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チタニア多成分会合コロイドの会合数分布シミュレーショ
ン
Monte-Carlo Simulation on Association Number ofTitania Multicomponent Colloidal Clusters
足立 聡 1, 山中 淳平 2, 奥薗 透 2, 豊玉 彰子 2
Satoshi ADACHI 1, Junpei YAMANAKA1, Tohru OKUZONO2 and Akiko TOYOTAMA2
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宇宙航空研究開発機構 きぼう利用センター，Japanese Experiment Module Utilization Center, JAXA
名古屋市立大学大学院薬学研究科，Graduate School of Pharmaceutical Sciences, Nagoya City University

1. Introduction
Multicomponent colloidal clusters experiments were carried out in the Japanese Experiment Module (JEM)‘Kibo’ in 2020. There are two kinds of microparticles, that is, positively charged ones and negatively charged
ones. Thus, colloidal clusters are formed by Coulomb attraction force. The particles are made of titania (TiO2).
Since the density of particles is much larger than that of water, the particles are rapidly precipitated on theground. Therefore, to make the sedimentation period long enough to form clusters, we used the microgravity
environment. The colloidal samples gelled by ultraviolet light on orbit to fix the particles coordinates. After
recovery of the samples, association number of clusters was investigated at the Yamanaka lab. To understandthe distribution of association number, Monte-Carlo simulation was executed. The simulation results arecompared with the experimental ones.
2. Experimental Results
The association number of titania clusters was investigated. All results are shown in another paper1). In thispaper, the population distribution of association number of m≥1 is shown in Figure 1 as a summary of thetitania cluster experiment. In this figure, the population of m=0 excluded since this position is not a cluster buta single particle.
The tendency of distribution monotonously decreases with increasing association number. However, thepopulation of m=4 in the case of NaCl concentration of 50 µM does not decrease and is the same population
as that of m=3. This is a good result because a titania tetrahedron is quite valuable from the engineering viewof point due to a minimum element of a diamond structure. The titania cluster with diamond structure istheoretically predicted as a perfect photonic material. Thus, we execute Monte Carlo simulation to know why
the population of m=4 and 50 µM is larger than the other cases

1 of 3Figure 1. Distribution of association number of m≥1. Cluster population of m≥1 is calculated byexcluding single particles.

3. Monte Carlo Simulation
We provided a Metropolis Monte Carlo simulation code. This is one of Markov chain Monte Carlo codes.
The Metropolis method is expressed as shown in Eq. (1).
⟨𝐴𝐴⟩ =𝑁𝑁

1𝐻𝐻(𝐫𝐫, 𝐩𝐩)1
𝐻𝐻(𝐫𝐫𝑖𝑖 , 𝐩𝐩𝑖𝑖 )� 𝑑𝑑𝒓𝒓 𝐴𝐴(𝐫𝐫, 𝐩𝐩) exp �−� ≃ � 𝐴𝐴(𝒓𝒓𝑖𝑖 , 𝒑𝒑𝑖𝑖 ) exp �−�,
𝑘𝑘𝐵𝐵 𝑇𝑇𝑍𝑍
𝑘𝑘𝐵𝐵 𝑇𝑇𝑍𝑍(1)𝑖𝑖where 𝐴𝐴 𝑍𝑍, 𝐻𝐻, 𝐫𝐫, 𝐩𝐩 are targeting physical quantity, a partition function, the Hamiltonian, coordinates, andmomentum, respectively. The partition function is described as follows.
𝑁𝑁𝐻𝐻(𝐫𝐫, 𝐩𝐩)
𝐻𝐻(𝐫𝐫𝑖𝑖 , 𝐩𝐩𝑖𝑖 )𝑍𝑍 = � 𝑑𝑑𝐫𝐫 exp �−� ≃ � exp �−�.
𝑘𝑘𝐵𝐵 𝑇𝑇
𝑘𝑘𝐵𝐵 𝑇𝑇(2)𝑖𝑖

To calculate the Hamiltonian, the DLVO potential is used as the potential energy and the virial theorem is used
to calculate the kinetic energy due to this system is not ideal gas.
4. Comparison
The simulation results are compared with the experimental ones in three cases, that is, NaCl concentrationof 0, 50, and 100 µM. The comparison is shown in Figure 2. It is found that the simulation results are consistent
with the experimental results. The population of association number of m=4 is largest in the 50 µM case inboth the simulation and experimental results.
To know whether the concentration of 50 µM indicates the largest population, the simulation is executed inseveral NaCl concentration cases. The result is shown in Figure 3. This figure shows that the largest population
exists at the concentration of 50 µM. The result also shows that the population is low in the range of 20 to 40µM. It is difficult to understand this behavior because the Coulomb interaction decreases with increasing theconcentration due to the shorter Debye length. More investigation is needed in future.

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Figure 2. Comparison of simulation results with experimental results. The tendency of simulation result
is consistent with the experimental one.

Figure 3. Population distribution of association number of m=4. The population of m=4 has the maximum
at the NaCl concentration of 50 µM.

5. Conclusions
To explain the experimental results in space, we conducted the Monte Carlo simulation. The simulation
results are compared with the experimental ones. The tendency of population distribution of simulation isconsistent with the experiments. The simulation also indicates that the population is largest at theconcentration of 50 µM.
References1)

H. Miki. T. Ishigami, J. Yamanaka, T. Okuzono, A. Toyotama, J. Mata, H. Komazawa, Y. Takeda, M. Minami, M.
Fujita, M. Doi, T. Higuchi, H. Takase, S. Adachi, T. Sakashita, T. Shimaoka, M. Nagai, Y. Watanabe and S. Fukuyama:
Clustering of charged colloidal particles in the microgravity environment of space, npj Microgravity, 9 (2023) 33, DOI:
https://doi.org/10.1038/s41526-023-00280-5.
© 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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