# Fileset [134_25154.pdf](https://mdr.nims.go.jp/filesets/2be65488-9490-4160-a9c7-cb5c8badd76d/download) ## Creator Haruto Morimoto, Daiki Sugihara, Koji Kimura, [Yohei Onodera](https://orcid.org/0000-0002-3080-6991), Qiao Xvsheng, Jens R. Stellhorn, Tomokatsu Hayakawa, [Shinji Kohara](https://orcid.org/0000-0001-9596-2680), Koichi Hayashi ## Rights [Creative Commons BY Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0/) ## Other metadata [Topology of planar corner-sharing network in B<sub>2</sub>O<sub>3</sub> glass at intermediate range](https://mdr.nims.go.jp/datasets/822e5f6a-03c0-48ee-8f12-7b2ac2e80b58) ## Fulltext Topology of planar corner-sharing network in B2O3 glass at intermediate rangeFULL PAPERTopology of planar corner-sharing network in B2O3 glassat intermediate rangeHaruto Morimoto1, Daiki Sugihara1, Koji Kimura1,2,³, Yohei Onodera2, Qiao Xvsheng3,Jens R. Stellhorn4, Tomokatsu Hayakawa5, Shinji Kohara2 and Koichi Hayashi11Department of Physical Science and Engineering, Nagoya Institute of Technology, Gokisocho, Showaku, Nagoya 466–8555, Japan2Center for Basic Research on Materials, National Institute for Materials Science (NIMS),1–2–1 Sengen, Tsukuba, Ibaraki 305–0047, Japan3State Key Laboratory of Silicon and Advanced Semiconductor Materials & School of Materials Science and Engineering, ZhejiangUniversity, Hangzhou 310058, China4Co-Creation Institute for Advanced Materials, Shimane University, 1060 Nishi-Kawatsu-cho, Matsue 690–8504, Japan5Department of Life Science and Applied Chemistry, Nagoya Institute of Technology, Gokisocho, Showaku, Nagoya 466–8555, JapanB2O3 glass is a non-tetrahedral network-forming glass whose structure consists of boroxol rings that cannot formin crystalline phases under ambient conditions. In this study, a three-dimensional structural model of B2O3 glasscontaining a large fraction of boroxol rings (³80%) was successfully constructed by reverse Monte Carlo(RMC) modeling on the basis of high-energy X-ray diffraction data. This achievement is notable given thatmaintaining such a large fraction of boroxol rings has traditionally been considered difficult in conventionalRMC modeling. Analyses of coordination number, bond angle distributions, and ring size distribution confirmedthat the boroxol rings were well preserved in the model. The ring size distribution and persistence diagramrevealed that B2O3 glass contains a large fraction of boroxol rings and a small fraction of larger rings withexceptionally high topological order, compared with typical tetrahedral network-forming glasses such as SiO2glass. Our modeling and topological analysis can be extended to various B2O3-based glasses to provide a firmbasis for understanding their physicochemical and structural properties at the atomic level.Key-words : Topology, Reverse Monte Carlo, High-energy X-ray diffraction, B2O3 glass[Received November 11, 2025; Accepted February 6, 2026; Published online March 19, 2026]1. IntroductionB2O3 is one of the representative glass formers. Unlikeother typical glass-forming oxides such as SiO2 and P2O5,which are characterized by networks of tetrahedralstructural units (SiO4 or PO4 tetrahedra), B2O3 glass iscomposed of triangular BO3 structural units. Furthermore,a significant fraction of B atoms in B2O3 glass (hereafterdenoted as f ) is incorporated into a larger structural unit,the B3O6 boroxol ring.1) The value of f has been estimat-ed to be 70–85% using various techniques, such asnuclear magnetic resonance,2–4) nuclear quadrupole reso-nance,5) X-ray diffraction (XRD),6) and neutron diffractionmeasurements.7)To date, numerous studies have been performed toconstruct a structural model of B2O3 glass.8–20) Such amodel is crucial for understanding the origin of the phys-icochemical properties, including the boron anomaly inalkali borate glasses,13,22,23) superior chemical durability ofborosilicate glasses,24) various optical properties of rare-earth-doped borate glasses,24–26) and low melting andglass-transition temperatures of pure B2O3 glasses.27,28)The