###### Abstract

In inertial microfluidics colloidal particles in a Poiseuille flow experience
the Segré-Silberberg lift force, which drives them to specific positions in
the channel cross section. Due to the Saffman effect an external force applied
along the microchannel induces a cross-streamline migration to a new
equilibrium position. We apply optimal control theory to design the time
protocol of the axial control force in order to steer a single particle as
precisely as possible from a channel inlet to an outlet at a chosen target
position. We discuss the influence of particle radius and channel length and
show that optimal steering is cheaper than using a constant control force.
Using a single optimized control-force protocol, we demonstrate that even a
pulse of particles spread along the channel axis can be steered to a target
and that particles of different radii can be separarted most efficiently.

###### keywords:

inertial microfluidics; optimal control; Saffman effect

xx 1 5 Received: date; Accepted: date; Published: date Optimal control of
colloidal trajectories in inertial microfluidics using the Saffman effect
Felix Rühle 1,*, Christian Schaaf 1 and Holger Stark 1,* Felix Rühle,
Christian Schaaf and Holger Stark Correspondence<EMAIL_ADDRESS>(F.R.),
<EMAIL_ADDRESS>(H.S.) 47.11.+j; 47.60.Dx 49M99; 65K10; 76D07

##  1 Introduction

The field of microfluidics is of utmost importance for numerous technological,
biochemical, and biomedical applications, especially for inexpensive lab-on-a-
chip applications and parallelized or automized studies Squires and Quake
(2005); Ismagilov et al. (2001); Whitesides (2006); Weibel and Whitesides
(2006). The experimental realization of high throughput has led to the
emergence of inertial microfluidic systems Di Carlo (2009); Amini et al.
(2014), that open up new possibilities. One important characteristic of the
regime of intermediate Reynolds numbers, where flow is still laminar, is the
breaking of Stokes reversibility. This leads to self-assembly, such as the
famous Segré-Silberberg effect discovered by its namesakes in 1961 Segré and
Silberberg (1961), where colloids travel to distinct lateral positions in the
channel cross section driven by inertial lift forces. Exploiting secondary
flow and inertial effects leads to many exciting applications Di Carlo (2009),
such as enhanced micromixing in curved channels Sudarsan and Ugaz (2006),
particle separation and filtration Bhagat et al. (2008); Sajeesh and Sen
(2014), or focusing and self-assembly Di Carlo et al. (2009); Lee et al.
(2010); Schaaf et al. (2019). An intriguing aspect is the reaction to external
forces pointing along the axial direction of the microchannel. Under creeping
flow conditions these forces do not cause lateral migration Bretherton (1962),
but do so in inertial microfluidic based on the so-called Saffman effect
Saffman (1965). Thus, the particles’ lateral positions in the cross section of
a microchannel can be manipulated with axial external forces, which drive the
particles, e.g., via electrophoresis Kim and Yoo (2009); Yuan et al. (2016);
Choudhary et al. (2019).

Parallel to the experimental progress there have been continuous and fruitful
efforts to tackle inertial microchannels via computer simulations Bazaz et al.
(2020). Here, lift forces and particle dynamics can be probed for different
channels and particle types Shin and Sung (2011); Asmolov et al. (2018);
Schaaf and Stark (2017), or in complex fluids Li et al. (2015); Raoufi et al.
(2019). Using external forces, optimal and feedback control has been applied
to particle separation and steering under inertial microfluidic conditions
Prohm et al. (2013); Prohm and Stark (2014), also together with thermal noise.
One example is the hysteretic control scheme, as applied by Prohm and Stark
Prohm and Stark (2014). Here, particles are periodically forced back to the
channel center using the Saffman effect, while the inertial lift force drives
them away from the center. Thus, the particle stays within a finite interval
around the channel center.

In this article, we present the theoretical concept to realize precise
particle steering with time-dependent axial control forces. The main idea is
to steer the particles to different outlets of a microchannel in order to
achieve particle separation and filtration. We use the Saffman effect and
optimal control theory Ioffe and Tihomirov (1979) to design the time protocol
of the axial control force in order to steer particles from an initial to a
target position in a microchannel (see Fig. 1). These positions are defined,
_e.g._ , by inlets and outlets of the microchannel. As an input for the
optimization, we employ the lattice-Boltzmann method to simulate particles in
Poiseuille flow in order to obtain a whole set of lift-force profiles,
depending on the axial control force. Then, we use analytical fit functions to
set up a system of ordinary differential equations that yield the particle
trajectories. The time-dependent axial control force for optimally steering
the particle from an inlet to an outlet follows by numerically minimizing a
cost functional with respect to the control force under the condition that the
target at the end of the channel is reached. We thoroughly discuss this method
of optimal steering and compare it with steering by a constant control force.
Using the optimal control-force protocol for a single particle, we demonstrate
that even a pulse of particles spread along the channel axis can be steered to
a target. Finally, we show how a single optimized control-force protocol can
separate particles of different radii. Here, we go beyond particle separation
with constant axial forces suggested in Ref. Prohm and Stark (2014). By using
the same time-dependent control force for both particles, one can further
increase the lateral particle distance at the end of the channel.

