P14
3 元系合金融体の表面張力の推算
Estimation of the Surface Tension for Molten Fe–Si–Cu
Ternary System Alloys
西村美咲 1, 吉崎隼人 1, 朝見海斗 1, 堀内豪暉 1, 清宮優作 1, 白鳥英 2, 小澤俊平 1
Misaki NISHIMURA1, Hayato YOSHIZAKI1, Kaito ASAMI1, Goki HORIUCHI1, Yusaku SEIMIYA 1,Suguru SHIRATORI2 and Shumpei OZAWA1
千葉工業大学, Department of Advanced Materials Science and Engineering, Chiba Institute of Technology, Chiba
Japan
2 東京都市大学, Department of Mechanical Systems Engineering, Tokyo City University, Tokyo Japan1

1. Introduction
Microgravity conditions at the International Space Station (ISS) offer an ideal environment for measuring
the surface tension of high-temperature melts. This is because the measurement does not require a containerthat might chemically react with the sample at elevated temperatures. The measurements are free from any
influences such as droplet deformation, gravitational convection, and hydrostatic pressure, which cancompromise the accuracy and precision of ground-based measurements. However, the opportunities formeasuring the surface tension at the ISS are still limited at present. To enhance the use of the measurement
results of the surface tension at the ISS, it is important to develop an estimation model for the surface tensionof high-temperature melts. The accurate and precise measurement results of surface tension for liquid puremetals at the ISS can serve as parameters for the model, while those for molten alloys can be used as benchmark
data for the model.
In this study, the surface tension of molten binary Fe–Si and ternary Fe–Si–Cu alloys was calculated using
the Butler model1), which often provides a comparatively accurate representation of the surface tension ofmolten binary alloys. The calculated results were then compared with the surface tension measured byoscillating droplet method using the electromagnetic levitation (EML)2). The purpose of this research was toextend the Butler model for estimating the surface tension of molten ternary alloys and to evaluate the validityof this extended model.
2. Extension of the Butler equation for ternary system alloys
A list of nomenclatures used throughout this study is provided in Table 1. The surface tension of a molten
multi-component alloy, σ, can be described by the Butler equation as follows,
𝜎𝜎 = σPure+𝑖𝑖

R𝑇𝑇 𝑛𝑛𝑖𝑖𝑆𝑆 𝐺𝐺𝑖𝑖𝐸𝐸.𝑆𝑆 − 𝐺𝐺𝑖𝑖𝐸𝐸.𝐵𝐵ln+𝐴𝐴𝑖𝑖
𝐴𝐴𝑖𝑖 𝑛𝑛𝑖𝑖𝐵𝐵(1)1 of 5

Table 1. nomenclature lists.
SymbolDefinition

Unit

𝜎𝜎Surface tension of molten multi-component alloys

N･m-1

𝜎𝜎𝑖𝑖Pure

Surface tension of liquid alloy element

N･m-1

TTemperature

KRGas constantmol-1𝐴𝐴𝑖𝑖

Molar surface area of alloy element

m2･mol-1𝑛𝑛𝑖𝑖𝑆𝑆

Moler fraction of alloy element at melt surface

𝐺𝐺𝑖𝑖𝐸𝐸.𝑆𝑆

Partial excess Gibbs free energy of alloy element at melt surface

J･mol-1𝐺𝐺𝑖𝑖𝐸𝐸.𝐵𝐵

Partial excess Gibbs free energy of alloy element in bulk melt

J･mol-1

𝑉𝑉𝑖𝑖

Molar volume of alloy element

m3･mol-1

𝑁𝑁0

Avogadro number

J･K-1 mol-1

βCoordination numbers between surface and bulk

𝐿𝐿

Interaction parameters used to calculate 𝐺𝐺𝑖𝑖𝐸𝐸.𝐵𝐵 for alloy

𝑛𝑛𝑖𝑖𝐵𝐵

Moler fraction of alloy element in bulk melt

J･mol-1

where the subscript i attached to the symbols represents the alloy element, while the superscripts S and B
represent the surface and bulk, respectively. 𝜎𝜎𝑖𝑖Pure is the surface tension of liquid alloy element, R is the gas

constant, T is the temperature, 𝐴𝐴𝑖𝑖 is the molar surface area of the molten alloy, 𝑛𝑛𝑖𝑖 is the molar fraction of thealloy element, and 𝐺𝐺 𝐸𝐸 is the partial excess Gibbs free energy. 𝐴𝐴𝑖𝑖 was deduced from the following equation
using the molar volume 𝑉𝑉𝑖𝑖 and the Avogadro number 𝑁𝑁0 .