reverse Monte Carlo (RMC) method29) is a powerfultool for constructing three-dimensional atomic configura-tions of glasses based on X-ray and neutron diffractiondata. RMC calculations have been applied to B2O3glass.18–21) Although the structural models obtained inthese studies successfully reproduced the experimentallyobtained structure factors, the large fraction of B atomsincorporated into boroxol rings could not be accountedfor in Refs. 18) and 21). In Refs. 19) and 20), the f valuewas not discussed in detail. Molecular dynamics (MD)simulations have also been extensively carried out.8–17)Whereas in the early days, the formation of boroxol ringswas not well reproduced by MD simulations,8) reasonablylarge values of f have been obtained in more recent studiesmainly because of the refinement of the interatomic poten-tials.12,14) Despite these advances, reproducing such aboroxol-ring-rich network remains difficult within stand-ard MD simulation procedures starting from random con-figurations. Moreover, diffraction data derived from the³ Corresponding author: K. Kimura; E-mail: kimura.koji@nitech.ac.jp‡ Preface for this article: DOI https://doi.org/10.2109/jcersj2.134.P4-1Journal of the Ceramic Society of Japan 134 [4] 257-263 2026DOI https://doi.org/10.2109/jcersj2.25154 JCS-Japan©2026 The Ceramic Society of Japan 257This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.https://doi.org/10.2109/jcersj2.134.P4-1https://doi.org/10.2109/jcersj2.134.P4-1https://doi.org/10.2109/jcersj2.25154https://creativecommons.org/licenses/by/4.0/MD-based structural models tend to show only moderateagreement with experimental results, particularly in thelow-wavenumber (Q) region, when compared with RMCmodelings. For example, Urata and Lodesani have recentlyreported the results of a combination of classical andab initio MD simulations,17) in which they controlled thefraction of boroxol rings in the initial configurations. Intheir study, the agreement of simulation data with diffrac-tion data is insufficient, particularly for the principal peak(PP) observed at Q ³ 3¡¹1 in the XRD data. As a result,reconciling diffraction data with a high f value has beenconsidered challenging.In addition to the techniques in the above studies, em-pirical potential structure refinement (EPSR)30) was alsoemployed to build a structural model of B2O3 glass.31) TheX-ray- and neutron-weighted structure factors derivedfrom the constructed structural model were in good agree-ment with experimental data with a maintaining large fvalue of ³75%. The key idea in Ref. 31) was the incor-poration of dummy atoms at the centers of boroxol rings toreinforce constraints and prevent their collapse during thestructural refinement. This idea can also be applied to theRMC method. Although Swenson and Börjesson18) con-cluded that it was not possible to construct a structuralmodel with f > 30% using the RMC method based onneutron diffraction data, the concept of using dummyatoms could improve the RMC modeling for B2O3 glass.In our previous studies, we have applied the RMCmodeling technique to various glass systems,32–34) and inthe present work, we aim to extend this method to addressthe long-standing issue of boroxol ring fractions in B2O3glass. In this work, we perform RMC modeling based onhigh-energy XRD data of B2O3 glass using a dummy-atomapproach and successfully construct a structural model thatreconciles the diffraction data with a large fraction ofboroxol rings. The validity of the obtained structuralmodel is confirmed in terms of the boroxol-ring fraction,diffraction data, coordination number, and bond-angle dis-tribution. Furthermore, the ring size distributions, persis-tence diagrams, and cavity-volume analyses are applied tothe constructed model, revealing an exceptionally highdegree of topological order in B2O3 glass.2. Experiment and modelingThe B2O3 glass was prepared by melting HBO2 at1300 °C in air for 90min and quenching the melt in a brassmold. The glass samples were sealed in a vacuum-packedplastic