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

We first consider a simple benchmark flow structure: uniform motion to the right in a confined chamber, with one active and five no-slip surfaces (Fig. 1E-F, see also Movie S3). The steady state on the boundary for one particular choice of parameters shows an accumulation of particles at the right wall (at the plus end of the actin filaments) and a depletion of particles on the left (Fig. 1E). The fluid at the right boundary is forced upwards, creating a steady state vortex in the $xz$-plane (Fig. 1F). Varying the Péclet number and detachment rates shows that higher streaming magnitudes occur for lower values of $Pe$ and an intermediate value of $P_{\text{off}}$ (Fig. 1G and Fig. S2; for a more detailed description of our choice in parameters and the parameter sweep range, refer to Sec. II of the SI). This optimum is explained as follows. On the one hand, attached particles will tend to accumulate at the chamber edges within a timescale $\tau_{w}\sim L/v_{w}$. Overly processive motors will therefore on average reach the opposite wall before they detach into the bulk, reducing the streaming velocity. This sets a lower bound on the detachment probability, $\tau_{detach}=1/P_{\text{off}}\leq\tau_{w}$, so we require that $P_{\text{off}}\geq v_{w}/L$ in order to establish nontrivial streaming velocities in confined volumes. On the other hand, particles that are not processive enough do not spend enough time at the surface to contribute significant momentum injection. A large diffusion coefficient (i.e. small Péclet number) helps to offset particle accumulation at edges and also homogenises density fluctuations, which increases the streaming strength and stability, allowing for the establishment of steady-state flow structures (Movies S2-3). This is maximised in the limit $Pe\to 0$, when diffusion dominates advection, as motors spread through the box uniformly.

In summary, the flow velocity can be optimised with a high diffusivity and an intermediate processivity. Note that experimental realizations place constraints on these parameters, as discussed in Supplementary Information Section II. We further note that the properties of the phase space diagram are particular to the geometry we have considered, and other cases (such as periodic boundary conditions) may yield very different results (Fig. S2, Movie S1). Moreover, uniform motion in confined chambers will lead to recirculating streamlines that can transport particles in the $\hat{\bm{z}}$ direction, despite that activity on the surface is directed uniquely along $\hat{\bm{x}}$ (Fig. 1F).

One way to engineer different fluid structures would be to manipulate the orientation of tracks on a surface, an experimental perturbation made difficult by the fact that these tracks are often permanent when laid out. Advances in optogenetics have enabled the dynamic control of active materials through applying an external perturbation with light. For example, a new class of engineered molecular motors are capable of changing their direction of motion along a filament in response to a light signal, providing a mechanism to reprogram the same surface by changing how the motors interact with it Nakamura et al. (2014). We now explore this regime to see how we can program and pattern bulk flow with variable surface light patterns (Fig. 2A,B).]

Consider the same configuration as before, with all tracks oriented along the $\hat{\bm{x}}$ direction. Suppose the $x>0.5$ (right) half of the box is illuminated, redirecting the optically controllable particles in that region toward the minus end (Fig. 2C). We term the resulting structure the “head-on” defect, since now two populations of motile particles are literally walking into each other at a given line defect defined by the light pattern. This gives rise to two distinct vortices on either side of the domain junction (Fig. 2D). Similarly, we can also choose to illuminate the $y>0.5$ half of the box, giving rise to a “shear” defect configuration (Fig. 2F,G). On a surface patterned with uniformly oriented tracks, the head-on and shear defects are the two fundamental modes that one can pattern with light.

These two flow structures have different properties that may be useful for different applications. Suppose, for example, that at subsequent stages of a chemical process two mixing procedures are needed. Figures 2E,H show that the shear and head-on defects preferentially mix tracer particles in different directions. The head-on defect is most effective at mixing along the $\hat{z}$-coordinate, as evidenced by the concentration profile of tracers initially in the bottom half of the box being spread uniformly along $\hat{z}$ by the end of the simulation (Fig. 2E). On the other hand, the shear defect is more effective at mixing along the $\hat{y}$ coordinate (Fig. 2H and Fig. S3). Note that in both cases the direction of motion along the boundary is strictly in the $\hat{x}$ direction – it is the geometry of the confined chamber (i.e. the location of walls and defects) that gives rise to recirculating streamlines.

A significant advantage of light-controlled surface patterning is the ease of transitioning from one flow structure to another. As a proof of principle, we conducted a simulation (see Movie S4) where we transition from a shear defect to a head-on defect, and back to a shear defect again, with period $2\times 10^{6}$ time steps, detachment rate $P_{\text{off}}=10^{-4}$ per time step and $Pe=1$ (the optimum in Fig. 1G, chosen for illustrating the phenomenon although experimental realizations would likely be at higher $Pe$; see Supplementary Table I). In our model, we assume that the behavior of the motile particles switch instantaneously with the external light pertubation. Figs. 2I,J show that with each transition, the flow relaxes to its unique steady state for each boundary condition after a short relaxation time. Interestingly, the shear defect generates a slightly lower streaming velocity despite having on average more particles attached to the boundary. Another characteristic of patterning with light is the ability to make continuous perturbations to the flow field. Fig. 2K depicts a continuous transformation from a shear to a head-on defect by smoothing varying the angle $\alpha$ of the light pattern with the $y$-axis, allowing for not only spatial but also precise temporal control of flow patterns (see also Movies S5-6).

