Glass formation from liquid has been studied extensively, and several theories of glass formation were established in the last century. Zachariasen1 and Sun2 proposed the basic concepts of glass formation by classifying constituents into glass formers, glass modifiers and intermediates. Furthermore, Angell3 introduced the concept of ‘fragility’ in glass-forming liquids (GFL) on the basis of the relationship between glass transition temperature and viscosity: liquids can be classified as ‘strong’ and ‘weak’ according to their glass-forming ability. Numerous structural studies on liquids and glasses have been performed both experimentally and theoretically4,5. The advent of advanced synchrotron/neutron sources and the development of high-performance computers have led to great progress in understanding liquid and glass structure4,5. The structural analysis of liquids with high melting points has been advanced significantly with the invention of the levitation technique6, especially in combination with diffraction techniques6. The structure of a typical non-GFL, liquid (l-) Al2O3 and its undercooled liquid have been studied extensively by X-ray diffraction7,8,9, neutron diffraction9,10 and molecular dynamics (MD) simulations8,9,11,12.

ZrO2 is one representative of non-glass formers and there are few reports about binary13 and ternary14 glass formation including ZrO2. Moreover, ZrO2 is commonly used as a refractory material and nucleating agent15 in the production of glass ceramics, suggesting that l-ZrO2 is indeed a non-GFL. As ZrO2 has an extremely high melting point (Tm=2,715 °C), the difficulties in handling the liquid at high temperatures lead to problems in selecting suitable container materials that avoid contamination effects. We have developed a beamline levitation furnace that enables us to perform precise synchrotron X-ray diffraction measurements of liquids at extremely high temperatures.

We report here the results of precise high-energy X-ray diffraction and density measurements on containerless levitated l-ZrO2. We also carry out large-scale density functional (DF)–MD simulations, to understand the liquid properties at the atomistic and electronic level, and we compare l-ZrO2 with other non-GFLs and a typical GFL, l-SiO2. The combination of experiment and theory allows us to identify trends in single-component non-glass-forming oxide liquids, with particular focus on short- and intermediate-range ordering, the electronic band gap, maximally localized Wannier functions (WFs) of the highest valence band orbitals, and viscosity. Furthermore, we compare features of single-component non-glass-forming oxide liquids with those of other systems.


Structure factors and real-space functions

Faber–Ziman16 total structure factors, S(Q), for l-ZrO2 at 2,600 °C–2,800 °C are shown in Fig. 1a. The structural change between the liquid at 2,800 °C and the undercooled liquid at 2,600 °C is very small. A sharp peak is observed at Q=2 Å−1. The total correlation functions, T(r), for l-ZrO2 (Fig. 1b) show subtle differences in real space as well. The first correlation peak observed at ~2.1 Å is assigned to Zr–O correlation and a significant tail to ~3 Å imply the formation of asymmetrical ZrOn polyhedra in the liquid. The second peak observed at ~3.7 Å can be assigned mainly to Zr–Zr correlations and the contribution of O–O correlation is unclear due to its small weighting factor (<10%) for X-rays. The Zr–O correlation length of 2.1 Å is significantly longer than those of Si–O (~1.63 Å17 at 1,600 °C–2,100 °C) and Al–O (~1.78 Å9 at 2,127 °C) due to substantial differences in ionic radius between silicon, aluminum and zirconium ions18.

Figure 1: X-ray diffraction data for l-ZrO2.
figure 1

(a) Faber–Ziman total structure factors, S(Q), for l-ZrO2 at 2,600 °C–2,800 °C together with the S(Q) derived from the DF–MD simulation at 2,800 °C. Both the experimental and DF–MD simulation data at 2,800 °C are displaced upward by 1 for clarity. (b) Total correlation functions, T(r), for l-ZrO2 at 2,600–2,800 °C. The sharp peak observed at Q=2.1 Å−1 in the S(Q) of l-ZrO2 can be assigned to the principal peak reported by Salmon et al.21

The observed correlation length of 2.1 Å for Zr–O agrees well with recent experimental data of Skinner et al.19 Furthermore, the increased cation–oxygen correlation length in l-ZrO2 indicates that the oxygen coordination number around zirconium is >4, because 2.1 Å is close to the sum of the ionic radii of oxygen (1.35 Å18) and sixfold zirconium (0.72 Å18). The intermediate-range structure of l-ZrO2 is then made up of large, interconnected polyhedral units and very different from those of l-SiO2 and l-Al2O3. This implies that the peak observed at Q=2 Å−1 in the S(Q) in Fig. 1a is not the first sharp diffraction peak (FSDP), which is typically associated with intermediate-range ordering in disordered materials, so that there is no such ordering in l-ZrO2.