bag to prevent moisture adsorption.The high-energy XRD experiments were performed atBL04B2 in SPring-8.35) To measure high-Q diffractiondata, XRD data were obtained at an incident energy of61.3 keV. The scattering angle 2ª ranged from 0.300 to48.9° using a combination of four CdTe and three Gedetectors. The corresponding Q ranged from 0.16 to25.72¡¹1. The measurements took approximately 1.5 h.A structural model of B2O3 glass was constructed byRMC modeling based on the XRD data using the RMC++code.36) The initial atomic configuration was constructedby dividing the oxygen atoms into two types, O(1) and O(2).The O(1) and O(2) atoms are defined by whether they wereincorporated into the boroxol ring. The dummy atoms Xwere placed at the centers of the boroxol rings to reinforcethe constraints for maintaining the shapes of the boroxolrings during the simulation. Here, the weighting factor ofthe dummy atoms for X-rays was set to 0. The length ofthe side of the simulation box was set to 33.45¡ to repro-duce the number density (0.0800¡¹3) of the sample.The initial atomic configuration in RMC modeling isoften obtained by MD simulations.32,33) However, con-structing the initial atomic configuration of B2O3 glassobtained by MD simulation is challenging because theboroxol rings easily collapse during the calculation inconventional RMC modeling. Therefore, the initial atomicconfiguration in this study was constructed as follows.First, 400 boroxol rings consisting of three B atoms andthree O(1) atoms were randomly distributed in the simu-lation box, taking into account reasonable distances be-tween the rings. Next, the dummy X atoms were placed atthe center of the 320 boroxol rings, whereas no dummyatoms were placed in the remaining 80 boroxol rings todeform their rings during the subsequent simulations.Hereafter, 240 O atoms belonging to the 80 boroxol ringswithout X atoms were treated as O(2) atoms. Then, 600 O(2)additional atoms were placed between two B atoms be-longing to two different boroxol rings. The initial atomicconfiguration, which contains 3320 particles (1200 B,1800 O, and 320 X atoms), was created by a hard-sphereMonte Carlo (HSMC) simulation. Two types of constraintwere applied to avoid physically unrealistic structures: theclosest atom–atom distance and the coordination number.The closest atom–atom distances for B–B, B–O, O–O, B–X, O–X, and X–X pairs were set to 2.10, 1.25, 2.10, 1.25,1.25, and 1.00¡, respectively, to avoid unreasonablespikes in the partial pair distribution functions. The con-straints on the B–O, O–B, X–B, and X–O coordinationnumbers were applied; all of the B atoms were coordinatedto three O atoms, all of the O atoms were coordinated totwo B atoms, and all of the X atoms were coordinated tothree B and three O atoms. Furthermore, the constraints forB–X–B and O–X–O bond angles at 120° were applied.The first coordination distances for constraints for coor-dination numbers and bond angles were set to 1.5¡. Theobtained initial atomic configuration is shown in Fig. 1. Inaddition to the boroxol rings bridged by the O(2) atoms,BO3 trigonal planars appear as a consequence of thecollapse of the boroxol rings in the absence of X atoms atthe center. After the HSMC simulation, the configurationwas refined by RMC simulation with the constraints of theB–O, O–B, X–B, and X–O coordination numbers, and theB–X–B and O–X–O bond angles.The bond angle distribution B(ª) was calculated as thenumber of bonds between ª + ¦ª, which depends on thesolid angle ¦³ £ sin(ª) subtended at that value of ª. Eachbond angle distribution was therefore plotted as B(ª)/sin(ª) to compensate for the effect of ¦³. The (B–O)n ringsize distribution of B2O3 glass was calculated from theMorimoto et al.: Topology of planar corner-sharing network in B2O3 glass at intermediate rangeJCS-Japan258RMC model using the R.I.N.G.S. code.37,38) The surfacecavity analysis of the RMC model of B2O3 glass wasconducted using the pyMolDyn code39) with a cutoffdistance rc = 2.5¡. Persistent homology analyses40) wereperformed