Supplemental Information for “Chemical herding as a multiplicative factor for top-down manipulation of colloids ”

In this section, we give the equations for the gradient vector field (GVF) used to set the trajectory of the herder. A GVF is a function $\bm{V}_{g}(\bm{y},\bm{y}^{*},\{\bm{x}_{i}\})$ that produces a direction for the herder to move at each time step. (Recall that $\bm{y}$ is the position of the herder, $\bm{y}^{*}$ is the target position for the herder, and $\{\bm{x}_{i}\}$ is the set of the positions of each follower.) We use a modified version of the GVF presented by Wilhelm and Clem to calculate the direction $\bm{V}_{g}/\tilde{V}_{g}$ that will move the herder to $\bm{y}^{*}$ while avoiding collisions with the follower particles.

The GVF is described by

$$\bm{V}_{g}=\bm{V}_{\mathrm{path}}+\sum_{i=0}^{n_{\mathrm{f}}}P_{i}\bm{V}_{\mathrm{obs},i},$$

(1)

where $\bm{V}_{\mathrm{path}}$ is the 2D vector that guides the herder to its optimal placement $y^{*}$, and $\bm{V}_{\mathrm{obs},i}$ is the vector that repels the herder from obstacle $i$. Also, $P_{i}$ is a decay function that determines how far from obstacle $i$ the repulsion persists, defined as

$$P_{i}=-\tanh\left(\frac{2\pi\tilde{d}_{ih}}{R_{i}}-\pi\right)+1,$$

(2)

where $\tilde{d}_{ih}=||\bm{y}-\bm{x}_{i}||$ (the distance between obstacle $i$ and the herder). For followers, we set $R_{i}=kR_{fh}$, where the value of $k=1.6$ was chosen by trial and error to ensure the particles do not overlap. In simulations with multiple herders, we set a different value of $R_{i}$ for herders to prevent them from moving too close to each other, settling on a value of ${R_{i}=16\mu}$m.

Each of the terms in Equation (1) is further divided into a convergence term and a circulation term. The convergence terms make the herder either approach its target or avoid an obstacle. The circulation terms help the GVF avoid singularities, or points where $||\bm{V}_{g}||=0$. Each term is weighted appropriately, giving

	$\displaystyle\bm{V}_{\mathrm{path}}=G_{\mathrm{path}}\bm{V}_{\mathrm{path}}^{\mathrm{conv}}+H_{\mathrm{path}}\bm{V}_{\mathrm{path}}^{\mathrm{circ}}$		(3)
	$\displaystyle\bm{V}_{\mathrm{obs},i}=G_{\mathrm{obs}}\bm{V}_{\mathrm{obs},i}^{\mathrm{conv}}+H_{\mathrm{obs}}\bm{V}_{\mathrm{obs,},i}^{\mathrm{circ}}$		(4)

which introduces the functions $\bm{V}_{\mathrm{path}}^{\mathrm{conv}}$,$\bm{V}_{\mathrm{path}}^{\mathrm{circ}}$,$\bm{V}_{\mathrm{obs}}^{\mathrm{conv}}$, and $\bm{V}_{\mathrm{obs}}^{\mathrm{circ}}$ that will be defined in the following paragraphs, as well as a set of constant weights. To calculate $\bm{V}_{\mathrm{path}}$, we used a convergence weight $G_{\mathrm{path}}=1$ and a circulation weight ${H_{\mathrm{path}}=0}$, meaning there is no circulation when the herder is far from any obstacle. To calculate each $\bm{V}_{\mathrm{obs},i}$ we used a convergence weight $G_{\mathrm{obs}}=-2$, meaning the herder will be repelled from obstacles, and a circulation weight ${H_{\mathrm{obs}}=0.5}$, which will bias the particle into clockwise movement around obstacles. These parameters were selected by trial and error.

The convergence and circulation terms of the portion of the GVF that make the herder approach its goal are

$$\bm{V}_{\mathrm{path}}^{\mathrm{conv}}=\frac{-1}{\sqrt{(\cos(\delta)x+\sin(\delta)y)^{2}}}\begin{bmatrix}x\cos^{2}(\delta)+\cos(\delta)\sin(\delta)y\\ y\sin^{2}(\delta)+\cos(\delta)\sin(\delta)x\end{bmatrix}$$

(5)

and

$$\bm{V}_{\mathrm{path}}^{\mathrm{circ}}=[\sin(\delta),-\cos(\delta)]^{T},$$

(6)

where $\delta$ is the angle between the desired path and the $x$-axis.