We introduce the theory of inertial microfluidics and the Saffman effect in
Sect. 2. We describe the setup of our system, the lift force profiles, and the
method of optimal control in Sect. 3. The results of our study for single and
multi-particle steering are presented in Sect. 4 and we conclude in Sect. 5.

##  2 Theory - Inertial Microfluidics

Figure 1: Sketch of the model system: A colloid flowing in the $x$-$z$ plane
experiences an axial control force $f_{\mathrm{ctl}}(t)$. The occuring Saffman
effect changes the lateral lift force $f_{\mathrm{lift}}$ and, thereby, the
colloid can be steered from the initial position $(z_{i},x_{i})$ to the target
$(z_{t},x_{t})$.

Segré and Silberberg first reported how colloidal particles self-organize on
an annulus under pipe flow conditions Segré and Silberberg (1961) that is
located approximately halfway between the channel center and the confining
walls. Since deterministic lateral motion for rigid particles is impossible
under strict creeping flow conditions, this migration results from the
inertial term of the Navier-Stokes equation. Hence, it was termed inertial
focusing and rationalized by a so-called inertial lift force Di Carlo (2009);
Asmolov (1999); Hood et al. (2015). For channels with rectangular cross
sections these equilibrium positions are either located on the main axes or
the diagonals of the cross section, which depends on the particle radius and
the cross-sectional aspect ratio (see, for example, Ref. Prohm and Stark
(2014)). If this ratio is sufficiently large, only two stable positions on the
short axes exist and it is sufficient to treat the flowing particle in a two-
dimensional plane as sketched in Fig. 1.

The Poiseuille-flow profile in a rectangular channel is known analytically
Bruus (2008). Since the flow field along the channel axis obeys
$\mathbf{u}=u(x,y)\mathbf{e}_{z}$, the convective term of the Navier-Stokes
equations vanishes, and for the stationary case the Stokes equations are
recovered. Restricting fluid flow in the cross section to $x\in(-w,w)$,
$y\in(-h,h)$ with $h>w$ and using no-slip boundary conditions at the channel
walls, $u(x=\pm w,y)=u(x,y=\pm h)=0$, one can write the solution as a Fourier
series expansion Bruus (2008)

$\mathbf{u}(x,y)=\dfrac{16w^{2}\Delta p}{\pi^{3}\eta
L}{{\sum_{n=0}^{\infty}}}(-1)^{n}\dfrac{1}{(2n+1)^{3}}\left[1-\dfrac{\cosh\left(\frac{(2n+1)\pi}{2w}y\right)}{\cosh\left(\frac{(2n+1)\pi}{2w}h\right)}\right]\cos\left(\frac{(2n+1)\pi}{2w}x\right).$
(1)

Here, a constant pressure gradient $\Delta p/L$ is used and the dynamic
viscosity of the fluid $\eta$. When we employ this analytical formula in our
numeric calculations, we truncate the series after $n=100$. The maximum flow
velocitiy $U_{m}$ is reached at the center $(x,y)=(0,0)$ of the channel. It is
determined by the choice of the Reynolds number $\mathrm{Re}=\rho
U_{m}2w/\eta$, where $\rho$ is the fluid density and $2w$ the width of the
channel.

Inertial effects become observable if a colloid is subjected to a Poiseuille
flow at finite Reynolds numbers $\mathrm{Re}$. This initiates a lift force
acting on the colloid, which can be controlled via the Saffmann effect by
applying an additional axial control force (see Fig. 1). We introduce those in
the following.

###  2.1 Lift force

Since the discovery of inertial focussing different scaling laws for the
dependence of the inertial lift force on particle radius $a$ and Reynolds
number have been derived Ho and Leal (1974); Schonberg and Hinch (1989);
Asmolov (1999); Di Carlo et al. (2009); Hood et al. (2015). For example, Ho
and Leal calculated the lift force for small particle radius ($a\ll w$) and
small particle Reynolds number $\mathrm{Re}(a/w)^{2}$. They arrived at the
scaling law $f_{\mathrm{lift}}\sim\mathrm{Re}^{2}(a/w)^{4}$ Ho and Leal
(1974), whereas numerical simulations at finite particle sizes arrived at
$f_{\mathrm{lift}}\sim(a/w)^{3}$ in the channel center for particle sizes
$a<w$ Di Carlo (2009). To correct for finite particle size and Reynolds
number, often the lift coefficient $f(a,Re)$ is introduced Di Carlo (2009). In
particular, it has been observed that the scaling exponent for the lift force
as a function of the particle radius depends on the lateral position in the
channel Di Carlo et al. (2009); Di Carlo (2009). Importantly, the fixed points
of the lift-force profiles indicating stable equilibrium positions change
considerably with the geometry of the channel cross section (see, for example,
Ref. Prohm and Stark (2014)). In the following, we calculate the lift-force
profiles numerically using lattice-Boltmann simulations as shortly explained
in Sec. 3.2. Typical examples for a zero axial control force are presented in
Fig. 2, left.

