Introduction

Determining the liquid structure is the first step in understanding the nature of glass-liquid transitions. However, a diffraction measurement of liquid provides very limited structural information because the liquid structure lacks long-range periodicity, and a Fourier transform of the diffraction data provides only pairwise correlations. Moreover, high-quality measurements are difficult to obtain at high temperatures.

Since glasses play an important role in technology, glass formation has been studied extensively. 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 (GFLs). He interpreted the strong and fragile behavior of liquids in terms of topological differences in potential energy hypersurfaces of the configuration space. Typical strong liquids are SiO2, GeO2, and B2O3. Their networks are covalently bonded, and the viscosities show an Arrhenius temperature dependence. In contrast, typical fragile liquids are chalcogenides and iron phosphates, the networks of which are mostly ionic and the viscosities of which deviate significantly from the Arrhenius behavior. Many experimental and theoretical structural studies of liquids and glasses have been performed, and with the advent of advanced synchrotron and neutron sources and the development of high-performance computers, they have led to great progress in our understanding of liquid and glass structures4,5. The structural analysis of liquids with high melting points has advanced significantly with the advent 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 diffraction7,8,9,10, neutron diffraction9,10,11, and molecular dynamics (MD) simulations9,10,11,12,13. In addition to the l-Al2O3 structure, several structures of molten pure oxides with high melting points (Tm) have been studied recently. For example, structures of UO214 and compounds in the UO2–ZrO2 system15 have been investigated for nuclear reactor accidents. The structures of ZrO216,17,18, HfO216, and lanthanide oxides17 have also been investigated to understand the fundamental properties of high-temperature liquids. Although these investigations are very important not only for materials science but also for preventing severe accidents, the research methods and data are still limited by the high melting points of the materials in question.

Er2O3 is a representative nonglass former that is commonly used as a refractory material and dopant for luminescent materials. Because Er2O3 has an extremely high melting point (Tm = 2686 K), the difficulties in handling the liquid lead to problems in selecting suitable container materials that do not contaminate the sample. To avoid contact with other materials, levitation furnaces have been developed that enable us to measure precise synchrotron X-ray diffraction and thermophysical properties for liquids at extremely high temperatures6.

This article presents the results of accurate high-energy X-ray diffraction and density measurements on containerless levitated l-Er2O3 using an electrostatic levitation furnace (ELF) at the International Space Station (ISS)19, as it is impossible to measure density data on the ground. We also perform reverse Monte Carlo – molecular dynamics simulations and obtain persistence diagrams from topological analyses to demonstrate liquid properties at the atomic level, comparing l-Er2O3 with other non-GFLs and a typical GFL, l-SiO2. Furthermore, a sample of l-Er2O3 is simulated for a short period with the density functional – molecular dynamics method to investigate the electronic structure and to obtain a realistic estimate of the viscosity above the melting point. The combination of an experiment and a simulation allows trends in single-component nonglass-forming liquid oxides to be identified, with a focus on atomic ordering and topology. Furthermore, the article compares the features of single-component nonglass-forming oxide liquids with those of other systems.

Materials and methods

Density measurement

The density of liquid (l-) Er2O3 was measured with an ELF at the ISS. A sample of 2 mm in diameter was prepared by melting Er2O3 powder with a purity of 99.99% and solidifying it in an aerodynamic levitator. It was charged by friction or contact with other materials in the ISS-ELF and then levitated to the center between six electrodes that applied a Coulomb force. The sample position was stabilized by tuning the voltages between electrodes at 1000 Hz and monitoring the image of the sample backlit by a He–Ne laser. The levitated sample was heated and melted by four 40 W semiconductor lasers (980 nm) under 2 atm of dry air. The temperature of the sample was measured by a pyrometer (1.45–1.8 μm). It was calibrated using an emissivity calculated from the plateau temperature at recalescence and the reference value of the melting point (2686 K). After melting, the nonspherical sample became spherical upon cooling after shutting off the lasers. During cooling, the sample image was observed by an ultraviolet back light and a CCD camera. The pixel size was calibrated against an image of 2.0 mm stainless steel spheres, which were recorded under the same conditions as the sample. The sample volume was calculated from its diameter, obtained from the image. Then, the density was calculated from the volume and weight.

