Engineering reconfigurable flow patterns via surface-driven light-controlled active matter

Surface-driven flows are ubiquitous in nature, from subcellular cytoplasmic streaming to organ-scale ciliary arrays. Here we model how confined geometries can be used to engineer complex hydrodynamic patterns driven by activity prescribed solely on the boundary. Specifically, we simulate light-controlled surface-driven active matter, probing the emergent properties of a suspension of active colloids that can bind and unbind from surfaces of a closed microchamber, together creating an active carpet. The attached colloids generate large-scale flows that in turn can advect detached particles toward the walls. Switching the particle velocities with light, we program the active suspension and demonstrate a rich design space of flow patterns characterized by topological defects. We derive the possible mode structures and use this theory to optimize different microfluidic functions including hydrodynamic compartmentalization and chaotic mixing. Our results pave the way toward designing and controlling surface-driven active fluids.

Figure 1

Experimental proposal for surface-driven flows in confined volumes. (a) In vitro reconstitution of cytoplasmic streaming as a basis for engineering flows. Load-carrying molecular motors walk along polarized and aligned actin filaments pinned to a surface. The viscous drag on the load (colloidal particles) imparts momentum in the surrounding fluid, and the action of many such particles can lead to emergent bulk flows. (b) The top shows that molecular motors have been genetically engineered to switch their direction of motion along a filament in response to blue light (see [49]). It is shown on the bottom that, by patterning the boundary with light, one can dynamically control the flow structure of a chamber with spatiotemporal precision.

Figure 2

Model for surface-driven flows in confined volumes. (a) Confined chamber of size $N_x\times N_y\times N_z=20\times 20\times 10$ that consists of a single active surface at $z=0$, uniformly coated with parallel and polarized actin filaments. The gray lines denote the orientation of particle trajectories on the surface. Active colloidal particles (green) bind and unbind from the surface with rates $P_{\text{on}}$ and $P_{\text{off}}$. Bound particles walk ballistically in a fixed direction along the tracks, imparting momentum on the surrounding fluid and creating macroscopic flows. Unbound particles are free to advect and diffuse in the bulk, with relative strength set by the Péclet number Pe. (b) The velocity at each grid point $v_{\partial \mathrm{\Omega }}\left(x\right)$ on the boundary increases linearly with particle concentration and saturates to the particle speed $V$. (c) The attachment rate $P_{\text{on}}$ decreases linearly to zero at the saturating concentration ${\rho }_{\text{sat}}$, a value which is set by the grid and particle size, while (d) the detachment rate $P_{\text{off}}$ is modeled as a constant. (e)–(g) Simulation results for a simple case: uniform flow in a closed chamber. All results are from snapshots of simulations recorded after simulating for $2\times {10}^6$ time steps to approach steady state, with each time step $dt\approx {10}^{-6}L/V$ s. (e) The magnitude of the velocity at each point on the boundary for a specific choice of parameters [the optimum in (g)] shows an accumulation of particles toward the right edge of the chamber. The color bar depicts the flow magnitude on the surface, scaled by $V$ to a maximum of one. (f) The existence of a wall forces the fluid at the edge upward, creating a 2D vortex in the $xz$ plane. (g) Phase diagram of $P_{\text{off}}$ and Pe, showing that high streaming velocities in a confined volume favor low Pe and an intermediate detachment rate. The average streaming velocity $\langle \left|u\right|\rangle /V$ is shown at $z/H=0.2$ averaged in the $x$ and $y$ directions across the chamber. Here Pe is varied by changing $D$ and keeping $L$ and $V$ constant. The $P_{\text{off}}$ is reported in units of probability per time step.

Figure 3

Static and dynamic surface-patterned defects using light-controlled active colloidal particles in confined volumes. (a) and (b) Schematic of a light pattern on the $z=0$ surface of a chamber with dimensions $N_x\times N_y\times N_z=20\times 20\times 10$. In (a) the $x>0.5$ half of the plane is illuminated by light and in (b) the $y>0.5$ half is illuminated. (c) and (d) Resulting steady-state flow patterns. In (c) the line defect on the active surface at $x=0.5$ causes fluid to be pushed upward, creating two distinct vortices. In (d) the line defect along $y=0.5$ creates vortices of opposite chirality in distinct regions of the chamber ($y<0.5$ and $y>0.5$). Confinement of the fluid additionally gives rise to recirculating streamlines in the $xy$ plane. (e) and (f) Head-on and shear defects preferentially mix passively advecting particles (${\text{Pe}}_{\text{tracer}}={10}^4$) in different directions. Plotted on the $y$ axis is the fraction of tracer particles in the initially empty portion of the box as a function of time. Here $N_{>}$ denotes the number of particles at $x,y,z>0.5,0.5,0.25$ for each of the three curves, respectively. (e) For the head-on defect, particles are nearly evenly distributed between the $z<0.25$ and $z>0.25$ halves of the box, but mostly remain in the $x<0.5$ and $y<0.5$ regions, showing that there is little mixing in $x$ and $y$. (f) On the other hand, the shear defect is less effective at mixing in $\widehat{z}$ but is able to mix particles along $\widehat{y}$ due to recirculation. (g) and (h) An optically controllable system allows easy temporal switching from one flow structure to another. As a proof of principle, we switch between head-on and shear defects, with period $2\times {10}^6$ time steps and parameters $P_{\text{off}}={10}^{-4}$ and $\text{Pe}=1$. (g) Distribution of motile particles, where time moves from blue to green to red. (h) Shown in red is the average streaming magnitude, computed as the spatially averaged fluid magnitude as a function of time, and in blue the average number of particles attached to the boundary as a function of time. Interestingly, the average streaming magnitude of the head-on defect is greater even though the number of attached particles on the boundary is slightly less than that of the shear defect.