The net inertial lift force is often described as a balance of two
contributions. They arise from a stresslet that is the leading force
distribution on the particle surface under a shear-flow gradient Ho and Leal
(1974); Amini et al. (2014): the reflection of the stresslet from the channel
wall induces a force, which pushes the particle away from the wall, whereas
the interaction with the shear gradient transports particles to regions of
larger shear, which is towards the wall in case of a Poiseuille flow.

###  2.2 Saffman effect

Applying an additional axial control force to the colloidal particle speeds up
or slows it down relative to the local Poiseuille flow velocity. This modifies
the slip velocity field close to the particle surface and at finite Reynolds
numbers creates an additional lateral contribution to the lift force,
described by Saffman Saffman (1965). It depends on the shear rate $\gamma$,
rather than the shear gradient, and was calculated to be $f_{S}\sim
va^{2}\gamma^{1/2}$ in bulk at small Reynolds numbers, where $v$ is the
difference between local flow field and particle velocity. Figure 2, right
demonstrates how the lift-force profile changes, when a control force is
applied. The stable fixed point ($f_{\mathrm{lift}}=0$ with a negative slope)
moves from the zero-force position ($f_{\mathrm{ctl}}=0$) either to the wall
or to the channel center depending on whether the control force is applied
along the flow direction ($f_{\mathrm{ctl}}<0$) or against it
($f_{\mathrm{ctl}}>0$), respectively. The respective stable equilibrium
positions are plotted in Fig. 3, left, also for different particle radii.
Because the inertial lift force grows more strongly with the radius than the
Saffman force, higher axial control forces are necessary to move the fixed
point for larger particles towards the center. Consequently, the curves in
Fig. 3, left become flatter for larger particle radii.

Figure 2: Left: Lift-force profiles along the positive $x$ axis for different
particle radii $a/w$ at zero axial control force, $f_{\mathrm{ctl}}=0$. Note
the larger strength of the lift forces for larger colloids and the shift of
the stable fixed point. The force unit $\rho\nu^{2}$ uses fluid density $\rho$
and kinematic viscosity $\nu=\eta/\rho$. Right: Lift-force profiles for a
colloid with radius $a=0.2w$ at different axial control forces and least-
square fits using eqs. (3.2) (solid lines). In both cases the Reynolds number
$\mathrm{Re}=10$ is used.

##  3 Methods

###  3.1 Setup

We consider a rectangular microchannel with Poiseuille flow at Reynolds number
$\mathrm{Re}=10$. The channel has an aspect ratio $w:h$ of 1:2, where
$x\in(-w,w)$ and $y\in(-h,h)$. The length of the channel is $L$ with
$z\in(0,L)$, which we vary in the following. At such an intermediate Reynolds
number, the regime of inertial microfluidics is reached and solid colloids in
a Poiseuille flow self-organize towards distinct lateral focus positions
Saffman (1965). For the aspect ratio chosen here, they equilibrate to the
plane $y=0$ and, therefore, we only consider the particle dynamics in the
$x$-$z$-plane Amini et al. (2014); Bhagat et al. (2008); Gossett et al.
(2012). Indeed in our lattice-Boltzmann simulations the position $y=0$ is
stable, i.e., a particle is immediately driven back once it leaves the center
plane. The stable fixed point at this location was numerically determined in
Ref. Prohm and Stark (2014). Switching on the axial force, the induced Saffman
force is strongest along the $x$ axis since in this direction the velocity
gradients are largest. Therefore, we expect the particle to stay in the center
plane. The stable equilibrium positions on the $x$ axis
($f_{\mathrm{lift}}=0$) depend on particle size as we show in Fig. 2, left.
There is also a dependence on $\mathrm{Re}^{2}$ Schaaf et al. (2019), which we
do not further explore here. These positions are reached after the colloid has
been advected for a sufficiently large axial distance $L_{f}$ without any
external forcing. Di Carlo and co-workers mention an estimate for this length
Di Carlo (2009); Amini et al. (2014),

$L_{f}=\dfrac{\pi\nu w^{2}}{f_{L}U_{m}a^{2}},$ (2)

with the maximum flow velocity $U_{m}=\nu\mathrm{Re}/(2w)$, kinematic fluid
viscosity $\nu$, and particle radius $a$. For our aspect ratio $w/h=0.5$, Ref.
Di Carlo (2009) gives a lift coefficient $f_{L}=0.05$. Furthermore, using
$\mathrm{Re}=10$, and $a/w=0.2$, in eq. (2), we obtain the focus length
$L_{f}\approx 314w$. Now, applying an additional lateral force along the $x$
direction, one can optimally steer particles to any position on the $x$ axis
as we showed in Ref. Prohm et al. (2013).

Figure 3: Left: Stable equilibrium positions (fixed points) as a function of
control force for different particle radii $a/w$. They were determined using
the analytical fits to the relevant lift-force profiles in Fig. 2, right.
Right: Hydrodynamic friction coefficients relative to the bulk value
$\xi_{\infty}=6\pi\eta a$ plotted versus the lateral particle position for
different particle radii. The solid lines are fits using eq. (4).

