered2,3, the maximum energy product ofNd-Fe-B magnets has almost reached its theoretical limit. However, thecoercivity of industrial Nd-Fe-Bmagnets, which quantifies the resistance ofa magnet to demagnetization in opposing magnetic fields, is still only20–25%of its potential, as givenby the anisotropyfield (μ0Ha~7.5 T)4–6. Thisdiscrepancy, known as the Brown paradox7, is mainly caused by micro-structural imperfections anddefects that promote the nucleation of reversedmagnetic domains atmagneticfieldsmuch lower thanHa. To achieve a highcoercivity, Nd-Fe-B magnets should comprise small Nd2Fe14B grains(D < 1 µm) that are well alignedwith c-axes and isolated from each other bya thin nonmagnetic intergranular phase (IGP), or at least by a weaklymagnetic one5,6. In addition, soft magnetic secondary phases should beeliminated, whereas grains on themagnet surface should be protected fromany damage or strengthened such as by Dy diffusion8. The laborious opti-mization of Nd-Fe-B magnets toward the desired microstructure andmagnetic performance is still in progress9–16. To accelerate this process,micromagnetic simulations are often used to elucidate the relationshipbetween the microstructural features and macroscopic magneticproperties6,17–35. Current micromagnetic models of Nd-Fe-B magnetsaddress two cases of high practical importance: sintered magnets6,8,20–23 andhot-deformed ones24–33.Nd-Fe-B sintered magnets are composed of large equiaxial grains ofsizes 3–5 µm. Laguerre tessellation is commonly used to create a 3Dmodelof such magnets. To make the model representative of a real sample, thetessellation parameters need to be adjusted, e.g., by varying until the powerspectra of 2D slices of themodel well reproduce the spectrum of a scanningelectron microscope (SEM) image of the sample20. The twins and crystal-lographic orientations of the grains canbe incorporated into themodel fromSEMvia electronbackscatter diffraction21.However,micromagneticmodelsshould have a spatial discretization of several nanometers to resolve the IGPand satisfy a constraint imposedby the exchange correlation length36. Such afine mesh is not feasible for sintered magnets even with state-of-the-artcomputational facilities. Either themodels are significantly downscaled, e.g.,by a ratio of 1:15 and even lower21, or the reduced order micromagneticapproach is used under certain assumptions, i.e., uniformly magnetized1Research Center for Magnetic and Spintronic Materials, NIMS, Tsukuba 305-0047, Japan. 2International Center for Young Scientists, NIMS, Tsukuba 305-0047,Japan. 3Graduate School of Science and Technology, University of Tsukuba, Tsukuba 305-8573, Japan. e-mail: bolyachkin.anton@nims.go.jpnpj Computational Materials |           (2024) 10:34 11234567890():,;1234567890():,;http://crossmark.crossref.org/dialog/?doi=10.1038/s41524-024-01218-5&domain=pdfhttp://orcid.org/0000-0003-0420-1806http://orcid.org/0000-0003-0420-1806http://orcid.org/0000-0003-0420-1806http://orcid.org/0000-0003-0420-1806http://orcid.org/0000-0003-0420-1806http://orcid.org/0000-0001-7046-2671http://orcid.org/0000-0001-7046-2671http://orcid.org/0000-0001-7046-2671http://orcid.org/0000-0001-7046-2671http://orcid.org/0000-0001-7046-2671http://orcid.org/0000-0002-7856-7897http://orcid.org/0000-0002-7856-7897http://orcid.org/0000-0002-7856-7897http://orcid.org/0000-0002-7856-7897http://orcid.org/0000-0002-7856-7897http://orcid.org/0000-0003-3548-1951http://orcid.org/0000-0003-3548-1951http://orcid.org/0000-0003-3548-1951http://orcid.org/0000-0003-3548-1951http://orcid.org/0000-0003-3548-1951mailto:bolyachkin.anton@nims.go.jpgrains are considered which switching conditions are defined based on theembedded Stoner–Wohlfarth model8,18,20.Micromagnetic models of hot-deformed Nd-Fe-B magnets are lessdemanding in terms of meshing owing to their significantly smaller grainsize24–33. One of the largest finite element models (FEM) of hot-deformedmagnetswas 0.4 × 0.4 × 0.4 µm3with an average lateral grain size of 77 nm28,whereas a finite difference model achieved 2 × 2 × 0.5 µm3 with a lateralgrain size of 127 nm but under the ignorance of IGP (weakened directexchange interaction between grains was prescribed instead)32,33. Thechallenge for hot-deformed magnets is attributed to their typical micro-structure – densely packed platelet grains with high aspect ratios –which ishard to mimic. Instead, a simplified layered stack of grains with Voronoitessellation within each layer is often considered. Another drawback is thatthe intergranular phase is usually introduced as a solid region with uniformproperties24–30, despite recent experimental observations – IGPs at facetsnormal to the c-axis have lower transition metal content than those at thelateral facets, and thus lowermagnetization9,13,34. It