High-energy synchrotron X-ray diffraction measurement

The high-energy X-ray diffraction measurement of l-Er2O3 was performed at the BL04B2 beamline20 of SPring-8 using an aerodynamic levitator21. The energy of the incident X-rays was 113 keV. The 2-mm Er2O3 sample was levitated in dry air and heated by a 200 W CO2 laser. The temperature of the sample was monitored by a two-color pyrometer. The background of the instrument was successfully reduced by shielding the detector and by optimizing a 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 procedures22. The corrected data sets were normalized to give the Faber–Ziman23 total structure factor S(Q), and the total correlation function T(r) was obtained by a Fourier transform of S(Q).

Molecular dynamics – reverse Monte Carlo simulation

To determine the atomic configuration of l-Er2O3, a molecular dynamics – reverse Monte Carlo (MD-RMC) simulation was performed with 5000 particles in a cube to reproduce the X-ray S(Q). The MD simulation was carried out with a Born-Mayer type of pairwise potential with a Coulomb interaction and a repulsive component, given by the following equation:

$$U_{ij}\left( r \right) = \frac{{e^2}}{{4\pi \varepsilon _0}}\frac{{Z_iZ_j}}{r} + B_{ij}\exp \left( { - \frac{r}{{\rho _{ij}}}} \right),$$
(1)

where r is the interatomic distance, Z is the effective charge, B is the repulsion, e is the elementary charge (ZEr = 2.1, ZO = −1.4), ε0 is the permittivity of the vacuum, and ρ is the softness parameter. The parameters used in the MD simulation are summarized in Table 1.

Table 1 The parameters for the Born-Mayer potential used in the MD simulation.
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The simulations were carried out for a system of 2000 Er and 3000 O atoms in the unit cell with a random atomic configuration. The cell volume was determined from the number densities of l-Er2O3 at the melting point, which were calculated with the density measured by the ISS-ELF. Periodic boundary conditions were used, and the long-range Coulomb interaction was treated with Ewald’s summation. A time step of 1 fs was used in the Verlet algorithm. First, the temperature of the system was maintained at 4000 K for 20,000 time steps and then cooled to 2923 K over 20,000 steps. The structural model was finally annealed at 2923 K for 150,000 steps. After the MD simulation, RMC refinement was conducted using the RMC++ code24. The benchmark RMC runs were performed using simulation boxes with 250, 500, 1000, and 3000 particles.

Density functional – molecular dynamics simulation

The simulations based on the density functional theory (DFT) of the electronic structure were performed with the projector augmented wave (PAW) method25, implemented in the VASP software26,27. The PAW potentials supplied within VASP for Er (5p, 5d, and 6s, with 11 4f electrons frozen in the core) and O (2s, 2p) were tested and used (see supplementary information for a comparison between the frozen core approximation and the treating of the Er-4f electrons explicitly, Fig. S1). For a liquid sample of 500 atoms, the energy cutoff of the plane waves was set to 400 eV, with a single Γ-point in the Brillouin zone. In comparison, the bulk crystalline (c)-Er2O3 unit cell was fully relaxed until the forces on all atoms were below 0.01 eV/Å with a Γ-centered 2 × 2 × 2 k-point grid and a plane-wave energy cutoff of 550 eV. The PBE functional28 was used for the geometry optimization and the molecular dynamics simulations, whereas the HSE06 hybrid functional29 was used to obtain the electronic densities of states (DOSs) and their projections to produce more realistic electronic band gaps and test the effect of the 4f electrons in c-Er2O3. The effective charges and atomic volumes were evaluated by Bader analysis30,31,32 using the PBE functional.

The density functional – molecular dynamics (DF-MD) simulations were performed with a Nóse-Hoover thermostat33 and a time step of 2 fs, with an initial atomic configuration given by the benchmark RMC model mentioned above with 500 atoms. The system was simulated at 2923 K (~2650 °C) for a total of 30 ps, where the last 25 ps were used for data collection (Fig. S2). The electron occupancy was described with a Fermi smearing corresponding to the kBT value at the target temperature. The mean-square displacements (MSDs) of the atoms show a liquid (diffusion) behavior where equilibrium is already achieved during the first few picoseconds.