Here, we propose an alternative strategy for optimal steering using the
Saffman effect. We apply an axial control force and thereby modify the lift-
force profile as demonstrated in Fig. 2, right. In Fig. 3 we show how the
stable equilibrium position now depends on the control force. Then, the idea
is to use a time varying axial control force, which can be realized, for
example, by electromagnetic fields Yuan et al. (2016), for optimal steering.
The goal is to optimally steer a particle from an inlet, which is located at
the start position $(z_{i},x_{i})$, towards a target $(z_{t},x_{t})$
fulfilling a criterion of optimality as we will outline below. To implement
this approach, we first need lift-force profiles
$f_{\mathrm{lift}}(x,f_{\mathrm{ctl}})$ for different control forces as well
as friction coefficients $\xi(x)$ for different particle sizes, which we
determined with the help of lattice-Boltzmann simulations, and approximate
them with appropriate fit functions (Sect. 3.2). They are then used in
dynamical equations for the particle motion, which we solve with explicit
Euler integration in order to determine the optimal steering path (Sect. 3.3).

###  3.2 Profiles for lift forces and friction coefficients

Our lattice-Boltzmann simulations (including the immersed-boundary method)
Chun and Ladd (2006); Dünweg and Ladd (2009) of single colloids in a
microchannel in the inertial regime are described in detail in Refs. Prohm and
Stark (2014); Schaaf et al. (2019), where we also explain how to determine
inertial lift forces for each particle position.

The simulated lift-force profiles for different particle radii and channel
Reynolds number $\mathrm{Re}=10$ are shown in Fig. 2, left. They display the
well-known behaviour of inertial focusing: Colloids are driven away from the
unstable fixed point at the origin and towards their stable equilibrium
positions (stable fixed points) between the channel center and the wall, which
depend on the particle radius. As in Ref. Prohm et al. (2013) we perform a
least-square fit of the lift-force profiles to a third-order polynomial of odd
degree together with a wall-repulsion term as the particle approaches the
walls. Additionally, we now also apply this fit to the dependence of the lift
force on the axial control force $f_{\mathrm{ctl}}$ using coeffients that are
second-order polynomials in $f_{\mathrm{ctl}}$. Thus, the functional form for
the fit of our lift-force profiles is as follows

$\displaystyle f_{\mathrm{lift}}(x,f_{\mathrm{ctl}})$ $\displaystyle=$
$\displaystyle\phi_{1}(f_{\mathrm{ctl}})x+\phi_{3}(f_{\mathrm{ctl}})x^{3}+\phi_{w}f_{w}(x)$
$\displaystyle\phi_{1}(f_{\mathrm{ctl}})$ $\displaystyle=$ $\displaystyle
a_{1}f_{\mathrm{ctl}}^{2}+b_{1}f_{\mathrm{ctl}}+c_{1}$
$\displaystyle\phi_{3}(f_{\mathrm{ctl}})$ $\displaystyle=$ $\displaystyle
a_{3}f_{\mathrm{ctl}}^{2}+b_{3}f_{\mathrm{ctl}}+c_{3}$ (3) $\displaystyle
f_{w}(x)$ $\displaystyle=$
$\displaystyle\dfrac{1}{x-(1+\delta)x_{w}}+\dfrac{1}{x+(1+\delta)x_{w}},$

where $x_{w}=w-a$ and we use $\delta=10^{-3}$ for numerical stability. As Fig.
2 demonstrates, the fit function works well, in particular, for non-zero
control force, as long as the region immediately at the wall is avoided. Note
that we did not attempt to include the dependence on particle radius in our
fit function, but instead perform a separate fit for each particle size. We do
this in order to limit the number of parameters.

We also determined the friction coefficients of the particles in the lattice-
Boltzmann simulations and plot their values as a function of the lateral
position in Fig. 3, right for three different colloidal radii. The presence of
the walls is clearly visible. We fit the position-dependent friction
coefficient $\xi$ by the function

$\xi(x)=\xi_{\infty}\Big{(}d_{1}+d_{2}\dfrac{a}{(w-a)-|x|}\Big{)},$ (4)

where $\xi_{\infty}=6\pi\eta a$ is the bulk friction coeffcient and $d_{1}$,
$d_{2}$ are fit parameters. The fits as solid lines are presented in Fig. 3,
right.

###  3.3 Dynamical system and optimal control

Using the axial control force $f_{\mathrm{ctl}}$, the fitted lateral lift
force $f_{\mathrm{lift}}$, and the friction coefficient $\xi$, the overdamped
motion of the steered particle in the Poiseuille flow profile $u(x)=u(x,y=0)$
is governed by the following differential equations disregarding any thermal
noise:

$\displaystyle\dot{z}$ $\displaystyle=$ $\displaystyle
u(x)+\dfrac{1}{\xi(x)}f_{\mathrm{ctl}}(t)$ (5) $\displaystyle\dot{x}$
$\displaystyle=$
$\displaystyle\dfrac{1}{\xi(x)}f_{\mathrm{lift}}\left(x,f_{\mathrm{ctl}}(t)\right).$
(6)

Here, $z$ is the coordinate along the channel and $x$ in lateral direction. As
described above, the size of the suspended particle influences the lateral
motion implicitly via the friction coefficient and the fitted function for the
lift force. In axial direction we do not consider that a force-free colloid is
slower than the streaming fluid but simply set this velocity to the Poiseuille
flow velocity $u(x)$. We note that the dynamics of the colloid is always
confined to one half of the channel. At $x=0$ the lift force - and hence the
total lateral force - is exactly zero, therefore it is impossible for the
colloid to cross the center line.