is difficult to incorporateboth realistic grain packing and IGP nonuniformities into current micro-magnetic models. The resulting accumulation of assumptions leads to acommonly observed offset between the simulated and experimental coer-civities for hot-deformed Nd-Fe-B magnets, as well as for sintered ones.In this paper, we present large-scale finite element models of hot-deformed Nd-Fe-B magnets developed based on tomographic data. Theproposed models can accurately reproduce the polycrystalline micro-structure of real magnets overcoming the current limitations. In contrast toprevious synthetic models, the intergranular phase was reconstructed as aset of thin individual regions localized between adjacent grains and isolatedfrom each other by triple junctions (TJs). It allowed us to demonstrate therole of triple junctions inmagnetization reversal and reach the stage atwhichthe coercivity of hot-deformed magnets can be well reproduced in simu-lations. Such tomography-based digital twins of hot-deformed Nd-Fe-Bmagnets will promote optimization of the magnets toward ultimatecoercivity.ResultsDevelopment of tomography-based modelsA hot-deformed Nd-Fe-B magnet with a nominal composition ofNd13.4Fe76.3Co4.5Ga0.5B5.3 (at.%) and the typical microstructure was chosento develop the tomography-based finite element model. The tomographicdata was acquired as a series of SEM images while the sample surface wassequentially polishedout using a focused ion beam (FIB)9,23. Thedirectionofcollecting such slices (x-axis in Fig. 1a) was perpendicular to the c-axiscrystallographic texture of the magnet, and the slicing step was 20 nm.An area of 1.54 × 1.54 µm2 was selected throughout the 78 FIB-SEMimages to process a cubic volume. All images weremanually segmented in avectorized format so that each grain was represented by a set of polygonalcontours in several slices (Fig. 1b; grains are encoded by individual colors).The contours were then interpolated with an interpoint distance of15–20 nm, which was comparable to the slicing step, while the last and firstcontours were additionally triangulated. The grains were thus convertedinto uniform point clouds distributed on their surfaces. The lists of neigh-boring grains were declared based on a 90 nm threshold for the minimuminterpoint distance. Note that the collected SEM images after the manualsegmentation provide a large dataset for training a deep learning modelwhich will be able to speed up the processing for new samples.The following routine was developed to construct 3D grains from theirpoint clouds. First, axis-alignedminimum bounding boxes were created foreach point cloud. These bounding boxes were scaled up by 1.2 with respectto their centers to fill artificial voids that may appear in the model along thex-axis due to the discrete step of FIB-SEM tomography (Fig. 1a). Second, asupport vector machine (SVM) with a linear kernel was applied to pairwiseseparate all neighboring grains. Each grain was then constructed in 3D bycutting its bounding box with planes obtained after SVM, hereafter referredto as cutting planes (CPs). This ensured a close packing of the grains.Additionally, SVM suppressed point cloud distortions caused by driftduring the FIB-SEM observations. One shortcoming was that only convexgrains can be created in such a way, however this assumption seems to beappropriate for most grains in hot-deformed Nd-Fe-B magnets. As shownin Fig. 1c, the polycrystalline microstructure generated following the pro-posed routine corresponds well with the segmented SEM images.It should be noted that the few-nanometer-thick intergranular phasecould not be resolved in the SEM images. Therefore, an algorithm for IGPreconstructionwasdeveloped (Fig. 2a). For eachpair of adjacent grains i andj, the corresponding CP with normal nij was shifted in parallel in the depthof both grains by t=2, where t is the IGP thickness. These shifted CPs wereused to trim the grains. The surfaces obtained after trimming were thenextruded along the outward normals by t resulting in two overlappingvolumes (Fig. 2a,middle image).Only the overlapwas retainedand assignedas the IGP between grains i and j. Figure 1c shows the model with thin IGPregions reconstructed in this way. Although the algorithm is simple, it offersvarious possibilities that have not been implemented previously. In contrastto the continuous IGP in synthetic models, the intergranular phase in ourmodel was composed of individual regions. Therefore, the micromagneticproperties and thickness of the IGP can be varied locally in this model.Those parameters can also be correlated with the crystallographic orienta-tions of contacting grains. Moreover, the space remaining after the con-struction of grains and IGP regions naturally introduced triple junctionsinto the model (inset of Fig. 1c).To proceed with micromagnetic simulations, the model should bediscretized in a high-quality mesh with a sufficiently small mesh size Δ toensure at least one layer of internal nodes in thin IGPs (Δ≈ffiffiffiffiffiffiffi3=8pt; see theinset in Fig. 1c). Therefore, small geometric entities should be eliminatedFig. 