OR3-7
⾼融点単原⼦液体の構造と物性
Structure and Properties of monatomic liquids with highmelting temperature
⽔野章敏 1,5、寺⾨修 1、鎌⽥アハメッド 1、中野彩花 1、尾原幸治 2、正⽊匡彦 3、渡邉学 4、⼩原真司 5
Akitoshi MIZUNO1,5, Osamu TERAKADO1, Ahmed KAMADA1, Ayaka NAKANO1, Koji OHARA2,Tadahiko MASAKI3, Manabu WATANABE4 and Shinji KOHARA5
函館⼯業⾼等専⾨学校，National Institute of Technology (KOSEN), Hakodate College,
島根⼤学，Shimane University,
3 芝浦⼯業⼤学，Shibaura Institute of Technology,
4 東京⼯業⼤学，Tokyo Institute of Technology,
5 物質・材料研究機構，National Institute for Materials Science12

1. Introduction
In recent years, machine learning using liquid metal properties and theoretical models as input has led to rapid
progress in the development of metallic glasses as well as many functional materials 1-3). For liquid and amorphous
metal oxides, a study was reported that use their liquid and amorphous structure data as input for machinelearning, which was applied to optimize model potentials and predict chemically realistic structures 4). It istherefore expected to become important in the future to use liquid structures as training data for machine learning
to predict glass-forming ability of metallic alloys. However, as metallic glasses are often multicomponent alloys,
it takes a lot of time to cover their liquid structures because of numerous experimental conditions with different
combinations of composition and temperature. Hence, it is important to predict the liquid structure of alloys frommonatomic liquid structures.
Recent advances in containerless techniques have made it possible to perform experimental studies of properties
and structure of elemental liquids with high melting temperature. Many experimental data on thermophysicalproperties such as density, viscosity and surface tension have been obtained by electrostatic levitation on theground and under microgravity conditions 5,6). Although fewer in number than the thermophysical properties,
experimental data on the structure of metallic liquids have been systematically compiled 7) and recently obtainedby combining synchrotron radiation X-ray diffraction (SR-XRD) with levitation techniques including aerodynamic
levitation (ADL) 8). However, there are very few reports on elements with melting temperature above 2500 K.
The aim of this study was to systematize liquid structure information by focusing on the main constituent
elements of metallic glasses and performing precise structure analysis of metallic liquids at high temperature
typically over 2000 K. In the present study, structural data for eight elements, including Hf with a melting pointof 2506 K, were measured using a combination of the SR-XRD and the ADL technique. In addition, persistenthomology (PH) analysis 9,10) was performed on 3D coordinates obtained from reverse Monte Carlo (RMC)
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modelling 11) to reproduce the measured structure data, and relationships with thermophysical property such as
entropy were investigated.
2.

Experimental

2.1. Aerodynamic levitation (ADL) furnace for synchrotron radiation X-ray diffraction (SR-XRD)
As the details of the experimental setup are similar to those in previous papers 12,13), only a brief summary isdescribed in this paper. To obtain the X-ray scattering data of chemically active liquids at high temperatures,
the ADL furnace was used as shown in Fig. 1. A conical nozzle was installed in a high vacuum chamber filledwith high-purity Ar gas. A spherical sample of approximately 2 mm diameter was levitated in the nozzle bycontrolling the gas flow rate with a mass flow controller. A CO2 laser with a maximum power of 200 W (Synrad,firester f200) was used to melt the sample and the temperature of the sample was monitored using a two-color
optical pyrometer (Impac, ISR 12-LO).
CO2 Laser（Ｍax.200W）

Pyrometer
Levitated sample(dia. ~2mm)

CCD camera

Scattered X-rays
2θ
3 Detectors

Incident X-ray beam

BL04B2＠SPring-8

Mass flow controller

Figure 1. Schematic diagram of the aerodynamic levitation apparatus constructed for synchrotron x-ray
diffraction (SR-XRD) experiments.

The SR-XRD experiments were carried out at the BL04B2 and BL08W beamlines of SPring-8. The
experimental conditions were optimized at the BL08W beamline, and the experiments were conducted under
optimum conditions at the BL04B2 beamline. Monochromatic X-rays with an energy of 112.7 keV wereirradiated into the sample and the scattered X-rays were detected by three Ge Solid state detectors (SSD) withthe scattering angle, 2θ, ranging from 0.3° to 25°. To confirm the reproducibility of the scattering intensity, themeasurement was repeated three or more times in the same angular range per sample. The temperature
conditions were just above the melting temperature of the measured element, including the supercooled liquidstate. The liquid static structure factor, S(Q), was extracted from the scattering intensity through the usual
corrections as a function of the scattering vector magnitude Q (=4πsinθ/λ, where λ is the wavelength of theincident X-ray).