Solving these equations for a given time protocol $f_{\mathrm{ctl}}(t)$ of the
axial control force, determines $x(t)$ and $z(t)$. We are looking for an
optimal protocol $f_{\mathrm{ctl}}^{*}(t)$, which steers a particle as close
as possible to the target $(z_{t},x_{t})$ at end time $T^{*}$. Thus, we define
the cost functional Ioffe and Tihomirov (1979)

$J[f_{\mathrm{ctl}}(t),T]=\dfrac{c_{x}}{2}\Big{|}x_{t}-x(T)\Big{|}^{2}+\dfrac{c_{z}}{2}\Big{|}z_{t}-z(T)\Big{|}^{2}+\dfrac{\varepsilon}{2}\int_{t_{0}}^{T}|f_{\mathrm{ctl}}(t)|^{2}dt,$
(7)

and obtain the optimal steering control force by minimizing the cost
functional with respect to $f_{\mathrm{ctl}}(t)$:

$f_{\mathrm{ctl}}^{*}(t)=\underset{f_{\mathrm{ctl}}}{\mathrm{arg\,min}}J\,.$
(8)

For the total duration $T$ of the trajectory, which is undetermined on the
right-hand side of eq. (7), the algorithm also finds an optimum $T^{*}$. In
concreto, we set $T^{*}=N\Delta t^{*}$, where $\Delta t^{*}$ is the time step
of our time discretization, choose a constant $N$, and determine $\Delta
t^{*}$ together with the force protocol by minimizing the cost functional. We
always choose a control force that is constant in time as our initial
function. When optimizing the single-particle trajectories, we choose $N=500$;
while when looking at a pulse of colloids in Sect. 4.3.1, we take $N=2500$
since a higher resolution in axial direction is required. The functional in
eq. (7) includes a regularization term, where we integrate over the square of
the total force because otherwise arbitrarily large forces would be
permissible. Keeping this regularization term low, decreases the energy cost
for steering the particle along a specific trajectory, for example,
electrophoretically by applying an electric field Yuan et al. (2016). We weigh
the cost of deviating from the target area differently for the $x$ and $z$
coordinates, because the velocities differ strongly. The control parameters
$c_{x},c_{z}$ and $\varepsilon$ have to be adapted manually to find the right
balance between the cost of higher forces versus the precision of steering.
When optimizing the single-particle trajectories, we choose $c_{x}=14000$,
$c_{z}=0.3$, and $\varepsilon=0.003$. For solving the differential equations,
we use explicit Euler integration. With this we optimize the discretized cost
functional $J[f_{\mathrm{ctl}}(t),T]$ using a sequential quadratic programming
(sqp) algorithm Nocedal and Wright (2006) provided by the package fmincon from
MathWorks’ software matlab (Release R2019b)
111https://mathworks.com/help/optim/ug/constrained-nonlinear-optimization-
algorithms.html. This robust and efficient method uses a Lagrangian
representation of a constrained problem: $\mathcal{L}=f(x)+\lambda^{T}c$,
where $x$ is the vector of unknowns, $c$ the vector of equality constraints
and $\lambda$ the vector of Lagrange multipliers. For the optimal solution
$x^{*}$, the gradient of $\mathcal{L}$ vanishes, which yields a nonlinear
equation. Here, $x^{*}$ can be approximated by iteration,
$x_{i+1}=x_{i}+d_{i}$, where the differences $d_{i}$ are determined in each
step by a simpler approximated quadratic problem, for which standard quadratic
solvers are available Nocedal and Wright (2006). We do not calculate the
necessary gradients to the functional ourselves but leave this to the software
matlab.

We also seek to maximize the lateral distance between two colloids, when
approaching the target with axial coordinate $z_{t}$. Therefore, in the cost
functional of eq. (7) we use $\dfrac{c_{x}}{2}\big{|}\Delta x(T)-\Delta
x_{f}\big{|}^{2}$ for the first summand, where $\Delta x(t)$ is the lateral
distance between two colloids at time $t$ and $\Delta x_{f}$ is the distance
aimed for. In the second term involving the $z$ coordinate we add up the
contributions from the two particles. The coefficient $c_{z}$ takes the same
values for both colloids. For the two-particle optimization we use
$c_{x}=500$, $c_{z}=30$, and $\varepsilon=0.01$.

We always obtained smooth solutions when solving the unconstrained problem.
Due to the diverging repulsive lift force close to the walls, it was not
necessary to use a numerical constraint for the lateral coordinate $x$ in
order to keep the particle within the channel. The sqp algoritm required $75$
iterations on average in order to converge for the single-particle steering.
For the separation of two particles, $161$ iterations were required on
average.