1 | Development of a tomography-based model. a Acquisition of a series ofFIB-SEM images for a hot-deformedNd-Fe-Bmagnet (cropped area of 0.8 × 0.8 µm2is shown). bProcessing of the images including 2D segmentation and the conversionof grain slices into point clouds. c Generation of close-packed 3D convex grainsisolated from each other by the intergranular phase. Triple junctions are madeinvisible except for a zoomed region showing the mesh around one of them.https://doi.org/10.1038/s41524-024-01218-5 Articlenpj Computational Materials |           (2024) 10:34 2before meshing, that are small curves (l<4, where l is the curve length),small surfaces (A<ffiffiffi3p 42, whereA is the surface area), and narrow surfaces.This is typically the most challenging part of developing a finite elementmodel. Furthermore, in the case of tomography-based models, such geo-metry revision should be gentle to preserve the microstructure retrievedfrom the FIB-SEM tomography as much as possible. In this study, wedeveloped a two-stage approach to realize that.The first stage is focused on the grains. It is applied when grains havebeen trimmed, but IGPs have not yet been introduced into the model. Let’sconsider a grain i with a small geometrical entity (Fig. 2b). To revise it, anappropriate CP should be selected first. For example, if the issue is a narrowsurface, the selection is organized as follows: the surface is pre-meshed, andthe worst triangular element with the highest aspect ratio (ratio of cir-cumcircle and incircle radii) is found; then, one of the CPs surrounding thesurface is randomly chosen with weight coefficients that are inversely pro-portional to the distances from the worst element to the CPs. After that, thenormal nij of the selected CP is tilted randomly by a small angle θh while itsorigin is shifted randomly by rh along nij – always with respect to the initialnij after tomography. We used zero-mean normal distributions with stan-dard deviations of 1–2° and 1–1.5Δ for θh and rh, respectively. The cor-rection is accepted, and the grains are updated if the total number of smallgeometric entities on the surfaces associated with the revising CP and sur-rounding CPs (red and green in Fig. 2b, respectively) in both the grain andits neighbor decreases. This procedure was repeated for all grains until allissues were resolved. When the grains were finalized, the IGPs were con-structed, imprinted on the grain surfaces, and merged.The second stage is dedicated to the revision of the IGPs. For a givenIGP, the surfaces of adjacent grains were pre-meshed, and all triangularelements with a high aspect ratio (>2:5) were identified. Then, mesh-basedprisms were iteratively subtracted from the IGP in the vicinity of the badelements to avoid the formation of small entities that were causing them.Figure 2c showsanexampleof sucheliminationof a small curve that appearsin the proximity of two IGPs. This procedure was performed for all IGPregions until a high-qualitymeshwas achieved for all grain surfaces. Finally,as the IGPs were revised, a triple junction region was created by subtractingthe grains and IGPs from the model volume.Microstructure and mesh of the modelsFollowing the workflow described above, we prepared several tomography-based models of hot-deformed Nd-Fe-B magnets. In this study, the samethickness of 3.5 nmwas prescribed for all IGPs. The largestmodelwas basedon the entire processed FIB-SEM tomography of 1.54 × 1.54 × 1.54 µm3. Itconsisted of 523 grains of their actual size and 2011 IGPs (Fig. 3a). Thismodel was representative of a real magnet, i.e., 41% of the grains wereinternal grains whose shapes were not affected by the model boundaries.Arbitrarily alignedminimum bounding boxes (tight bounding boxes, TBB)were found for all grains to quantify their platelet shapes (inset of Fig. 3b).Thegrainwidthandheight (maximumandminimumTBBedges), aswell astheir aspect ratio had mean values of 363 nm, 97 nm, and 3.9, respectively,with prominent standard deviations (Fig. 3b). TBBswere also used to assigneasy magnetization axes (EAs) to grains since they are usually compressedalong their c-axes after hot pressing. The details of this procedure aredescribed in Supplementary Note 1.To fit our computational facilities, five smaller models of0.8 × 0.8 × 0.8 µm3 volume each were cropped from different places of theentire tomography (Fig. 3a). For example, Model 1 was discretized into146 × 106 tetrahedral elements. The grains, IGPs, andTJs occupied 93.3, 3.9,and 2.8 vol.