The total structure factor, S(Q), obtained from the DF–MD simulations at 2,800 °C is shown in Fig. 1a as a magenta curve. The agreement with experimental data is excellent. Additional insight into the intermediate-range ordering of l-ZrO2, in comparison with l-SiO2 and l-Al2O3, can be found by calculating the Bhatia–Thornton20 number–number partial structure factor, SNN(Q),

where Sij(Q) is a Faber–Ziman partial structure factor (see Supplementary Fig. 1) and ci denotes the atomic fraction of chemical species i21. Figure 2a shows SNN(Q) (see Supplementary Fig. 2) of l-ZrO2 at 2,800 °C compared with those of l-Al2O3 at 2,127 °C9 and l-SiO2 at 2,100 °C22. Only l-SiO2 exhibits FSDP at QrAX=2.7 (rAX is the atomic cation (A)–anion (X) distance in AX polyhedra to normalize Q). Neither l-Al2O3 nor l-ZrO2 show an FSDP in the SNN(Q), whereas it is present in the total X-ray and neutron S(Q) of l-Al2O3 (see Supplementary Fig. 3)9 due to the contribution of weighting factors for X-rays and neutrons. As Bhatia–Thornton SNN(Q) can eliminate the weighting factors, the absence of FSDP in the SNN(Q) of l-ZrO2 is a signature of non-glass-forming behaviour.

Figure 2: Comparison of the structural data for oxide liquids.
figure 2

(a) The Bhatia–Thornton number–number partial structure factor, SNN(Q), for l-ZrO2 at 2,800 °C derived from the DF–MD simulation (black curve) in comparison with those of l-Al2O3 at 2,127 °C (red curve)9 and l-SiO2 at 2,100 °C (blue curve)22. The momentum transfer Q was scaled by rAX, where rAX is the first coordination distance between A and X in the real-space function. The Bhatia–Thornton concentration–concentration partial structure factor, SCC(Q), and number–concentration partial structure factor, SNC(Q), are shown in Supplementary Fig. 2 together with the detailed note (Supplementary Note 1). (b) The coordination number distribution of oxygen around the cations in l-ZrO2 at 2,800 °C, l-Al2O3 at 2,127 °C9 and l-SiO2 at 2,100 °C22. (c) The polyhedral connections in l-ZrO2 at 2,800 °C, l-Al2O3 at 2,127 °C9 and l-SiO2 at 2,100 °C22. CS, corner sharing of oxygen; ES, edge sharing of oxygen; FS, face sharing of oxygen.

The partial pair correlation functions, gij(r), of l-ZrO2 derived from the DF–MD simulations are presented in Supplementary Fig. 4, together with those of the high-temperature phase of crystalline (c-) ZrO2 (ref. 23). The first correlation peak of gZrO(r) for l-ZrO2 is broad and shows a tail up to 2.8 Å. A very broad first maximum in gOO(r) overlaps the first correlation peaks of gZrO(r) and gZrZr(r), indicating that the oxygen coordination is very different from those of l-SiO2 and l-Al2O3, where corner-sharing tetrahedra are predominant.

Analysis of three-dimensional atomic arrangement

The average coordination number of oxygen around zirconium, NZrO, calculated up to 2.8 Å for l-ZrO2 is 5.9 for the DF–MD configuration (NOZr~3), significantly lower than 8 in c-ZrO2. Recently, Skinner et al.19 reported that NZrO=6.1 at 2,897 °C, which is close to our value (5.9 at 2,800 °C), although the estimated density in ref. 19 is smaller than our experimentally measured value. The oxygen coordination number around zirconium is significantly larger than NSi-O=3.9 in l-SiO2 at 2,100 °C22 and NAl-O=4.4 in l-Al2O3 at 2,127 °C9, due to large differences between the ionic radii of Si, Al and Zr. Similarly, large oxygen numbers reported on other non-GFLs (NY-O=5.5 in l-Y2O3 at 2,597 °C19 and NLa-O=5.9 in l-La2O3 at 2,497 °C19) support our argument that a large oxygen coordination in l-ZrO2 is a signature of non-GFLs and is associated with the absence of FSDP.