##  4 Results and Discussion

In the following we investigate steering strategies for single and multiple
particles that make use of the aforementioned inertial lift forces. First, we
discuss steering with a constant axial control force and then the outcome of
our optimal control scheme. Using the results from single-particle steering,
we investigate the implications for a particle pulse spread along the channel
axis. Finally, we use optimal control to find control forces that maximize the
separation of two particles so that they can be carried off at different
outlets of a channel.

###  4.1 Steering with constant axial control forces

Exploiting the Saffman effect, the easiest approach to steer a particle to a
target position is to use constant axial control forces. They shift the stable
fixed point of the lift-force profile and thereby, in principle, the
equilibrium position of a colloidal particle can be adjusted arbitrarily, as
shown in Fig. 3. In the same way, also colloids of different sizes can be well
separated within the microfluidic channel using a constant control force Prohm
et al. (2013). As Fig. 3 shows this is achieved by choosing the control force
such that the smaller particle (_e.g._ , $a=0.2w$) is pushed to the center
while the larger particle still keeps a noticeable distance from the center.

Figure 4: Inertial focusing of a colloid with radius $a=0.2w$ starting from
different initial positions and using constant control forces. Lateral
position versus time is plotted. Left: for constant control forces
$f_{\mathrm{ctl}}/(\rho\nu^{2})=-1.5,-1.0,0,1.0,1.5$ (which give an increasing
equilibrium $x$ position or stable fixed point). Right: for adjusted constant
control forces such that the stable fixed point assumes the values
$x/w=0.1,0.2,...,0.7$.

In Fig. 4 the lateral positions for the moving colloid are plotted versus time
for specific control forces (left) and when the forces are adjusted to give
specific final positions (right). Negative forces point along the direction of
the channel flow and thus drive the equilibrium position closer to the wall,
whereas positive forces slow down the colloid and thereby induce motion
towards the channel center. A closer inspection of the two plots shows that
the necessary travel time for focusing varies with the initial position.
However, more pronounced is the dependence on the final position as Fig. 4,
right demonstrates. Fixed points closer to the channel center are reached
later than those closer to the wall. This is consistent with the fact that the
Saffman effect (or the shear-induced lift force) increases with the shear
rate, which is larger close to the walls. Thus, using the Saffman effect for
steering requires less time the closer the final position is situated to the
wall. In Sect. 3.1 we evaluated the focus length $L_{f}\approx 314w$ for our
setup. Indeed, it gives a good estimate for the focus length in our
simulations at zero control force. When plotting the lateral positions of Fig.
4 versus the traveled axial distance $z$ instead of time, the curves look very
similar to the ones in Fig. 4, even though particles closer to the center
(small $x$) should flow faster and the curves should stretch even farther
close to the centerline. However, to reach these targets, the control force
has to act against the flow and therefore decreases the axial flow velocity of
the particles.

In conclusion, in order to relax to the adjusted equilibrium position with the
constant-force strategy, considerably longer travel times and axial distances
are required for targets that lie close to the channel center. Therefore, long
enough channels need to be used in order for the strategy to work. Thus, it is
not possible to use a single channel length to steer particles to different
lateral target positions using the constant-force strategy. Furthermore,
steering with a constant axial force means, it has to be maintained for the
whole trajectory. We compare the cost of this constant-force scheme with the
optimal control scheme in the next section, which also allows to operate with
channels of one length.

###  4.2 Optimal control of single colloids

We now turn to the optimization problem for the cost functional
$J[f_{\mathrm{ctl}}(t),T]$ of eq. (7) set up in Sec. 3.3. This will provide us
with a time-dependent control force and the particle trajectory in the $x$-$z$
plane. In the following, we provide optimal solutions of the cost functional
for different start and end positions in the channel. The optimization
procedure is applied to two axial target positions at $z_{t}=300w$ and
$z_{t}=500w$. Furthermore, three particle sizes with radii $a=0.2w$, $0.25w$,
and $0.3w$ are considered. Since the initial position is always set at
$z_{i}=0$, we call $z_{t}$ channel length for short.

Figure 5: Optimal control-force protocols (left) and particle trajectories in
the $x$-$z$ plane (right) found for steering colloids with radius $a=0.2w$
from a set of initial positions to a set of targets, which both assume the
same values $\\{0.1,0.2,0.3,0.4,0.5,0.6,0.7\\}$. The same color refers to
trajectories ending at the same target position $x_{t}$. The vertical dashed
line indicates $z_{t}$.

In Fig. 5 we plot the optimal force protocol (left) and the trajectories
(right) resulting from different start and target positions using the particle
radius $a=0.2w$ and channel length $z_{t}-z_{i}=500w$. Interestingly, the
regularization term generates solutions, where the force is zero at first,
meaning that all colloids travel towards the equilibrium position at zero
control force except when both $x_{i}$ and $x_{t}$ are close to the channel
center. In particular, for $x_{i}=x_{t}=0.1w$, the algorithm chooses a nearly
constant force protocol, clearly recognizable in Fig. 5, left. In all other
cases, the control force increases or decreases monotonously starting around
$t=300w^{2}\nu^{-1}$, which corresponds to a traveled distance between
$z=250w$ and $300w$. At the end of the trajectory the target is reached with
high precision at $z=z_{t}$.

