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[Shinji Kohara](https://orcid.org/0000-0001-9596-2680)

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[Hyperordered structures in silica polymorphs](https://mdr.nims.go.jp/datasets/fd6f2e58-dc7d-4d87-be32-6b878af277cb)

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Hyperordered structures in silica polymorphsREVIEWHyperordered structures in silica polymorphsShinji Kohara1,³1Center for Basic Research on Materials, National Institute for Materials Science, Tsukuba, Ibaraki 305–0047, JapanWe have been exploring with the quest hyperordered structures in terms of order within disorder in silica (SiO2)polymorphs. In this article, we review and discuss our recent findings on this topic comprehensively incomparison with those of previous studies. We chose several SiO4 tetrahedral corner-sharing crystalline silicaand siliceous zeolites of various density. Furthermore, we attempted to control the intermediate-range orderingof glass by tuning of the density of silica glass under high pressures and temperatures. We extracted the density-driven modification of the topology of tetrahedral silica polymorphs in a wide density range. Our state-of-the-artanalyses revealed two descriptors for hyperordered structures in silica polymorphs. The first descriptor ofhyperordered structures in silica glass can be expressed by the position and height of diffraction peaks observedin X-ray and neutron diffraction data. This descriptor is not new, but we can systematically understand thedensity-driven behaviour of diffraction peaks in silica glass. The second descriptor of hyperordered structures ina series of silica polymorphs can be expressed in terms of topological characteristics: ring size distribution, cavitydistribution, ring shape, and tetrahedral order. We found an unusually large cavity volume in ¢-cristobalitewhich was attributable to the formation of highly symmetrical –Si–O– sixfold rings, and highly symmetricaleightfold and twelvefold rings in coesite even though most of the small rings were significantly buckled, whichwas due to coesite having the highest density of coesite among the series of silica polymorphs. Moreover, wefound a topological similarity between glass and siliceous zeolite (MFI), in which fivefold and sevenfold rings areobserved. It is concluded that both diffraction measurement and topological analysis provide us crucialinformation on hyperordered structures in silica polymorphs.Key-words : Densified silica glass, Silica polymorphism, Siliceous zeolite, Topology, Diffraction[Received March 13, 2025; Accepted April 21, 2025; Published online September 1, 2025]1. IntroductionSilica (SiO2) polymorphisms, which are the most impor-tant solid oxides in materials and earth sciences, have beenwidely studied using many experimental and simulationtechniques.1,2) In contrast to crystalline silica, the structuralinformation of the glass and liquid is insufficient owing tothe limited information that can be accessed by exper-imental and simulation techniques. However, the recentadvent of advanced quantum beam (X-rays, electrons, andneutrons) and computer simulation techniques has madefeasible to obtain dependable atomic configurations.3–9)Quantum beam diffraction techniques can provide usdirect information on the atomistic structure of glassy andliquid materials. In particular, the combination of hardX-ray diffraction measurements at synchrotron radiationsources and neutron diffraction measurements at both reac-tor and spallation sources is powerful because neutronsare sensitive to oxide atoms, whereas X-rays are sensitiveto heavy elements.1,2) Silica glass exhibits a three-peakstructure: the first sharp diffraction peak (FSDP, Q1 ³1.5¡¹1),10–13) in X-ray or neutron structure factor S(Q),the second principal peak (PP, Q2 ³ 3¡¹1)12,13) in neutronS(Q), and Q3 at Q ³ 5¡¹1 in X-ray or neutron S(Q). FSDPcan be understood in terms of a periodicity of 4¡ (2³/Q1)and a coherence