P04
HZ 内の温度差と濃度差に起因するマランゴニ対流に及ぼすマランゴニ数の時間変化の影響
Effect of time dependency of Marangoni number onMarangoni convection due to temperature andconcentration differences in a HZ
島袋史栄 1, 水口尚 2, 岡野康則 3,
Shiei SHIMABUKURO1, Hisashi MINAKUCHI2 and Yasunori OKANO3
琉球大学大学院理工学研究科,Graduate School of Engineering and Science, University of the Ryukyus, Okinawa,Japan
2 琉球大学工学部, Faculty of Engineering, University of the Ryukyus, Okinawa, Japan
3 大阪大学大学院基礎工学研究科, Graduate School of Engineering Science, Osaka University, Osaka, Japan1

1. Introduction
One of the production methods Si/Ge bulk single crystals is a Floating Zone (FZ) method, which is
performed in a microgravity environment. In the FZ method, the polycrystalline raw material is fed fromthe top and passed through a heater to create a melt section with the seed crystal, and the melt section iscooled to obtain the desired single crystal. However, the FZ method has the problem that Marangoni
convection generated in the melt adversely affects crystal growth. Marangoni convection is a flow generated
by surface tension differences due to temperature and concentration differences. Therefore, controlling
Marangoni convection leads to quality crystal growth. This study uses the half-zone model, which isfrequently used in studies of Marangoni convection (see, for instance, Refs1-2). The strength of convection isexpressed in terms of a dimensionless number called the Marangoni number , and the critical value of theMarangoni number is treated as the critical Marangoni number3). The Marangoni number is defined by thephysical properties of the melt, the temperature difference.
There have been many analyses of Marangoni convection caused by temperature differences. However, inthe growth of alloys such as SixGe1−x, it is necessary to consider not only the thermal Marangoni convection
but also the solutal Marangoni convection due to surface tension differences of the components of the alloy.
Therefore, in order to shed light on the relative contributions of thermal and solutal Marangoni convections
occurring in the melt of a half-zone of the FZ system, we have carried out three-dimensional numericalsimulations4-5). Especially analysis was performed considering time dependency of temperature andconcentration differences in the present study.

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2.Analysis Method
In this study, Figure 1 was used as the model for the analysis. The sides are free surface, with the top
surface hot and the bottom surface cold. The analysis area was the melt only, the free surface height [L] was
5 mm radius [a] was 10 mm, and As = a/L = 0.5 was used. In addition to assuming a microgravity
environment, the gas-liquid and solid-liquid surface shapes were flat. The sampling point is(r,θ,z)=(0.99a,0,0.5L). The basic analytical equations are the continuous equation (1), the Navier-Stokes
equation (2), the energy equation (3), and the diffusion equation (4), which are discretized using the finitevolume method. The equations are shown below.
𝛁⋅𝒖=0(1)𝜕𝒖
1+ 𝒖 ∙ 𝛁𝒖 = − ∇𝑝 + 𝜈∇2 𝒖
𝜕𝑡
𝜌

(2)

𝜕𝑇+ 𝒖 ∙ 𝛁𝑇 = 𝛼∇2 𝑇
𝜕𝑡(3)𝜕𝐶+ 𝒖 ∙ 𝛁𝐶 = 𝐷∇2 𝐶
𝜕𝑡(4)The boundary condition for temperature was set constant at the lowest temperature on the top surface andthe highest temperature on the bottom surface, and adiabatic at the free interface on the side boundarycondition. The boundary condition for concentration was the same as the boundary condition fortemperature. The boundary condition for velocity was the no slip condition for the top and bottom surfaces,
while the Marangoni convection caused by the temperature and concentration differences was considered
for the side surfaces. Thermal and solutal Marangoni numbers and sum of thermal and solutal Marangoni
numbers, which indicates the intensity of convection, are then given below.
𝜕𝜎 ∆𝑇𝐿𝜕𝜎 ∆𝐶𝐿𝑀𝑎 𝑇 = − 𝜕𝑇 𝜇𝜐 , 𝑀𝑎𝐶 = 𝜕𝐶 𝜇𝜐 , 𝑀𝑎 = 𝑀𝑎 𝑇 + 𝑀𝑎𝐶(5)~(7)

