Orientation control and nonlinear trajectory tracking of colloidal particles using microfluidics

Suspensions of anisotropic Brownian particles are commonly encountered in a wide array of applications such as drug delivery and manufacturing of fiber-reinforced composites. Technological applications and fundamental studies of small anisotropic particles critically require precise control of particle orientation over defined trajectories and paths. In this work, we demonstrate robust control over the two-dimensional center-of-mass position and orientation of anisotropic Brownian particles using only fluid flow. We implement a path-following model predictive control scheme to manipulate colloidal particles over defined trajectories in position space, where the speed of movement along the path is a degree of freedom in the controller design. We further explore how the external flow field affects the orientation dynamics of anisotropic particles in steady and transient extensional flow using a combination of experiments and analytical modeling. Overall, this technique offers new avenues for fundamental studies of anisotropic colloidal particles using only fluid flow, without the need for external electric or optical fields.

Figure 1

Characterization of particles used for trapping and manipulation experiments. (a) Scanning electron microscopy (SEM) images of spherical polystyrene particles. (b) SEM images of anisotropic rodlike polystyrene particles.

Figure 2

Overview of trapping method and anisotropic particle orientation. (a) Schematic of the experimental setup. Inlet/outlet ports of the microfluidic device are connected to fluidic reservoirs that are pressurized by regulators controlled by a custom LabVIEW code. (b) Schematic showing the top view of a $\mathit{M}=4$ channel microfluidic device for manipulating a single anisotropic particle at the center of the cross-slot. (c) Schematic of the orientation angle $\varphi$ of an anisotropic particle, which is measured with respect to the $x$ axis as shown in Fig. 2 in anticlockwise direction. The tangent vector $\mathit{t}$ along a line connecting the points $C$ and $T$ and the normal vector $\mathit{n}$ defines the particle orientation. The rod half-length is L.

Figure 3

Simultaneous control over position and orientation of anisotropic colloidal particles. (a) Trajectory of angular displacements of a trapped Brownian rod over a period of 170 s. (Inset, upper) Probability distribution function of particle orientation; Gaussian fit shown in red. The standard deviation of rotational displacement $\sigma$ is equal to $2.1^{\circ }$. (Inset, lower) Sequential images of a trapped Brownian rod while controlling the COM position and orientation. At $t=0$, the rod is initially oriented at $\varphi ={10}^{\circ }$ with respect to the $x$ axis. At $t=0.033$ s, the target orientation is changed to $\varphi ={80}^{\circ }$. The final desired state is achieved within 6.7 s, yielding an average angular velocity of $\approx$ 0.16 rad/s for the rod. (b) Power spectral density of orientation angle fluctuations for the trajectory shown in panel (a). The PSD is fit to a Lorentzian function, enabling determination of the corner frequency $f_c$ along the $\varphi$ direction. (Inset, upper) Power spectrum of the $x$ component of translational displacement with Lorentzian fitting. The corner frequency is 0.58 Hz for the $x$ component of COM fluctuations. (Inset, lower) Power spectrum of the $y$ component of translational displacement with Lorentzian fitting. The corner frequency is 1.55 Hz for the $y$ component of COM fluctuations.

Figure 4

Simultaneous position and orientation control of Brownian rods. (a) Translational displacement histogram for the $x$ component of displacement during the rod manipulation experiment. (b) Translational displacement histogram for the $y$ component of displacement during the rod manipulation experiment.

Figure 5

Simultaneous manipulation of 2D center-of-mass position and orientation angle for single anisotropic particles using fluid flow. (a) The position of a rodlike Brownian particle was controlled to trace the letter “I” while maintaining a constant orientation angle of $\varphi ={130}^{\circ }$ throughout the path. The actual trajectory of particle is shown in colormap, while the reference path is shown in black dotted markers. (b) Comparison between the active control and no control over rod orientation angle during the 2D center-of-mass manipulation experiment.

Figure 6

Reference and actual path of a rod when tracing the letter “I.” (a) $x$ coordinate of the rod and reference path as a function of time. (b) $y$ coordinate of the rod and reference path as a function of time.

Figure 7

Characterization of trap response as a function of rotational Peclet number ${\text{Pe}}_r$. (a) Tightness of confinement along the translational directions as a function of ${\text{Pe}}_r$ for different values of trapping parameters: ${\text{RMS}}_x$ (red squares), ${\text{RMS}}_y$ (blue squares) at ${\alpha }_x=1,{\alpha }_y=1,{\alpha }_{\varphi }=1000, {\text{RMS}}_x$ (red diamonds), ${\text{RMS}}_y$ (green diamonds) at ${\alpha }_x=2,{\alpha }_x=2,{\alpha }_{\varphi }=2000$, and ${\text{RMS}}_x$ (magenta circles), ${\text{RMS}}_x$ (black circles) at ${\alpha }_x=3,{\alpha }_y=3,{\alpha }_{\varphi }=3000$. (b) Tightness of confinement ${\text{RMS}}_{\varphi }$ along the angular direction at varying ${\text{Pe}}_r$ for different values of trapping parameters: (red squares) at ${\alpha }_x=1,{\alpha }_y=1,{\alpha }_{\varphi }=1000$, (blue diamond) at ${\alpha }_x=2,{\alpha }_y=2,{\alpha }_{\varphi }=2000$, (magenta circles) at ${\alpha }_x=3,{\alpha }_y=3,{\alpha }_{\varphi }=3000$.

Figure 8

Trajectory control for manipulating a spherical Brownian particle ($2.2\mu \mathrm{m}$ diameter bead) over a “figure-eight” curve. The positional history of the particle is shown with a green line. MPC control parameters were set to ${\alpha }_x=1,{\alpha }_y=1, {\alpha }_{\varphi }=0, \lambda =0.001,$ and ${\mathrm{\Phi }}_{\text{max}}=0.05$ for the these experiments.

Figure 9

Actual and reference paths for a spherical Brownian particle tracing a “figure-eight” curve. The reference trajectory is shown in blue, with the red circles marking the particle position over the course of the trajectory.

Figure 10

Reference path and the actual path of an isotropic particle during trajectory control. (a), (b) $x$ and $y$ coordinates of the particle and the reference trajectory as a function of time, respectively. (c) Offset error between the $x$ coordinate of the particle and the $x$ coordinate of the reference trajectory as a function of time. (d) Offset error between the $y$ coordinate of the particle and the $y$ coordinate of the reference trajectory as a function of time.

Figure 11

Trajectory control for manipulating an isotropic Brownian particle ($2.2\mu \mathrm{m}$ diameter spherical bead) over a complex parametric curve. The positional history of the particle is shown with the green line. The particle closely follows a complex parametric path and traverses a linear distance of several hundred microns in only $\approx 9$ s.

Figure 12

Orientation dynamics of a confined anisotropic particle. (a) Comparison of probability distribution function from experiments, and theory. (b) Steady-state distribution of the anisotropic particle at different Peclet number from theory. (c) Order parameter of the particle as a function of ${\text{Pe}}_r$ using an analytical model.

Figure 13

Transient orientation dynamics of an anisotropic Brownian rod in extensional flow. (a) Transient orientation angle $\varphi$ of a trapped rod during the reorientation process. (b) Characteristic reorientation time for a single rod following a step change in orientation angle set-point.