length of 10¡ (2³/¦Q1) by the peakposition and width, respectively.14) Indeed, FSDP is asignature of periodicity arising from the succession of cagestructures formed by corner-sharing SiO4 tetrahedra.15,16)We have been working on the synthesis of densifiedsilica glass via hot (up to 1500 °C at 7.7GPa) and cold(RT/20GPa) densifications to understand the behaviour ofFSDP and PP on the basis of X-ray and neutron diffractiondata. Note that the density of glass recovered at>1200 °C/7.7GPa by hot densification was identical to that of cold-densified glass. We observed that the height of FSDP andthe density of the glass densified at 1300 °C/7.7GPa weremaximum, whereas the height of FSDP of the cold-densified glass was minimum.17,18)We have recently applied a series of topological analysistechniques for ring size distribution, cavity distribution,ring shape, and tetrahedral order to silica polymorphs. Inthis work, we chose several SiO4 tetrahedral corner-sharing crystalline silica and siliceous zeolites (MFI, SOD,and FAU) of various densities to understand the density-³ Corresponding author: S. Kohara; E-mail: KOHARA.Shinji@nims.go.jp‡ Preface for this article: DOI https://doi.org/10.2109/jcersj2.133.P9-1Journal of the Ceramic Society of Japan 133 [9] 488-496 2025DOI https://doi.org/10.2109/jcersj2.25038 JCS-Japan©2025 The Ceramic Society of Japan488This 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/jcersj2.133.P9-1https://doi.org/10.2109/jcersj2.133.P9-1https://doi.org/10.2109/jcersj2.25038https://creativecommons.org/licenses/by/4.0/driven modification of topology in a wide density range(1.33–2.91 g cm¹3).19)In this article, we review our recent findings as men-tioned above and discuss the origin of hyperordered struc-tures in silica polymorphs, and the capability of diffrac-tion measurements and a series of topological analysistechniques.2. Data analysisTo understand the topology of silica polymorphs, thering size distribution, cavity distribution (surface cavityvolume was calculated using a cut off distance rcut =2.5¡), and ring shape [one-dimensional persistence dia-gram (PD),20,21) which shows the birth and death of ringsin silica polymorphs], were calculated using the R.I.N.G.S.code,22,23) pyMolDyn code,24) and HomCloud package,25)respectively.The tetrahedral order parameter q for Si-centric tetrahe-dra is expressed by26)q � 1� 38X3i¼1X4k¼iþ113þ cos ªijk� �2; ð1Þwhere ªijk is the angle between the central Si atom j and itsneighbouring Si atoms i and k. This parameter was de-signed to be unity in a regular tetrahedron and have a meanvalue of zero in a perfect gas.The details of a series of analyses are described in ourprevious publications.27–29)3. Results and discussion3.1 Hot- and cold-densified silica glassesFigure 1(a) shows the temperature-dependent densitiesof the hot-densified silica glass at 7.7GPa (black) and thecold-densified silica glass (RT/20GPa, cyan), and thoseof ¡-cristobalite (green), ¡-quartz (magenta), and coesite(grey).17) As can be seen in the figure, the density of hot-densified silica glass does not change monotonically; thedensity increases rapidly up to 600 °C/7.7GPa, but in-creases slowly thereafter. Figure 1(b) shows the X-ray(upper) and neutron (lower) total structure factors S(Q)of densified silica glasses.17) It is well known that X-raysare more sensitive to the silicon–silicon correlation atQ1 ³ 1.5¡¹1, whereas neutrons are more sensitive to theoxygen–oxygen correlation at Q2 ³ 3¡¹1 [no PP is ob-served in X-ray S(Q)],5,17) suggesting that we should focuson the FSDP in X-ray S(Q) and the PP in neutron S(Q). Ascan be seen in Fig. 1(b), the FSDP of X-ray S(Q) shifts tothe high-Q side and its height decreases at 400 °C/7.7GPabut increases at 1200 °C/7.7GPa. This peak height changereflects the ordering of Si–Si correlation at the FSDP posi-tion according to the MD-RMC model for hot densifica-tion.17) We suggest that the rapid increase in density isassociated with the reduction in cavity volume17) and