Topological analysis using a persistent homology

The homology of atomic configurations has been investigated using the persistence diagram D1, which consists of two-dimensional histograms showing a persistent homology. The details of the analysis are described elsewhere34. The persistence diagram D1 of a set of atoms is given by the following thickening process of spheres: (1) place a sphere with a radius r at the center of each atom, (2) increase the radii of the spheres from 0 to a sufficiently large value, and (3) encode the pair of birth and death radii (bi, di) for each ring ci consisting of a set of spheres. The persistence diagram is then constructed by the two-dimensional histogram on the birth and death plane obtained by the pairs for independent ci, i = 1,…, K. Here, the birth (death) radius is defined as the radius of spheres at which the ring ci first appears (disappears). The birth radius has information about the distances between atoms of the ring ci, and the death radius has information about the size of the ring. The persistence diagram provides statistical information on the shapes of all independent rings and thereby provides insight into intermediate ordering in the liquid structure. The rings and cavities detected by this process are recorded for the computation of the persistence diagrams; hence, their geometric shapes can be identified for further analysis. The persistence diagrams were calculated using the HomCloud package35.

Results and discussion

Density data

Figure 1 shows the density of l-Er2O3 as a function of temperature, which exhibits a linear temperature dependence. The least-squares fit to the data is given by the following equation:

$$\rho \left( T \right) = {\rho_{\mathrm{m}}}\left[ {1 - \alpha \left( {T - {T_{\mathrm{m}}}} \right)} \right]\left( {{\mathrm{kg}}/{\mathrm{m}}^3} \right),$$
(2)

where ρm is the molten density at Tm (8170 kg/m3) and α (=1/ρm[dρ(T)/dT]) is the thermal expansion coefficient and is assumed to be constant (1.0 × 10−4 K−1) at any temperature of the liquid. The correlation coefficient of this fitting is 0.98. The uncertainty in the measurements is estimated to be 2% from the image resolution (640 × 480 pixels) and from the uncertainty in the mass measurement (±0.1 mg).

Fig. 1
figure 1

Density of l-Er2O3 as a function of temperature.

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The density and the expansion coefficient for l-Er2O3, together with those for l-SiO236 and other non-GFLs18,37, are compared in Table 2. Although the density trends increase with increasing cation atomic number, they do not show a clear relation. On the other hand, the thermal expansion coefficients show a similarity as each value approaches 1 × 10−4 K−1. The thermal expansion coefficient of l-Er2O3 is especially close to those of l-SiO2 and l-Al2O3.

Table 2 Densities at Tm and the thermal expansion coefficients.
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Structure factors and real-space functions

The Faber-Ziman X-ray total structure factors, S(Q), for l-Er2O3, l-SiO238, l-Al2O311, and l-ZrO218, together with the results of the MD-RMC simulation for l-Er2O3, are compared in Fig. 2a. It is noted that the scattering vector Q is scaled by multiplying by rA-X (distance between the center and corners of the polyhedron). The experimental S(Q) of l-Er2O3 (solid cyan curve) is well reproduced by the MD-RMC simulation (dotted black curve) using the liquid density measured by the ISS-ELF shown in Fig. 1. A well-defined first sharp diffraction peak (FSDP)39 is observed only for l-SiO2 (GFL) at QrA-X = 2.6, while a principal peak (PP)39 is observed in both the l-ZrO2 and l-Er2O3 data at QrA-X ~ 4.5. On the other hand, l-Al2O3 gives rise to a small peak between the FSDP and PP, suggesting that the structure of l-Al2O3 is intermediate17 between l-SiO2 and l-ZrO2/l-Er2O3. It is well known that the PP reflects the packing of oxygen atoms in neutron diffraction data40, since neutrons are sensitive to oxygen. For the same reason, a PP is not observed in the X-ray S(Q) for l-SiO2 (see Fig. 2a), and the origin of the PP in l-ZrO2 and l-Er2O3 is ascribed to the packing of cations. The X-ray total correlation functions T(r) for l-Er2O3, l-SiO238, l-Al2O311, and l-ZrO218 are shown in Fig. 2b. The first peak observed at approximately 2.2 Å is assigned to the Er–O correlation, and a tail to ~3 Å implies the formation of distorted ErOn polyhedra in the liquid. The second peak, observed at 3.7 Å, can be assigned mainly to the Er–Er correlation, and the O–O correlation peak is unclear due to its small weighting factor for X-rays. The Er–O correlation length of 2.2 Å, as well as that of Zr–O (2.1 Å), is longer than those of Si–O (~1.63 Å at 2373 K) and Al-O (~1.78 Å at 2400 K) owing to substantial differences between the ionic radii of the elements. The increased cation-oxygen correlation length in the liquid phases of Er–O and Zr–O suggests that the oxygen coordination number around cations is higher than 4 because the Er–O correlation length (2.2 Å) or Zr–O correlation length (2.1 Å) is close to the sum of the ionic radii41 of oxygen (1.35 Å) and six-fold erbium (0.89 Å) or zirconium (0.72 Å), respectively. The structures of l-Er2O3 and l-ZrO2 therefore consist of large interconnected polyhedral units and are very different from those of l-SiO2 and l-Al2O3. This behavior is consistent with the fact that the peaks observed at QrA-X ~ 4.5 in Fig. 2a are not the FSDP, which is typically associated with intermediate-range ordering in oxide glasses and liquids; thus, there is no such ordering in l-Er2O3 and l-ZrO2 due to a very densely packed structure.

