Orientational order and morphology of clusters of self-assembled Janus swimmers

Due to the combined effect of anisotropic interactions and activity, Janus swimmers are capable to self-assemble in a wide variety of structures, many more than their equilibrium counterpart. This might lead to the development of novel active materials capable of performing tasks without any central control. Their potential application in designing such materials endows trying to understand the fundamental mechanism in which these swimmers self-assemble. In the present work, we study a quasi-two-dimensional semidilute suspensions of two classes of amphiphilic spherical swimmers whose direction of motion can be tuned: either swimmers propelling in the direction of the hydrophobic patch or swimmers propelling in the opposite direction (toward the hydrophilic side). In both systems we have systematically tuned swimmers' hydrophobic strength and signature and observed that the anisotropic interactions, characterized by the angular attractive potential and its interaction range, in competition with the active stress, pointing toward or against the attractive patch gives rise to a rich aggregation phenomenology.

Figure 1

(a) WP Janus swimmers. (b) AP Janus swimmers. The attractive patch is represented in blue (dark gray), and the nonattractive patch is in green (light gray). The swimming direction $\widehat{e}$ is the red arrow and direction of the attractive patch $\widehat{p}$ is the green one. The center-to-center distance between two swimmers $\overrightarrow{r_{ij}}$ is the black vector.

Figure 2

XY projection of a squirmer at $\beta =-1$ (pusher, left panel), $\beta =0$ (neutral, middle panel), and $\beta =1$ (puller, right panel). The red arrow represents the swimming direction, the black arrows the velocity directions of the fluid, the color code the speed of the fluid normalized by the Stokes velocity ($v_s=2/3B_1$). Velocities are computed in the squirmer frame of reference.

Figure 3

Time evolutions of the mean cluster size for squirmers when $\xi =0.1$ (pullers: red squares and black circles; neutral: blue up-triangles; pushers: pink down-triangles and gray diamonds). (a) AP squirmers with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, (c) WP squirmers with $r_c=2.5\sigma$, and (d) $r_c=1.5\sigma$. Note the different scale on the $y$ axis.

Figure 4

Time evolutions of the mean cluster size for squirmers when $\xi =1$ (pullers: red squares and black circles; neutral: blue up-triangles; pushers: pink down-triangles and gray diamonds). (a) AP squirmers with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, (c) WP squirmers with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Note the different scale on the $y$ axis.

Figure 5

Time evolutions of the mean cluster size for squirmers when $\xi =10$ (pullers: red squares and black circles; neutral: blue up-triangles; pushers: pink down-triangles and gray diamonds). (a) AP squirmers with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, (c) WP squirmers with $r_c=2.5\sigma$, and (d) $r_c=1.5\sigma$. Note the different scale on the $y$ axis.

Figure 6

Mean cluster size for suspensions with (a) $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$ in steady state for WP ($\xi =10, \xi =1$, and $\xi =0.1$, square symbols) and AP ($\xi =10$ and $\xi =1$, circular symbols)

Figure 7

Snapshots of different types of self-assembly for $r_c=2.5\sigma$. The attractive hemisphere is represented in blue, the repulsive in green and the fixed orientation vector is shown in red. Panels (a), (b), and (c) are AP squirmer suspensions with $\xi =\{0.1,1,10\}$, respectively, while panels (d), (e), and (f) are WP squirmer suspensions with $\xi =\{0.1,1,10\}$, respectively. (a) Coarsening with $\beta =3$. (b) Clustering with $\beta =3$. (c) Gas system with polar order (polar gas) with $\beta =0$. (d) Chains system with $\beta =0$. (e) Trimers state with $\beta =-3$. (f) Gas case with nematic order (nematic gas) with $\beta =0$.

Figure 8

CSD for $\xi =1$. AP squirmers (a) with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$. WP squirmers (c) with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Gray dashed line in (a) and (b) represents the analytical function for a CSD given by Eq. (11) with $B=0$, plotted as guide to the eye, with $s_0=4, {\gamma }_0=5/4$. In (c) and (d) a vertical dashed line is shown as a reference for $s=3$ (trimers).

Figure 9

CSD for $\xi =10$. AP squirmers (a) with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$. WP squirmers (c) with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Gray dashed line in (a) and (b) represents the analytical function for a CSD given by Eq. (11) with $B=0$, plotted as guide to the eye, with $s_0=4, {\gamma }_0=5/4$.