theslow increase in density thereafter associated with thestructural ordering expressed by the height of FSDP in X-ray S(Q). This ordering is largely affected by the evolution131211109872.82.72.62.52.42.32.23210109876543210 RT/7.7 GPa 400 /7.7 GPa 1200 /7.7 GPa RT/20 GPaQ (Å–1)S(Q)Q1 (FSDP) Q2 (PP) Q3(a) (c)Density (g cm–3)Coherence length (Å)2.92.82.72.62.52.42.32.210005000Temperature ( )Density (g cm–3)(b)RT400 1200 Fig. 1. (a) Temperature-dependent densities of the hot-densified silica glass at 7.7GPa (black) and cold-densified silica glass (RT/20GPa, cyan), and those of ¡-cristobalite (green), ¡-quartz (magenta), and coesite(grey).17) (b) X-ray (upper) and neutron (lower) total structure factors S(Q) of densified silica glasses.17)(c) Coherence length (2³/¦Q1) of the hot-densified silica glass and cold-densified glass (cyan).17)Journal of the Ceramic Society of Japan 133 [9] 488-496 2025 JCS-Japan489of the Si–Si correlation towards relaxation to the crystal-line phase because no such a clear behaviour is observedin the FSDP of neutron S(Q) owing to the difference inweighting factors for X-ray and neutron diffraction data.Although the diminishment of FSDP was observed inmany previous studies using in situ X-ray diffraction,30–34)in situ neutron diffraction,35) and X-ray diffraction mea-surements of hot- and cold-densified glasses,36) our findingis the first to show that the evolution of FSDP in hot-densified glass. Note that the density of hot-densified glasssynthesized at 1200 °C/7.7GPa is identical to that of cold-densified glass [cyan in Fig. 1(a)]. Nevertheless, the heightof FSDP of the latter is much lower than that of the former,suggesting that we synthesized two densified silica glasseswith the same density, but different structures. Figure 1(c)shows the coherence length (2³/¦Q1) values of the hot(black)- and cold (cyan)-densified silica glasses,17) dem-onstrating the modification of the glass structure in theintermediate-range scale by pressure and temperature. Thisbehaviour suggests that a glass structure rejuvenates upto 400 °C/7.7GPa and then relaxes to a coesite structurethereafter because hot-densified glass transforms into coes-ite when the temperature is increased. Moreover, we mea-sured the density and X-ray diffraction data of hot- andcold-densified glasses and found that cold-densified glassis not permanently densified, which is inconsistent withprevious studies.37,38)3.2 Amorphous silica with a zeolite liketopologyFigure 2 shows the X-ray total structure factors S(Q) ofcold-densified silica glass17) and cold-densified amorphousMFI obtained by cold densification of a MFI single crys-tal.39,40) Note that the densities of two samples are both2.7 g cm¹3. The X-ray S(Q) of both samples are identicaland exhibit an FSDP at Q = 2¡¹1, although amorphousMFI has a tiny peak at around Q ³ 0.6¡¹1, as indicatedby an arrow, which corresponds to a trace of (101) and(020) reflections in crystalline MFI. Figure 3 shows theprimitive41,42) ring size distributions (left) and atomic con-figurations (right) of (a) cold-densified silica glass and (b)cold-densified amorphous MFI.40) The ring size distribu-tion of silica glass shows a broad (threefold to tenfold)distribution, although (Si–O)6 sixfold rings, which are theonly rings of cristobalite,19) are predominant. This broaddistribution is topologically disordered according toCooper.43) Amorphous MFI also shows a broad (fourfoldto tenfold) distribution, but the fraction of fivefold rings islarge, which is a trace of crystalline MFI because we chosethe crystal structure as an initial atomic configuration forthe RMC modelling of amorphous MFI. Indeed, we couldnot fit the experimental diffraction data by RMC modellingwhen we started from a random configuration.39) The dif-ference in ring size distribution is found to be well reflect-ed in the distributions in cavities; cavities of silica glass arerandomly distributed, whereas those of amorphous MFIhas a crystalline topology. This