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[Seiichiro Ii](https://orcid.org/0000-0003-3999-384X), Ken-ichi Ikeda, [Toru Hara](https://orcid.org/0000-0002-9715-6444)

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[Multi-Dimensional Quantitative Analysis of Precipitates in the Hot-Rolled Al-1%Mn Alloy](https://mdr.nims.go.jp/datasets/b6abbd40-18cd-46aa-b898-75d2aade1ce8)

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Multi-Dimensional Quantitative Analysis of Precipitates in the Hot-Rolled Al-1%MnAlloy+1Seiichiro Ii1,+2, Ken-ichi Ikeda2,+2 and Toru Hara11Research Center of Structural Materials, National Institute for Materials Science (NIMS), Tsukuba 305-0047, Japan2Division of Materials Science and Engineering, Faculty of Engineering, Hokkaido University, Sapporo 060-8628, JapanPrecipitates formed during the hot-rolling process of Al-1%Mn alloy were quantitatively characterized by 3-dimensional (3-D) evaluation.The 3-D image was reconstructed using a serial sectioning technique by combining the focused ion beam and scanning electron microscopy(SEM). In the reconstructed 3-D volume, more than 10000 precipitates were extracted, and the log-normal distribution explained the sizedistribution well. We also quantitatively evaluated the precipitates from the 2-dimensional (2-D) SEM image taken during the serial-sectioningand from the scanning transmission electron microscopy (STEM) observation that the specimen was fabricated from the serial-sectioned sample.According to the data evaluated from the 2-D and the 3-D images, the average size of the precipitation measured by the 3-D image was largerthan that of the 2-D images of SEM and STEM. That is presumed by the morphological anisotropy owing to hot rolling. We also discussed theeffect of the number of the analyzed precipitates on the average size and distribution. The average size significantly depends on the number ofanalyzed precipitates. Besides, we evaluated the normality of the logarithmic distribution of the size by the Quantile-Quantile plot.[doi:10.2320/matertrans.MT-L2026006](Received April 17, 2026; Accepted May 21, 2026; Published August 25, 2026)Keywords: quantitative analysis, precipitate, serial-sectioning, FIB-SEM, Al–Mn alloy1. IntroductionThe mechanical and functional properties of materials aregoverned by their microstructure. Recently, combined studieshave been performed to improve the understanding andprediction of microstructure; for example, such studiesinclude the prediction of microstructure through computa-tional simulation and experimental validation, as well as theintegration of experimental results and modeling using atheoretical approach. Consequently, the importance ofcomputational simulations has increased substantially. Olsonemphasized the importance of designing materials on thebasis of optimum computational simulations, which dependon the target structure [1]. Subsequently, Raabe summarizedand introduced computational simulation techniques span-ning from the atomic level to the macrostructural level, alongwith their underlying physical principles [2]. Computationalsimulations and modeling have also been widely appliedto aluminum alloys. In particular, a modeling technique forthermomechanical processing is used to optimize themicrostructure of aluminum alloys to achieve desirableproperties [3]. Owing to improvements in computationalperformance and advances in modeling techniques, theaccuracy of such models has improved considerably.However, most computational simulations require inputparameters related to materials and processing conditions,and further refinement of these parameters is important forachieving high-precision modeling.Microstructural parameters, such as grain size, that is, themicrostructural factors, are obtained through characterizationusing microscopy and X-ray analysis. In microscopy, the areafraction of second-phase particles [4] and the grain size [5]can be directly obtained from cross-sectional observations.The obtained two-dimensional (2D) data has been proposedto be converted into three-dimensional (3D) space as avolume fraction. This