Hydrodynamic Capture and Release of Passively Driven Particles by Active Particles Under Hele-Shaw Flows

Abstract

The transport of active and passive particles plays central roles in diverse biological phenomena and engineering applications. In this paper, we present a theoretical investigation of a system consisting of an active particle and a passive particle in a confined micro-fluidic flow. The introduction of an external flow is found to induce the capture of the passive particle by the active particle via long-range hydrodynamic interactions among the particles. This hydrodynamic capture mechanism relies on an attracting stable equilibrium configuration formed by the particles, which occurs when the external flow intensity exceeds a certain threshold. We evaluate this threshold by studying the stability of the equilibrium configurations analytically and numerically. Furthermore, we study the dynamics of typical capture and non-capture events and characterize the basins of attraction of the equilibrium configurations. Our findings reveal a critical dependence of the hydrodynamic capture mechanism on the external flow intensity. Through adjusting the external flow intensity across the stability threshold, we demonstrate that the active particle can capture and release the passive particle in a controllable manner. Such a capture-and-release mechanism is desirable for biomedical applications such as the capture and release of therapeutic payloads by synthetic micro-swimmers in targeted drug delivery.

Appendices

Appendices

Dynamic Equivalence for Opposite Rotational Mobilities

Here, we demonstrate via a transformation of coordinates, in the relative frame of reference given by (4), the interaction between an active particle with $\nu >0$ and a passive particle is dynamically equivalent to the interaction between an active particle with $\nu <0$ and a passive particle. Namely, we start by considering the equations of motion for the case of $\nu >0$, and show that the equations have the same form as for the case of $\nu <0$ after the transformation. For convenience, we rewrite (4) for the case of active particle with $\nu >0$ in polar coordinate, $\beta =R\mathrm {e}^{ \mathrm {i}\theta }$,

$$\begin{aligned} \begin{aligned} \dfrac{\hbox {d}}{\hbox {d}t}(R\mathrm {e}^{ -\mathrm {i}\theta })&=U\mathrm {e}^{ -\mathrm {i}\alpha } -\mu \frac{a^2 U\mathrm {e}^{ \mathrm {i}(\alpha - 2\theta ) }}{R^2}+\frac{2 f_0(2a)^{13}\mathrm {e}^{ -\mathrm {i}\theta }}{R^{13}}, \\ \dot{\alpha }&=\left| \nu _o \right| \text {Re}\left[ \frac{2a^2(1-\mu )V}{R^3}\mathrm {i}\mathrm {e}^{ \mathrm {i}(2\alpha -3\theta ) }\right] , \end{aligned} \end{aligned}$$

(9)

where we set $\nu =\left| \nu _o \right|$ for the case of $\nu >0$ and $\nu =-\left| \nu _o \right|$ for the case of $\nu <0$ . We consider the transformation $\theta =\theta ^{*}+\pi$, $\alpha =\alpha ^{*}+\pi$. Direct substitution into (9) gives

$$\begin{aligned} \begin{aligned} \dfrac{\hbox {d}}{\hbox {d}t}(R\mathrm {e}^{ -\mathrm {i}\theta ^{*} })&=U\mathrm {e}^{ -\mathrm {i}\alpha ^{*} } -\mu \frac{a^2 U\mathrm {e}^{ \mathrm {i}(\alpha ^{*}- 2\theta ^{*}) }}{R^2}+\frac{2 f_0(2a)^{13}\mathrm {e}^{ -\mathrm {i}\theta ^{*} }}{R^{13}}, \\ \dot{\alpha }^{*}&=-\left| \nu _o \right| \text {Re}\left[ \frac{2a^2(1-\mu )V}{R^3}\mathrm {i}\mathrm {e}^{ \mathrm {i}(2\alpha ^{*}-3\theta ^{*}) }\right] , \end{aligned} \end{aligned}$$

(10)

which take the same form as for the case of $\nu <0$. This indicates that the interaction of an active particle with $\nu >0$ and a passive particle under the initial conditions $(R(0),\theta (0),\alpha (0))=(R_o,\theta _o+\pi ,\alpha _o+\pi )$ is dynamically equivalent to the interaction of an active particle with $\nu <0$ and a passive particle under the initial conditions $(R(0),\theta (0),\alpha (0))=(R_o,\theta _o,\alpha _o)$.

$\theta$–$\alpha$ Phase Space for Capture and Non-capture Events

Figure 11 depicts the $\theta$–$\alpha$ phase diagrams for capture and non-capture events for different external flow intensities V. It is observed that capture region (represented by the white space) lies along the line $\theta = \pi + \alpha$, which corresponds to the scenario of an active particle with a propulsion direction pointing directly toward the center of the passive particle. The capture region enlarges with an increased external flow intensity V. When the external flow intensity is sufficiently high (Fig. 11d), we remark that capture can occur for all $\theta$ for certain values of $\alpha$, enhancing the capture process.

Fig. 11

a The active particle points directly toward the passive particle when $\theta =\pi +\alpha$, which facilitates the capture process. b–d $\theta$–$\alpha$ phase diagrams for capture at different external flow intensities: b $V=5$; c $V=10$; d $V=20$. Parameter values are $U=1$, $a=1$, $\mu =0.5$, $\nu =0.5$, $R(0)=20$. The white and black spaces denote the regions for capture and non-capture, respectively. The dashed lines indicate the parameter values for $\theta =\pi +\alpha$