difference is a reason whysilica glass is not permanently densified, whereas amor-phous MFI formed by cold densification is stable for atleast ten years17) and has a different amorphous structurecompared to cold-densified silica glass.3.3 Topological analyses of silica polymorphsFigure 4 shows the crystal structures of ¡-cristobalite,¢-cristobalite, ¡-quartz, coesite, FAU, SOD, and MFItogether with the atomic configuration of silica glassobtained by a combination of molecular dynamics andreverse Monte Carlo8,44) (MD–RMC) modelling.40) Notethat the density of silica glass (2.21 g cm¹3) is the same asthat of ¢-cristobalite, and those of siliceous zeolites (FAU:1.33 g cm¹3; SOD: 1.66 g cm¹3; MFI: 1.84 g cm¹3) arelower than that of silica glass. On the other hand, thedensities of the remaining three crystalline phases (¡-21020151050 Cold-densified silica glass Cold-densified amorphous MFIS(Q)Q (Å 1)Fig. 2. X-ray total structure factors S(Q) of cold-densified silicaglass17,40) and cold-densified amorphous MFI.39,40)silica glass(cavity volume ratio: 9.9%)n-fold ringFraction(a)(b) amorphous MFI(cavity volume ratio: 6.1%)n-fold ringFraction0.50.40.30.20.10.01514131211109876543Fig. 3. Primitive ring size distributions (left) and atomic con-figurations (right) of (a) cold-densified silica glass and (b) cold-densified amorphous MFI.40) Red spheres: oxygen; blue spheres:silicon. Cavities are shown in green. Note that the atomic con-figuration of cold-densified silica glass was obtained by MD–RMC simulation.17)Kohara: Hyperordered structures in silica polymorphsJCS-Japan490cristobalite: 2.33 g cm¹3; ¡-quartz: 2.65 g cm¹3; coesite:2.92 g cm¹3) are higher than that of silica glass. As can beseen in Fig. 4, the intertetrahedral oxygen–oxygen corre-lations up to 3.2¡, which are indicated by green sticks, areobserved only in coesite, silica glass, and MFI. The be-haviour of MFI might be the reason why densified amor-phous MFI was permanently densified for at least 10 yearsby cold densification, as discussed in sect. 3.2.A schematic of intra- and intertetrahedral oxygen–oxygen correlations in silica glass is shown in Fig. 5(a) asan example.19) These correlations are observed only incoesite, silica glass, and MFI. Since coesite has the high-est density among silica polymorphs, an intertetrahedraloxygen–oxygen correlation is formed at a high density. Inthe case of silica glass, an intertetrahedral oxygen–oxygencorrelation is the result of disorder. However, it is difficultto form an intertetrahedral oxygen–oxygen correlation inMFI because density is very low compared with those ofcoesite and silica glass.Here, we discuss the origins of FSDP and PP.Figure 5(b) shows the schematic illustrations of ¢-cristo-balite with the assignment of FSDP and PP proposed byBenmore and Wilding.45) As discussed in sect. 3.1, theperiodicity estimated by the position of FSDP in silicaglass is 4¡ (2³/Q1), which corresponds to that of dFSDPin Fig. 5(b). They assigned the coherence length of 2¡(2³/Q2) to the distance from the base to the apex of aSiO4 tetrahedron (dPP) as suggested by Salmon andZeidler.13) However, this interpretation cannot explain thereason why the PP of silica glass evolved under highpressures at room temperature,35) where the Si–O coordi-nation number remained to be four. We suggest that theordering of intertetrahedral oxygen–oxygen correlationunder high pressures associated with the reduction incavity volume, as illustrated in Fig. 5(c), is the origin ofthe evolved PP under high pressures. This hypothesis is inline with the density increase shown in Fig. 1(a). The ori-gin of Q3 was discussed in Ref. 14, in which the succes-sion of nearest neighbour correlations can reproduce Q3.Figure 6 shows the oxygen–oxygen total correlationfunctions TOO(r) (a) and intratetrahedral oxygen–oxygentotal correlation functions T intraOO(r) (b) of hot-densifiedsilica glasses obtained by MD-RMC simulation. Note thatT intraOO(r) were calculated by using only the