technique is known as “QuantitativeMicroscopy” [6] and is widely used for the quantitativecharacterization of microstructures. Quantitative character-ization of metastable and stable precipitates, which areamong the most important microstructural factors inaluminum alloys, has often been performed by X-ray small-angle scattering (SAXS) [7–10]. SAXS enables the character-ization of nanosized particles in a bulk sample. To date,SAXS has been applied not only to static characterization butalso to the study of the dynamics of nanostructural evolutionwith aging [10]. The characteristic structural size rangeaccessible by standard SAXS is from a few to tens ofnanometers; therefore, SAXS is a powerful tool forcharacterization of clusters in aluminum alloys. Clusterevaluation has also been performed using 3D atom probe(3DAP) [11]. 3DAP is a powerful tool for chemical analysisand is used to characterize fluctuating chemical structures.However, in 3DAP, only a very small specimen is analyzed,making characterization of precipitates with sizes rangingfrom a few tens to hundreds of nanometers difficult.Recently, microstructures have been characterized from 2Dto 3D representations through the development of advancedobservation techniques and computational visualizationmethods [12–24]. The 3D microstructure has been charac-terized using optical microscopy (OM) [12–14], scanningelectron microscopy (SEM) [15, 16], and transmissionelectron microscopy (TEM) [17–19]. In addition, 3Dstructures can be observed using X-rays at synchrotronfacilities, enabling the experimental and quantitative evalua-tion of microstructural factors [20–24]. Using OM, therecrystallized grain structures of pure iron [12] and low-carbon steel [13] have been characterized in 3D by serialsectioning and discussed in relation to the grain growthbehavior. Wang et al. characterized the microstructure of+1This Paper was Originally Published in Japanese in J. JILM 75 (2025)144–150.+2Corresponding authors, E-mail: II.Seiichiro@nims.go.jp, ikeda.ken-ichi@eng.hokudai.ac.jpMaterials Transactions, Vol. 67, No. 9 (2026) pp. 1462 to 1468©2026 The Japan Institute of Light Metalshttps://doi.org/10.2320/matertrans.MT-L2026006perlite in 3D and quantitatively analyzed its features, such asthe curvature of the ferrite/cementite phase boundary [14].Based on these results, they discussed the relationshipbetween microstructural evolution and cementite spheroid-ization. For 3D characterization using SEM, serial sectioningcombined with a focused ion beam (FIB) has been employed.Among these studies, Hara et al. succeeded in 3Dobservation of the microstructure of precipitates in the heat-resistant steel and ω-phase in Ti alloys at a nanoscale [15,16]. In TEM, 3D tomography is widely used and is based onthe acquisition of images over a wide tilt angle rangefollowed by computational reconstruction. As an exampleof TEM-based 3D tomography in Al alloys, Kaneko et al.characterized individual Si and Ge precipitates in aged Al–Siand Al–Ge alloys [17, 18]. Furthermore, X-ray tomography isadvantageous because it is a nondestructive method andhas been applied to the study of microstructural evolutionduring crack propagation [21] and to the evaluation of poreand void formation and growth [22]. Quantitative 3Dcharacterization has thus been widely performed, dependingon the spatial resolution of the probe. However, quantitativeanalysis of precipitates in aluminum alloys has mainly beenconducted for the clusters nucleated at the early stage ofprecipitation using SAXS. Although case studies in whichthe size of the precipitates in bulk samples has been evaluatedby X-ray tomography have been reported [23, 24], to thebest of the authors’ knowledge, no previous studies havequantitatively and statistically characterized individualprecipitates.In this study, we report the quantitative analysis of theprecipitates nucleated during hot rolling in an Al–Mn alloyusing a serial sectioning technique combined with orthogo-nally arranged FIB-SEM. Tanaka et al. investigated thesubstructure of Al–Mn alloys formed by the plane straincompression method and