#### 4.3.1. Steering a Pulse of Colloids

In the following, we consider the situation where a pulse of colloids is injected at the inlet of a microchannel. We assume that they all have the same initial lateral position

${x}_{i}$, but are spread along the axial direction according to a Gaussian distribution, as shown in the inset of

Figure 10, right. At time

$t=0$ the center of the Gaussian is at

${z}_{i}=0$. Now, we ask which final lateral position

${x}_{f}$ the colloids attain when reaching the axial target position

${z}_{t}$ under the action of the optimal control force

${f}_{\mathrm{ctl}}^{*}\left(t\right)$. For the latter, we use the optimal force protocol calculated for the central initial position at

${z}_{i}=0$. It is switched on at

$t=0$ and switched off at the time

${T}^{*}$ when the particle moving on the original optimized trajectory with

${z}_{i}=0$ has reached the axial target position

${z}_{t}$. Because all particles start on the same initial lateral position

${x}_{i}$ at

$t=0$, they move on replicas of the opimized trajectory, but shifted along the

z direction by

${z}_{i}$. Colloids with

${z}_{i}>0$ precede the original optimized trajectory and therefore experience the control force until they have reached

${z}_{t}$. However, for colloids lagging behind the optimized trajectory (

${z}_{i}<0$), we simply turn off the control force once the optimized time period

${T}^{*}$ has passed and wait until they have reached

$z={z}_{t}$. During this time, the lateral motion is completely determined by inertial focusing without any Saffman force, where the focusing position is

${x}_{\mathrm{eq}}^{0}$. This allows for two scenarios: if the lateral target position

${x}_{t}$ is closer to the channel center than the focusing position (

${x}_{t}<{x}_{\mathrm{eq}}^{0}$), then we know from

Section 4.2 that the optimal trajectory approaches the target from above. Therefore, the preceding and lagging colloids will both reach a final lateral position

${x}_{f}>{x}_{t}$. In contrast, for targets closer to the channel wall than the focusing position (

${x}_{t}>{x}_{\mathrm{eq}}^{0}$), both leading and lagging colloids end up closer to the center than the target (

${x}_{f}<{x}_{t}$).

The qualitative description is confirmed by

Figure 10, left, where we plot the final lateral positions

${x}_{f}$ versus

${z}_{i}$ for different target positions

${x}_{t}$. They follow piecewise linear functions, where the two arms have different slopes, since different mechanisms determine the final positions of preceding and lagging colloids. Only for the target position

${x}_{t}=0.6$ do the final positions

${x}_{f}$ deviate from the linear course for very negative

${z}_{i}$. We note that the deviations of

${x}_{f}$ from the target remain small, even when the axial particle positions are spread over

$30w$ to both sides of

${z}_{i}=0$.

Taking the Gaussian distribution of initial axial positions in the inset of

Figure 10, right, with standard deviation

${\sigma}_{z}=10w$ and assuming the linear dependence

${x}_{f}\left({z}_{i}\right)$ for the final lateral positions, one can readily write the distribution

$p\left({x}_{f}\right)$ of final lateral positions. It is a superposition of two Gaussian functions, where only one half is used from each Gaussian (see below). The resulting distributions for the different target positions are presented in

Figure 10, right. Although the axial width of the initial distribution is ca.

$50w$, the final positions only deviate a little from the target position. The distribution

$p\left({x}_{f}\right)$ for

${x}_{t}=0.4w$ (green curve) is the sharpest, since

${x}_{t}=0.4w$ is closest to the zero-force focusing position

${x}_{\mathrm{eq}}^{0}$. The distributions become broader when

${x}_{t}$ is moved towards the wall or the channel center, respectively. Thus, here we demonstrate that a pulse of colloidal particles fairly spread in the axial direction can be focused into one target position at the channel outlet using one control-force protocol for all of the particles.

At the end, we shortly present the derivation of the distribution

$p\left({x}_{f}\right)$ of final positions at the channel outlet. Because particles with a specific initial axial position

${z}_{i}$ move to a specific

${x}_{f}$, one can directly derive

$p\left({x}_{f}\right)$ from the distribution

${p}_{z}\left({z}_{i}\right)$ and obtain:

Here,

${x}_{f}={f}_{\pm}\left({z}_{i}\right)={a}_{\pm}{z}_{i}+{x}_{t}$ is the piecewise linear function from fitting the curves in

Figure 10, left,

${z}_{i}={f}_{\pm}^{-1}\left({x}_{f}\right)$ is its inverse function, and

${\left({f}_{\pm}^{-1}\right)}^{\prime}\left({x}_{f}\right)=1/{a}_{\pm}$. Taking a Gaussian distribution for