%, respectively, in this model. The triple junctions formed arather complicated net-like structure (see Supplementary Fig. 2). Althoughthe cropped models are smaller, their microstructural statistics are still ingood agreement with those of the large model (Fig. 3b).Micromagnetic simulationsWe analyzed the dependence of coercivity (Hc) on the saturation magne-tization (Ms) of the intergranular phase using the developed tomography-based models of hot-deformed Nd-Fe-B magnets (Fig. 4a). Although theferromagnetic nature of IGP is justified, the actual value of µ0Ms is not wellestablished; different experimental techniques provided estimations in therange of 0.4–1.1 T for sintered magnets6,37–40. Therefore, we varied the IGPmagnetization between 0–1.5 T to cover several possibilities in the simula-tions, while the IGP exchange stiffness was assumed to scale asA / M2s17,34.To compare the simulated and experimental coercivities, the IGP µ0Ms of0.8 T was selected as a potential reference. This Ms value is close to theFig. 2 | Geometry revisions for the tomography-basedmodels. aReconstruction ofan intergranular phase (IGP) region between adjacent grains i and j (from top tobottom): parallel shifts of the cutting plane (CP) with the normal nij in the depth ofthe grains by t=2, where t is the IGP thickness; trimming of the grains and extrusionof the obtained surfaces toward each other by t; creating IGP as an overlap of theextruded volumes. b 2D schematic showing the correction of a grain i: nij of a CP istilted randomly by a small angle θh while its origin is shifted by rh. The correction isaccepted if the total number of small geometric entities in surfaces associated withthe revised CP (red) and all surrounding CPs (green) in both the grain and itsneighbor decreased. c Elimination of a small entity that appeared due to theproximity of IGPs: IGPs are pre-meshed; then, mesh-based prisms are iterativelysubtracted to avoid the formation of the small element.https://doi.org/10.1038/s41524-024-01218-5 Articlenpj Computational Materials |           (2024) 10:34 3magnetization of Nd-Fe thin films containing approximately 70 at.% ofFe41; such films can be considered as IGP counterparts to some extent.Two principal cases were analyzed for triple junctions. In an ideal caseof nonmagnetic triple junctions, coercivity decreases monotonically from3.1 ± 0.1 T to 1.8 ± 0.2 T as the IGP µ0Ms increases up to 1.5 T (Fig. 4a). Theerror bars represent the statistical uncertainties evaluated for differentmodels. The coercivity is 2.2 ± 0.2 T for the IGP µ0Ms of 0.8 T. This isdistinctly higher than the experimental coercivity of 1.54 T measured forthis sample (Fig. 4a). In the second case, we considered ferromagnetic triplejunctions whose magnetization followed the IGPMs. In other words, IGPsand TJs were united in one phase with uniform magnetic properties (asusual IGP in synthetic models). It significantly changed the coercivitydependence that decreased rapidly from 3.0 ± 0.3 T to 0.7 ± 0.2 T with amore pronouncedasymptotic behavior. The coercivitywas 1.0 ± 0.1 T at theIGP µ0Ms of 0.8 T, which is significantly lower than the experimental value.Thick triple junctions with highmagnetization act as strong nucleation sitesthat promote magnetization reversal under relatively low magnetic fields.Although such a scenario is quite artificial for real magnets (unless they donot have a high content of soft magnetic secondary phases), it demonstratesthe importanceof realizing individual IGPs insteadof theuniformone in thetomography-based models of Nd-Fe-B magnets.Furthermore, the experimental coercivity confined between thesesimulated extreme cases led us to hypothesize that some triple junctions inNd-Fe-Bmagnets, especially the thin ones, are ferromagnetic withMs lowerthan that of the IGP. For example, if we consider the IGP µ0Ms of 0.8 T andthe twice reduced magnetization of TJs, 0.4 T, the simulated coercivity of1.7 ± 0.1 T approaches the experimental coercivity of 1.54 T (Fig. 4a). Figure4b shows the demagnetization curves for this case and the extreme alter-natives (TJs µ0Ms of 0.8 T and nonmagnetic one) for one of themodels. TJswith twice reduced Ms still resulted in a remarkable decrease in thenucleation field from 2.22 T to 1.3 T. This was accompanied by appearedpinning of reversed magnetic domains. The micromagnetic configurationFig. 