suggested the importance ofdynamic precipitates formed during hot deformation forsubstructure stability [25]. In addition, precipitates areimportant microstructural factors that govern interactionswith dislocations, and their size and distribution are importantparameters for thermomechanical modeling. Therefore, wefocused on precipitates in Al–Mn alloys formed throughhot rolling. The 2D characterization was conducted usingSEM and TEM on samples thinned from those used for serialsectioning, and the obtained features were compared.2. Experimental ProcedureThe material used in this study was an Al–Mn alloyconsisting of pure Al with a composition equivalent to 1050Al and 1mass% Mn. The alloy composition is listed inTable 1. The ingot was cast using a standard semicontinuousdirect-chill technique and hot-rolled at 773K. The hot-rolledplate was cut into rectangular pieces a few centimeters in sizeand mechanically and electrically polished to achieve amirror surface. For the serial sectioning, a cubic specimenwith a side length of approximately 30 µm was extractedfrom the plate using a FIB. Serial sectioning was performedusing an orthogonally arranged FIB-SEM (Hitachi High-TechScience Co., SMF 1000). A schematic of the conventionaland orthogonally arranged FIB-SEM, together with thespecimen, is shown in Fig. 1. As shown in Fig. 1(a), in theconventional system, the FIB and SEM were arranged at anangle of 60°. In contrast, in the system used in this study,the FIB and SEM were arranged at 90° shown in Fig. 1(b),such that the incident beam is always perpendicular to thespecimen surface. High-quality SEM images were acquiredthrough serial sectioning. In this study, room-temperatureFIB slicing was performed using Ga+ ions at an accelerationvoltage of 1 kV, a beam current of 2 nA, and a slice pitch of5 nm. SEM images are acquired at a resolution of 2000 ©2000 pixels over a field of view of 20 © 20 µm2 at anacceleration voltage of 1 kV. Accordingly, a voxel in thereconstructed 3D data, representing the minimum unit forthe quantitative analysis, corresponds to 10 © 10 © 5 nm3.The image contrast was optimized by combining signals fromsecondary and backscattered electrons to observe theprecipitate clearly. Approximately 1000 SEM images wereacquired during serial sectioning to construct the 3D image.The 3D image was reconstructed using commerciallyavailable software (Image Pro, Media Cybernetics, Ver. 10),and the precipitates were extracted on the basis of the contrastvalue of each voxel using the same software. The sizes ofindividual precipitates were measured. The serially sectionedspecimen was further sliced using a FIB to prepare a thinfilm for scanning TEM (STEM) (JEOL Co., JEM-2800)observation. STEM was operated at an acceleration voltageof 200 kV. Energy dispersive X-ray spectroscopy (EDS)analysis was performed to chemically characterize theprecipitates.Table 1 Chemical composition of the specimen used in this study.(mass%)Fig. 1 Schematic drawing of typical configurations of the FIB-SEMsystem. (a) Conventional arrangement and (b) Orthogonally configuredsystem. The geometry of the specimen in the serial sectioning conductedin this experiment is also indicated in (b).Multi-Dimensional Quantitative Analysis of Precipitates in the Hot-Rolled Al-1%Mn Alloy 14633. Results and Discussion3.1 FIB-SEM serial sectioning and 3D image recon-structionA 3D image of the hot-rolled Al–Mn alloy obtained byserial sectioning followed by the image reconstruction isshown in Fig. 2(a). X, Y, and Z directions correspond to therolling direction (RD), normal direction (ND), and transversedirection (TD), respectively. The upper region of the 3Dimages exhibits a dark contrast, whereas the lower regionappears bright. This contrast difference arises from the crystalorientation during data acquisition. In addition, markedlybright features elongated along the RD are observed in thecentral region of Fig. 2(a). These features correspond tosecondary phases formed during the casting process. Incontrast, precipitates exhibiting relatively weak contrast areuniformly distributed. A white-contrast region