Fig. 2: High-energy X-ray diffraction data for high-temperature oxide liquids.
figure 2

a Faber-Ziman X-ray total structure factors, S(Q), for l-Er2O3, l-SiO238, l-Al2O311, and l-ZrO218 together with that of l-Er2O3 derived from the MD-RMC simulation. The l-Er2O3 and l-Al2O3 data are displaced upward by 2 for clarity. b Total correlation functions, T(r), for l-SiO238, l-Al2O311, and l-ZrO218. The l-Er2O3 and Al2O3 data are displaced upward by 5 for clarity.

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Coordination number distributions from the simulation

The coordination number distributions, NA-X and NX-A, for l-Er2O3, l-SiO242, l-Al2O311, and l-ZrO218 obtained from the simulation are compared in Fig. 3a, b, and their average values are summarized in Table 3. The Er–O coordination number (up to 3.0 Å) is found to be 6.1 from our combined MD-RMC simulation, which is rather close to the crystalline phase43, and the O–Er coordination number can be estimated to be 4.1. It is suggested that the cations are tetrahedrally coordinated in l-SiO2 (GFL), while they are octahedrally coordinated in l-ZrO2 and l-Er2O3 (non-GFLs), and the cation-oxygen coordination number in l-Al2O3 is intermediate17 between GFL and l-ZrO2/l-Er2O3, although l-Al2O3 is a non-GFL. This behavior is consistent with that of the first correlation peaks in experimental real-space functions (see Fig. 2b) and with the fact that the viscosity of l-ZrO2 is approximately one-tenth of that in l-Al2O318. Another interesting behavior is observed for the oxygen-cation coordination numbers. It is demonstrated that oxygen is twofold in l-SiO2, which is a signature of the formation of a sparse network, while triclusters (XA3) are dominant in l-Al2O3 and l-ZrO2. The formation of tetraclusters (XA4) is confirmed in l-Er2O3, suggesting that this behavior is a distinct feature of this liquid. We suggest that the behavior of the coordination numbers in a series of oxide liquids is affected by both the composition and the ionic radii between the constituent anions and cations. For instance, the ionic radii of Si and Al are small, which results in tetrahedral coordination, although the Al-O coordination number is greater than four on average. The tetracluster formation is caused by the ratio of Er and O in Er2O3.

Fig. 3: Coordination number distributions in l-Er2O3, l-SiO242, l-Al2O311, and l-ZrO218.
figure 3

a NA-X, b NX-A.

