Hydrodynamic interaction of trapped active Janus particles in two dimensions

The dynamics of a pair of identical artificial microswimmers bound inside two harmonic traps, in a thin sheared fluid film, is numerically investigated. In a two-dimensional Oseen approximation, the hydrodynamic pair coupling is long-ranged and proportional to the particle radius to film thickness ratio. On increasing such ratio above a certain threshold, a transition occurs between a free regime, where each swimmer orbits in its own trap with random phase, and a strong synchronization regime, where the two swimmers strongly repel each other to an average distance larger than both the trap distance and their free orbit diameter. Moreover, the swimmers tend to synchronize their positions opposite the center of the system.

Figure 1

Trapped active Janus particles in a thin sheared free-standing film of thickness $h$ and width $y_L$: (a) Ideal experimental setup. A pair of identical Janus particles of radius $a$ diffusing in a planar Couette cell are trapped by orthogonal optical tweezers of equal strength $k$ (orange cylinders). $v_0$ and $\pm v_s$ are the speeds respectively of the self-propelling particles and the Couette cell walls. (b) Illustration of the hydrodynamic effects on the pair dynamics: small-radius particles orbit inside harmonic traps, placed a distance $d_0$ apart, with average radius $r_0=v_0/k$ and random phases; large-radius particles repel one another and synchronize their phases, keeping an average distance, $r_{\infty }$, much larger than $d_0$.

Figure 2

Contour plots of $P_i(x,y)$ for the harmonically trapped active JP 1 (upper panels) and JP 2 (lower panels) at zero shear, $\mathrm{\Omega }=0$. The JPs' radius, $a$, and the centers of the JP 1 and 2 traps, $(0,\pm y_0)$, are reported in the legends; for $y_0=0$, (a)–(c), the contours of $P_i(x,y)$ coincide, for $y_0>0$, (e)–(g), are center-symmetric. Other simulation parameters are: $L=50$, $h=1$, $v_0=1$, $k=1$, and $D_{\varphi }=0.05$. The simulation parameters in (d) and (h) are the same as, respectively, in (c) and (g), except for $D_{\varphi }=0.5$.

Figure 3

Active JP's in harmonic traps with $\pm {\mathbf{d}}_0=(0,\pm y_0)$, zero shear, $\mathrm{\Omega }=0$, and different $D_{\varphi }$: (a) average pair distance $\langle r\rangle$ vs. $a$; (b) angular synchronization factor $R_{\theta }$ vs. $a$ (see text). Other simulation parameters are: $L=50$, $h=1$, $v_0=1$, and $k=1$. The horizontal dashed curves in (a) represent our predictions for $\langle r\rangle$ at low $a$, $r_0$. Our estimates of $\langle r\rangle$ for $y_0=0$ and large $a$ in the regime of high, ${\overline{r}}_{\infty }\left(a\right)$, and low angular noise, $r_{\infty }\left(a\right)$, are plotted, respectively, as dashed and solid curves. For a given $y_0$, $r_0$ and ${\overline{r}}_{\infty }\left(a\right)$ cross at the critical coupling, $a_c$.

Figure 4

Contour plots of $P_i(x,y)$ for two identical harmonically trapped active particles, JP 1 (right) and JP 2 (left), in a Couette shear flow with $\mathrm{\Omega }=0.1$. The particles' radius, $a$, and the traps' centers, $(\pm x_0,0)$, vary as in the legends. Other simulation parameters are: $L=50$, $h=1$, $v_0=1$, $k=1$, and $D_{\varphi }=0.05$. The simulation parameters in (d) and (h) are the same as, respectively, in (c) and (g), except for $D_{\varphi }=0.5$.

Figure 5

Harmonically trapped active JP's in a Couette shear with $\mathrm{\Omega }=0.1$: $\langle r\rangle$ (a) and $R_{\theta }$ (b) vs. $a$ for ${\mathbf{d}}_0\left|\right|\widehat{\mathbf{x}}$ (filled symbols) and ${\mathbf{d}}_0\left|\right|\widehat{\mathbf{y}}$ (empty symbols). Other simulation parameters are: $L=50$, $h=1$, $v_0=1$, $k=1$, and $D_{\varphi }=0.01$.

Figure 6

Trajectories of two active JP's bound to the same harmonic trap centered at the origin, $d_0=0$, in the absence of angular noise, $D_{\varphi }=0$, and subjected to a Couette shear with $\mathrm{\Omega }=0.1$. The initial conditions are the same in all panels, $x_1\left(0\right)=0$, $y_1\left(0\right)=0$, ${\varphi }_1\left(0\right)=0$, and $x_2\left(0\right)=1$, $y_2\left(0\right)=0$, ${\varphi }_2\left(0\right)=0.045\pi$. whereas $a$ has been varied as reported in the legends. Other simulation parameters are: $L=50$, $v_0=1$, and $k=1$; the trajectories of JP1 and JP2 are drawn, respectively, in blue and black.