is clearlyvisible in the upper part of Fig. 2(a). As shown in Fig. 2(b),the precipitates analyzed in this study were selected fromthis upper region of Fig. 2(a). For ease of recognition, theprecipitates were visualized as dark contrasts. Althoughprecipitates were present throughout the entire image, asshown in Fig. 2(a), the contrast between the matrix andprecipitates in the lower part of Fig. 2(a) was similar, makingidentification of precipitates difficult. Therefore, this regionwas excluded from the present analysis, and only a partof Fig. 2(a) was used for the quantitative analysis ofprecipitates. The size of the analyzed volume wasapproximately 16.0 © 5.7 © 4.6 µm3, and the number ofanalyzed precipitates was 11186.A histogram of the sizes of 11186 precipitates is shownin Fig. 3. The Feret diameter, defined as the length of thecircumscribed boundary of a precipitate, was used, and theaverage of the longest and shortest Feret diameters was takenas the precipitate size. Assuming a log-normal distributionof the histogram, the fitted curve agrees well with theexperimental histogram. Therefore, the size distribution ofthe precipitates can be explained by a log-normal distribu-tion. As mentioned previously, regarding the size anddistribution of precipitates, Harkness et al. [8] and Osamuraet al. [9] have reported metastable Guinier–Preston (GP)zones with sizes of a few nanometers using SAXS. However,to the best of our knowledge, this is the first study in whichthe size of the stable phase with dimensions of a few hundrednanometers has been evaluated through individual observa-tion and statistical analysis. A slight deviation of the fittedcurve from the experimental data was observed atapproximately 20 nm. This deviation arises from the SEMresolution of 10 nm/pixel. Noise smoothing over a singlepixel can extend over a few pixels, corresponding toapproximately 20 nm, which may lead to erroneous countingof precipitates.3.2 STEM observation and EDS analysisA characteristic feature of serial sectioning using anorthogonally arranged FIB-SEM system is that a thin filmfrom the same region can be fabricated by further slicing afterserial sectioning. Therefore, nanoscale structures can beobserved by TEM following the 3D observation. The annulardark field (ADF) image is shown in Fig. 4(a), and the EDSmaps obtained by STEM–EDS are shown in Figs. 4(b)–(d).As shown in Figs. 4(b)–(d), Mn, the main constituent, andFig. 2 (a) 3D volume of the hot-rolled Al–Mn alloy reconstructed fromthe series of images taken by the serial sectioning technique, and (b)precipitates extracted from the 3D volume in (a).Fig. 3 Histogram of the diameter of precipitates observed in Fig. 2(b). Thedashed line is the fitting profile of the histogram based on the log-normaldistribution.Fig. 4 STEM–EDS analysis of Al–Mn alloy. The specimen was taken fromthe serial-sectioned piece. The electron beam is parallel to the TDdirection. (a) STEM–ADF image. (b)–(e) EDS maps showing thedistribution of (b) Mn, (c) Si, (d) Fe, and (e) overlayed Mn, Fe, and Si,respectively. (online color)S. Ii, K. Ikeda and T. Hara1464Fe and Si, which are typical impurities in Al, were detected.The EDS map in which these three elements are super-imposed is shown in Fig. 4(e). The ellipsoidal precipitatesexhibiting bright contrast contain both Fe and Si in additionto Mn; therefore, these precipitates are presumed to bethe quaternary α-phase compound Al(Mn, Fe)Si, which isfrequently observed in the low-purity Al–Mn alloy. Li et al.have reported the existence of the α-phase in 3003 Al alloysannealed at 773K [26]. The observation of the α-phase in thepresent study is reasonable considering the thermomechan-ical conditions employed. In addition, the existence of theα-phase has also been reported by Tanaka et al. [27].3.3 Quantitative analysis of precipitatesAs shown in Fig. 3, the quantitative analysis of precipitatesize was successfully performed in 3D, and the sizedistribution was well described by a log-normal distribution.To date, particle size distribution have been generallymeasured in 2D, even when the particles are distributed