Next, we consider the challenge of designing bulk flow patterns and consider the breadth of design space available in this problem. To speed up our simulations, we continue in the optimal limit $Pe\to 0$ where the Péclet number tends to zero, when active particles cover all surfaces uniformly (or for a uniform carpet of cilia), and simulate the steady state flow structure in chambers of size $N_{x}\times N_{y}\times N_{z}=40\times 40\times 40$. Hence, we no longer simulate the particle dynamics explicitly and study the steady-state flow structures obtained by patterning surfaces directly with constant slip velocities. This indeed limits the class of patterns that might be feasible, but allows us to understand the limit of vanishing Péclet number where active particles can reach surfaces in abundance. Though these surface velocities can be patterned arbitrarily, the resulting bulk flows must still obey the constraints set by the Stokes equations and incompressibility. The question arises how the optogenetic design can be optimized to alleviate those constraints in terms of transport and streamline connectivity.

For surface patterning, we utilize the language of defects that refer to zones where surface bound active particles dramatically change behavior. Starting with a single active surface with uniformly oriented filaments, different regions of left- or right-moving fluid can be created with light, as depicted for the head-on, shear, and patch defects (Fig. 3A, i-iii). Again, all three patterns are interchangeable by dynamically changing the pattern of light. We can add an additional handle by no longer subjecting the filaments to be uniformly oriented. Alternating orthogonal patches of tracks, in tandem with a light pattern on the surface, can give rise to flow structures such a vortex (Fig. 3A, iv).

Integrated streamlines of the patch defect in Fig.  3C,i highlights the separatrix formed by a small region of oppositely moving flow. Streamlines shown in red traverse clockwise (CW) and are centered on top of the patch, whereas all other streamlines travel counterclockwise (CCW). The head-on and patch defects therefore have a compartmentalizing effect, with regions of streamlines that do not mix. Conversely, Fig. 3C,iv shows that the streamlines of the vortex defect traverse the $xy$-plane as well as a distance of over half the height of the grid in $z$. The shear and vortex defects are therefore effective fluid mixers. These countervailing properties of compartmentalizing and mixing can guide the design of numerous functions useful in self-driven microfluidics context.

We can further build upon the complexity of our designs by patterning multiple surfaces at once. Fig. 3B approaches this systematically by considering only head-on (i-iv) or shear defects (v-viii) on 2 or 4 surfaces. Interpreting the resultant flow structures created by head-on defects is straightforward: two stable vortices will form on either side of a defect, where the flow either moves toward ($\rightarrow\leftarrow$) or away ($\leftarrow\rightarrow$) from each other. Furthermore, the integrated streamlines of the 8 vortex structure (Fig. 3C,iii) shows that the streamlines are two dimensional. Each vortex can therefore be considered as its own compartment.

On the other hand, patterning with shear defects can lead to mixing within each compartment (Fig. S4). Fig.  3B,vii depicts four consecutive active surfaces, with each pair of opposite faces patterned with shear defects of opposite signs ($\uparrow\downarrow$ and $\downarrow\uparrow$). In the $xz$-plane, this gives rise to head-on defects at the four corners, creating four stable vortices. However, recirculation in the $yz$ and $xy$-planes cause the streamlines within the four vortices to traverse along the $y$-coordinate, as depicted in Fig. 3C,v. The expectation that shear defects give rise to three dimensional streamlines is a general but not very robust rule, however. Fig. 3B,viii depicts a surface pattern similar to Fig. 3B,vii, except with the patterns on the two faces with surface normal $\hat{x}$ flipped. The result is that the head-on defects in the $xz$-plane are completely eliminated, leading to a continuous current that runs CCW in the $y<0.5$ region of the box and CW in the other. Interestingly, the resulting streamlines (Fig. 3C,vi) are again co-planar.

Given the quickly increasing complexity of the resulting flows and the myriad possibilities of surface patterns, it is clear that adopting a more analytical approach to understanding the design space of boundary-driven flows is required. For this we turn to solutions of the squirmer model on a sphere Blake (1971); Reigh et al. (2017); Pedley (2016); Pak and Lauga (2014); Fadda et al. (2020). Initially adopted to study the external flow fields of microswimmers, we invert the problem and study instead the flow structure within the sphere Happel and Brenner (1983), subject to the condition that flows normal to the surface vanish on the boundary. The resulting general solution for incompressible Stokes flow inside sphere is:

where $P_{m}^{n}$ are the associated Legendre polynomials indexed by integers $m,n$, and $R$ the radius of the sphere. For the sake of simplicity we set $b_{mn}=\tilde{b}_{mn}$ and $c_{mn}=\tilde{c}_{mn}$, which fixes the phase of $\phi$ whilst keeping the topology the same. The modes are thus denoted by two variables, $b_{mn}$ and $c_{mn}$.