Figure 4

Design space of surface-driven flows using mainly head-on and shear defects. All flow structures are solved on an $N_x\times N_y\times N_z=40\times 40\times 40$ grid. (a) Each panel depicts a pattern of light on a single active boundary (top) and the corresponding 3D flow structure (bottom). Note that (i)–(iii) are interchangeable by light, whereas (iv) introduces additional complexity where the orientation of actin on the surface is no longer fully uniform. (b) Panels (i), (ii), and (v) pattern two or four surfaces with head-on defects. Since the resulting streamlines are effectively two dimensional, only $xz$ cross sections of the flow structures are plotted. Panels (iii), (iv), and (vi) consider the same but with shear defects and with the full 3D flow structure to highlight the recirculating streamlines. (c) Select streamlines from (a) and (b) are plotted to enhance visualization of the flow structures. Note the red streamlines in (i), which highlights the particles trapped in the clockwise vortex established by the oppositely moving patch in Fig. 3. Contrasting (iii) with (iv) shows the effect of the recirculating streamlines: Particles in the former remain advected in the $xz$ plane whereas particles in the latter circulate in $xy$.

Figure 5

Fundamental modes of the squirmer model: an analytical approach to boundary-driven flows. (a)–(d) Comparisons between theory (left) and simulations (right) for the first four axisymmetric modes. The green $\pm$ symbols at the corners of the box indicate the polarity of the actin filaments patterned on the boundary. The top row of each panel depicts the surface patterning and the bottom row depicts the internal flow structures taken at some cross section. (a) and (c) The surface patterns of the $b_{10}$ and $b_{20}$ modes lie on longitudinal tracks. The $b_{10}$ mode consists of uniform (in direction) motion from the north to the south pole, while the $b_{20}$ mode naturally encodes a line of head-on defects at the equator. All interior flow structures are taken at the cross section $y=0$. (b) and (d) Conversely, the surface patterns of the $c_{10}$ and $c_{20}$ modes lie on lines of latitude. Similar to the $b_{20}$ mode, the $c_{20}$ mode naturally encodes a shear defect along its equator. The interior cross sections of $c_{10}$ for both the sphere and the box are taken at $x=0$, while the pair of cross sections of $c_{20}$ are taken at $\theta =\pi /3,3\pi /4$ for the sphere and at $z=0.25,0.75$. Cross sections of the sphere and box in all four panels show that the interior flow structures are analogous. This shows that the language of head-on and shear defects is intrinsically built into the squirmer model.

Figure 6

Emergence of chaotic mixing. (a) Surface streamline structure of the modes (i.e., the fluid velocity at $r/R=1$) plotted as a function of the spherical angles $\theta$ and $\varphi$. Note that the $b$ modes give rise to patches of oppositely moving flow, whereas the $c$ modes comprise closed vortices. (b) Examples of ten separate trajectories of the $b_{21}$ mode. The blue circles denote the starting points of two particles, spaced $dr/R={10}^{-6}$ apart, a distance that cannot be resolved by eye. The green circles denote the ending positions after integrating for $t=500$, which are still very closely spaced, showing that the two trajectories do not substantially diverge. (c) Example of a single chaotic trajectory of the $b_{21}+c_{21}$ mode. Note the positions of the green circles, which now span a distance comparable to the size of the droplet. (d) Poincaré section at $x=0$ for the $b_{21}$ mode computed from 1000 trajectories. Note that the plane is only sparsely populated. (e) The Poincaré section of $b_{21}+c_{21}$, on the other hand, nearly fills the entire plane. (f) Evolution in time of the logarithmic displacement between pairs of particles initially spaced $dr/R={10}^{-6}$ apart, averaged over 1000 randomly seeded trajectories. These plots show that of the three modes in (a), only the superposed mode shows evidence of chaotic mixing on the scale of the droplet size. (g) Exponents of the time evolution of $dr$ after integrating for $t=500$ for various superpositions of modes, indicated by the row $r$ and column $c$ of the element in the matrix. For example, $(r,c)=(2,3)$ denotes the $c_{21}+b_{22}$ mode. The matrix shows that the only modes that show evidence of chaotic mixing are the $b$ and $c$ superposed modes. However, not all such modes are chaotic, as evidenced by the $b_{21}+c_{22}$ mode.