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Table 3 Average coordination numbers derived from the simulations.
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Very sharp principal peak (PP) in l-Er2O3

As shown in Fig. 2a, the PP of l-Er2O3 is very sharp compared to that of l-ZrO2. The FWHM of the PP in l-Er2O3 is 0.4299, in comparison to 0.7669 in l-ZrO2 (see Fig. 4). A simulation box with 501 particles was used in the previous RMC – density functional (DF) simulation for l-ZrO218, where a good agreement was observed between the experimental data and simulation (see Fig. 4a). However, as can be seen in the inset data of Fig. 4b, a simulation box of 500 particles is insufficient to reproduce the sharp PP in l-Er2O3; larger atomic models are needed to reproduce this feature. Insight into the structure of l-Er2O3, in comparison with those of l-SiO2 and other non-GFLs, can be obtained by calculating the Faber-Ziman partial structure factors, Sij(Q), and the Bhatia-Thornton44 number – number partial structure factor, SNN(Q), which indicates the topological order in a system:

$${S_{\mathrm{NN}}}\left( Q \right) = {c_{\mathrm{A}}^2}{S_{\mathrm{AA}}}\left( Q \right) + {c_{\mathrm{X}}^2}{S_{\mathrm{XX}}}\left( Q \right) + 2c_{\mathrm{A}}c_{\mathrm{X}}S_{\mathrm{AX}}\left( Q \right),$$
(3)

where Sij(Q) is a Faber-Ziman partial structure factor and ci denotes the atomic fraction of chemical species i. Moreover, it is possible to compare data for the four liquids while ignoring the difference in the sensitivity of elements to X-rays because the weighting factors for X-rays are eliminated in SNN(Q). The Sij(Q) values calculated from the simulation models for l-Er2O3, l-SiO242, l-Al2O311, and l-ZrO218 are shown in Fig. 5a. It is confirmed that a very sharp PP in l-Er2O3 can be assigned to the Er-Er correlation. The SNN(Q) for l-Er2O3 and those for l-SiO2 and other non-GFLs are compared in Fig. 5b. As mentioned above, only l-SiO2 exhibits an FSDP at QrA-X = 2.6. The QFSDP position arises from an underlying periodicity of 2π/QFSDP that originates, for example, from the formation of pseudo-Bragg planes with a finite correlation length of 2π/ΔQFSDP in l-SiO2, while neither l-Al2O3, l-ZrO2, nor l-Er2O3 show an FSDP in SNN(Q), as discussed in Kohara et al.18. Since the Bhatia-Thornton SNN(Q) can eliminate the weighting factors for X-rays, the absence of an FSDP in SNN(Q) is characteristic of a non-GFL. Another important feature in SNN(Q) is that l-SiO2 and l-Al2O3 exhibit a second PP at QrA-X ~ 5, while a PP is not distinct in the l-ZrO2 or l-Er2O3 data.

Fig. 4: X-ray total structure factors, S(Q), obtained from high-energy X-ray diffraction measurements and simulations.
figure 4

a l-ZrO218, b l-Er2O3. The inset shows an enlarged principal peak.

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Fig. 5: Partial structure factors for l-Er2O3, l-SiO242, l-Al2O311, and l-ZrO218.
figure 5

a Faber-Ziman partial structure factors. b Bhatia–Thornton number – number partial structure factors, SNN(Q). The scattering vector Q is scaled by multiplying by rA-X (distance between the center and corner of the polyhedron).

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The absence of an FSDP in the l-ZrO2 and l-Er2O3 data suggests that both cations and oxygen are densely packed. To confirm this in real space for l-Er2O3, the partial pair distribution functions, gij(r), of l-Er2O3 are compared with those of l-SiO2 in Fig. 6a. The atomic distance r is scaled by dividing by rA-X (distance between the center and corners of the polyhedron). It is found that the scaled first A-A and X-X correlation distances of l-Er2O3 are much shorter than those of l-SiO2, demonstrating that l-Er2O3 has a much more densely packed structure, manifested by the formation of the OEr4 tetracluster network shown in Fig. 6b. This network cannot be found in l-Al2O3 nor in l-ZrO2, suggesting that the very sharp PP in l-Er2O3 is a specific signature of the formation of a tetracluster network with long-range periodicity.

Fig. 6: Analyses in real space.
figure 6

a Partial pair-distribution functions, gij(r), for l-Er2O3 and l-SiO242. b Visualization of the OEr4 tetracluster network in l-Er2O3. Pink, oxygen; blue, erbium. c Visualization of the nearly linear arrangements of Er–O–Er.