3 | Microstructure of the tomography-based Nd-Fe-B models. a Side view ofthe large 1.54 × 1.54 μm2 model and several cropped 0.8 × 0.8 μm2 models used formicromagnetic simulations. bHistograms of the following microstructural featuresfor the large model and one of the cropped models: the width and height of grainaccording to its 3D tight bounding box (TBB), resulting aspect ratio, and easymagnetization axis (EA) inclination from the c-axis texture (EA of a platelet grain isdefined based on its TBB; see the inset). Mean values (μ) and standard deviations (σ)are given for both models.Fig. 4 | Micromagnetic simulations on hot-deformed Nd-Fe-B magnets.a Simulated coercivity vs. intergranular phase (IGP)magnetization for themodels inwhich triple junctions (TJs) are either nonmagnetic or ferromagnetic as IGP. Theexperimental coercivity is indicated by a dashed line. Solid lines are eye guides.b Demagnetization curves simulated for different magnetizations of TJs while theIGP magnetization is fixed. cMicromagnetic configurations before and after coer-civity for the case when the magnetization of TJs was 0.4 T.https://doi.org/10.1038/s41524-024-01218-5 Articlenpj Computational Materials |           (2024) 10:34 4with such a pinned reversed domain is demonstrated in Fig. 4c, where thecolor code corresponds to the magnetization projection onto the appliedmagnetic field (z-axis). To further substantiate the proposed hypothesisregarding triple junctions and their role inmagnetization reversal, scanningtransmission electron microscopy (STEM) and atom probe tomography(APT) were conducted on the studied sample.Chemical compositions of the intergranular phase and triplejunctionsFigure 5a shows the high-angle annular dark field (HAADF)-STEM imageand superimposed energy-dispersiveX-ray spectroscopy (EDS)map for twoNd2(Fe,Co)14B grains separated by a thin amorphous intergranular phase.The (001) lattice planes were visible in the right grain. The composition lineprofile in Fig. 5b was extracted from the STEM-EDS map across the IGPinterface. Such data were collected for eight different IGP regions, yieldingthe average concentrations of Nd, Fe, Co, and Ga of 33(3), 57(4), 6(1), and4(1) at.%, respectively (standard deviations in parentheses). The inter-granular phasewas enriched inNdandGaanddepleted in transitionmetals,although 63(4) at.% of Fe+Co is high enough to keep IGP ferromagnetic atroom temperature41. Note that light elements, such as boron, could not bedetected by STEM-EDS, leading to a minor overestimation of theseconcentrations.Figure 6 shows a 3D atomic distribution map of Fe and Ga in thevicinity of a thin triple junction and two IGPs separating a few grains in thehot-deformed Nd-Fe-B magnet. The average concentrations of Nd, Fe, Co,and Ga in the intergranular phase were 25(2), 65(3), 6.0(0.2), and 1.2(0.2)at.%, respectively (statistics over five IGP regions in Fig. 6(i) andSupplementary Fig. 3). The concentrations of Nd, Fe, Co, and Ga averagedover the triple junction volume were 51(3), 35(4), 9(1), and 5(1) at.%,respectively. The presence of B was negligible. Therefore, STEM-EDS canprovide an unbiased evaluation of triple junctions that can be directlycompared with APT data.APT suggested a higher fraction of transition metals in IGP thanSTEM-EDS, whereas both methods were in good agreement regarding thespatial distribution of elements in thin triple junctions. A high-resolutionSTEM image of one of the TJs in the regionwhere it was attached to an IGPis shown in Supplementary Fig. 4, along with EDS line profiles takenequidistantly over the region. TheNd content increased from33 at.%with agradient of 0.3 at.% nm−1 while moving from the IGP into the depth of theTJ, where the average concentrations of Nd, Fe, Co, and Ga become 50(3),33(4), 10(1), and 7(1) at.%, respectively.To enrich the statistics, a large area with various triple junctions wasacquired using HAADF-STEM (Fig. 7a), and the chemical compositionswere probed at the locations indicated by crosses in the correspondingSTEM-EDS map (Fig. 7b). In Fig. 7c, the obtained compositions arerepresented by a bar chart ranked depending on the Nd content, whosevalues are given, as well as the atomic concentrations of Fe+Co. One cansee that some thick triple junctions (e.g., at locations 1 and2)were rich inNd(>70 at.%). Such thick triple junctions are likely to be paramagnetic at roomtemperature since the Fe+Co content is below30 at.%,which is consideredas a threshold41. They can act as pinning sites for domain wall propagation.At the same time, a dozenrelatively thin triple junctionshave amoderateNdcontent whereas Fe+Co occupies 40–58 at.