intrateraheraloxygen–oxygen correlations. A prominent correlationpeak is found at 2.6¡, and a subtle second oxygen–oxygen correlation peak is observed at ³3.5¡, whoseheight increases with temperature (density), whereasT intraOO(r) remains the same, suggesting that only theintertetraheral oxygen–oxygen correlation evolved by hotdensification. This behaviour is in line with the behaviourof the PP shown in Fig. 1(c), in which the PP shifts to thelow-Q side caused by the evolution of the height of thepeak observed at 3.5¡ in TOO(r) by hot densification.Figure 7 shows the cavity distributions in silica poly-morphs19) and Table 1 summarizes the densities and cav-SOD (cubic)d = 1.66 g cm 3MFI (orthorhombic)d = 1.84 g cm 3-quartz (trigonal)d = 2.65 g cm 3coesite (monoclinic)d = 2.92 g cm 3glassd = 2.21 g cm 3FAU (cubic)d = 1.33 g cm 3-cristobalite (cubic)d = 2.21 g cm 3-cristobalite (triclinic)d = 2.33 g cm 3Fig. 4. Crystal structures of ¡-cristobalite, ¢-cristobalite, ¡-quartz, coesite, FAU, SOD, and MFI together withthe atomic configuration of silica glass obtained by MD–RMC simulation.19) Red spheres: oxygen; blue spheres:silicon. The intertetrahedral oxygen–oxygen correlations are highlighted by green sticks.Journal of the Ceramic Society of Japan 133 [9] 488-496 2025 JCS-Japan491ity volume ratios of silica polymorphs. It is found that thecavity volume ratio of ¢-cristobalite (54%), whose densityis identical to that of silica glass, is higher than those ofsilica glass (33%) and MFI (47%). On the other hand, weobserve no cavities in ¡-cristobalite, ¡-quartz, and coesite.Moreover, FAU has the highest cavity volume ratio (69%)because it has the lowest density among the silica poly-morphs. A series of analyses suggest that the cavity vol-ume ratio of ¢-cristobalite is comparable to those ofsiliceous zeolites, which agrees with Gaskell and Wallis’sargument,46) on the basis of which they reported thestructural similarity between silica glass and ¢-cristobalitein terms of the peak position in diffraction data.Figure 8 shows the O-centric PDs for ¡-cristobalite, ¢-cristobalite, ¡-quartz, coesite, FAU, SOD, MFI, and silicaglass.19) The O-centric PD is important in understandingthe packing fraction of oxygen atoms47) in oxide materials.All PDs exhibit prominent profiles at bk ³ 1.6¡2. A similarbehaviour between ¡- and ¢-cristobalites is found in theSi-centric PDs,19) but no such similarity is observed in theO-centric PDs, although the O-centric PD is identical be-tween ¡- and ¢-quartz. A long lifetime (= dk ¹ bk)28)profile observed at bk/dk = 1.45¡2/6.20¡2 in the O-centric PD for ¢-cristobalite shows that the sixfold rings arevery symmetrical in ¢-cristobalite. It is also suggested thatthis long lifetime profile is the reason for the high cavityvolume ratio in ¢-cristobalite. As can be seen in Fig. 8,the O-centric PD for SiO2 silica glass shows a prominentvertical profile along with the dk axis at bk ³ 1.7¡2. Forsiliceous zeolites, no systematic change can be observed inthe PDs; the O-centric PD for FAU has a long lifetimeprofile at bk/dk = 1.4¡2/24.5¡2, indicating that a large15105054321 RT/7.7 GPa 400 /7.7 GPa 1200 /7.7 GPa15105054321 RT/7.7 GPa 400 /7.7 GPa 1200 /7.7 GPar (Å)T OO(r)r (Å)T intraOO(r)(a) (b)Fig. 6. Oxygen–oxygen total correlation functions TOO(r) (a) and intratetrahedral oxygen–oxygen totalcorrelation functions T intraOO(r) (b) of hot-densified silica glasses obtained by MD–RMC simulation.2.59 Å 2.54 Åintratetrahedraloxygen–oxygen correlationintertetrahedraloxygen–oxygen correlation(a)                      (b)dPPdFSDPdPPdFSDPsilica glassβ-cristobalite(c)Fig. 5. (a) Schematic of intra- and intertetrahedral oxygen–oxygen correlations in silica glass.19) Blue: Si; red:O. Schematic illustrations of ¢-cristobalite (b) and silica glass (c) with the coherence lengths of FSDP and PP.Kohara: Hyperordered structures in silica polymorphsJCS-Japan492symmetrical cage is formed. MFI has an intermediate-sizecage, as can be seen in the O-centric PD at bk/dk ³ 1.4¡2/13.0¡2, although the overall features of PDs for MFI areidentical to that of SiO2 silica glass. On the other hand, sucha long lifetime profile can hardly be observed in the O-centric PD for SOD (Fig. 8). This feature for SOD can beseen in Fig. 4, in which no well-defined large cage struc-ture is observed.Figure S1 shows the tetrahedral order parameter qvalues of SiSi4 tetrahedra for a series of silica polymorphs.