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Topology and homology in l-Er2O3

To reveal the origin of the very sharp PP in l-Er2O3, we calculated the bond angle distributions of the liquid and crystal43 and summarized them in Fig. 7. A pronounced difference was found between the liquid and crystal data for the O–Er–O and Er–O–Er distributions. The O–Er–O bond angle distribution exhibits two peaks at 80° and 140°, suggesting that ErO6 polyhedra are highly distorted in the liquid. Another interesting feature is that the Er–O–Er bond angle distribution exhibits a peak at ~180° in addition to the peak at ~90°, which is not observed for the crystal43 nor in l-ZrO218. This two-peak structure in the Er-O-Er bond angle distribution indicates the formation of a distorted OEr4 tetracluster network, whereas tetraclusters are symmetric (comprising regular tetrahedra) in the crystalline phase. This behavior suggests that the coordination of OEr4 tetraclusters is more octahedral-like and hence tolerant of disorder even in the liquid due to the distortion, providing a linear arrangement manifested by a prominent peak observed at 180° in the Er–O–Er bond angle distribution. This is clearly visible in Fig. 6c, where linear atomic arrangements are highlighted by the magenta lines.

Fig. 7: Bond angle distributions for Er2O3. Black line, liquid; red line, crystal43.
figure 7

The B(θ)/sinθ data for the crystalline data have been scaled by a factor of 20 for clarity.

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To shed light on the similarity in topology between the crystal and liquid phases, we calculated the persistence diagram for l-Er2O3 and compared it with the crystal data in Fig. 8. The figures show the similarity between the crystal43 and liquid phases. In particular, both the Er-centric and O-centric persistence diagrams for l-Er2O3 do not show a vertical profile along the death axis, which is a pronounced feature in a typical GFL, such as l-SiO242. The short lifetime of the profile manifested by the small death value demonstrates that both the crystal and liquid phases exhibit a very densely packed structure associated with the formation of tetraclusters in both phases. We suggest that this similarity is a signature of non-GFL behavior.

Fig. 8: Persistence diagrams of Er2O3.
figure 8

a Er-centric persistence diagram, b O-centric persistence diagram.

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Electronic structure and viscosity of l-Er2O3

As previously mentioned, a 500-atom RMC model of l-Er2O3 appeared to be too small in reproducing the very sharp PP accurately (see Fig. S3). Nevertheless, we used this model as a starting structure in our DF-MD simulations to study the electronic structure and atomic diffusion in the liquid phase. The electronic density of states (DOS) and effective charges were calculated for snapshots of l-Er2O3 atomic structures and a fully DFT-relaxed (0 K) c-Er2O3 unit cell. The DOSs of l- and c-Er2O3 are shown in Fig. 9a, b. The valence band consists mainly of O-2p states, while the conduction band consists mainly of Er-5d states. As a measure of the orbital localization, we also present the inverse participation ratios (IPRs) for l-Er2O3. The IPRs show increased weight at the valence band maximum (VBM), indicating stronger localization, whereas the states around the conduction band minimum (CBM) are delocalized. The obvious broadening of the valence and conduction bands in l-Er2O3 is caused by the distortion of the ErOn polyhedra at elevated temperatures. The DOS for l-Er2O3 reveals a band gap of 0.57 eV in comparison to the substantially large band gap of 5.46 eV in c-Er2O3. Previously, a vanishing band gap was reported for l-ZrO218, but we ascribe this effect to the PBE functional, which is known to underestimate the real value, whereas the hybrid HSE06 functional used here predicts a more correct band gap.

Fig. 9: The electronic structure of Er2O3.
figure 9

a Calculated atom-resolved DOS (top) and IPR (bottom) for l-Er2O3. b Calculated DOS of c-Er2O3 for comparison. c Visualization of the highest occupied molecular band (HOMO). d Visualization of the lowest unoccupied molecular band (LUMO). Er and O atoms are shown in green and red, respectively. Yellow and cyan correspond to different signs of the wavefunction.

