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2 FIG. 1: Construction of new B2O3 polymorphs: Vertices of sp - carbon structures are replaced either by the BO3 triangle or the B3O6 (boroxol) ring as illustrated here in the case of a graphene layer. The T0 and T0-b polymorphs were obtained by stacking the BO3- and B3O6-substituted layers respectively.

FIG. 2: Examples of nanoporous structures: The BO3-made T7 and dehydration of orthoboric acid (H3BO3)[22] or by seeding a B O melt with borophosphate [23]. its corresponding 3 6-made T7-b structures: large channels, of di- ameter approximately 8.0 A˚ and 11.0 A˚ respectively, are clearly seen. The glassy state has also remarkable anomalies: 1) with only two experimentally known polytypes, well separated inal structures, 13 topologically different B2O3 polymorphs in energy, the ease of vitrification of B2O3 seems at odds were generated which are labeled T0 to T13 in the following. with the correlation mentioned before: a poor polymorphism Among them, T13 is the known B2O3-I crystal. In T8 and should reflect in a poor glass-former; 2) the glass density (1.84 T10, 50 % of the boron atoms belong to three-fold rings, i.e. g.cm−3) is significantly lower than that of the known crys- boroxol rings. Further to expand the search and to investigate talline polymorphs, by a magnitude of 30 and 40 % (2.55 −3 −3 the role of the boroxol ring as a structural motif, 13 additional g.cm for B2O3-I and 3.11 g.cm for B2O3-II); 3) the structures, labeled T0-b to B2O3-I-b, were generated by re- glass medium-range order is markedly different from that of placing the BO3 units by the B3O6 ones. In this way, struc- the crystalline polymorphs: in the glass, a majority of the tures which are made of 100 % boroxol units were obtained, BO3 molecular units form three-fold rings, B3O6, known as taking advantage of the self-similarity between a BO3 and a boroxol rings (Fig. 1) [6, 18, 19]. Strangely, these rings, B3O6 unit (Fig. 1). All structures were then relaxed using which can be viewed as superstructural units, are totally ab- first-principles calculations within the density functional the- sent in both B2O3 crystalline polymorphs. Boroxol rings are ory framework. however found in large amounts in certain borate crystals such −3 The resulting B2O3 polymorphs have low (1.0-2.0 g.cm ) as K3B3O6 or Cs2O-9B2O3 [21]. to very low (< 0.7 g.cm−3) densities. Most of the new crys- The seemingly poor polymorphism of B2O3 compared to silica and the fact that pressure is required to crystallize tals are microporous due to cage- or channel-like structures. The shape of these channels is either circular, rectangular or B2O3-I from the anhydrous melt raise the question whether triangular with a minimum length of typically 6 to 17 A˚ (see B2O3-I is indeed the equilibrium crystalline phase at ambient pressure. Although unknown polymorphs have been proposed Fig. 2 for an example). in the literature [20, 24], none of them can explain consistently Fig. 3a shows the cohesive enthalpy versus density for all the mentioned anomalies since the predicted structures appear the obtained crystals. Very strikingly, many of them fall in a at too high energies in the CEL. This motivated the present narrow energy range which includes the known B2O3-I poly- systematic exploration of the B2O3 CEL. morph. This situation, with many minima in the CEL, is ex- This was done by exploiting the isomorphism between the pected to prevent the nucleation of a given crystal by favoring trigonal network of B2O3 and other three-fold coordinated a disordered structure instead. Therefore, this explains the networks such as sp2-carbon structures (see Methods). We glass anomaly 1) (ease of vitrification). investigated all possible networks (13) with up to six vertices All new predicted structures have densities lower than in the unit cell, as obtained from an exhaustive search origi- B2O3-I and the majority of them fall in the range 1.1-1.7 nally applied to carbon polymorphs [25]. We also considered g.cm−3, i.e. close to the liquid density above and near the −3 a layered structure based upon the topology of graphite (Fig. glass transition temperature (ρliquid(2000 K) ∼ 1.5 g.cm −3 1). By placing BO3 triangle units at the vertices of the orig- and ρglass(540 K) ∼ 1.8 g.cm ). This is consistent with the 3

FIG. 3: Enthalpy as a function of the density for the obtained B2O3 polymorphs: a) at P= 0.0 GPa b) at P = 1.0 GPa, on the same scale as a). The enthalpy reference, H0, is that obtained for B2O3-I and is shown by the horizontal dashed line. The vertical dashed line indicates the glass density. The different symbols and colors refer to the proportion of boroxol rings in the polymorphs. observed amorphisation in a low density glass as the temper- phisation. The degeneracy of B2O3-I with the competing ature is decreased from the liquid state. This thus explains the phases can be lifted by the application of a modest pressure glass anomaly 2). which leaves B2O3-I sufficiently separated in energy from Another result of noteworthy relevance is the fact that the other