The first few axisymmetric modes are shown to match with the simulated flow structures on a grid (Fig. 3). We observe that the $b$-modes are aligned longitudinally across the sphere, whereas the $c$-modes run along lines of latitude and form closed streamlines. Cross sections of the box and sphere reveal the similarities between the internal flow structures. The topological equivalence between a sphere and a box dictates that these flow structures should be compatible. We do note, however, that corner effects can give rise to eddies that are unique to the geometry of a box Dauparas and Lauga (2018). Most interestingly, the $b_{20}$ and $c_{20}$ modes have built-in defects analogous to topologies proposed earlier: the former consists of a line of head-on defects at the equator, while the latter consists of two oppositely rotating hemispheres, giving rise to a shear defect along the equator. The spherical solutions therefore present a natural framework in which to embed the design space of surface-driven flow structures.

Plots of higher order modes and their streamlines (Fig. 5 and Figs. S5-9) show that these general rules of thumb still apply: $b$ modes give rise to patches of oppositely moving flow on the surface, while $c$-modes give rise to closed vortices. Higher order modes give rise to more patches or more vortices, corresponding to smaller compartments in which tracer particles can traverse (Figs. S8-9). The $b$-modes are better at mixing particles radially within their compartments due to streamlines that redirect particles toward the $z=0$ axis, whereas particles move along concentric closed curves at approximately constant radius in $c$-modes (Fig. S5-7). These properties can be combined and used when designing microfluidic devices that require specific bulk flow patterns.

Inspired by previous work showing that Stokes flows within droplets can give rise to chaotic streamlines Bajer and Moffatt (1990); Stone et al. (1991); Ward and Homsy (2006), we ask whether our surface-driven flow patterns can do the same. Indeed, we find evidence of chaotic mixing in the superposed $b$ and $c$ modes (Fig. 5). Whereas the individual $b$ and $c$ modes do not feature chaos, we do find evidence of chaotic mixing in superpositions of these modes (Fig. 5). To quantify this, we consider the trajectories of tracer particle pairs that are initially spaced a distance $dr_{0}=10^{-6}$ apart (in dimensionless units). Fig. 5B depicts 10 such trajectories subject to the $b_{21}$ flow field only. The blue dots denote the starting positions of one pair of particles, which cannot be visually resolved. The green dots denote their final positions after integrating to time $t=500$. The red curve in Fig. 5F plots the separation $dr$ as a function of time, averaged over 1000 randomly seeded trajectories, and shows that the particle pairs remain close together throughout their trajectories, with a final separation less than $dr=10^{-3}$. Similarly, the blue curve in Fig. 5F suggests that the $c_{21}$ mode is not chaotic.

Trajectories of the $b_{21}+c_{21}$ modes combined, however, will on average diverge rapidly to a distance comparable to the system size $(R=1$). Fig. 5C depicts a single pair of trajectories that begin at the blue dot near the center of the sphere. After a time $t=500$, the green dots show the separation of these two particles at a distance comparable to the diameter of the sphere. This chaotic mixing is further illustrated by the Poincare sections at $x=0$ for the $b_{21}$ and $b_{21}+c_{21}$ modes (Fig. 5D-E), each built from 1000 randomly seeded trajectories. The Poincare section of $b_{21}$ is notably sparser than that of $b_{21}+c_{21}$, showing that the superposition of modes is far more effective at mixing.

To understand this better, we map out the entire phase space of mixing potential by superpositions of the $n=2$ modes (Fig. 5G). This heat map shows the exponent of $dr$ for each pair of superposed modes. Again, only the superposed modes show signs of mixing. We note, however, that only the modes consisting of $b$ superposed onto $c$ lead to chaotic advection, suggesting that fundamental properties from each mode are necessary ingredients. Somewhat surprisingly, even the axisymmetric mode $b_{20}$ showed moderate signs of mixing when superposed with the $c_{21}$ and $c_{22}$ modes, but the $b_{21}+c_{22}$ mode combination did not. Though we can predict the flow structures of simple surface patterns, it is not obvious, for example, which of the more complex topologies will lead to chaotic mixing and which do not. Prior theoretical work has shown that fluid properties such as stretching, twisting, and folding are essential for chaotic mixing Bajer and Moffatt (1990); Stone et al. (1991); Ward and Homsy (2006); Smith et al. (2019); Ottino and Ottino (1989), but further work is required to understand how such concepts may arise in the devices proposed here.