Chapter 3 Collective motion of self-propelled particles driven by external torque 47
3.6 Appendix
3.6.1 Active particle systems with exclusion-volume interactions . 60
In this appendix, we introduce the simple active particles with excluded volume interactions to show that the polar symmetry of the spatial velocity correlation function arises purely from the hydrodynamic interactions.
Consider an active particle described by the equation of motion shown below;
mdvi dt =
α−β|vi|2
vi+Õ
j,i
Fi j, (3.21)
whereFi j is the force due to the excluded volume interaction described by WCA potential.
In this model, the active particles try to keep propelling at a constant speed v0 = p α/β.
Such active particles behave like a gas, since only the repulsive interactions due to excluded volume effects contribute to the direct momentum exchange of the particles.
3.6 Appendix 61
Fig. 3.9 The spatial velocity correlation function for the simple active particles with excluded volume effect. The angular direction is labeled as zero in the propulsion direction and the radial coordinate means the distance between particles.
The spatial velocity correlation function for this model of an active particle system is shown in Fig. 3.9. The spatial velocity correlation function shows a spatially isotropic distribution, and we cannot observe the anisotropic distribution with respect to the direction of propulsion with polar symmetry seen in the case with the hydrodynamic interactions.
This is due to the fact that the excluded volume effect is an isotropically symmetric repulsion that satisfies the action-reaction law.
63
Chapter 4
General conclusion and outlook
In this thesis, we provide a simulation framework for collective phenomena caused by self-propelled particles in viscous fluids through simple modeling and direct numerical calculations. The topic addressed in this thesis was inspired by experiments on artificial self-propelled particles.
In Chapter 2, we treated the chemically active particles at an air-liquid interface. In the modeling, hydrodynamic interactions, self-driving forces due to Marangoni effect, Marangoni flow, and capillary interactions are introduced as physical elements. By intro-ducing these physical elements analytically into the model, we can avoid direct numerical calculations of the field variables such as concentration field and the flow velocity field in the viscous fluid. Then, we can consider the effects of each element separately. From this modeling based on physical elements, the interaction between particles is derived. This interaction, which shows attraction and repulsion depending on the distance between the particles, is shown to contribute to the realization of the crystalline state, spreading state and liquid-like state in many particle systems. In addition, by introducing the hydrodynamic interactions, the spatial velocity correlations in liquid-like states shows a polar symmetry.
Furthermore, by introducing a self-driving force that depends on the particle velocity and the velocity field of the fluid, worm-like cluster state is realized as one of the collective motions. These diverse behaviors have not been understood by the existing simple mathe-matical models, e.g. boids or Vicsek model. In particular, we believe that the worm-like cluster state reproduced by our model is unique to chemically active particles and may be applied to a possible control of artificial colloidal particles and droplets.
In Chapter 3, by direct numerical calculations using smoothed profile method, we have simulated a self-propelled particle system driven by an external torque. The fluid flow
around the particle driven by torque was confirmed to be the rotlet flow with a polar sym-metry. With respect to the collective motion, the spatial velocity correlation is found to have a polar symmetry as well as the symmetry of this hydrodynamic interaction. In addition, the acceleration effect occurred during a local collision between self-propelled particles, and the distribution of the particle speed was obtained by our numerical simulation. The distribution of the particle speed reproduced the distribution obtained in the experiment.
In the future, based on the findings in the present study, we would like to develop a simpler coarse-grained model to understand the formation and the relationship between the symmetry of the interaction of individual particles and the symmetry of their collective behavior. In particular, the symmetry in the formation of the collective behavior and the symmetry on the spatial correlation of the interaction of individual particles is an interesting issue. Additionally, we hope to elucidate the connection between dynamic self-assembly and interactions acting on the self-propelled particles of animals, such as flight of birds with a school of V-shape and vortex-like behavior of school of fish.
65
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List of Publications and Awards
List of publications
1. Shun Imamura and Toshihiro Kawakatsu, “Modeling of chemically active particle on the air-liquid interface”,manuscript in preparation
Presentations
1. S.Imamura and T.Kawakatsu, Collective motion of self-propelled droplets by Marangoni effect ，Active Matter Workshop 2018, 講演番号:ST1 2., 京都大学 福井謙一記念研究センター，2018.01.19.（査読有）
2. 今村舜，川勝年洋， マランゴニ効果により自己駆動する液滴の集団運動の解 析 ，第7回ソフトマター研究会，ポスター番号:P03，京都大学 北部研究総合棟 益川ホール＆理学研究科セミナーハウス，2017.10.23.（査読無）
3. 今村舜，川勝年洋， マランゴニ効果により自己駆動する液滴の集団運動 ，日本 物理学会 第73回年次大会，ポスター番号:25aPS-92，東京理科大学 野田キャン パス，2018.03.25.（査読無）
4. 今村舜，川勝年洋， マランゴニ効果により自己駆動する液滴の集団運動II ，日 本物理学会2018年秋季大会，ポスター番号:12aPS-112，同志社大学 京田辺キャ ンパス，2018.09.12.（査読無）
5. S.Imamura and T.Kawakatsu, Collective motion of self-propelled droplets by
Marangoni effect , APEF2018, ポスター番号: A29，東京大学本郷キャンパ
ス，2018.11.12. / 2018.11.13. (査読無)
6. 今村舜，大山倫弘，John Jairo Molina，山本量一， 外部トルクによって駆動され るコロイド粒子系の直接数値シミュレーション ，日本物理学会 第75回年次大 会（2020年）現地開催中止（査読無）