1/3

𝐴𝐴𝑖𝑖 = 1.091𝑁𝑁0 𝑉𝑉𝑖𝑖

2/3

(2)

When assuming that the dependence of temperature and alloy composition on 𝐺𝐺𝑖𝑖E.S is the same as that of𝐺𝐺𝑖𝑖E.B, as proposed by Speiser et al.3,4), it can be expressed as follows,𝐺𝐺𝑖𝑖E.S = 𝛽𝛽𝐺𝐺𝑖𝑖E.B(3)where 𝛽𝛽 represents the ratio of coordination number on the surface to that in the bulk for pure substances,taking into account the relaxation of the surface. In this study, we adopted 𝛽𝛽＝0.83, which was proposed byTanaka et al.5) from the relationship between surface tension of liquid metals and binding energy at the surface.

In a ternary alloy consisting with components labeled as 1, 2 and 3, 𝐺𝐺𝑖𝑖𝐸𝐸.𝐵𝐵 can be expressed using the Redlich-Kister polynomial as follows,𝐺𝐺𝑖𝑖𝐸𝐸.𝐵𝐵 =33𝑀𝑀3𝐾𝐾𝑘𝑘𝑚𝑚� � � 𝐿𝐿𝑚𝑚𝑖𝑖𝑖𝑖 𝑛𝑛𝑖𝑖 𝑛𝑛𝑗𝑗 (𝑛𝑛𝑖𝑖 − 𝑛𝑛𝑗𝑗 ) + 𝑛𝑛1 𝑛𝑛2 𝑛𝑛3 � � 𝐿𝐿123 𝑛𝑛𝑖𝑖
𝑖𝑖=1 𝑗𝑗>𝑖𝑖 𝑚𝑚=0(4)𝑖𝑖=1 𝑘𝑘=0

where L is the interaction parameter, subscript j attached to the symbol denotes alloy components, just as idoes, and superscripts m and k denote the numbers of interaction parameters determined by the combinationof alloy elements. In this study, we used the values of L from Table 2 for the combinations of Fe, Si, and Cu6-9).

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Table 2. Interaction parameter L used in this study to calculate the partial excess Gibbs energy
𝐺𝐺𝑖𝑖𝐸𝐸.𝐵𝐵 for molten ternary Fe–Cu–Si alloys.
Interaction parameter L [J･mol-1]
L0Fe,Si = -164434.6 + 41.9773 T
L1Fe,Si = -21.523 T
L2Fe,Si = -18821.542 + 22.07 T
L3Fe,Si = 9695.8
L0Si,Cu = -36835 + 6.92 T
L1Si,Cu = -46727 + 19.94 T
L2Si,Cu = -37886 + 19.27 T
L0Fe,Cu = 35625.8 - 2.19045 T
L1Fe,Cu = 1529.8 - 1.15291 T
L2Fe,Cu = 12714.4 - 5.18624 T
L3Fe,Cu = -1177.1
L0Fe,Si,Cu = 19000 - 22.5 T
L1Fe,Si,Cu = 19000 - 22.5 T
L2Fe,Si,Cu = 19000 - 22.5 T

Surface Tension σ, 10-3 N･m-1

2100Ref.

6)

7)

8)

9)

Calculated by Butler model
Measured by EML
Brooks and Cameron(EML)1823K10)
Monma and Suto(SD)1873K11)
Dzhemilev et al.(SD)1823K12)
Kawai et al.(SD)1823K13)

1900
1700
1500
1300
1100
900
700
Fe010203040 50 60at.%Si708090100
SiFigure 1. Surface tension of molten Fe–Si alloys at 1850 K calculated by the Butler
equation, along with the results by EML. Literature data are also shown for comparison.
3. Results
Figure 1 shows the surface tension of the molten Fe-Si binary alloy at 1850 K as a function of sample
composition calculated by the Butler equation, along with the results measured by EML. Literature data arealso shown for comparison 10-13). The calculated surface tension decreases as the silicon composition increases
to about 70%, and then nearly plateaus, approaching the value for pure silicon. This behavior is in very good
agreement with the experimental results, including our measurements with EML, which are shown as plots.