in3D space [4–6]. Because the precipitates were directlymeasured in 3D in this study, we compared the sizedistributions obtained from the 3D SEM image, a 2D SEMimage extracted from the 3D image that is shown in Fig. 5,and the 2D STEM image in Fig. 4(a).The size distribution of the precipitates obtained from the(a) 2D STEM and (b) 2D SEM images is shown in Fig. 6.The precipitates were extracted from each image using thesame procedure as that applied to the 3D SEM image. A totalof 147 and 203 precipitates were identified in the 2D STEMimage and 2D SEM image, respectively. Because theresolutions of 2D STEM and SEM images are 3.38 nm/pixeland 10 nm/pixel, respectively, features smaller than 3.38 nm(STEM) and 10 nm (SEM) were excluded from the analysisas precipitates. Consequently, 129 precipitates in the 2DSTEM image and 160 in the 2D SEM image werequantitatively analyzed. Although both datasets could befitted with a log-normal distribution, the fitted curves andexperimental data show greater scatter than the 3D databecause of the smaller number of analyzed precipitates. Theresults are summarized in Table 2. The average sizemeasured from the 3D SEM image was 81.24 nm; however,the average sizes obtained from the 2D images wereapproximately 53 nm and showed no dependence on theobservation method. To clarify the origin of the differencesin average sizes, we examined the reconstructed 3D SEMimage from different viewing directions. The reconstructedmicrostructure viewed from the (a) Y-direction (Z–X plane)and (b) the X-direction (Y–Z plane), corresponding to theND and RD of the rolled plate, respectively, is shown inFig. 7. Notably, the SEM image in the RD was rotated 90°anticlockwise so that the Z-direction is oriented consistentlyin both (a) and (b). Moreover, the uppermost image of theZ–X plane and the leftmost image of the Y–Z plane inFig. 2(b) are shown in Figs. 7(a) and (b), respectively. AsFig. 5 The image of the TD plane on the surface of the reconstructed 3Dvolume viewed from the Z-direction in Fig. 2(b).(a)(b)Fig. 6 Histograms of the diameter of precipitates observed in (a) STEMimage in Fig. 4(a), (b) SEM image shown in Fig. 5. The dashed lines arethe fitting profile based on the log-normal distribution.Table 2 Statistic data of the precipitates obtained from the images of 3D-SEM in Fig. 2(b), 2D-SEM in Fig. 5, and 2D-STEM in Fig. 4(a),respectively. N is the number of analyzed precipitates, �D, ·, and SE arethe average diameter, standard deviation and standard error measuredfrom the data, respectively.Fig. 7 (a) ND (Z–X) and (b) RD (Y–Z) planes of the 3D volume shownin Fig. 2(b), each plane corresponds to (a) Y = 0 and (b) X = 0,respectively.Multi-Dimensional Quantitative Analysis of Precipitates in the Hot-Rolled Al-1%Mn Alloy 1465shown in Figs. 7(a) and (b), a few precipitates that elongatedalong the Z direction can be observed. The precipitates in the3D SEM image were measured by considering the length inthe Z direction, whereas this dimension was not captured inthe 2D SEM and STEM images. Therefore, the precipitatesmeasured in the 3D SEM images were larger than thosemeasured in the 2D SEM and STEM images. Based on theseresults, because the microstructure of the rolled plate isexpected to exhibit anisotropy, observing it from variousdirections is important.Thus far, we have analyzed the precipitates observed in the3D SEM image and compared the results with those obtainedfrom 2D SEM and STEM images, thereby clarifying themicrostructural features such as the average size and datascatter. However, the number of precipitates analyzeddiffered substantially: 11186 in the 3D SEM image and 160and 129 in the 2D SEM and STEM images, respectively,differing by more than two orders of magnitude. Therefore,we examined the dependence of the results on the number ofanalyzed precipitates. The histograms obtained for differentnumbers (N) of precipitates selected from the 3D SEMimages are shown in Fig. 8. The analyzed precipitates wereidentified based on their center positions and selected fromZ = 0. This procedure corresponds to the analysis of theprecipitates