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DARTS 公開データを使用した FACET 実験のデータ解析
Data Analysis of FACET Experiment by Use of Open
Data Stored in Kibo Experiment Data Archive of DARTS
足立 聡
Satoshi ADACHI
宇宙航空研究開発機構，Japan Aerospace Exploration Agency

1. Introduction
FACET experiment was carried out in 2009 and 2010 on orbit 1). The experimental data are opened and areavailable on the DARTS (Data Archives and Transmission System) website 2). The data can be freely used foracademic and educational purposes.
The objectives of FACET experiment are to investigate conditions causing faceted morphological instability
and to investigate mechanisms of faceted morphological instability. Faceted cellular growth of anisotropic
materials is a target phenomenon in the FACET experiment. The similar instability in the isotropic materials
is well known as the Mullins-Sekerka morphological instability. However, it is not clarified whether thefaceted instability is the same mechanism as the Mullins-Sekerka instability or not.
Thus, a two-wavelength interferometer and an amplitude modulation microscope, which are installed in theSolution Crystallization Observation Facility (SCOF), are used to achieve the objectives. Temperature andconcentration distributions are obtained from the interferometer measurements.
There are several issues to fully understand the instability mechanisms, that is, dependency of growth rateon supersaturation to identify the growth mechanisms, microscopic observation to find the beginning locationof instability, description of instability model to understand the instability mechanisms. The dependency ofgrowth rate on supersaturation is investigated in this paper as the first step.
2.

Experimental Setup
The SCOF is located on the ‘Kibo’ module of the international space station (ISS). The SCOF has a two-wavelength Mach-Zehnder interferometer and an amplitude modulation microscope. Magnification factors
are two and four. The sizes of the field of view are 2.4 mm×3.6 mm and 1.2 mm×1.8 mm, respectively. TheSCOF optical observation devices are schematically shown in Figure 1(a). The experiment cell shown in Figure1(a) is schematically shown in Figure 1(b). Mixture of phenyl salicylate (Salol) and tert-butyl alcohol (tbutanol) is put into the cell and sealed.
Two cells are installed in one cartridge. Initial target concentrations of t-butanol are 4 mol% and 8 mol%,respectively. After the experiment, the cartridge is returned on the ground. The t-butanol concentrations aremeasured and is found that those are 1.15 mol% and 2.58 mol%, respectively. The concentration decrease may
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be caused by the vaporization of t-butanol during the sample preparation. Although the concentration differsfrom the target concentration, this is not a big problem because the known concentration is the most importantto obtain concentration distributions. In this paper, run number 9 (named as FACET2-07) is used as the typicalexperimental result. The temperature gradient and the cooling rate are 15 K/cm and 2 K/min, respectively.

(a) Schematic view of SCOF optical observation devices(b) Schematic view of FACET experiment cell
Figure 1. Schematic view of SCOF optical observation devices and FACET experiment cell.

3.

Data Analysis Method
The main data are interferometric images. The temperature and concentration distributions are obtained by2 of 7

analyzing the interferometric images. A two-dimensional Fourier transform is used in this paper to analyze
the interferometric image. This method has features of no need of additional images and reasonable calculation
time. The temperature and concentration are calculated by using the following equations.
𝜆𝜆01
𝜆𝜆0 2𝑏𝑏 − Δ(Δ𝜙𝜙)2𝑏𝑏
2𝜋𝜋𝐿𝐿𝑠𝑠 22𝜋𝜋𝐿𝐿𝑠𝑠 1
𝑇𝑇 =+ 𝑇𝑇(0) and𝐿𝐿𝑔𝑔
𝐿𝐿𝑔𝑔
�𝑎𝑎1 + 𝑎𝑎3 � 𝑏𝑏2 − �𝑎𝑎2 + 𝑎𝑎4 � 𝑏𝑏1
𝐿𝐿𝑠𝑠
𝐿𝐿𝑠𝑠
Δ(Δ𝜙𝜙)1𝐶𝐶 =𝛥𝛥(𝛥𝛥𝛥𝛥)1𝜆𝜆01
𝜆𝜆0 2𝐿𝐿𝑔𝑔
𝐿𝐿𝑔𝑔
�𝑎𝑎 + 𝑎𝑎4 � − 𝛥𝛥(𝛥𝛥𝛥𝛥)2
�𝑎𝑎 + 𝑎𝑎3 �2𝜋𝜋𝐿𝐿𝑠𝑠 2𝐿𝐿𝑠𝑠
2𝜋𝜋𝐿𝐿𝑠𝑠 1
𝐿𝐿𝑠𝑠+ 𝐶𝐶(0)
𝐿𝐿𝑔𝑔
𝐿𝐿𝑔𝑔
�𝑎𝑎2 + 𝑎𝑎4 � 𝑏𝑏1 − �𝑎𝑎1 + 𝑎𝑎3 � 𝑏𝑏2
𝐿𝐿𝑠𝑠
𝐿𝐿𝑠𝑠