to derive the B/O-centric one-dimensional per-sistence diagram using the HomCloud Package.40,41) Forthe analyses of the ring size distribution, cavity, andpersistent homology, the dummy X atoms were removed.3. ResultsFigure 2(a) shows the structure factors S(Q) of B2O3glass obtained from the RMC calculations [SRMC(Q)] andthe XRD measurements [S exp(Q)]. Excellent agreementbetween calculation and experimental results is confirmed.In particular, the first sharp diffraction peak (FSDP) ob-served at Q ³ 2¡¹1 and the PP observed at Q ³ 3.5¡¹1in the S exp(Q) are well reproduced by the RMC modelingin terms of both peak position and height. The R-factorreached a value of 11.3%, which indicates that the three-dimensional structure constructed by RMC modeling ac-curately reflects the experimental data. Note that the PP isclearly observed in XRD data unlike in the case of SiO2glass, because the atomic number of B is smaller than thatof O; consequently, the positive contribution from the O–Opartial S(Q) is larger than the negative contribution fromthe B–O partial S(Q) at the position of the PP.19,20) Theagreement with the experimental XRD data demonstratesthe overall structural consistency, while the fraction ofboroxol rings is examined in detail below.Figure 2(b) shows the pair distribution functions g(r) ofB2O3 glass derived by the Fourier transforms of SRMC(Q)and S exp(Q), denoted as gRMC(r) and gexp(r), respectively.A Lorch modification function was applied to the dataup to Qmax = 25¡¹1. Here, we can confirm a very goodagreement between the gRMC(r) and gexp(r). The peaksobserved at 1.35 and 2.36¡ correspond to the B–O bondsand B–B/O–O correlations, respectively. Their positionsare in accordance with the previously reported values of1.35–1.37 and 2.36–2.38¡.1,7,31)Figure 3 shows a slice of the atomic configuration ofB2O3 glass obtained by RMC modeling. It is confirmedthat a number of boroxol rings maintain their hexagonalshapes around the dummy X atoms. Figure 4 shows thepartial pair distribution functions, gij(r) of B2O3 glassderived from the RMC model. The peak positions of B–O,B–B, and O–O correlations in gij(r) are in agreement withthe interatomic distances in B2O3 glass reported in a pre-vious study.1) Note that the O–X, and B–X peaks areobserved at around 1.35¡, which is consistent with the B–O distance, demonstrating that the symmetric shape of theboroxol rings is maintained during the RMC modeling.The results of the coordination number analysis aresummarized in Table 1. The B–O coordination number of2.99, which is close to 3.00, indicating that the basicFig. 1. Initial atomic configuration of B2O3 glass.Fig. 2. (a) Total structure factor S(Q) and (b) pair distribution function g(r) obtained by RMC modelingtogether with the experimental high-energy XRD data for B2O3 glass.Fig. 3. Atomic configuration of B2O3 glass obtained by RMCmodeling (7-¡-thick slice).Journal of the Ceramic Society of Japan 134 [4] 257-263 2026 JCS-Japan259structural units of B2O3 glass, i.e., boroxol rings and thetrigonal planar units, are properly formed in the model.Furthermore, the X–B and X–O coordination numbers areboth 3.00, demonstrating that the shape of the 320 boroxolrings, which corresponds to f = 80%, has been maintainedwithout significant distortion.The bond angle distributions B(ª)/sin ª of B2O3 glassare shown in Fig. 5. The O–B–O and B–O–B bond angledistributions exhibit peaks at approximately 120° inFigs. 