%. Triple junctions with suchcompositions are expected to be ferromagnetic, with a reduced Ms com-pared to IGP. If their thickness is greater than the doubled exchange cor-relation length, they act as prominent nucleation sites and deteriorate thecoercivity. Therefore, the STEM and APT observations are consistent withthe conclusions from the tomography-based micromagnetic simulationsregarding the magnetism of thin triple junctions.Note that the STEM-EDS observations for some triple junctions mayhave a biased interpretation if they overlap with the grains. Such overlapscan be clearly observed when the grains have inclined facets (e.g., to the leftof locations 8 and 11) – the contrast for Nd varies gradually in a diffusivemanner. Therefore, the probing points shown in Fig. 7b were selected nearthe interfaces with prominent contrast changes.DiscussionThis study developed a large-scale tomography-based finite elementmodel of hot-deformed Nd-Fe-B magnets that minimizedmicrostructure-related assumptions. Consequently, the micromagneticsimulations can be performed with a significantly lower bias and uncer-tainty. This improvement is essential for addressing the problem of esti-mating the magnetic properties of the intergranular phase, e.g., byapproximating the macroscopic hysteretic properties of magnets, such ascoercivity. When the individual IGP regions and triple junctions weredistinguished in the models, it was found that in addition to IGP, themagnetism of the triple junctions should also be considered to explain thecoercivity of hot-deformed magnets. The content of Fe and Co in therelatively thin triple junctions is sufficiently high to keep them ferro-magnetic at room temperature. Such triple junctions can be widely dis-tributed in a magnet and serve as prominent nucleation sites formagnetization reversal– they arenew factors in the search for “weak links”to coercivity18. At the same time, thick triple junctions are likely to beparamagnetic and contribute as pinning sites for the domain walls. Thus,assuming the triple junctions to be magnetic with 0.4 T saturation mag-netization while the IGP µ0Ms was set to 0.8 T, we reproduced theexperimental coercivity in micromagnetic simulations. However, a rea-sonable question arises as to whether it is just a quantitative match or thetomography-based model can describe the coercivity mechanism of hot-deformedmagnets in general. To answer this question, we also performedsimulations of the angular dependence of coercivity, which is commonlyused to reveal the coercivity mechanism6,22,42–46.Fig. 5 | Composition of the intergranular phase. aHigh-resolutionHAADF-STEMimage and superimposed STEM-EDSmap for a typical intergranular phase region inthe hot-deformedNd-Fe-Bmagnet for which FIB-SEM tomographywas performed.b Composition line profile across the interface.https://doi.org/10.1038/s41524-024-01218-5 Articlenpj Computational Materials |           (2024) 10:34 5Figure 8 shows the simulated angular dependencies of coercivity,Hc(θ), for two cases: nonmagnetic triple junctions and weakly magneticones with 0.4 T saturation magnetization. Magnetization of the inter-granular phase was set to 0.8 T in both cases. Here, θ denotes an anglebetween the c-axis texture and an applied external magnetic fieldH (insetin Fig. 8). Each angular dependence was normalized to the coercivity atθ = 0°. Solid lines indicate analytical dependencies attributed to pinning-and nucleation-type magnets within the Kondorsky47,48 andStoner–Wohlfarth (SW)49 models, respectively. Following Ref. 35, bothanalytical dependencies were adjusted to account for the distribution ofEAswithin 2σ cone (18°, see Fig. 3b). ExperimentalHc(θ) for thisworkandRef. 35 are marked with crosses.The nucleation-type angular dependence of coercivity with dHc/dθ < 0at θ = 0° and the minimum at θ ≈ 30° was obtained for the case of non-magnetic triple junctions. This dependence diverged from the pinning-typeexperimental Hc(θ) with dHc/dθ ≥ 0 at θ = 0°. When triple junctions wereassumed to beweakly ferromagnetic, the angular dependencies of coercivitycollected over different models for the statistics (Fig. 3a) split into twodistinct trends – Fig. 8 shows the corresponding twoHc(θ) curves averagedover the models. The three models demonstrated no remarkable pinningFig. 6 | 3D atomic distribution map near a triple junction. Atomic map of Fe andGa obtained from the hot-deformed magnet with the nominal composition ofNd13.4Fe76.3Co4.5Ga0.5B5.3 and compositional profiles of the constituent elementscalculated from volumes (i) and (ii) across the intergranular phase and a thin triplejunction, respectively.Fig. 7 | Compositions of the triple junctions. a Low-magnificationHAADF-STEMimage and b STEM-EDS map for the hot-deformed Nd-Fe-B magnet. Chemicalcompositions in different triple junctions were probed at locations indicated bycrosses and ranked in c bar plots. Element content is proportional to the corre-sponding bar length, while the atomic concentrations of Nd and Fe+ Co are given.https://doi.org/10.1038/s41524-024-01218-5 Articlenpj Computational Materials |           (2024) 10:34 6during magnetization reversal. Their angular dependencies also followedthe nucleation type, although theywere closer