¢-cristobalite, which shows perfect hyper-tetrahedral coor-dination, has a q value larger than those of ¡-cristobalite,¡-quartz, and coesite. On the other hand, siliceous zeolitesexhibit an opposite behaviour. MFI has the largest qamong these zeolites and significant broad distributions ofq are observed in both the MFI and silica glass profiles,β-cristobalite glassSODFAU MFIStructure Density(g cm 3)Cavity volume ratio (%)FAU 1.33 69SOD 1.65 65MFI 1.84 47glass 2.21 33β-cristobalite 2.21 54α-cristobalite 2.33 0α-quartz 2.65 0coesite 2.92 0Table 1.  Density and cavity volume ratio of silica polymorphsFig. 7. Cavity distribution in a series of silica polymorphs.19) Cavities are shown in green. Red sphere: oxygen;pink sphere: silicon.-cristobalited = 2.21 g cm–3-cristobalited = 2.33 g cm–3coesited = 2.92 g cm–3-quartzd = 2.65 g cm–3Death dk(Å2 )SODd = 1.66 g cm–3FAUd = 1.33 g cm–3glassd = 2.21 g cm–3MFId = 1.84 g cm–3Death dk(Å2 )Birth bk (Å2)Birth bk (Å2)Fig. 8. O-centric PDs for a series of silica ploymorphs.19) The prominent profile observed at bk/dk ³ 1.6¡2/2.2¡2 arises from threefold O–O–O rings in SiO4 tetrahedra.Journal of the Ceramic Society of Japan 133 [9] 488-496 2025 JCS-Japan493which have comparable average q values of 0.89 and 0.84,respectively. This behaviour suggests that both MFI andsilica glass are intermediate between crystalline silica andother siliceous zeolites. Small q values of FAU and SODare presumably due to the large fraction of twelvefold ring(See Fig. S2).Figure 9 shows typical even-numbered rings (cycles)obtained by a combination of ring size distribution andpersistent homology analyses together with the lifetime ofrings (cycles).19) This method enables us to measure thelifetime of each ring (cycle) on the basis of the Si–Si cy-cles, O–O cycles, and Si–O bonds. The rings (cycles) arerepresented by Si–Si bonds (orange), O–O bonds (cyan),and Si–O bonds (grey). The three values in parentheses arethe lifetimes of each ring (cycle) calculated as dk ¹ bk. Themost universal feature is that oxygen atoms are buckled in(Si–O)n rings because the lifetime (shown in orange) isvery short except in some symmetrical even-numberedrings, such as the sixfold rings in ¢-cristobalite, and largerings, such as the eightfold and twelvefold rings in coesite,tenfold rings in MFI, and twelvefold rings in SOD/FAU.Another remarkable feature is that the oxygen cycles offour-, six-, and eightfold ring have short lifetimes in MFI,which is the reason why MFI has the intetertrahedral O–Ocorrelation despite its low density. On the other hand,the shapes of fourfold rings in coesite, FAU, SOD, andsilica glass are identical. The (Si–O)6 sixfold ring in ¢-cristobalite is very symmetrical owing to its long lifetime,which is different from the asymmetrical shape of the O–Ocycle in ¡-cristobalite. We stress that this difference,manifested by the large dk in the O-centric PD, is related tothe formation of cavity with a small density difference be-tween ¡-cristobalite (d = 2.33 g cm¹3) and ¢-cristobalite(d = 2.21 g cm¹3). Note that the sixfold rings in ¡-quartzare significantly buckled, those in MFI are slightly buck-led, and those in coesite are rather symmetrical andsquarish. On the other hand, the sixfold rings in FAU andSOD are very symmetrical, and those in glass are also verysymmetrical similarly to ¢-cristobalite, although glassexhibits various ring shapes, as indicated by the broadprofiles in the O-centric PDs. A similar trend is observedin the eightfold rings in ¡-quartz, FAU, MFI, and glass,but the