Figure 10

Radius of gyration normalized by the particle radius as a function of the cluster size for suspensions with $\xi =1$. Top panels are AP squirmers with (a) $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers with (c) $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Gray dashed lines in all panels are the function $R_g=K{s}^{0.64}$.

Figure 11

Radius of gyration normalized by the particle radius as a function of the cluster size for suspensions with $\xi =10$. Top panels are AP squirmers (a) with $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers (c) with $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$. Gray dashed lines in all panels are the function $R_g=K{s}^{0.64}$ and most of the curves fit with this power law.

Figure 12

Polar order as a function of cluster size for $\xi =1$. Top panels: AP squirmers with (a) $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers with (c) $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$.

Figure 13

Polar order as a function of cluster size for $\xi =10$. Top panels: AP squirmers with (a) $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers with (c) $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$.

Figure 14

Nematic order as a function of cluster size when $\xi =1$. Top panels represent results for AP squirmers: with (a) $r_c=2.5\sigma$ and (b) $r_c=1.5\sigma$, whereas bottom panels are WP squirmers with (c) $r_c=2.5\sigma$ and (d) $r_c=1.5\sigma$.

Figure 15

Nematic order as a function of cluster size for suspensions with $\xi =10$. (a) AP squirmers with $r_c=2.5\sigma$. (b) AP squirmers with $r_c=1.5\sigma$. (c) WP squirmers with $r_c=2.5\sigma$. (d) WP squirmers $r_c=1.5\sigma$.

Figure 16

Global polar $P_{\infty }$ and nematic ${\lambda }_{\infty }$ order parameters. Top panels represent results for AP squirmers: (a) polar order parameter $P_{\infty }$ and (b) nematic order parameter ${\lambda }_{\infty }$. Bottom panels represent results for WP squirmers: (c) polar order parameter $P_{\infty }$ and (d) nematic order ${\lambda }_{\infty }$.

Figure 17

Graphic representation of the total potential $V\left(r_{ij}\right)$ with $\varphi =1$, using the Lennard-Jones potential with a smooth transition function given in Eq. (4) as the attractive part of $V\left(r_{ij}\right)$ (red curve), the repulsive soft sphere potential at short separation distance with $h_0=(2.2^{1/6}-1)\sigma$ (red points) is using with this cut-off potential. The blue curve is using Eq. (3), with the repulsive soft sphere potential at short separation distance with $h_0=(2^{1/6}-1)\sigma$ (blue asterisks).

Figure 18

Graphical representation of the angular dependence of two different Janus potentials. In blue the symmetric potential used in our simulations given by Eq. (5), while in red the asymetric potential used in Refs. [37, 59] and depicted in Eq. (B1). In order to compare both angular functions, we set ${\theta }_j=0$ in both cases and ${\theta }_{\max }=\pi /2$ which corresponds to the symmetric case.

Figure 19

Radial distribution functions $g\left(r\right)$ with $\xi =1$ for aligned with the patch interaction (left) and aligned against the patch interaction (right).

Figure 20

Left: Time evolution of the mean cluster size. Right: Cluster size distribution.

Figure 21

Simulation zoomed snapshots of WP squirmers with $\xi =0.1$ and $r_c=2.5\sigma$ snapshots are taken at the last time step of the simulations. The attractive hemisphere is represented in blue, while pure repulsive is in green and the fixed orientation vector is shown in red. (a) $\beta =3$ and (b) $\beta =-3$.

Figure 22

Simulation zoomed snapshots of WP squirmers with $\xi =0.1$ and $r_c=1.5\sigma$ snapshots are taken once the simulations reach a steady state given by the mean cluster size. The attractive hemisphere is represented in blue, while pure repulsive in green and the fixed orientation vector is shown in red. (a) $\beta =3$ and (b) $\beta =-3$.

Figure 23

Cluster analysis of WP squirmers with $\xi =0.1$ and $r_c=1.5\sigma$. (a) Cluster size distribution. (b) Radius of gyration as a function of cluster size. (c) Local polar order and (d) local nematic order parameters. Pullers curves are red and black, while pushers are pink and gray and $\beta =0$ in blue. Note the different scale on the $y$ axis.