(1)(2)

where Δ(Δ𝜙𝜙)1 = Δ𝜙𝜙1 (𝑡𝑡) − Δ𝜙𝜙1 (0) , Δ(Δ𝜙𝜙)2 = Δ𝜙𝜙2 (𝑡𝑡) − Δ𝜙𝜙2 (0) , 𝐿𝐿𝑔𝑔 , 𝐿𝐿𝑠𝑠 , 𝜆𝜆01 , 𝜆𝜆0 2 , Δ𝜙𝜙1 , Δ𝜙𝜙2 , 𝑇𝑇(0) , and 𝐶𝐶(0)represent phase change at the wavelength of the first laser (wavelength of 532 nm), phase change at thewavelength of the second laser (wavelength of 780 nm), the thickness of glass cell, sample thickness,wavelength of the first laser in vacuum (532 nm), and wavelength of the second laser in vacuum (780 nm),phase change at the wavelength of the first laser, phase change at the wavelength of the second laser, thetemperature at 𝑡𝑡 = 0, and the concentration at 𝑡𝑡 = 0, respectively. The parameters 𝑎𝑎1 , 𝑎𝑎2 , 𝑎𝑎3 , 𝑎𝑎4 , 𝑏𝑏1 , and 𝑏𝑏2

are expressed as

𝜕𝜕𝑛𝑛1
𝜕𝜕𝑛𝑛2
𝜕𝜕𝑛𝑛1
𝜕𝜕𝑛𝑛2
𝜕𝜕𝑛𝑛1
𝜕𝜕𝑛𝑛2
𝑎𝑎1 = �� , 𝑎𝑎2 = �� , 𝑎𝑎3 = �� , 𝑎𝑎4 = �� , 𝑏𝑏1 = �
� , and 𝑏𝑏2 = ��,𝜕𝜕𝜕𝜕 𝑠𝑠𝜕𝜕𝜕𝜕 𝑠𝑠𝜕𝜕𝜕𝜕 𝑔𝑔𝜕𝜕𝜕𝜕 𝑔𝑔𝜕𝜕𝜕𝜕 𝑠𝑠𝜕𝜕𝜕𝜕 𝑠𝑠(3)

where 𝑛𝑛1 and 𝑛𝑛2 are refractive indices at the wavelength of the first and second lasers, respectively. Since

denominators of Eqs. (1) and (2) are subtractions of multiplications, the parameters 𝑎𝑎1 to 𝑏𝑏2 should havehigh precision at least four digits of accuracy. This is the reason why the two-wavelength interferometer is notwell familiar. Since it is not easy to obtain such precision due to easy vaporization of t-butanol, the refractive
indices of sample have been measured several times. As a result, the parameters 𝑎𝑎1 , 𝑎𝑎2 , 𝑏𝑏1 , and 𝑏𝑏2 aredetermined as�𝜕𝜕𝑛𝑛1
� = −3.8843 × 10−4 [1⁄K],𝜕𝜕𝜕𝜕 𝐶𝐶�

𝜕𝜕𝑛𝑛2
� = −2.3867 × 10−4 [1⁄K],𝜕𝜕𝜕𝜕 𝐶𝐶�

𝜕𝜕𝑛𝑛2
� = −6.3917 × 10−4 [1⁄mol%].
𝜕𝜕𝜕𝜕 𝑇𝑇𝜕𝜕𝑛𝑛1�� = −6.7583 × 10−4 [1⁄mol%], and𝜕𝜕𝜕𝜕 𝑇𝑇(4)

Since temperature dependency of refractive indices of fused quartz has weak dependency on wavelength, 𝑎𝑎3

and 𝑎𝑎4 are the same value and are expressed as�

𝜕𝜕𝑛𝑛1
𝜕𝜕𝑛𝑛2� ≃�� ≃ 8 × 10−6 [1⁄K].
𝜕𝜕𝜕𝜕 𝑔𝑔𝜕𝜕𝜕𝜕 𝑔𝑔(5)

Thus, Eqs. (1) and (2) are calculated.
4.

Experimental Results and Discussion
The typical results of stable growth are shown in Figure 2. All images are obtained under the condition of
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the magnification factor of 2. Figure 2(a) is the amplitude modulation microscopic image, Figure 2(b) is theinterferometric image with the wavelength of 532 nm, and Figure 2(c) is the interferometric image with thewavelength of 780 nm. It is found that the fringes are curved near the interface. Since the temperature
distribution is much more linear than the concentration distribution, the curved fringe shows theconcentration is piled up near the interface due to segregation.