The convergence and circulation terms that prevent the herder from colliding with an obstacle centered at $(x_{c},y_{c})$ are

$$\bm{V}_{\mathrm{obs}}^{\mathrm{conv}}=\frac{-1}{\bar{x}^{2}+\bar{y}^{2}}\begin{bmatrix}2\bar{x}^{3}+2\bar{x}\bar{y}^{2}\\ 2\bar{y}^{3}+2\bar{x}^{2}\bar{y}\end{bmatrix}$$

(7)

and

$$\bm{V}_{\mathrm{obs}}^{\mathrm{circ}}=[2\bar{y}-2\bar{x}]^{T},$$

(8)

where $\bar{x}=x-x_{c}$ and $\bar{y}=y-y_{c}$.

The GVF, as given so far, encounters a problem when the followers are close together. When an unchased follower is too close to the chased follower, treating that unchased follower as an obstacle may prevent the herder from ever reaching its optimal placement $\bm{y}^{*}$. Our heuristic for solving this problem is that we consider only those followers that are located further than $R_{fh}$ from $\bm{y}^{*}$ as obstacles to be included in the sum in Equation (1). Collisions with obstacles closer than that are deemed necessary.

In Equation (1), $\bm{V}_{g}$ gives the direction for the herder to move. We would also like the herder to move at the maximum speed $v_{\mathrm{max}}$ allowed by the physical system. However, since the controller is discrete in time, care must be taken to ensure the herder does not overshoot its target. Thus the force $\bm{F}_{\mathrm{ext,h}}$ applied to the herder dynamics is set to be

$$\bm{F}_{\mathrm{ext,h}}=\gamma_{h}\frac{\bm{V}_{g}}{\tilde{V}_{g}}\min\left(v_{\mathrm{max}},\frac{\tilde{e}_{h}}{\Delta t_{\mathrm{control}}}\right),$$

(9)

where $\tilde{e}_{h}=||\bm{y}-\bm{y}^{*}||$ is the vector from the herder to its target position and $\Delta t_{\mathrm{control}}$ is the timestep of the controller. This velocity ensures that the herder moves at a speed of $v_{\mathrm{max}}$ until it is within the distance from its optimal position that it can move in a single timestep, then it slows to avoid overshooting. We used a value of $\Delta t_{\mathrm{control}}=0.1$ s.

In this section, we use dimensional analysis to show that a pseudo-steady state approximation can be used to find the concentration field around a herder. We start with the reaction-diffusion equation

$$\frac{\partial C_{s}}{\partial t}=D_{s}\nabla^{2}C_{s}+g_{h}\delta(\bm{X}-\bm{y}).$$

(10)

We then nondimensionalize using $\tilde{\bm{X}}=\bm{X}/R_{fh}$, $\tilde{\bm{y}}=\bm{y}/R_{fh}$, $\tilde{C_{s}}=C_{s}/C_{\infty}$, and $\tilde{t}=t/t^{*}$. For the time scale $t^{*}$, we use the length of time it takes the herder to move a chased particle a distance of $R_{fh}$, which, from Equation 39 of the main paper, is given by $t^{*}=R_{fh}^{3}/k_{\mathrm{diff}}$. Then, substituting in the dimensionless variables, we have

$$\frac{R_{fh}^{2}}{D_{s}t^{*}}\frac{\partial\tilde{C_{s}}}{\partial\tilde{t}}=\tilde{\nabla}^{2}\tilde{C_{s}}+\frac{g_{h}R_{fh}}{D_{s}C_{\infty}}\delta(\tilde{\bm{X}}-\tilde{\bm{y}}).$$

(11)

If the Peclet number $Pe=R_{fh}^{2}/D_{s}t^{*}$ is small, then diffusion will dominate over the motion of the herder, and the $\partial C_{s}/\partial t$ term of Equation (10) will be negligible. Using the values of parameters given in the main paper, we have $Pe\approx 2\times 10^{-3}$. Since this is much smaller than unity, we conclude that a pseudo-steady state approximation is valid. We have also verified the validity of the pseudo-steady state solution numerically to our satisfaction.