Note, the control-force protocols are similar for the same target position
(same color). However, the increase/decrease from zero starts earlier if the
initial lateral position is closer to the channel center. This is because flow
velocity is larger and thus particles travel faster downstream.

Figure 6: Solid lines: Optimal particle trajectories in the $x$-$z$ plane
found for steering colloids with radius $a=0.2w$ from the initial position
$x_{i}=0.2w$ to a set of targets $\\{0.1,0.2,0.3,0.4,0.5,0.6,0.7\\}$. Dashed
lines: Sequence of fixed-point positions of the lift-force profiles resulting
from the optimal control-force protocols $f_{\mathrm{ctl}}(t)$ (same colors
represent the same force protocol).

From the explanation so far, one could assume that the particle
instantaneously follows the stable fixed points of the lift-force profiles
associated with the optimal control-force protocol $f_{\mathrm{ctl}}(t)$. We
plot the sequence of fixed points as dashed lines in Fig. 6 in the $x,z$ plane
together with the realized particle trajectories starting at $x_{i}=0.2w$ and
ending at different target positions. Clearly, the particle does not follow
the sequence of stable fixed points, since migrating there is hindered by
viscous friction. For target positions close to the channel center, the
control force induces a fixed point at the center ($x=0$) for the last part of
the trajectory to accomplish the trajectories bend downward in Fig. 6.

####  4.2.1 Comparison with constant force strategy

Figure 7: Comparison between the constant-force strategy (solid lines) and the
optimal control-force strategy (dashed line), for different inital lateral
positions and a channel length $z_{t}=500w$. We show a semi-logarithmic plot
of the cost functional $I$ of the axial control force versus lateral target
position $x_{t}$.

We compare the costs of the constant-force strategy and the optimal control-
force scheme using the cost functional
$I:=\int_{0}^{T^{*}}|f^{*}_{\mathrm{ctl}}|^{2}(t)dt$. It integrates the square
of the control force along the particle trajectory, where lateral initial and
target positions are equal for both strategies. Note that $I$ is a measure for
the energy costs needed to realize the control schemes. To compare both
strategies, we decided to work with a constant channel length $z_{t}$ as a
typical situation in experiments. While the optimal-control scheme can be
adjusted to such a specific axial target $z_{t}$, for the constant-force
strategy the necessary channel length varies depending on the lateral target
position, as we discussed in Sect. 4.1. In Fig. 7, left we plot the cost
functional $I$ for both strategies versus target position $x_{t}$ for
different initial lateral positions $x_{i}$. The channel length is always
$z_{t}=500w$. Interestingly, the curves for each strategy are all very similar
and therefore independent of $x_{i}$. The reason is the control-force profiles
for different $x_{i}$ but same $x_{t}$ in Fig. 5 have all very similar shape
and are mostly shifted relative to each other. Clearly, the optimal-control
scheme is less costly than the constant-force strategy up to an order of
magnitude, besides for the smallest target position $x_{t}/w=0.1$. Here, we
note that for the constant-force strategy the channel length $z_{t}=500w$ is
not sufficient for reaching targets with $x_{t}/w\leq 0.4$. Taking into
account the need for longer channels, the costs of the constant-force strategy
goes up. In contrast for large $x_{t}\geq 0.5$ a channel length smaller than
$500w$ is sufficient, which reduces the costs. But those do not fall below the
costs of the optimal-control scheme.

####  4.2.2 Dependence on particle size

Figure 8: Optimal control-force protocols (left column) and particle
trajectories in the $x$-$z$ plane (right column) found for steering colloids
with radius $a=0.25w$ (top row) and $a=0.3w$ (bottom row) from a set of
initial positions to a set of targets, which both assume the same values
$\\{0.1,0.2,0.3,0.4,0.5,0.6\\}$. The same color refers to trajectories ending
at the same target position $x_{t}$. The vertical dashed line indicates
$z_{t}$.

Increasing the particle size, strongly increases the strength of the inertial
lift force ($f_{\mathrm{lift}}\propto(a/w)^{4}$ for very small particles).
Thus, the fixed-point position at the initially zero control force is reached
faster, as a comparison of the force protocols and the partcle trajectories in
Figs. 5 and 8 for different radii shows. The control force remains longer at a
zero value before it steers the particle to its target position. However, for
the larger particles this then also requires larger control forces to steer
them laterally because they experience a higher drag force and it is therefore
harder to move them relative to the external flow. Furthermore, as before for
target positions further away from the zero-force fixed-point position, larger
control forces are necessary for steering, which makes sense. Finally, as we
already noted in Sect. 2.2, since the Saffman force ($f_{\mathrm{S}}\propto
a^{2}$) grows less strongly with the radius than the focussing inertial lift
force, one again needs larger control forces for particle steering, which also
drives up the whole costs. Thus, in all trajectories the particles utilize
inertial focussing at first to relax towards the zero-force equilibrium
positions and then the control force is switched on.