(a) Amplitude modulation microscopic image(b) Interferometric image with wavelength of 532 nm(c) Interferometric image with wavelength of 780 nm
Figure 2. Typical experimental results of stable growth

The phase distribution is calculated from the interferometric images to calculate the temperature andconcentration. The noise reduction is required as the first step of calculation. Two-dimensional Fouriertransform is executed. There should be some peaks, but the peak positions are unclear due to noises. To reduce
the noises, the amplitude is calculated. Amplitude clearly shows the peaks. The peak at the center mainly
consists of fringe information. Therefore, the next peak on the right side of the center peak is selected as thesignal.
The inverse Fourier transform of this peak is executed. The amplitude of the inverse Fourier transform
clearly shows that the fringes are fully eliminated. To obtain phase distribution, the following equation is used.
𝐼𝐼𝐼𝐼(𝐼𝐼𝐹𝐹 )𝜙𝜙 = tan−1 ��.
𝑅𝑅𝑅𝑅(𝐼𝐼𝐹𝐹 )(6)

The calculated results of Eq. (6) are shown in Figure 3. Figure 3(a) represents the phase change from − 𝜋𝜋⁄2to 𝜋𝜋⁄2 at the wavelength of 532 nm. Figure 3(b) is an unwrapped phase distribution. It is found that the phaseunwrapping is smooth and successful though a few phase singularities exist. Figure 3(c) is the phase change
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at 780 nm, and Figure 3(d) is the unwrapped phase distribution. Thus, Eqs. (1) and (2) are calculated by using
obtained two phase changes Δ𝜙𝜙1 and Δ𝜙𝜙2 . Since there are no data on refractive indices of the solid phase ofphenyl salicylate / tert-butanol alloy, phase unwrapping and phase calculation in the liquid phase are correct.

(a) Phase change at 532 nm(b) Unwrapped phase at 532 nm(c) Phase change at 780 nm(d) Unwrapped phase at 780 nm

Figure 3. Calculated phases at 532 nm and 780 nm

By calculating Eqs. (1) and (2), the temperature and concentration distributions in the liquid phase areobtained and are shown in Figure 4. Figure 4(a), Figure 4(b), Figure 4(c), and Figure 4(d) represent
temperature in stable growth, concentration in stable growth, temperature in faceted cellular growth, andconcentration in faceted cellular growth, respectively. It is found that the temperature decreases with timeadvancement and the concentration piles up over time. This behavior seems to be reasonable.
The dependency of growth rate on supersaturation is investigated from the interferometric measurements
to investigate not only the growth mechanisms but also measurement accuracy, especially a measurement
offset. The dependency is shown in Figure 5. Red circles and blue circles represent growth rates at a vertex
location of faceted cell and growth rates at a valley location of faceted cell, respectively. It seems that the offset
is about 1 % or less. It is found that three kinds of growth mechanisms exist, that is, adhesive growth, spiralgrowth, and two-dimensional nucleation growth. It is also found that both growth rates at the vertex and at
the valley are sit on the same line. This means that the growth mechanisms at the vertex and at the valley arethe same.
To confirm this hypothesis, the time evolution of interface movement is investigated and is shown in Figure6(a). The result clearly shows that the interface movement at the vertex is faster than that at the valley during
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the time from 3.6 to 4.7 min, but the interface movement is almost the same after that time. Figure 6(b) shows
the time evolution of supersaturation. This shows the supersaturation at the vertex is larger than that at thevalley from 3.6 to 4.7 min, but the supersaturation at the vertex is almost the same or smaller than that at thevalley except for that time. This is the reason why the growth rate at the vertex is similar to that at the valley
after 4.7 min. As a conclusion, no special growth mechanism exists at the vertex.

(a) Temperature in stable growth(b) Concentration in stable growth(c) Temperature in faceted cellular growth(d) Concentration in faceted cellular growth

Figure 4. Temperature and concentration distributions in stable growth and in faceted cellular growth

Figure 5. Growth rate dependency on supersaturation
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Figure 6. Time evolution of interface movement and supersaturation

5.

Conclusions
Data analysis of faceted cellular growth, FACET experiment, is carried out by using the open data being

available on the DARTS. The dependency of the growth rate on the supersaturation is investigated tounderstand the growth mechanisms. The offset of supersaturation may be 1 % or less from the dependency.
The dependency shows that there are three mechanisms, that is, adhesive growth, spiral growth, and twodimensional nuclear growth. No new growth mechanisms exist. This means that the trigger of morphologicalinstability should be the supersaturation variation.
References1)
2)

Y. Inatomi: Rapid Communication: Crystal Growth Experiments in KIBO, J. Jpn. Soc. Microgravity Appl., 27 (2010)32.
https://darts.isas.jaxa.jp/iss/kibo/

Acknowledgements
FACET data are obtained from Kibo Experiment Data Archive, which is a part of DARTS (Data Archivesand Transmission System). The author is grateful to all concerned with DARTS for their efforts to open
valuable data and to keep opening them.

© 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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