eightfold rings in coesite are symmetrical despitehaving the highest density among the silica polymorphs asmentioned above. On the other hand, the twelvefold ringsin FAU and SOD are very symmetrical and that in FAUexhibits the longest lifetime among the silica polymorphs.It is found that the shapes of the tenfold rings in MFI andglass are similar, although the fraction of such rings is notvery large in the glass.Figure 10 shows typical odd-numbered rings (cycles)in silica glass and MFI obtained by a combination of ringsize distribution and persistent homology analyses togetherwith the lifetime of each ring (cycle) based on the Si–Sicycles, O–O cycles, and Si–O bonds.19) It is found that theshapes of the fivefold and sevenfold rings in MFI are-quartzglass-cristobalite coesite FAU(5.13)(2.52)(3.63)(6.07)(4.75)(5.74)(3.12)(1.60)(2.66)(3.86)(1.93)(3.03)(4.28)(5.99)(4.61)(1.89)(1.22)(2.32)(2.60)(3.78)(3.72)(12.64)(8.13)(9.21)(10.45)(10.02)(10.24)(2.30)(1.50)(2.56)(7.04)(2.99)(4.07)(8.37)(5.80)(6.88)(30.44)(23.22)(24.30)MFI(2.36)(1.03)(2.11)(5.34)(3.09)(4.18)(15.76)(12.11)(13.19)SOD(2.51)(1.73)(2.81)(7.53)(4.97)(6.04)(20.93)(18.34)(19.42)(2.03)(1.61)(2.76)(12.51)(12.49)(12.48)(8.79)(7.13)(8.67)(6.86)(4.89)(6.07)(3.80)(3.94)(3.87)-cristobalite4R 4R 4R 4R4R6R6R6R 6R6R6R6R6R8R8R8R8R12R12R10R10R12R8R8RFig. 9. Typical even-numbered (Si–O)n rings extracted from topological analyses of a series of silicapolymorphs. Red sphere: oxygen; blue sphere: silicon. The conventional ring size distributions of a series of silicapolymorphs are shown in Fig. S2.19)Kohara: Hyperordered structures in silica polymorphsJCS-Japan494identical. Furthermore, similar shapes of rings can be observed in silica glass. It is also found that oxygen atoms inboth the fivefold and sevenfold rings are significantlybuckled, which contributes to the local disorder in thesestructures.A series of our topological analyses suggest that silicaglass is crystallographically an analogue to ¢-cristobalitein terms of the diffraction peak position, as suggested byGaskell and Wallis.46) However, ¢-cristobalite is topolog-ically ordered because it shows only sixfold rings,19) incontrast to the topological disorder in silica glass. In thissection, we compared silica crystals and glass with a seriesof siliceous zeolites in terms of topology and concludedthat silica glass is topologically an analogue to MFI. Asreported by Onodera and coworkers,14,17) a series oftopological analyses can clarify various hidden topologies,which cannot be detected in diffraction data. We con-sider that such hidden topologies are correlate with thefunctionality of oxide materials in terms of order withindisorder.48)4. ConclusionsIn this article, we chose several SiO4 tetrahedral corner-sharing motif crystalline silica and siliceous zeolites inaddition to the densified glasses of various densitiers.Using our approach, we found two descriptors for hyper-ordered structures in silica polymorphs.The first descriptor of the hyperordered structure insilica glass can be expressed by the position and height ofFSDP and PP. We investigated the behaviour of FSDP andPP by tuning pressure and temperature to obtain a sys-tematic understanding of hyperorder. The second descrip-tor of hyperordered structures in silica polymorphs can beexpressed in terms of topological characteristics: ring sizedistribution, cavity distribution, ring shape, and tetrahedralorder without crystallographic information. We found anunusually large cavity volume in ¢-cristobalite and highlysymmetrical eightfold and twelvefold rings in coesite. Wealso found a topological similarity between glass and MFI,in which fivefold and sevenfold rings are observed. Aseries of analyses demonstrate that the way to controltopology is by tuning density through temperature andpressure treatments even where the synthesis processesfor siliceous zeolites49) are completely different from thosefor other silica polymorphs. 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