5(a) and 5(b). These peaks primarily originate fromthe O–B–O and B–O–B bond angle distributions withinthe boroxol rings, as evidenced by the these distributionscalculated from bond angles exclusively in the boroxolrings (dashed curves), which produce a sharp distribution.This feature indicates that the symmetric shape of theboroxol rings is preserved during the RMC modeling. Thissymmetric hexagonal shape is further supported by thepeaks at 60° in the B–B–B and O–O–O distributions[Figs. 5(c) and 5(d)], and at 30 and 90° in the O–B–B andO–O–B distributions [Figs. 5(e) and 5(f )]. Moreover, asshown in Fig. S1 in Supporting Information, the B–X–O,B–O–X, and X–B–O distributions exhibit peaks at around60°, the O–X–O and B–X–B distributions at around 120°,and the X–B–B distribution at around 30°. Additionally,the B(ª)/sin ª value increases as ª approaches 180° in theX–B–O distribution. These features correspond well to thesymmetric shape of the hexagonal boroxol rings.The results of the coordination numbers and bond angledistributions together with the excellent agreement be-tween SRMC(Q) and Sexp(Q) described above consistentlydemonstrate that the structure of B2O3 glass with a largenumber of boroxol rings has been successfully reproduced,highlighting the capability of the present method to con-struct a realistic structural model of borate glass. Impor-tantly, this large fraction of boroxol rings is achievedwithout compromising the agreement with the experimen-tal XRD data shown in Fig. 2.4. DiscussionFigure 6(a) shows the primitive42) (B–O)n ring sizedistribution obtained from the RMC model of B2O3 glass.As can be seen in the figure, it has a large fraction ofthreefold rings, indicating that the boroxol rings are thedominant structural unit in the glass. The total number ofboroxol rings incorporating 972 B atoms is 324. Therefore,the fraction f is estimated to be 972/1200 = 0.81. In addi-tion, as clearly shown in the inset of Fig. 6(a), a certainfraction of the larger rings (n > 3) is present, which isattributed to larger rings formed by the linkage of multipleFig. 4. Partial pair distribution functions, gij(r) obtained fromthe RMC model for B2O3 glass.Table 1. Coordination numbers in B2O3 glass obtained from theRMC modelAtomic pair Coordination numberB–O 2.99X–B 3.00X–O 3.00Fig. 5. Bond angle distributions of (a) O–B–O, (b) B–O–B, (c) B–B–B, (d) O–O–O, (e) O–B–B, and (f ) O–O–B obtained from the RMC model for B2O3 glass. Dashed curves in (a) and (b) indicate the bond angledistributions within the boroxol rings.Morimoto et al.: Topology of planar corner-sharing network in B2O3 glass at intermediate rangeJCS-Japan260boroxol rings via O(2) atoms. Figure 6(b) shows the ringsize distribution calculated on the basis of King’s crite-rion.43) It exhibits 324 boroxol rings, as observed in theprimitive ring size distribution. However, compared withthe primitive rings, it shows a clearer ring size distributionranging from 4- to 23-fold [inset of Fig. 6(b)]. The differ-ence arose from different algorithms in the two criteria; theprimitive ring size distribution enumerates all the shortestclosed paths along with B–O bonds within a maximumsearch depth, exhibiting a broader distribution includinglarger rings (n > 24). In contrast, King’s criterion enablesthe finding of a combination of boroxol (threefold) ringsand other rings, which cannot be detected by the primitivering criterion, because it cannot detect a decomposable(nonprimitive) ring. [see Refs. 37) and 38) for details ofring size statistical analysis]. Indeed, the presence of thelarger rings, excluding the boroxol rings, demonstrates theformation of a topologically disordered network, which isa characteristic of glasses.44,45) Nevertheless, the B2O3glass is exceptionally topologically ordered compared withtypical glasses such as SiO2 glass,46,47) owing to the dom-inant fraction of boroxol rings. Notably, this coexistence isnot imposed by explicit constraints on ring sizes in theRMC modeling, but instead emerges naturally from thediffraction-based structural refinement.Figure 7 shows the persistence diagram obtained fromthe RMC model for B2O3 glass. We can observe a distinctvertical profile along with the death axis at the birth b ³0.5¡2, corresponding to half of the B–O bond. Region I(highlighted in blue) around b ³ 0.5¡2 and within1.4¡2 < death (d) < 1.9¡2 corresponds to the birth anddeath of boroxol rings. This prominent profile is a signa-ture of the topologically ordered feature of B2O3 glass, asmentioned above. On the other hand, region II (high-lighted in red) around b = 0.5¡2 and within 2.2¡2