OR3-12
コロイド結晶形成シミュレーションにおけるポテンシャルの影響
Potential Influence on Simulation of Colloidal Crystal
Formation
足立 聡 1, 山中 淳平 2
Satoshi ADACHI 1 and Junpei YAMANAKA2
1
2

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
Several models related to colloidal crystals formation exist, for example, DLVO potential, Sogami potential1),
depletion force, impurity influence, and so on. To understand the mechanisms of colloidal crystals formation,a microgravity experiment was planned to suppress particle sedimentation. In the experiment plan, confocal
fluorescence microscopy is a candidate of main diagnostics. Therefore, we investigated whether three
dimensional coordinates of particles can be obtained by use of the microscope.
The sample conditions, that is, particle diameters, surface density of charge, and volume fraction of particles,
should be decided so that which model being correct could be clarified. To determine the sample conditions,
we carried out Monte Carlo simulation and summarize results as phase diagrams. The diagram based on the
DLVO potential and that based on the Sogami potential are compared to find the condition causing phasedifference.
2. Microscopic Observation Results
In the observation experiments, we used fluorescent polystyrene particles since the density is close to thatof water. The prepared samples are shown in Figure 1. The colloidal sample was provided from the Yamanaka
Lab. and was packed in soft bags at JAXA. The sample was observed by use of a confocal fluorescencemicroscope, which is basically the same model as the microscope on orbit (COSMIC). The typical observation
result with void is shown in Figure 2. The figure clearly shows the void and particles. The void is a region
without particles. To form the void, attractive force should exist.
The particle coordinates can be obtained by vertical scanning. The pair distribution function is calculated
from the three dimensional coordinates. Important parameters such as the nearest neighbor inter-particle
distance, and phase are obtained from the pair distribution function.

1 of 4

Figure 1. Fluorescent polystyrene colloidal sample bags. The colloidal sample was provided by the
Yamanaka Lab. The sample was packed in soft bags at JAXA.

Figure 2. Typical observation result of void. Void is a region without particles. Void is clearly observed
by a confocal fluorescence microscope.

3. Monte Carlo Simulation
To obtain phase diagrams in different potentials, Monte Carlo simulation is executed. A metropolis method
is used in this simulation. The metropolis method is described as
𝑁𝑁

1
𝐻𝐻(𝐫𝐫, 𝐩𝐩)1
𝐻𝐻(𝐫𝐫𝑖𝑖 , 𝐩𝐩𝑖𝑖 )⟨𝐴𝐴⟩ = � 𝑑𝑑𝒓𝒓 𝐴𝐴(𝐫𝐫, 𝐩𝐩) exp �−� ≃ � 𝐴𝐴(𝐫𝐫𝑖𝑖 , 𝐩𝐩𝑖𝑖 ) exp �−�,
𝑘𝑘𝐵𝐵 𝑇𝑇𝑍𝑍
𝑘𝑘𝐵𝐵 𝑇𝑇
𝑍𝑍

(1)

𝑖𝑖

where 𝐴𝐴, 𝑍𝑍, 𝐻𝐻, 𝐫𝐫, 𝐩𝐩 are targeting physical quantity, a partition function, the Hamiltonian, coordinates, andmomentum, respectively.

The typical result of pair distribution function is shown in Figure 3. In this case, the DLVO potential as

shown in Eq. (2) is used.
𝜙𝜙𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 (𝑟𝑟) =2

𝑄𝑄1 𝑄𝑄2 exp(𝜅𝜅𝜅𝜅) exp(−𝜅𝜅𝜅𝜅)��
+ 𝜙𝜙𝑣𝑣𝑣𝑣𝑣𝑣 ,4𝜋𝜋𝜖𝜖𝑟𝑟 𝜖𝜖0 1 + 𝜅𝜅𝜅𝜅
𝑟𝑟

(2)

where 𝜅𝜅, 𝑄𝑄1 , 𝑄𝑄2 , 𝑎𝑎, 𝜙𝜙𝑣𝑣𝑣𝑣𝑣𝑣 are the reciprocal of Debye length, charge of a particle, charge of another particle, a particle

diameter, van der Waals potential, respectively.

2 of 4

Figure 3. Typical pair distribution function calculated from simulation result. The DLVO potential isused in this case. In this calculation, the conditions of charge density of 1×10‒7 c/cm2, volume
fraction of 1.262 vol%, particle diameter of 700 nm are used.

It is found that the structure of colloidal crystal represented in Figure 3 is bcc, but the peak and bottom positions, andcoordination number slightly differ from those of ideal values. This means the lattice is distorted. From the simulation
results under various conditions, the phase diagrams are obtained.
4. Phase Diagram
Phase diagrams in cases of the DLVO and Sogami potentials are shown in Figure 4. In those phase diagrams,
the horizontal axis is the reciprocal of Debye length multiplied by Wigner-Seitz radius, 𝜅𝜅𝑎𝑎𝑤𝑤𝑤𝑤 , and the verticalone is the Coulomb coupling parameter, Γ𝑂𝑂𝑂𝑂𝑂𝑂 . These parameters are generally used in complex plasmas andare usable to the colloidal suspension due to the similar system. By comparing the diagram in DLVO potential

with that in Sogami potential, the particle conditions for clarifying the potential acting in colloidal suspension
are determined.

(a) DLVO Potential(b) Sogami Potential

Figure 4. Phase diagrams in cases of (a) DLVO and (b) Sogami potentials. The phases represent bydifferent colored closed circles. There is no void region in the case of DLVO potential.

5. Conclusions
It is found that the void is clearly observed, and particle coordinates are obtained by using the confocal
fluorescence microscope. The coordinates are used for calculating the pair distribution function, which is used
3 of 4

to obtain the nearest neighbor inter-particle distance and crystal structure. To obtain the phase diagrams todetermine the particle conditions for observation experiments, the Monte Carlo simulation is executed. By
comparing the diagram in the DLVO potential with that in the Sogami potential, the region, in which thedifferent phase appears in each potential, is clarified. This is useful for determination of particle conditions forobservation experiments.
References
1)

I. Sogami: Effective Potential between Charged Spherical Particles in Dilute Suspension, Phys. Lett., 96A (1983) 199,
DOI: https://doi.org/10.1016/0375-9601(83)90704-1.
© 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/).

4 of 4