The coordination number distributions of l-SiO2 (ref. 22), l-Al2O3 (ref. 9) and l-ZrO2 calculated from the structural models are shown in Fig. 2b. SiO4 tetrahedra are predominant in l-SiO2 (ref. 22), while l-Al2O3 comprises AlO3, AlO5 and AlO6 units, as well as fourfold Al. For l-ZrO2, the most common configurations are ZrO5, ZrO6 and ZrO7. Although ZrO2 and Al2O3 have different stoichiometries, this comparison supports our view that the variety of oxygen coordination around cations in l-ZrO2 is another characteristic feature of non-glass-forming behaviour, because it can disturb the evolution of intermediate-range ordering.

To obtain structural features beyond the first coordination distance, a polyhedral connection analysis for l-SiO2 (ref. 22), l-Al2O3 (ref. 9) and l-ZrO2 has been carried out. Figure 2c shows the fraction of corner sharing, edge sharing and face sharing of polyhedral units in the liquids. Corner sharing of oxygen is prevalent in l-SiO2 (ref. 22), which is a unique feature of GFLs according to Zachariasen1. However, both l-Al2O3 and l-ZrO2 exhibit a considerable oxygen edge sharing, so that the variety of polyhedral connections is a further characteristic feature of single-component, non-glass-forming oxide liquids.

The bond angle distributions for l-ZrO2 at 2,800 °C derived from the DF–MD simulations are shown in Fig. 3. The peak at ~105° in the Zr–O–Zr distribution is very different from that at ~145° in the Si–O–Si distribution of l-SiO2 (ref. 22), but similar to the Al–O–Al distribution in l-Al2O3 (ref. 9). The O–Zr–O distribution shows a principal peak at 75° and a small peak at 150°. They are very different from the angle of a typical AX4 tetrahedron (109.47°), but similar to the O–Al–O distribution in l-Al2O3 (ref. 9). This indicates that the wide variation of ZrOn polyhedra (ZrO5, ZrO6 and ZrO7) is another characteristic feature of l-ZrO2. The distributions of Zr–Zr–Zr, Zr–Zr–O and O–O–Zr are also similar to those of l-Al2O3 (ref. 9). The O–O–O distribution of l-ZrO2 shows a maximum only at 60°, while the O–O–O distribution of l-Al2O3 (ref. 9) has a prominent peak at 60° and a small but distinct peak at ~110° (signature of an anion in tetrahedral coordination). However, l-ZrO2 does not have this feature, implying that its oxygen coordination is different.

Figure 3: Bond angle distributions for l-ZrO2 at 2,800 °C and c-ZrO2.
figure 3

(a) The bond angle configurations Zr–Zr–Zr, (b) Zr–O–Zr, (c) Zr–Zr–O, (d) O–Zr–O, (e) O–O–Zr and (f) O–O–O. The B(θ)/sinθ data for c-ZrO2 has been scaled by a factor of 20 for clarity.

The bond angle distributions for the high-temperature phase of c-ZrO2 (ref. 23) are shown in Fig. 3 as red lines. Although there are similarities in the liquid and crystal, the most striking difference is in the O–O–O distribution: l-ZrO2 has a prominent peak at 60°, while c-ZrO2 shows intense peaks at 90° and 174°. This difference is caused by the variation of short-range ordering in the ZrOn polyhedral units.

Analysis of electronic structure

The electronic structure analysis was performed in terms of the electronic density of states (DOS), WFs and effective charges for snapshots of the high-temperature phases of c-ZrO2 and l-ZrO2. The DOS (above −20 eV) of c- and l-ZrO2, and its projections (P-DOS) for l-ZrO2 are shown in Fig. 4a. The P-DOS plots reveal that this part of the electronic spectrum is associated mainly with oxygen (O-2p orbitals) and the Zr semicore states corresponding to the atomic Zr-4s and Zr-4p orbitals are deeper (below −20 eV, not shown). The zirconium d-component dominates in the conduction band of l-ZrO2. The effect of high temperature on the distorted ZrOn polyhedra in l-ZrO2 is evident as a broadening of the energy bands and the gap between the valence and conduction bands has disappeared (the calculated band gap is 3.26 eV for c-ZrO2).

