Gravity induced formation of spinners and polar order of spherical microswimmers on a surface

We study numerically the hydrodynamics of a self-propelled particle system consisting of spherical squirmers sedimented on a flat surface. We observe the emergence of dynamic structures, due to the interplay of particle-particle and particle-wall hydrodynamic interactions. At low coverages, our results demonstrate the formation of small chiral spinners: two or three particles are bound together via near-field hydrodynamic interactions and form a rotating dimer or trimer, respectively. The stability of the self-organized spinners can be tuned by the strength of the sedimentation. Increasing the particle concentration leads more interactions between particles and the spinners to become unstable. At higher area fractions we find that pusher particles can align their swimming directions, leading to a stable polar order and enhanced motility. Further, we test the stability of the polar order in the presence of a solid boundary. We observe the emergence of a particle vortex in a cylindrical confinement.

Figure 1

(a) The time evolution of the angle $\phi$ between the particle axis and the wall plane when the particle sediments towards the surface. The sedimentation strength is $u_s/u_0=2.4$. The inset shows the definition of $\phi$ and the gap-size $h$ used in the text. (b) Floating of a self-propelled particle near the surface. The final stable distance $h$ between the particle surface and flat wall is plotted as a function of the sedimentation strength. The simulation box is $L_X=12R,L_Y=12R$, and $L_Z=12R$.

Figure 2

The calculated flow fields for a $\beta =-1$ pusher, a neutral swimmer ($\beta =0$), and for a $\beta =+1$ puller floating above a no-slip surface ($u_s/u_0\approx 2.4$). The color code corresponds to the magnitude of the fluid flow.

Figure 3

(a) A typical configuration of rotating dimer with $u_s/u_0=6.4$ for $\beta =-1$ pushers. The definition of the chirality $\alpha$ is shown in the figure. The black arrows show the projections of the particle axis in $Y-Z$ plane. (b) The $Y$ positions of each particle in the dimer are plotted as a function of time $u_{0}t/R$.

Figure 4

(a) A typical configuration of rotating trimer with $u_s/u_0=8$ for $\beta =-1$ pushers. The definition of the chirality $\alpha$ is shown in the figure. The black arrows show the projections of the particle axis in the $Y\text{−}Z$ plane. (b) The $Y$ positions of each particle in the trimer are plotted as a function of time $u_{0}t/R$.

Figure 5

(a) The rotation frequency $\omega R/u_0$, (b) the chirality $\alpha$ of the dimer structure, (c) the gap size $h$, and (d) the orientation angle $\phi$ for a dimer as a function of the sedimentation strength.

Figure 6

(a) The rotation frequency $\omega R/u_0$, (b) the chirality $\alpha$ of the trimer structure, (c) the gap-size $h$, and (d) the orientation angle $\phi$ for a trimer with $\beta =-1$ as a function of the sedimentation strength.

Figure 7

A typical case showing a rotor motion of a rotating dimer when the pure spinning motion becomes unstable ($\beta =-1$ and $u_s/u_0=4.8$). (a) A snapshot of the configuration of the dimer. The black arrows show the projections of the particle axis in the $Y\text{−}Z$ plane. (b) The trajectories of the particles in the dimer. The initial positions and final positions are shown by the black solid and hollow circles, respectively. The final configuration is given in a. (c) The gap size $h$ and (d) the orientation angle $\phi$ of each particle in the dimer are plotted as a function of time $u_{0}t/R$.

Figure 8

(a) Schematic of the simulation setup. Particles are initially randomly positioned above a solid surface. The sedimentation force $F_s$ is then turned on, leading to the formation of a monolayer of particles near the surface with an area fraction $\varphi$. (b) Different phases observed when the area fraction $\varphi$ and the squirmer parameter $\beta$ are varied. The corresponding configurations for each phase are shown in (c)–(g). (c) Phase I: rotating dimers. (d) Phase II: isolated particles. (e) Phase III: erratic motion. (f) Phase IV: crystal. (g) Phase V: polar order. The black arrows in (c)–(g) show the velocities of the particles. The cases in (c)–(g) use the simulation box of $L_X=6R,L_Y=80R$, and $L_Z=80R$.

Figure 9

Dynamics of particles in a polar state ($N=3200$, $\varphi \approx 40$% and $L_X=6R,L_Y=160R$, and $L_Z=160R$). (a) An example of a two-dimensional particle velocity distribution in the polar state. (b) The polar order parameter $\langle cos\varkappa \rangle$ as a function of time $t{u}_0/R$. The calculated particle (c) pair correlation function $g\left(r\right)$ and (d) the number fluctuations $\mathrm{\Delta }{N}^2$ as a function of the average number of particles $N$.

Figure 10

(a) The average velocity $\langle \overline{v}\rangle /u_0$ and (b) the polar order parameter $\langle \overline{cos\varkappa }\rangle$ as a function of the area fraction $\varphi$ for $\beta =-1$ pushers. The average velocity and (d) the polar order parameter as a function of $\beta$ for $\varphi \approx 40$%. The shaded areas indicate the range of the polar state.

Figure 11

(a) A snapshot of the particle vortex in a cylindrical boundary. The color code and the arrows show the velocities of the particles. (b) Time-averaged velocity field of the particles. (c) The local area fraction ${\varphi }_l$ along the radius for different size of the cylindrical boundary. (d) The azimuthal velocity $V_{\theta }/u_0$ along the radius. (The black lines are guide to eyes.)