to the experimental data. Thetwo models with pinning at the reversal (e.g., see Fig. 4c) had remarkablepinning-type angular dependencies of coercivity, which were in goodagreement with the experimental data. Apparently, the micromagneticsimulations of the larger model would have a much higher chance ofencountering pinning; therefore, the obtained pinning-type Hc(θ) is morerepresentative of a real magnet between these two trends. In previoussimulations, such a pinning-type angular dependence was only possibleafter introducing a defective grain, e.g., a grain with a significantly reducedmagnetic anisotropy constant6,22. Thismagnetically softened grain acted as alow-field nucleation center, thereby enabling a domain wall pinning in themodel.Weakly ferromagnetic thin triple junctions are supposed to be bettercandidates for this role. In this case, the pinning-typeHc(θ) can be naturallyreproduced without artificial defects and be in good agreement with theexperimental results.In summary, we developed an approach for constructing large-scalefinite element models of polycrystalline materials based on FIB-SEMtomography. In addition to reproducing the grain geometry and packing,special efforts were devoted to reconstructing thin intergranular phaseregions separated by triple junctions. Such models with realistic micro-structural features are crucial for accurately simulating the magnetic,mechanical, thermal, and other physical properties of a wide range ofmaterials.This approachwas applied toultrafine-grainedhot-deformedNd-Fe-Bpermanentmagnets to address the long-standingproblemof the remarkablediscrepancy between simulated and experimental coercivities. Usingmicromagnetic simulations on such tomography-based models, we quan-titatively reproduced the coercivity of hot-deformed Nd-Fe-B magnets andits pinning-type angular dependence. The key insight that allowed us toachieve good agreement with experiment is that thin triple junctions wereconsidered as weakly ferromagnetic at room temperature. When thethickness of such junctions exceeded twice the exchange correlation length,they contributed to magnetization reversal as prominent nucleation sites,which also enabled domain wall pinning. The chemical compositions ofvarious triple junctions were investigated by APT and STEM-EDS, whichshowed relatively high Fe+Co contents in the thin triple junctions(40–58 at.%), supporting the above hypothesis.Therefore, we established tomography-based digital twins of poly-crystallinematerials and demonstrated their benefits for hot-deformedNd-Fe-Bmagnets. Such digital twins ofNd-Fe-Bmagnets can be further used todesign theirmicrostructure towardultimate performance,while the conceptitself can be adapted to other functional materials to enhance theiroptimization.MethodsSample preparation and magnetic properties measurementA hot-deformed Nd-Fe-B magnet with the nominal composition ofNd13.4Fe76.3Co4.5Ga0.5B5.3 (at.%) was produced by induction melting of theconstituent elements followed by melt-spinning, hot-pressing, and die-upsetting routines. Details of the synthesis are described elsewhere50. TheM(H) demagnetization curves were measured at room temperature using aBH tracer after prior magnetization in a 7 T pulsed magnetic field. Thesemeasurements were performed at different tilting angles between theappliedmagnetic field and the c-axis crystallographic texture of themagnet.The coercivity at each angle was evaluated as the field at which themagneticsusceptibility (dM/dH) approached its maximum22,42.Microstructure characterizationA field emission scanning electron microscope Carl Zeiss CrossBeam1540EsBwasused to collect a series of 78backscattered electronSEMimageswhile the sample surface was sequentially polished out by a Ga FIB in stepsof 20 nm9,23. Such slicing was performed in a direction perpendicular to thec-axis crystallographic texture of the magnet.The local chemical composition of the magnet was investigated bySTEM and APT. The former was performed using an aberration-correctedFEI Titan G2 80-200 transmission electron microscope. A CAMECALEAP5000 XS laser-assisted local electrode atom probe microscope wasused for the APT. The samples for STEMandAPTwere prepared using thelift-out technique with a FEI Helios G4-UX dual ion beam system.Tomography-based finite element modelThe FIB-SEM tomography was segmented using CorelDRAW software, sothat each grain was represented by a set of 2D polygonal contours. Thesedata were transformed into uniform point clouds using a Python script.Then, lists of neighbors were obtained and the neighboring point cloudswere separated by a linear support vector machine (Scikit-learn library). Allthe data were further used in the Coreform Cubit software, where the 3Dpolycrystalline microstructure was