Figure 4: The electronic structure of c-ZrO2 and l-ZrO2.
figure 4

(a) DOS and its projections onto atomic orbitals for the higher valence bands and conduction band, together with the Wannier function spreads (occupied states). (b) The partial pair correlation functions, gij(r), of WF centres, Zr–Cw (black), O–Cw (red) and CwCw (blue). (c) A visualization of the HOMO state (KS orbital) in l-ZrO2. (d) Two WFs corresponding to HOMO and HOMO-1 of l-ZrO2. Zr and O atoms are shown in cyan and red colour, respectively. Yellow and blue isosurfaces denote different signs of the wavefunction nodes.

The difference between the electronegativities of Zr (1.3) and O (3.5) indicates that the chemical bonding between the two elements is mainly ionic and this is supported by the significant weight on oxygen P-DOS for the highest valence band. Table 1 summarizes the atomic charges and volumes of Zr cations and O anions in c- and l-ZrO2 obtained by the Bader method24. For l-ZrO2, the evaluated effective charges are +2.62e and −1.31e for Zr and O, respectively, and reflect the ionic bonding. The atomic charges in l-ZrO2 are very similar to those in the crystalline phase, which is in accordance with our previous studies on MgO–SiO2 (ref. 25) and CaO–Al2O3 (ref. 26) glasses. The associated atomic volumes imply that the increased oxygen volume in l-ZrO2 compensates for the decreased oxygen coordination, and this results in comparable atomic charges for the two phases. Similar behaviour has been found for MgO–SiO2 glass25.

Table 1 Atomic charges and volumes obtained by the Bader method.

The maximally localized WF can be considered as the natural generalization of ‘localized molecular orbitals’ in solids and they provide valuable insight into chemical bonding27,28. The WFs have been obtained from the occupied Kohn–Sham (KS) orbitals by a unitary transformation where the spatial extension (spread) of the WF orbitals is minimized. For each WF orbital, the Wannier centre (Cw) location denotes the most probable point for locating an electron (or electron pair in case of a spin-degenerate orbital) and the corresponding Wannier spread is a direct measure of the degree of localization. The distribution of Wannier spreads and Cws in terms of pair correlation functions are shown in Fig. 4a,b, respectively. The gij(r) for Zr–Cw, O–Cw and CwCw (Fig. 4b) have maxima well below 1 Å, showing that Cws are close to Zr and O atoms. The ionic character of chemical bonding is clearly visible along Zr–O bonds, where the associated WF centres (electron pairs from higher valence bands) are always close to oxygen and there are four Cws around each O. In the high-temperature phase of c-ZrO2, the corresponding Cws are symmetrically aligned along Zr–O bonds, as O is tetrahedrally coordinated by Zr (Fig. 4b, O–Cw partial). The oxygen coordination is smaller (NOZr~3) and less regular in the liquid, and there is considerable scatter in both the Cw locations and spreads. The latter show (Fig. 4a, bottom panel) that WFs are considerably less localized than the crystalline reference value of ~2.9.

The highest occupied molecular orbital (HOMO) has been also visualized in Fig. 4c, where the KS orbital (HOMO) is delocalized over a group of atoms highlighted by a dashed circle, while the transformed WFs for HOMO and HOMO-1 (‘molecular orbitals’, Fig. 4d) are each localized over one Zr–O bond. The WF shapes of these examples are very similar to those in the high-temperature phase of c-ZrO2, but there are also cases with considerable deviation, as can be expected from the scatter of WF spreads (Fig. 4a).