competitors. majority of the low energy polymorphs are those incorporat- These results reveal a much richer polymorphism in B2O3 ing large amounts (50 or 100 %) of boroxol rings. Indeed, than expected before. In this sense, the behavior of B2O3 the BO3 to B3O6 substitution tend to give polymorphs with appears now to be much more similar to that from other well- lower enthalpies (by ∼ 2.0 kcal/mol on average, up to ∼ 5.5 known glassy systems such as silica. The slight differences kcal/mol in the T3 case) with only two exceptions, T8 and in energy between the polytypes comes from the variability of T10, which already incorporate 50 % of three-fold rings in the the B-O-B angle linking the structural units, just as in silica starting structures. Interestingly, B2O3-I-b, the boroxol-made Si-O-Si angles bridging the tetrahedra. equivalent of B2O3-I, is found among the lowest energy poly- An important issue raised by the present work is whether types. These findings, thus, confirm that boroxol rings are the predicted B2O3 crystals can be experimentally synthe- essential to stabilize low density structures [19] and explain sised and if so, how. All the structures investigated fall within the glass anomaly 3). We note that in the T10 structure, the the energy range of thermodynamically accessible poly- dihedral angle distribution between adjacent ring and non-ring morphs (a debatable quantity, of order of 3 kcal.mol−1) [15]. units is well in line with the one measured in the glass by two- Of course, kinetical or experimental difficulties may however dimensional NMR[26]: this supports the idea that the glass prevent their observation. We note that there are at least two shares structural similarities with the predicted polymorphs in reports in the literature of low density polymorphs[27, 28], the region of energy-density degeneracy. both obtained by dehydration of H3BO3. In the first one[27], The enthalpy penalty with respect to pressure is larger for made in the early attempts to obtain B2O3-I, the structure is a −3 the new polymorphs than it is for B2O3-I as a consequence cubic phase of density 1.805 g.cm . However, in this case, of their low densities. As a result, the new polymorphs be- the fact that others have not been able to repeat the experi- come less stable than B2O3-I at relatively low pressures (Fig. ment remains puzzling. Part of the difficulty may well be due 3b). The enthalpy separation between B2O3-I and the sec- to the fact that in this density region, one needs to avoid the ond lowest polymorph rapidly increases with pressure (∼ 0.8 glass transition resulting from the polymorphic degeneracy in kcal/mol at 0.5 GPa and ∼ 1.3 kcal/mol at 1.0 GPa), gradu- the energy-density diagram. The second[28], much more re- ally leading to a situation in which a monomorphicbehavior is cent, reports an hexagonal phase of density 0.69 g.cm−3, in- expected [15]. This is fully in line with the experimental ob- correctly assigned to B2O3-I and which could well be one of servation that a threshold pressure (0.4-1.0 GPa) is required to our very low density polymorphs (< 0.7 g.cm−3, Fig. 3a). In observe B2O3-I crystallisation from the melt [7–9]. this case, the low dehydration rate used[28] was most proba- Thus, the present results unravel the mystery of the crys- bly a key ingredient. tallisation anomaly: in B2O3, the crystallisation is avoided In light of our predictions, further experimental studies ded- at ambient pressure as a result of the existence of several icated to crystal synthesis would be very much indicated. competing phases which eventually induces the system amor- Among the most promising directions towards the synthesis 4 of low-density structures are sol-gel techniques [29]. Start- proposed in the literature appear at higher energies [24] than ing from an alkyl (R) substituted metaboric acid, R3B3O6, the upper scale of Fig. 3. boroxol-boroxol linkages could be formed via hydrolysis fol- Acknowledgments This work was performed using HPC lowed by either alcohol or water condensation [21]. resources from GENCI-CINES/IDRIS (Grant x2010081875). Our results confirm that the presence of small rings is a very We thank Ph. Depondt and E. Lacarce for critical reading of desirable feature to stabilize low density frameworks [19], the manuscript. This paper is dedicated to the memoryof J.-D. a correlation which was first observed in the context of ze- Pinou. olites [30]. The propensity of boron atoms to form three- membered rings is therefore quite appealing as illustrated by the recent discovery of crystalline covalent organic frame- works (COF), a novel family of high porosity and low density materials [31]: COF incorporate small rings such as borox- [1] M. H. Cohen and D. Turnbull, Nature 189, 131 (1961). ines, i.e. boroxol rings terminated by a radical such as alkyl, [2] M. H. Bhat et al., Nature 448, 787 (2007). alkoxy or triaryl. Thus, our work opens interesting new per- [3] H. Shintani and H. Tanaka, Nat. Phys. 2, 200 (2006). spectives for the search and design of boron-basednanoporous [4] H. W. 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