reconstructed using the Python-writtenroutines described in the Results section. In this way, five models of0.8 × 0.8 × 0.8 µm3 volume were developed from different locations of theprocessed tomography. The mean grain width and height were 363(176) nm and 97 (42) nm, respectively. The thickness of the intergranularphase was set to 3.5 nm. Each model was discretized in 136–153 × 106 tet-rahedral elements. The mesh size for the intergranular phase and triplejunctions was 2.3 nm, whereas the mesh size for the grains graduallyincreased from 2.3 nm at the grain surfaces up to 5.0 nm toward theircenters19.Micromagnetic simulationsDemagnetization curves of the tomography-based models were simulatedusing the ‘b4vex’ code, which performs free energy minimization by anonlinear conjugate gradient method51. All grains were considered to havemagnetic properties of the Nd2Fe14B phase at room temperature as follows:saturation magnetization µ0Ms = 1.61 T, uniaxial magnetic anisotropyconstantK1 = 4.36MJm−3, and exchange stiffnessA = 8 pJm−1 52. The easymagnetization axes were declared for the grains according to their spatialorientations, as described in Supplementary Note 1. The saturation mag-netizations varied for both the intergranular phase and triple junctions,while their exchange stiffnesses were assumed to scale as A / M2s17,34 withFig. 8 | Coercivity mechanism in hot-deformed Nd-Fe-B magnets. Simulatedangular dependencies of coercivity for themodels with nonmagnetic triple junctions(TJs) and with TJs having a low magnetization of 0.4 T. The intergranular phase(IGP) magnetization is fixed at 0.8 T in both cases. Hc(θ) for weakly ferromagneticTJs splits into two trends depending on whether or not domain wall pinning occursduring the magnetization reversal. Solid lines are guides to the eye, except for thetheoretical Hc(θ) given by the Stoner–Wohlfarth (SW) and Kondorsky models.Experimental Hc(θ) is indicated by crosses.https://doi.org/10.1038/s41524-024-01218-5 Articlenpj Computational Materials |           (2024) 10:34 7respect to the Nd2Fe14B phase; their magnetic anisotropy was neglected(K1 = 0). 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Tang, Xin, Sepehri-Amin,H.,Matsumoto,M., T.Ohkubo, T. &Hono,K.RoleofCoon themagneticpropertiesofCe-substitutedNd-Fe-Bhot-deformed magnets. Acta Mater. 175, 1–10 (2019).51. Fischbacher, J. et al. Nonlinear conjugate gradient methods inmicromagnetics. AIP Adv. 7, 045310 (2017).52. Hirosawa, S. et al.Magnetization andmagnetic anisotropyof R2Fe14Bmeasured on single crystals. J. Appl. Phys. 59, 873 (1986).AcknowledgementsThis work was supported in part by the MEXT Program: Data Creation andUtilization-Type Material Research and Development Project (DigitalTransformation Initiative Center for Magnetic Materials;JPMXP1122715503) and JSPS KAKENHI Grant (JP23H01674). A.B. andX.T. acknowledge the International Center for Young Scientists (ICYS) atNIMS for providing ICYS fellowships.Author contributionsX.T. performed FIB-SEM tomography and measured the magnetic proper-ties of the Nd-Fe-Bmagnet. N.K., E.D., and A.B. segmented and processedthe SEM images. A.B. conceptualized the development of a tomography-based finite element model, then it was realized in code by joint efforts ofA.B., E.D., and N.K. TEM observations were performed by H.S. Micro-magnetic simulationswere performedbyA.B., then hewrote themanuscriptwhile all the authors critically reviewed it. H.S., T.O., and K.H. supervisedthe work.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41524-024-01218-5.Correspondence and requests formaterials should be addressed to AntonBolyachkin.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing,adaptation, distribution and reproduction in anymedium or format, as longas you give appropriate credit to the original author(s) and the source,provide a link to the Creative Commons licence, and indicate if changeswere made. 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To view a copy of thislicence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2024https://doi.org/10.1038/s41524-024-01218-5 Articlenpj Computational Materials |           (2024) 10:34 9https://doi.org/10.1038/s41524-024-01218-5http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/ Tomography-based digital twin of Nd-Fe-B permanent magnets Results Development of tomography-based�models Microstructure and mesh of the�models Micromagnetic simulations Chemical compositions of the intergranular phase and triple junctions Discussion Methods Sample preparation and magnetic properties measurement Microstructure characterization Tomography-based finite element�model Micromagnetic simulations Data availability Code availability References Acknowledgements Author contributions Competing interests Additional information