The origin of FSDP associated with the formation of intermediate-range ordering in oxide glasses and liquids remains controversial, because the inherent disorder complicates the ability of AX polyhedral connections to form an A–X network. SiO2 has an exceptionally high glass-forming ability and the origin of FSDP in SiO2 has often been studied. The results are summarized in ref. 29. The random network model of Zachariasen1 and modified for an oxide glass modified in refs 30, 31 (illustrated in Fig. 7 of ref. 29) suggests that the intermediate-range ordering arises from the periodicity of boundaries between successive small cages in the network formed by connected, regular SiO4 tetrahedra with shared oxygen atoms at the corners. It has also been demonstrated that small cages are topologically disordered32, resulting in a broad distribution of ring sizes from 3-fold to 12-fold rings centred at 6-fold rings25,33. This is reflected in the SNN(Q) of l-SiO2 (Fig. 2a), where the FSDP width is broader than that of the corresponding Bragg peak in the crystalline phase (β-cristobalite, c-SiO2), where only a sixfold ring donates. Figure 5a,b show three-dimensional atomic configurations and schematic illustrations for c-SiO2 and l-SiO2, respectively. The crystalline phase exhibits only sixfold rings of six SiO4 tetrahedra, yielding a long-range periodicity (dashed cyan lines in Fig. 5a). However, some pseudo Bragg planes (dashed cyan lines in the left panel of Fig. 5b) can be recognized in l-SiO2. Although the introduction of different ring sizes can easily modify the crystalline topological order (Fig. 5b), the interconnection of regular SiO4 tetrahedra with shared oxygen at corners only yields the broadened Bragg peak as FSDP.

Figure 5: The atomic structure of oxide liquids.
figure 5

(a) c-SiO2 (for reference), (b) l-SiO2, (c) l-Al2O3 and (d) l-ZrO2. Colour code (right panels): Si, magenta and blue spheres; Al, red; Zr, black; and O, white. The periodicity of cage boundaries is highlighted by cyan dashed lines and curves.

In l-Al2O3, a considerable fraction of AlO5 units associated with the formation of OAl3 triclusters34 and the contribution of edge-sharing atoms (see Fig. 5c)9 are necessary to compensate the negative charge of AlO4, because the nominal charge of the Al cation is three. We suggest that the variety of oxygen coordination (see Fig. 2b) and polyhedral sharing (Fig. 2c) disturb the formation of intermediate-range ordering in l-Al2O3. This is apparent in the absence of FSDP in the SNN(Q) of l-Al2O3 in Fig. 2a, despite the similarity between l-SiO2 and l-Al2O3 due to the predominant AlO4 units and corner sharing of oxygen in l-Al2O3 (note that the stoichiometry of Al2O3 is different from that of SiO2). The three-dimensional atomic configuration and schematic illustration of l-Al2O3 are illustrated in Fig. 5c, where the periodicity of boundaries is less obvious, due to the large contribution of AlO5 (purple polyhedra) and edge sharing of oxygen.

As can be seen in Fig. 2a, an FSDP is absent in the SNN(Q) of l-ZrO2, because the variety of short-range structural units with large oxygen coordination, ZrO5, ZrO6 and ZrO7, and the large contribution of oxygen edge sharing prevents the formation of intermediate-range ordering. A similar feature can be expected in l-Y2O3 and l-La2O3, because their Faber–Ziman partial structure factors, Sij(Q), do not contribute to the expected Q position of~1 Å−1 for a FSDP19. Short-range structural disordering in l-ZrO2 is further demonstrated in the three-dimensional atomic configuration and the schematic illustration of l-ZrO2 (Fig. 5d). The periodicity of boundaries (dashed cyan lines) is suppressed by the formation of edge-sharing of oxygen associated with the formation of ZrO5 (white polyhedra), ZrO6 (grey polyhedra) and ZrO7 (black polyhedra). Although ZrO2 forms a network structure by interconnecting AX polyhedra in the liquid phase, we have shown that the various short-range structural units and their connectivity cause disorder at the intermediate range and prevent the evolution of a FSDP in the liquid. Our results demonstrate that the absence of FSDP in SNN(Q) can be an important indicator for single-component non-glass-forming oxide liquids, but it does not necessarily apply similarly to other non-GFLs.

