OR1-2
微小重力環境における Fe–Cu 合金融体の表面張力測定結果を利用した Butler モデルの評価
Evaluation of the Butler Equation Using Surface Tensionof Fe-Cu Alloys Measured under Microgravity
Conditions
西村美咲 1, 吉崎隼人, 朝見海斗, 堀内豪暉 1, 清宮優作 1, 白鳥英 2, 小澤俊平 1
Misaki NISHIMURA1, Hayato YOSHIZAKI, Kaito ASAMI, Goki HORIUCHI1, Yusaku SEIMIYA1,Suguru SHIRATORI2 and Shumpei OZAWA1
12千葉工業大学, Chiba Institute of Technology
東京都市大学, Tokyo City University

1.

Introduction

Space provides an ideal environment for measuring the thermophysical properties of high-temperaturemelts. In this unique environment, the absence of containers eliminates any risk of sample contamination fromthe container, even at elevated temperatures. In addition, the measurements are conducted free from theeffects of natural convection and hydrostatic pressure, which often degrade the accuracy and precision ofmeasurements on Earth. As a result, several space missions have been conducted on the International Space
Station (ISS) to measure the thermophysical properties of various high-temperature melts using theElectrostatic Levitation Furnace aboard the KIBO (ELF-ISS) and European Space Agency Materials Science
Laboratory Electromagnetic Levitator (MSL-EML). These missions aim to optimize materials processes andenhance our understanding of phenomena. However, it would be short-sighted to use these results solely for
their initial intended purposes. These measurements can also serve as benchmark data to validate and improveterrestrial methods and predictive models for measuring thermophysical properties.
This study utilizes space-based measurements of the surface tension of molten Fe-Cu binary alloys fromthe “Thermal Storage” mission to validate the effectiveness of terrestrial electromagnetic levitation (EML) for
surface tension measurements and the Butler model 1) for predictions. Based on these results, we have extended
the Butler model to the ternary Fe-Si-Cu alloy system and evaluated its effectiveness.
2.
2.1.

Experimental procedure
Measurement of Surface tension using EML

A piece of Fe-Si master alloy was placed on a quartz holder within the EML coil, alongside a high-purity
iron ingot. The alloy was electromagnetically levitated with a piece of the iron ingot and then melted uniformly
under a high-purity mixture of argon and helium gas flowing at 2 L/min. The sample temperature wascontrolled by adjusting the partial pressures of argon and helium gases, which have different thermalconductivities, using a two-color pyrometer. After the droplet temperature stabilized, the oscillation behaviorof the droplet was monitored from above at 500 fps for 16 sec using a high- speed camera. The frequencies ofthe surface oscillations of m = 0, ±1, and ±2 for the l = 2 mode, and those of the center of gravity were analyzed
from time-sequential data of the HSV images. The surface tension of the droplet was calculated from thefrequencies using the Rayleigh equation 2) and the Cummings and Blackburn calibration 3).
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2.2.

Butler model

Butler derived an equation to calculate the surface tension of a liquid solution by considering the activities
at the surface and in the bulk phase, assuming equilibrium between a bulk phase and a hypothetical surface
phase. The equation is given by:
𝜎𝜎 = 𝜎𝜎𝑖𝑖Pure +𝑅𝑅𝑅𝑅
𝐴𝐴𝑖𝑖

ln𝑛𝑛𝑖𝑖S

B +𝑛𝑛𝑖𝑖

𝐺𝐺𝑖𝑖E.S − 𝐺𝐺𝑖𝑖E.B

(1)

𝐴𝐴𝑖𝑖

where 𝜎𝜎𝑖𝑖Pure is the surface tension of the liquid alloying element, R is the gas constant, and T is thetemperature. A is the molar surface area of the molten alloy calculated from the molar volume V as 𝐴𝐴𝑖𝑖 =1/3 2 /3

1.091𝑁𝑁0 𝑉𝑉𝑖𝑖 . 𝑛𝑛𝑖𝑖S and 𝑛𝑛𝑖𝑖B are the molar fractions of the alloying element at the surface and in the bulk,respectively. 𝐺𝐺𝑖𝑖E.S and 𝐺𝐺𝑖𝑖E.B are the partial excess Gibbs free energies at the surface and in bulk, respectively.
Following the approach by Speiser et al., if we assume the dependence of temperature and alloy composition
on 𝐺𝐺𝑖𝑖E.S is identical to that on 𝐺𝐺𝑖𝑖E .B, it can be expressed as follows,(2)𝐺𝐺𝑖𝑖E .S = 𝛽𝛽𝐺𝐺𝑖𝑖E.B

where 𝛽𝛽 is the ratio of coordination number on the surface to those in the bulk for pure substances, reflectin gsurface relaxation. In this study, we adopted 𝛽𝛽＝0.83, as proposed by Tanaka et al.4), which was derived fromthe relationship between surface tension of liquid metals and binding energy at the surface.
For a ternary alloy consisting of components labeled as 1, 2 and 3, 𝐺𝐺𝑖𝑖E.B can be expressed using the RedlichKister polynomial:

𝐺𝐺𝑖𝑖E.B =33𝑀𝑀

3𝐾𝐾𝑘𝑘𝑚𝑚� � � 𝐿𝐿𝑚𝑚
𝑖𝑖𝑖𝑖 𝑛𝑛𝑖𝑖 𝑛𝑛𝑗𝑗 (𝑛𝑛𝑖𝑖 − 𝑛𝑛𝑗𝑗 ) + 𝑛𝑛1 𝑛𝑛2 𝑛𝑛3 � � 𝐿𝐿123 𝑛𝑛𝑖𝑖

𝑖𝑖=1 𝑗𝑗>𝑖𝑖 𝑚𝑚=0(3)𝑖𝑖 =1 𝑘𝑘=0

where L is the interaction parameter, subscript i and j attached to the symbol denote alloy components,
superscripts m and k represent the numbers of interaction parameters determined by the combination of alloy
elements. Due to the nonlinear nature of these equations, solutions were computed using the Newton-Raphson
method.

3.

Results and Discussion

Figure 1 shows the surface tension of molten Fe-Cu alloys at 1800 K, measured on Earth using EML andcalculated using the Butler equation. The figure also includes results from the “Thermal Storage” mission
using ELF-ISS and comparative literature data. As the Cu composition increases to 20at%, the measured
surface tension of molten Fe-Cu decreases dramatically and approaches that of molten copper, indicating
surface segregation of the copper atoms. The results calculated using the Butler equation closely align withthese measurements. Additionally, the data obtained with ELF-ISS are consistent, supporting the validity of
the Cumming and Blackburn approach for calibrating droplet deformation by gravitational acceleration andelectromagnetic force to droplet oscillation in terrestrial experiment. Furthermore, these findings support theassumption in the Butler model that the bulk phase is thermodynamically equivalent to a hypothetical surface
phase, at least within the context of the Fe-Cu alloy system.
Figure 2 displays the surface tension of molten Fe-Si-Cu ternary alloys at 2100 K, as measured using EMLand calculated with the Butler equation. Although the measurements exhibit some scatter, they consistently
show a gradual decrease in surface tension with increasing Cu composition. The results calculated with theButler equation are in comparative agreement with the experimental data, demonstrating that the Butler
model is effectively practical for estimating surface tension amidst observed scatter. The observed scatter may
be attributed to a strong phase separation tendency affecting surface segregation of atoms. To further validate
the Butler equation for these ternary alloys, a more detailed study at higher temperatures, which promoted
atomic mixing, will be necessary.
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Surface tension σ, 10-3 N･m-1

2000
1900
1800
1700
1600
1500
1400
1300
1200
1100
1000

T = 1800K― Butler model● Measured values using EML▲ ISS-ELF△ Brillo (EML) 5)△ Kasama (SD) 6)△ Dyson (SD) 7)△ Nogi (SD) 8)

0 10 20 30 40 50 60 70 80 90 100
Feat%Cu
Cu

Figure 1. Surface tension of molten Fe-Cu alloy estimated with the extended Butler equationand measurement plots obtained using EML.

Surface tension σ, 10-3 N･m-1

1800

T = 2100K― Butler model● Measured values using EML

1700
1600
1500
1400
1300
1200
1100
1000010 20 30 40 50 60 70 80 90 100
at%Cu
CuFigure 2. Surface tension of molten Fe-Si-Cu alloy estimated with the extended Butler
equation and measurement plots obtained using EML.
Acknowledgement
This research was partially funded by the Japan Society for the Promotion of Science (JSPS) KAKENHI
under Grant No. P20H02453, and by the Grant-in-Aid for Front Loading Research, supported by the Advisory
Committee for Space Utilization Research at the Institute of Space and Astronautical Science, Japan Aerospace
Exploration Agency (ISAS/JAXA). We extend our thanks to Prof. K. Sugioka (Toyama Prefectural University),Prof. T. Ishikawa (JAXA), and H. Kobatake (Doshisha University) for their insightful discussions.
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2)

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D. L Cummings, and D. A. Blackburn: Oscillations of magnetically levitated aspherical droplets. J. Fluid.
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K.Nogi, W.B.Chung, A.McLean and W.A.Miller: Surface Tension of Liquid Fe–(Cu, Sn, Cr) and Ni–(Cu,Sn) Binary Alloys Mater, Trans., JIM, 32, (1991), 164© 2024 by the authors. Submitted for possible open access publication under theterms and conditions of the Creative Commons Attribution (CC BY) license(http://creativecommons.org/li censes/by/4.0/).

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