within an arbitrary volume in the 3D imageprojected onto the X–Y (TD) surface. As shown in Fig. 8(a),the histogram for N = 250 is not smooth and is scatteredfrom the log-normal fitting. In contrast, the experimental dataand fitted curve become similar to each other with increasingN from 500 to 2500 (Figs. 8(b)–(d)), and for N = 5000(Fig. 8(e)), it is in good agreement, except for the data forsizes of approximately 20 nm. The relationship between theaverage precipitate size and N is shown in Fig. 9. With anincrease in the number of analyzed precipitates, the averagesize increased and eventually saturated at the average sizeof all precipitates analyzed in this study, for N exceeding1000. Therefore, quantitative analysis of the precipitation sizerequires evaluation of more than 1000 precipitates. However,further analysis is necessary to assess the effectiveness ofusing 1000 precipitates as the number of objects.3.4 Normality of the size of precipitatesAs discussed in the previous sections, dissimilarities wereobserved among the analytical results obtained under variousconditions. In particular, variations in the experimental dataand the fitted profile were qualitatively described on the basis(a) (b) (c)(d) (e)Fig. 8 Effect of the number of analyzed precipitates, N, on the histogram. In each histogram, N is (a) 250, (b) 500, (c) 1000, (d) 2500, and(e) 5000, respectively.81.24 nmFig. 9 Effect of the number of analyzed precipitates, N, on the averagediameter of the precipitates. The average diameter of all precipitatesdetected in the 3D volume shown in Fig. 2(b), 81.24 nm, is indicated inthis graph with the dashed line. Error bars represent a standard error.S. Ii, K. Ikeda and T. Hara1466of a log-normal distribution. In this section, the normality ofthe data is evaluated and discussed quantitatively. Normalityis assessed by comparing the experimentally obtained datawith theoretically expected data under the assumption thatthe experimental dataset has a normal distribution using aquantile–quantile (Q–Q) plot [28, 29]. If the measured datafollows a normal distribution, the Q–Q plot exhibits a linearrelationship with a slope of 1 that passes through the origin,that is, y = x. In this study, because the measured precipitatesizes were described by a log-normal distribution, thenormality of the logarithmic precipitate size was examined.The Q–Q plots obtained from the data shown in Figs. 3, 6,and 8 are shown in Fig. 10. In these plots, the x and y axesrepresent the theoretical and experimental values, respec-tively. In addition, because statistical parameters such as theaverage and standard deviation differ among the datasets, thedeviations from the averages, ¦x and ¦y, are plotted in theQ–Q plots. The Q–Q plots for all datasets are shown inFig. 10(a). The plots near ¦x = 0 for all datasets are close tozero; however, the upper and lower edges, which correspondto the data far from the average, deviate from linearity. Theenlarged Q–Q plots near the origin are shown in Fig. 10(b).The Q–Q plots for smaller N values deviate from the linearrelationship, indicating that normality does not depend onthe dimensions. In addition, the plots approach a linearrelationship as N increases. To quantitatively evaluate thesedeviations, we show the relationship between the definitioncoefficient, R2, and the number of analyzed precipitates inFig. 11. Even for the data obtained from the analysis of 2DSTEM image analysis, in which the minimum number ofprecipitates was 129, the R2 was more than 0.94. However,R2 increases with increasing N, which is the same trendobserved for the statistical analysis of the average size shownin Fig. 9.In this study, we observed the precipitates in the Al–Mnalloy by serial sectioning using a FIB-SEM system andstatistically and quantitatively evaluated the precipitate sizes.Although the present study reports results from only onespecimen, comparison of material dependence would enableanalysis not only of specific parameters, such as the averagesize, but also of the size distribution. As future workprogresses, the method developed in this study will beapplied to other microstructures. By obtaining additionalmicrostructural parameters, such as shape, number density,and interparticle distance, we expect to further clarify thenature of the microstructure and contribute to improvementsin modeling.4. Concluding RemarksIn this study, we observed precipitates in hot-rolled Al-1mass% Mn using multidimensional characterization andevaluated their sizes quantitatively. The results are summa-rized as follows.