Although ZrO2 and Al2O3 have different stoichiometries, the absence of FSDP in the SNN(Q) suggests that they are both very ‘fragile’ liquids3. This suggestion is supported by a comparison with l-ZnCl2, which is recognized as an intermediate case between a ‘strong’ and ‘fragile’ liquid. l-ZnCl2 shows a well-defined, but not sharp, FSDP in the SNN(Q) by the contribution of corner sharing of ZnCl4 tetrahedra, while edge sharing is also found35. This behaviour of ‘fragile’ glass former is very similar to l-GeSe2 where the FSDP in SNN(Q) is weak, and a considerable fraction of edge sharing of GeSe4 tetrahedra contribute36, as in glassy (g-) GeSe2 (ref. 37). We suggest that the liquid fragility increases with the contribution of edge sharing of tetrahedra, as discussed in ref. 38. However, the typical glass former g-GeO2 shows the contribution of only corner sharing of GeO4 tetrahedra33 that results in a sharp FSDP, while g-Ge-Te systems do not exhibit FSDP due to co-existing octahedral and tetrahedral Ge–Te polyhedra. Here the Te–Ge–Te bond angle distribution is peaked around 90°, quite different from 109.47° of regular tetrahedra in g-GeO2 (refs 39, 40, 41). We conclude that the magnitude of the FSDP is sensitive to the variety in atomic coordination and polyhedral connections, which are connected in turn to the difference in ionic radii between constituent anions and cations.

The fragility of l-ZrO2 is confirmed by comparing the thermodynamic parameters of l-ZrO2 and l-Al2O3. Recent MD simulation for l-Al2O3 at 2,227 °C42 reported a zero-frequency viscosity of 2.5 × 10−2 Pa s−1, while the viscosity of l-SiO2 at 1,652 °C (a typical ‘strong’ liquid3) is 6.12 × 106 Pa s−1 (ref. 43). The zero-frequency viscosity value of l-Al2O3 is comparable to the results of the recent inelastic X-ray scattering measurement44 and the macroscopic shear viscosity of 3.3 × 10−2 Pa s−1 at 2,213 °C45, confirming that it is a ‘fragile’ liquid. The self-diffusion coefficients for Zr and O in l-ZrO2 derived from our DF–MD simulations are 3.6 × 10−5 and 7.1 × 10−5 cm2 s−1, respectively, at 2,800 °C. The viscosity of l-ZrO2, estimated by assuming spherical particles and applying the Stokes–Einstein equation, is ~2 × 10−3 Pa s−1 at 2,800 °C, indicating that l-ZrO2 is an extremely fragile liquid. This conclusion is further supported by the Zr–O bond lifetime analysis of DF–MD simulations (Supplementary Note 2), which shows that 50% of the bonds break within 185 fs at 2,800 °C (Supplementary Fig. 5). The Zr–O bond lifetime is extremely short when it is compared with the observation that the exchange-rate between bridging and non-bridging oxygen atoms in a silicate melt is within a nanosecond-to-microsecond time scale46. This behaviour of Zr–O bonds is closely related to the variety of ZrOn polyhedral units and polyhedral connections with a reduced electronic band gap and increased delocalization in the ionic Zr–O bonding.

We study the atomic and electronic structure of non-glass-forming l-ZrO2 with an extremely high melting point by using a combination of containerless processing, synchrotron X-ray diffraction, density measurements and DF–MD simulations. Although a sharp peak is observed in the X-ray total structure factors, we find that FSDP is absent in the Bhatia–Thornton SNN(Q). We show that the variety of short-range structural units with large oxygen coordination and their associated distortion due to edge sharing are signatures of single-component non-glass-forming oxide liquids. The absence of FSDP is ascribed to the variety of ZrOn polyhedral units induced by the large ionic radius of Zr cation. This is reflected in the short lifetime of Zr–O bonds (and polyhedral units), which prevents the evolution of intermediate-range ordering. These structural features are coupled to irregularity and reduced localization in the ionic Zr–O bonds with short lifetime, yielding a reduced electronic band gap in l-ZrO2 and a low viscosity of the liquid. By comparing our results for l-ZrO2 with other GFLs, non-GFLs and glasses, we conclude that the absence of FSDP in the SNN(Q), associated with a short lifetime of Zr–O bonds and extremely low viscosity, is a feature of single-component non-glass-forming oxide liquids, although this does not necessarily apply to all non-GFLs. The DF–MD simulation results support the observed absence of FSDP and suggest that l-ZrO2 is an extremely fragile liquid. Finally, the containerless preparation and measurement techniques open up fresh capabilities to study new features in extremely high-temperature liquids, and we demonstrate the importance of combining experiment and theory to understand the nature of liquids at the atomistic (structure and dynamics) and electronic (chemical bonding) level.