${p}_{z}\left({z}_{i}\right)$, the final distribution

$p\left({x}_{f}\right)$ is a sum of two shifted and rescaled Gaussians with means at

${x}_{t}$. However, because the value range of

${f}_{\pm}$ is either

$(-\infty ,{x}_{t}]$ or

$[{x}_{t},\infty )$, the end result is a sum of two half-normal distributions, either to the left (

${x}_{t}>{x}_{\mathrm{eq}}$) or to the right (

${x}_{t}<{x}_{\mathrm{eq}}$) of the mean

${x}_{t}$ of the full Gaussian. This is readily seen in

Figure 10, right. To have a quantitative measure for the width of the distribution

$p\left({x}_{f}\right)$, we calculate its mean value:

where the plus sign applies to

${x}_{t}<{x}_{\mathrm{eq}}^{0}$ and vice versa. The deviation from

${x}_{t}$ provides a measure for the width of

$p\left({x}_{f}\right)$. It is determined by the slopes

${a}_{\pm}$ of the linear fits to

${x}_{f}={x}_{f}\left({z}_{i}\right)$. Because they are also small, the width is small and it decreases when

${x}_{t}$ approaches

${x}_{\mathrm{eq}}^{0}$. Ultimately, this small width comes from the fact that drift velocities in lateral channel direction are much smaller than the axial flow velocity. This means that inertial transport is much weaker than axial transport due to Poiseuille flow.

#### 4.3.2. Separation of Particles

In the end, we examine the case where two particles of different size are steered using the same control-force protocol. Thus, we assume here that both particle types experience the same external force independent of their sizes. From

Figure 3, we already know that this is possible: a properly chosen constant axial force can drive the smaller particle to the center while the larger particle stays at a finite distance from the center. Here, we aim to maximize the lateral distance after both particles have traveled the distance

${z}_{t}$ in axial direction. At the end of

Section 3.3, we already formulated the appropriate cost functional for maximizing the lateral distance between both particles. In

Figure 11, we show the resulting trajectories (left) and force protocols (right) for two particles with radii

${a}_{1}=0.2w$ and

${a}_{2}=0.3w$ and the axial target

${z}_{t}=500w$. Without control force, these particles would arrive at very similar positions, because their zero-force equilibrium positions are very close to each other. We present results for two cases where both particles start at the same initial position at either

${x}_{i}=0.2w$ or

${x}_{i}=0.5w$. Again, we assume that they do not interact. Interestingly, the control-force protocols for both cases look rather different in the beginning. However, in both cases, the smaller particle (solid lines in

Figure 11, left) is pushed towards the centerline, while the larger particle (dashed lines) moves towards the channel wall during the second half of the trajectories. The separation reached at the end is

$\Delta x=0.45w$ for

${x}_{i}=0.2w$ and

$\Delta x=0.43w$ for

${x}_{i}=0.5w$, which is not attainable with any passive method.

To develop a better understanding for the optimal control-force protocols of

Figure 11, right, we show in

Figure 12 the momentary stable fixed points for the smaller and larger particles corresponding to the momentary axial control force, when the particles are at position

z. The path of the momentary fixed points reflects the particle trajectories of

Figure 11, left, where the algorithm attempts to steer the smaller particle (solid lines) to the channel center and the larger particle (dashed lines) towards the wall. For initial position

${x}_{i}=0.5w$ (blue lines), the fixed point at zero control force is closer to the channel center (around

$0.4w$); therefore, the control force close to zero is sufficient to move, in particular, the smaller particle towards the center. Subsequently, it rises noticeably, bringing the smaller particles to the center, as documented by the course of the momentary fixed point. In contrast, the initial position

${x}_{i}=0.2w$ (orange color) is already closer to the channel center. Thus the control force is switched on immediately to push the smaller particle (orange solid line) to the channel center, where the momentary fixed point is located. As we know from

Figure 3, left, the fixed point of the large particle does not react so strongly to the control force. It is only shifted towards the center, but hardly reaches it. Nevertheless, this initial behavior causes the minima in the dashed trajectories of

Figure 11, left. Afterwards, in both cases the axial control force identified by our algorithm becomes strongly negative and the momentary fixed points are pushed towards the wall, even stronger for the smaller particles. Nevertheless, because the lift force scales with the particle radius

${a}^{2}$, the larger particles are pushed towards the wall, while the smaller particles stay close to the channel center and hardly move away from it (see

Figure 11, left).