3 of 51500
Surface Tension σ, 10-3 N･m-1

Calculated by Butler model
Measured by EML

1400
1300
1200
1100
1000
01020304050 60
at.%Cu708090100Cu

Figure 2. Surface tension of molten Fe-Si-Cu alloys at 1800 K, with a fixed molar ratio ofFe to Si of 3:2, calculated using the extended Butler equation. The measurement result of thealloy with a copper composition of 10at% is also shown.
Since it has been confirmed that the Butler equation expresses the surface tension of molten Fe–Si binary
alloys well, we calculated the surface tension of molten Fe–Si–Cu ternary alloys using the extended Butler
model. As an example, Fig. 2 displays the calculated result of the alloy at 1800 K in which the molar ratio of
Fe to Si is fixed at 3:2. The calculated surface tension decreases as the Cu content is increased to 40at%, followed
by a plateau up to about 80at% before a sudden increase. Although the measurement plot, we now have isonly one for the alloy with a copper composition of 10at%, it agrees well with the calculated result. Further
experimental measurements will confirm the validity of our extended Butler equation for ternary alloys.
4.

Summary
Using the Butler equation, we calculated the surface tension of molten Fe–Si binary alloys. We then extended

the Butler equation to calculate the surface tension for molten Fe–Si–Cu ternary alloys. The calculated results
for these molten alloys align well with measurements obtained using EML.
Acknowledgement
Part of this work was financially support by JSPS KAKENHI under Grant No. P20H02453. This work waspartly supported by the Grant-in-Aid for Front Loading Research from the Advisory Committee for SpaceUtilization Research in ISAS/JAXA.
References
1) J. A. V. Butler: The Thermodynamics of the Surfaces of Solutions. Proceedings of the Royal Society A, 135(1932) 348, DOI: https://doi.org/10.1098/rspa.1932.0040.

4 of 52)

S.Ozawa, Y. Nagasaka, M. Itakura, K. Sugisawa, and Y. Seimiya: Influence of oxygen adsorption fromatmosphere on surface tension of liquid silicon. Journal of Applied Physics, 130 (2021) 135101-1, DOI:
https://doi.org/10.1063/5.0062062.

3)

K. S. Yeum, R.Speiser and D. R. Poirier: Estimation of the Surface Tensions of Binary Liquid Alloys.
Metallurgical Transactions B, 20 (1989) 693, DOI: https://doi.org/10.1007/BF02655927.

4)

R. Speiser, D. R. Poirier and K. Yeum: Surface tension of binary liquid alloys. Scripta Metallurgica, 21(1987) 687, DOI: https://doi.org/10.1016/0036-9748(87)90385-1.

5)

T. Tanaka, K. Hack and T. Iida: Application of Thermodynamic Databases to the Evaluation of Surface
Tensions of Molten Alloys,Salt Mixtures and Oxide Mixtures. Zeitschrift für Metallkunde, 87 (1996) 380,DOI: https://hdl.handle.net/11094/26520.

6)J. LACAZE and B. SUNDMAN: An Assessment of the Fe–C–Si System. Metallurgical Transactions A, 22(1991) 2211, DOI: https://doi.org/10.1007/BF02664987.

7)

L. Soldi, A. Laplace, M. Roskosz, S. Gosse: Phase diagram and thermodynamic model for the Cu–Si andtheCu–Fe–Sisystems.

JournalofAlloysandCompounds,803(2019)

61,DOI:

https://doi.org/10.1016/j.jallcom.2019.06.236.
8)

Q. Chen, Z. Jin: The Fe–Cu System: A Thermodynamic Evaluation, Metallurgical and Materials
Transactions A, 26 (1995) 417, DOI: https://doi.org/10.1007/BF02664678.

9)

M. Hino, T. Nagasaka and T. Washizu: Phase diagram of Fe–Cu–Si ternary system above 1523 K. Journalof Phase Equilibria, 20 (1999) 179, DOI: https://doi.org/10.1361/105497199770335695.

10) R. Brooks and A. Cameron: Measurements of the surface tension of the Iron–Silicon system using
electromagneticlevitation.

ISIJInternational,

40(2000)

157,

DOI:

https://doi.org/10.2355/isijinternational.40.Suppl_S157.
11) 門 間改三 , 須藤 一: 置 換型 2 元 系合金 の表面 張力の 測定. 日本金 属学会 誌, 24 (1960) 163, DOI:
https://doi.org/10.2320/jinstmet1952.24.3_163.

12) N. K. Dzhemilev, S. I. Popel and B. V. Tsarewskiy: Isotherm of the Drnsity and Surface Energy of Iron–Silicon Melts.

Fiz. metal. metalloved., 18 (1964) 83, DOI:

13) 川合 保治, 森 克己, 岸本 誠, 石倉 勝彦, 下田 俊郎: 溶融 Fe-Si-C 合金の表面張力. 鉄と鋼, 60 (1974)
29, DOI:© 2023 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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