Density measurement

The density measurement of l-ZrO2 was performed with an aerodynamic levitator6,47,48. A small ZrO2 sample whose diameter was around 2 mm was set in a shallow nozzle where the sample was aerodynamically levitated. The levitated sample was then heated by a 100-W CO2 laser and a 500-W Nd:YAG laser. The temperature of the sample was measured by a single colour pyrometer. The weight of the recovered sample was measured. The temperature was calibrated using the given melting temperature (Tm=2,715 °C) in density measurements. The details of measurement can be found in the Supplementary Note 3 and typical image of the levitated specimen and the density data is shown in Supplementary Figs 6 and 7, respectively.

High-energy X-ray diffraction measurement

The high-energy X-ray diffraction experiments of l-ZrO2 were carried out at the BL04B2 and the BL08W beamlines49 at the SPring-8 synchrotron radiation facility, using the aerodynamic levitation technique6,47,48,49. The energy of the incident X-rays was 113 keV (BL04B2) and 116 keV (BL08W). The ZrO2 sample of 2-mm size was levitated by dry air and heated by a 100-W CO2 laser. The temperature of the sample was monitored by a two-colour pyrometer. As can be seen in Supplementary Fig. 8, the background of the instrument was successfully reduced due to adequate shielding of the detector and the optimization of beam stop. The measured X-ray diffraction data were corrected for polarization, absorption and background, and the contribution of Compton scattering was subtracted using standard analysis procedures49. The corrected data sets were normalized to give the Faber–Ziman16 total structure factor S(Q) and the total correlation function T(r) was obtained by a Fourier transformation of S(Q).

DF–MD simulation

The combined DF and MD simulations were performed with the CP2K programme package ( The CP2K method uses two representations for the KS orbitals and electron density: localized Gaussian and plane wave basis sets. For the Gaussian-based (localized) expansion of the KS orbitals, we used a library of contracted molecularly optimized valence double-zeta plus polarization basis sets51 and the complementary plane wave basis set for electron density has a cutoff of 400 Ry. The generalized gradient approximation of Perdew, Burke and Ernzerhof52 (PBE) was adopted for the exchange-correlation energy functional and the valence electron–ion interaction was based on the norm-conserving and separable pseudopotentials of the form derived by Goedecker et al.53 We consider the following valence configurations: Zr (Cl4s24p25s25d2) and O (2s22p4). Periodic boundary conditions were used, with a single point (k=0) in the Brillouin zone. Effective charges of individual atoms were evaluated from electron density by integrating electronic charge inside the corresponding atomic volumes24. For reference, electronic structure of the high-temperature phase of c-ZrO2 (T>2,370 °C) is computed for a sample of 324 atoms.

The initial atomic configuration is created by a reverse Monte Carlo (RMC) simulation with high-energy X-ray diffraction data on 501 atoms in a cubic box of 18.98 Å (experimental density, 0.0733 atoms per Å3). The RMC++ programme code54 was used. The Born–Oppenheimer MD simulations were performed with a Nóse–Hoover thermostat55 and time steps of 2 fs (initialization) and 1 fs (data collecting). The system was simulated at 3,100 K (~2,800 °C) for a total of 30 ps, where the last 10 ps were used for data collection55. The corresponding mean-square displacement of atoms shows clearly a liquid behaviour (diffusion). The comparison of the partial pair correlation functions, gij(r), between the initial RMC configuration (start) and the DF–MD simulation is shown in Supplementary Fig. 9. The sharp shape of O–O gij(r) in the RMC model is artificially sharp due to small weighting factor for X-rays, while the shape of O–O gij(r) is reasonable in the DF–MD simulations. The system lost its memory of the initial (RMC) starting structure within a few picoseconds (Zr–O bond lifetimes ~185 fs).

Additional information

How to cite this article: Kohara, S. et al. Atomic and electronic structures of an extremely fragile liquid. Nat. Commun. 5:5892 doi: 10.1038/ncomms6892 (2014).