(1) A total of 11186 precipitates were selected from the 3Dimage obtained by FIB-SEM serial sectioning andreconstruction. The size distribution was well describedby a log-normal distribution.(2) The precipitate size measured from the 3D image was81.24 nm. However, the sizes measured from the 2Dimage obtained from SEM and STEM observationswere approximately 53 nm. Examination of the 3Dimage viewed from the ND and RD revealed that theprecipitates were elongated along the TD. Because(a)(b)Fig. 10 (a) Q–Q plot of the distribution of the measured diameter, y, andthe theoretical one, x, based on the assumed log-normal distribution withthe average and standard deviation obtained from measurement. Theenlarged graph around the original point is also shown in (b). In thesegraphs, vertical and horizontal axes represent the difference from theaverage value of �x and �y, respectively. (online color)Fig. 11 Relationship between the coefficient of determination, R2, obtainedfrom the least squares fitting of the Q–Q plot in Fig. 10 and the number ofanalyzed precipitates, N.Multi-Dimensional Quantitative Analysis of Precipitates in the Hot-Rolled Al-1%Mn Alloy 1467observation from the ND and RD is not possible in 2Dimages, the use of 3D analysis accounts for thedifferences in the measured sizes.(3) The dependence of precipitate size distribution on thenumber of analyzed precipitates was also investigated.The average precipitate size increased with theincreasing number of analyzed precipitates and reacheda constant value when more than 1000 precipitateswere included. This finding indicates that quantitativestatistical analysis requires more than 1000 precipitates.(4) The normality of the size distribution was evaluatedusing a Q–Q plot. The regression analysis of the Q–Qplots yielded a high R2 value, with the lowest being0.94 at N = 129. However, R2 increased with increasingnumbers of analyzed precipitates.(5) 3D observation is highly effective and important formicrostructural characterization. In particular, a quanti-tative analysis of the anisotropic microstructures, suchas those in as-rolled plate, which need to be observedfrom various directions, was demonstrated experimen-tally.AcknowledgmentsThese results were obtained through the activity of theresearch subcommittee on “Static/dynamic microstructuredevelopment prediction in thermo-mechanical process.” Theauthors acknowledge the members of the committee for theirfruitful discussions. The 3D observations were supportedby the “Advanced Research Infrastructure for Materials andNanotechnology in Japan (ARIM)” of the Ministry ofEducation, Culture, Sports, Science and Technology(MEXT), under Proposal Number JPMXP1223NM5406. Inparticular, the authors are grateful to Ms. A. Nakamura forher technical support.Open AccessThis paper is open access and licensed under a CC-BY-NC-ND license. You are free to share or adapt the materialsas long as you follow the license term: Attribution,NonCommercial, and NoDerivatives. To view a copy of thislicense, visit https://creativecommons.org/licenses/by-nc-nd/4.0/.REFERENCES[1] G.B. Olson: Computational Design of Hierarchically StructuredMaterials, Science 277 (1997) 1237–1242.[2] D. Raabe: Computational Materials Science, (Wiley-VCH VerlagGmbH, Weinheim, 1998). Online ISBN 9783527601943.doi:10.1002/3527601945.[3] J. Hirsch: Virtual Fabrication of Aluminum Products, (WILEY-VCHVerlag GmbH & Co. KGaA, Weinheim, 2006). ISBN 978-3-527-311363-X.[4] E.E. Underwood: Particle-size Distribution, Quantitative Microscopy,ed. by R.D. DeHoff and F.N. Rhines, (McGraw-Hill Book Inc., NewYork, 1968) pp. 149–200.[5] F. 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