The thermal and solutal Marangoni numbers were changed with time considering the time dependency oftemperature and concentration differences. Using the three routes shown in Figure 2. 𝑀𝑎𝐶 and 𝑀𝑎 𝑇 are
increased between 20000s. Route I is increased to 𝑀𝑎𝐶 =0~1786 and then to 𝑀𝑎 𝑇 =0~3572. Route II isincreased 𝑀𝑎𝐶 =0~1786 and 𝑀𝑎 𝑇 =0~3572 simultaneously. Route Ⅲ is increased to 𝑀𝑎𝑇 =0~3572 and then to𝑀𝑎 𝑇 =0~1786.
The physical properties of Si/Ge (SixGe1-x) used in this study are listed in Table 1.
Table 1. Physical properties of SixGe1-x
𝝊Kinematic viscosity

Thermal Diffusion Coefficient
Diffusion coefficient

𝑫𝜶

1.4×10-7[m2/s]

2.2×10-5[m2/s]

1.0×10-8[m2/s]

Prandtl number

𝑷𝒓

6.37×10-3

Schmidt number

𝑺𝒄

14.0

[-]
[-]

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OpenFOAM, a free and open source CFD software, is used. The governing equations (eq. 1-4) are
discretized using the finite volume method, and the unsteady algorithm, PISO method, is used.

Figure 1. Geometry of the analytical liquid bridge.

Figure2. Route diagram.

3.Results and discussion
Figures 3(a), (b) and (c) show the convective structure changes at each route. The wave number (m) was m= 0→4→6→7 at route I. It was m = 0→4→5→7 at route II. It was m = 0→3→3’ at route III. Figure 4 shows
the time dependency of the heightwise velocity at a sampling point (r, θ, z)=(0.99a, 0, 0.5L). As theMarangoni number (time) increased, the velocity of each route increased. However, results at routes I andII transitioned to oscillating flow, while it at route III remained steady flow. Figure 5 shows the relationshipbetween sum of thermal and solutal Marangoni numbers and wave number. Even with the same Ma
number, the final wave number was depended on the route. Comparing routes I and II, the criticalMarangoni number at route I was MaC=1250 and MaT=0. At route II, the critical Marangoni number was
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MaC=696 and MaT=1393. Therefore, the critical solutal Marangoni number of thermo-solutal Marangoni
convection was smaller than that of only solutal Marangoni convection.

Figure 3(a). Convective structure changes at route I.

Figure 3(b). Convective structure changes at route II.
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Figure 3(c). Convective structure changes at route III.

Figure 4. Tiem dependency of the heightwise velocity at a sampling point.

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Figure 5. Relationship between sum of Marangoni numbers and wave number.

References
1)

M.Sakurai, N.Ohishi, A.Hirata: Oscillatory thermocapillary convection in a liquid bridge: Part 1-1g

Experiments, Journal Crystal of Growth, 308 (2007) 352. Doi: 10.1016/j.jcrysgro.2007.07.026.
2)

I.Ueno, Y.Abe, K.Noguchi, H.Kawamura: Dynamic particle accumulation structure (PAS) in half-zone

liquid bridge - Reconstruction of particle motion by 3-D PTV, Advance in Space Research., 41 (2008) 2145.
Doi: 10.1016/j.asr.2007.08.039.
3) K.Li, S.Matsumoto, N.Imaishi, W.R.Hu: Marangoni flow in floating half zone of molten tin,
INTERNATIONAL JOURNAL OF HEAT AND MASS TRANSFER, 83 (2014) 575, Doi:
10.1016/j.ijheatmasstransfer.2014.12.016.
4) H.Minakuchi, Y.Okano, S.Dost: Effect of thermo-solutal Marangoni convection on the azimuthal wave
number in a liquid bridge, Journal Crystal of Growth, 468 (2017) 502. Doi: 10.1016/j.jcrysgro.2016.09.028
5) K. Harada, A. Misato, H. Minakuchi, Y. Okano: Proceedings of the JASMAC-34 (2022) P28.

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