2.2. Liquid structure analysis
Since the S(Q) is information in reciprocal space, it can be analyzed as information in real space by estimating
the pair distribution function (PDF), g(r), by the following Fourier transform relation:
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g (r ) - 1 =1Q max

2p 2 r0 r òQ min

Q ( S (Q) - 1) sin QrdQ ,(1)

where 𝜌! is the average number density. As the PDF represents the three-dimensional atomic arrangement asa one-dimensional distribution, the structural information obtained is limited to the distance between nearest
neighbor atoms and the coordination number. Therefore, the RMC method 11) was used to reproduce the threedimensional atomic coordinates based on the S(Q) obtained from experiments. In the present study, the RMC
was performed for 10000 particles in a cubic box to reproduce the S(Q) obtained from the SR-XRD
experiments. The volume of the cubic box was estimated from the experimental density 14).

2.3. Persistent homology (PH) analysis
In recent years, PH analysis, one of the topological data analysis methods, has been developed as a structure
analysis method for amorphous materials and glasses 9,10). PH analysis can be applied to atomic arrangements
obtained by the RMC method to extract and visualize hierarchical structures. As shown in Fig. 2, the radius ofan atom in 3D coordinates is simultaneously increased, and the radius when a ring (hole) appears after
overlapping with neighboring atoms is assigned as “birth” time, and the radius when the ring disappears after
further increase is assigned as “death” time. A persistent diagram (PD), with the radius at “birth” time on thehorizontal axis and the radius at “death” time on the vertical axis, allows the extraction of crystalline, liquidand glassy structural features from the distribution 9).
The main feature of this method is that it can analyze the shape of voids (homology) without defining thebonds between atoms, thus eliminating the problem of selecting the cut-off range of interest in structure
analysis, such as Voronoi polyhedral analysis. In this study, the HomCloud package 10), an application for PH
analysis, was used.

birth time (of ring 2)

birth time (of ring 1)

death time (of ring 1)

Figure 2. Schematic illustration showing the rings on birth time and death time defined in the Persistent
Homology (PH) analysis.

2.4. Configuration entropy
The relationship between the structure and physical properties of liquids has long been a topic ofexperimental and theoretical research, and in particular, the following relationship have been proposed for
the PDF, g(r), and entropy S:
3 of 7S=S(1)+S( 2)+S(3)(2)+ ,{}S (2) = -2pr0 ò g (r )ln g (r ) - [g (r ) -1] r 2dr ,(3)where S(1), S(2) and S(3) represent the entropy due to ideal gas, two-body (pair) and three-body (triplet)correlations, respectively, and the table equation takes many-body correlations into account 15,16). However, it
is difficult to obtain correlations for more than three bodies, and many approximate models have been
proposed. In the present study, since the focus is on pair correlations, the two-body term S(2) was calculatedusing the PDF g(r) estimated from the experimental S(Q).

3.

Results and Discussion

3.1. Liquid structure factor S(Q)The elements for which liquid S(Q)s were successfully obtained in this study are shown in Fig. 3. As themain components of metallic glasses are mostly transition elements framed in the figure, measurements werealso carried out on Si, a high-melting metalloid, as a comparison. The elements for which liquid S(Q) could
not be obtained in the present study were due to insufficient laser power (maximum power: 200 W) or the low
absorption rate of the CO2 laser (laser emission wavelength: 10.6 µm). Therefore, it is necessary to use a laser
with even higher power or a semiconductor laser with emission wavelengths that are highly absorbed bymetallic elements.
123456789101112

131415161718

12

Melting temperature:unit in Kelvin
Hatched elements: Measured in the present study

3421Sc22Ti
1814 1941

23V
24Cr
25Mn
2183 2180 1519

26Fe
1811

27Co
1768

28Ni
29Cu
1728 1358

539Y
40Zr
1799 2128

41Nb
42Mo
43Tc
2750 2896 2430

44Ru
2607

45Rh
2237

46Pd
47Ag
1829 1235

657~
71

73Ta
74W
3290 3695

76Os
3306

77Ir
2739

78Pt
79Au
2041 1337

72Hf
2506

75Re
3459

14 Si
1685

7Figure 3. Elements for which liquid S(Q) was measured in the present study.