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The difference in electronegativities of Er (1.24) and O (3.44) suggests predominantly ionic chemical bonding between the two atoms. This becomes clear from the partial DOS in Fig. 9a, and the calculated atomic charges are summarized in Table 4. The effective charges for l-Er2O3 are +1.96 e and –1.31 e for Er and O, respectively, similar to those in c-Er2O3 (see Table 4). These values are in agreement with previous works on l-ZrO218, glassy MgO-SiO245, and CaO-Al2O346 and are consistent with the nominal charges Er3+ and O2- (systematically scaled down by a factor of \(\sim \frac{2}{3}\)). As for ZrO218, the increased atomic volume of oxygen in the transition from c- to l-Er2O3 compensates for the corresponding decreased oxygen coordination, which results in similar atomic charges for the two phases. Note here also the similarity with the charges used in the classic MD force field (+2.1 and −1.4 e).

Table 4 Average atomic charges and volumes calculated using the Bader method.
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A real space visualization of the highest occupied molecular band (HOMO) is shown in Fig. 9c, where the orbital is found to be distributed over a group of atoms. Locally, the shape of the HOMO is similar to that of c-Er2O3 (Fig. S4); however, some deviations occur caused by the aforementioned distorted tetracluster network in l-Er2O3. Figure 9d shows a real space visualization of the lowest unoccupied molecular band (LUMO), where most of the orbital is distributed over nonbonding regions in-between the ErOn polyhedra. The orbital projections for each of the bands reveal that the HOMO band shows mainly an O-2p character, while the LUMO band consists mainly of Er-5d states and some Er-5p character, in agreement with the DOS and IPR in Fig. 9a.

The DF-MD simulations above the melting point at 2650 °C show that the atoms move rapidly, breaking old and forming new Er-O bonds on a picosecond time scale. The MSD analysis (Fig. S2) shows a consistent linear behavior as a function of time, the evaluated self-diffusion constants are 2.40 × 10−5 and 5.84 × 10−5 cm2/s for Er and O, respectively, and the average self-diffusion constant is 4.64 × 10−5 cm2/s. By using the average value and assuming spherical particles in the Stoke-Einstein relation, one can estimate the viscosity, and the obtained value is ~3 × 10−3 Pa s−1 for l-Er2O3 in comparison to the previously reported value for l-ZrO2, i.e., ~2 × 10−3 Pa s−1 at 2800 °C18. Since these values are an order of magnitude smaller than that for l-Al2O3 (fragile liquid) and 9–10 orders of magnitude smaller than that for l-SiO2 (strong liquid), we characterize l-Er2O3 as an “extremely fragile” liquid18.

We combined an aerodynamic levitation technique and a synchrotron high-energy X-ray diffraction and density measurement at the ISS for l-Er2O3 to reveal the structure of A2X3-type non-GFLs. As the main finding, we observed a very sharp PP in the diffraction data. The Er-O coordination number was estimated to be 6.1 from a combined MD-RMC simulation, which is comparable to that of another nonglass-forming liquid, l-ZrO2, and to that of crystalline Er2O3. The formation of distorted OEr4 tetraclusters in the liquid is confirmed, while OZr3 triclusters are dominant in l-ZrO2 and OAl3 triclusters dominate in another A2X3 non-GFL, l-Al2O3. Apparently, the formation of a distorted tetracluster network in l-Er2O3 gives rise to a long periodicity, yielding the sharp principal peak. This long-range periodicity originates from the significantly increased weight at ~180° in the Er-O-Er bond angle distribution, suggesting that the arrangement of distorted OEr4 tetraclusters involves nearly linear connections, which are not observed in other oxide liquids. Furthermore, persistent homology suggests that l-Er2O3 is homologically similar to the crystalline phase and that both phases are very densely packed, in contrast to a typical GFL such as l-SiO2. This similarity is presumed to be the signature of the liquid, and a considerable difference between two A2X3-type oxide liquids, l-Al2O3 and l-Er2O3, is uncovered. The additional DF-MD simulations demonstrate that l-Er2O3 has an electronic band gap of 0.6 eV, which is considerably lower than that of c-Er2O3 due to angular distortions and nearly linear connections within the polyhedral network. The dynamics of atoms show pronounced mobility, as evidenced by the MSDs, resulting in a very low viscosity, and thus place l-Er2O3 within the regime of extremely fragile liquids.