Active colloidal molecules in activity gradients

We consider a rigid assembly of two active Brownian particles, forming an active colloidal dimer, in a gradient of activity. We show analytically that depending on the relative orientation of the two particles the active dimer accumulates in regions of either high or low activity, corresponding to, respectively, chemotaxis and antichemotaxis. Certain active dimers show both chemotactic and antichemotactic behavior, depending on the strength of the activity. Our coarse-grained Fokker-Planck approach yields an effective potential, which we use to construct a nonequilibrium phase diagram that classifies the dimers according to their tactic behavior. Moreover, we show that for certain dimers a higher persistence of the motion is achieved similar to the effect of a steering wheel in macroscopic devices. This work could be useful for designing autonomous active colloidal structures which adjust their motion depending on the local activity gradients.

Figure 1

A sketch of an active colloidal dimer consisting of two active Janus particles. The orientation vectors ${\mathit{n}}_1$ and ${\mathit{n}}_2$ of the active particles are shown in blue. The angles ${\varphi }_1$ and ${\varphi }_2$ are the angles between ${\mathit{n}}_1$ and ${\mathit{n}}_2$ and the vector connecting the centers of the particle.

Figure 2

Density for different dimers (see insets) relative to the bulk density ${\rho }_b={\int }_0^{L}dx\rho \left(x\right)/L$, where $L=25$ is the simulation box with periodic boundary conditions. The orientations of the particles in the dimer are indicated in the figure. The symbols show the simulation of Eqs. (3) and (4), the solid line show the theoretical prediction [Eq. (15)], and the red dashed line in panel (a) shows the shape of the swim-force profile $f_s\left(x\right)=8[1+sin(2\pi x/L+3\pi /2)]$. The orientation of the particles in the dimer can be used to control wether the dimer accumulates in regions where $f_s$ is small [panels (a), (b), (e), and (f)], or in regions where $f_s$ is large [panels (c) and (d)].

Figure 3

Flux perpendicular to swim-force gradients. The bulk density is ${\rho }_b={\int }_0^{L}dx\rho \left(x\right)/L$ with $L=25$. These transverse fluxes are reminiscent of, for instance, chemotactic sea urchin sperm swimming in the presence of a chemical source [83]. The swim force is $f_s\left(x\right)=8[1+sin(2\pi x/L+3\pi /2)]$ (same as Fig. 2). The orientations of the particles in the dimer are shown in the figures.

Figure 4

Phase diagrams for the tactic behavior of dimers for a value of the swim force of $f_s=0.5$ in (a) and $f_s=5$ in (b). Every point in the ${\varphi }_1\text{−}{\varphi }_2$ plane corresponds to a different dimer structure. The blue region corresponds to antichemotactic dimers, which experience an effective force down the swim-force gradients ($U^{\prime }>0$). The red region corresponds to chemotactic dimers ($U^{\prime }<0$). These tactic regions are separated by white boundaries which correspond to dimers which show no preferential accumulation ($U^{\prime }=0$). The phase behavior is dependent on the magnitude of the swim force implying that the same dimer can be chemotactic or antochemotactic depending on the magnitude of the swim force.

Figure 5

Top: Density relative to the bulk density for a dimer with ${\varphi }_1=\pi /2$ and ${\varphi }_2=0$ (see inset) for different values of the swim force. This is the same dimer as Fig. 2. The bulk density is ${\rho }_b={\int }_0^{L}dx\rho \left(x\right)/L$, where $L=25$ is the simulation box with periodic boundary conditions. The swim force is $f_s\left(x\right)=f_s^0[1+sin(2\pi x/L+3\pi /2)]$, with the value of $f_s^0$ as indicated in the legend. Bottom: Derivative of the effective potential $U^{\prime }=dU/d{f}_s$. Wherever $U^{\prime }<0$ ($U^{\prime }>0$) the dimer is chemotactic (antichemotactic). The density profile changes qualitatively when the swim force increases. In this case there is a single peak in the density where $f_s$ is large for $f_s^0=1/4$, as the swim force increases ($f_s^0=1/2$) the peak split in two, and if the swim force is increased further ($f_s^0=1$) a third peak appears.