The representatives of measured S(Q) of the monatomic liquids in comparison with the RMC results ofS(Q) and the corresponding PDF, g (r), are shown in Fig. 4. The use of three detectors enabled SR-XRD
data to be acquired in a shorter time than in the past experiments, and S(Q) could be obtained over a wide
Q range. For the transition elements, an asymmetric shape, called a “shoulder”, is observed on the highQ side of the second peak of S(Q) around 50 nm-1. On the other hand, for liquid Si, a shoulder is observed
more clearly on the high Q side of the first peak of S(Q) around 35 nm-1. This feature of the S(Q) of liquid
Si is thought to be related to its covalent nature.
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(a)

(b)

(c)

Figure 4. Comparison of experimental liquid S(Q) with RMC results and corresponding PDF g (r) for(a) Fe at 1820 K (Tm: 1811 K), (b) Hf at 2510 K (Tm: 2506 K), (c) Si at 1700 K (Tm: 1685 K).

As shown in the upper part of Fig. 5, the distribution of free volume around each atom can be visualised
for the 3D coordinates of the atomic configuration obtained by the RMC method. The free volume distributionof Si is considerably more spread out than that of Fe and Hf, indicating that the structure of the Si liquid issparse, whereas the Fe and Hf liquids are closer to a densely packed structure.

(a)

(b)

(c)

Figure 5. Comparison of free volume distributions (upper side) and PD2 (lower side) obtained from 3D
coordinates by RMC modelling for liquid S(Q), for (a) Fe at 1820 K (Tm: 1811 K), (b) Hf at 2510 K (Tm: 2506 K),(c) Si at 1700 K (Tm: 1685 K).

The results of the PH analysis of the atomic configurations obtained by the RMC method are also shownin lower part of Fig. 5 as two-dimensional PD (PD2), which can be considered to represent the distribution of
the radii of formation and annihilation of voids (free volume). The values of Birth and Death radii of the5 of 7

annihilation of voids are not widely scattered from the PD of Fe and Hf. On the other hand, the values of bothBirth and Death radii for Si are widely distributed up to large values. Therefore, it can be interpreted that thevoids in the atomic configuration are large and widely distributed. This is consistent with the results for thedistribution of free volumes presented in the upper part of Fig. 5, indicating that PH analysis is effective in
capturing the feature of atomic-scale packing in liquids.
To obtain more insight into the relationship between the results of the structural analysis and thethermophysical properties, the entropy S(2) of the two-body correlation calculated from the PDF are shown inFig. 6. In the present study, due to the small number of data points and the narrow temperature range, thetemperature dependence was approximated as a linear equation. Equations (2) and (3) represent that for thetwo-body term S(2), the smaller its value, the smaller the total entropy S. Therefore, the temperaturedependence shows a reasonable trend. In addition, transition metals have a smaller S(2), which suggests thatthey have more ordered liquid structures than Si.
0.00

SiEntropy S(2) /NkB-0.50-1.00

Ti-1.50
-2.00

NiVFe

Hf-2.50-3.00
1500

Pd
1900

2300

2700

Temperature T /KFigure 6. Temperature dependence of the entropy S(2) of the two-body correlation calculated from thePDF.

4.

Conclusion

In the present study, liquid structure factors S(Q)s for eight elements, including Hf with a melting pointof 2506 K, were measured using a combination of ADL and SR-XRD. PH analysis was performed on 3D
coordinates obtained from RMC modelling to reproduce the obtained S(Q). The PH analysis and the freevolume distribution estimated from the RMC results indicate a dense liquid structure of transition metals anda sparse liquid structure of Si. Furthermore, it was shown to be qualitatively related to macroscopic
thermophysical properties such as entropy calculated from the PDF.
Acknowledgement
This work was supported by JSPS KAKENHI (Grant Number JP 20H02496, 23K04464) and the synchrotron radiation
experiments were performed at the BL08W and the BL04B2 of SPring-8 with the approval of the Japan Synchrotron
Radiation Research Institute (JASRI) (Proposal No. 2021A1445, No. 2021B1280).

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