Figure 9: Optimal control-force protocols (left column) and particle
trajectories in the $x$-$z$ plane (right column) found for steering colloids
with radii $a/w=0.2$, $0.25$, and 0.3 (top, middle, and bottom row) to an
axial target position $z_{t}=300w$. The vertical dashed line indicates
$z_{t}$.

####  4.2.3 Dependence on channel length

Since the control forces obtained in Fig. 5 and 8 all remain at zero at the
beginning of the particle trajectories, it should be possible to further
decrease the channel length. Indeed, we managed to obtain numerically stable
solutions for a channel length of $z_{t}=300w$, which we show in Fig. 9. Here,
the trajectories do not ($a/w=0.2$ and $0.25$) or only shortly ($a/w=0.3$)
stay on the lateral focus position, and thus the control force is always non-
zero or zero for a short time. Interestingly, the algorithm chooses relaxation
towards the equilibrium position for the largest radius $a=0.3w$. Again, the
inertial lift force increases strongly with the particle radius and,
therefore, it is too costly to compete against it with a non-zero control
force over the whole simulation time. Instead, the algorithm chooses to drive
up the control force to high absolute values but for a shorter time period at
the end of the trajectory.

###  4.3 Controlled steering of multiple colloids

We use our model to steer two or more particles to their respective targets.
As a first approximation we neglect here two-particle interactions. It is
known that the lift-force profiles of two particles are influenced due to
secondary flows, when their axial distance is smaller or of the order of the
channel width Amini et al. (2014); Schaaf et al. (2019). In the following, we
first consider the steering of a pulse of equal-sized particles, and then
investigate the lateral separation of two particles with different radii under
the same control-force protocol.

####  4.3.1 Steering a pulse of colloids

Figure 10: Left: Final lateral position $x_{f}$ for particles with initial
axial position $z_{i}$ when steered by the axial control force
$f_{\mathrm{ctl}}^{*}(t)$, which is opimized for steering the central particle
with $z_{i}=0$ and $x_{i}=0.2$ to different target positions $x_{t}/w=0.2$,
$0.3$, $0.4$, $0.5$, and $0.6$ at the axial target position $z_{t}=500w$. The
linear fits by solid lines are hardly visible. Right: Distributions of final
lateral positions $x_{f}$ for the Gaussian distribution of initial axial
positions $z_{i}$ shown in the inset. The colors indicate different target
positions $x_{t}/w$ as in the plot on the left.

In the following we consider the situation where a pulse of colloids is
injected at the inlet of a microchannel. We assume 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 Fig. 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
$f_{\mathrm{ctl}}^{*}(t)$. 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 $z_{t}$. Since 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 ($x_{t}<x_{\mathrm{eq}}^{0}$), we know from Sec. 4.2 that
the optimal trajectory approaches the target from above. Therefore, both the
preceding and lagging colloids will 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 Fig. 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
Fig. 10, right with standard deviation $\sigma_{z}=10w$ and assuming the
linear dependence $x_{f}(z_{i})$ for the final lateral positions, one can
readily write the distribution $p(x_{f})$ 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 Fig. 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 $p(x_{f})$ for $x_{t}=0.4w$ (green
curve) is 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 we
demonstrate here, that a pulse of colloidal particles fairly spread in the
axial direction can be focussed into one target position at the channel outlet
using one control-force protocol for all the particles.

At the end we shortly present the derivation of the distribution $p(x_{f})$ of
final positions at the channel outlet. Since particles with a specific initial
axial position $z_{i}$ move to a specific $x_{f}$, one can derive $p(x_{f})$
directly from the distribution $p_{z}(z_{i})$ and obtain:

$p(x_{f})=p_{z}(f^{-1}_{+}(x_{f}))|(f^{-1}_{+})^{\prime}(x_{f})|+p_{z}(f^{-1}_{-}(x_{f}))|(f^{-1}_{-})^{\prime}(x_{f})|.$
(9)

Here, $x_{f}=f_{\pm}(z_{i})=a_{\pm}z_{i}+x_{t}$ is the piecewise linear
function from fitting the curves in Fig. 10, left, $z_{i}=f^{-1}_{\pm}(x_{f})$
is its inverse function, and $(f^{-1}_{\pm})^{\prime}(x_{f})=1/a_{\pm}$.
Taking a Gaussian distribution for $p_{z}(z_{i})$, the final distribution
$p(x_{f})$ is a sum of two shifted and rescaled Gaussians with means at
$x_{t}$. However, since 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 Fig. 10, right. To have a quantitative measure for the
width of the distribution $p(x_{f})$, we calculate its mean value:

$\mu=x_{t}\pm\frac{\sigma_{z}}{\sqrt{2\pi}}(|a_{+}|+|a_{-}|),$ (10)

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 $p(x_{f})$. It is
determined by the slopes $a_{\pm}$ of the linear fits to $x_{f}=x_{f}(z_{i})$.
Since they are small also 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, inertial transport is much weaker than
axial transport due to Poiseuille flow.

####  4.3.2 Separation of Particles

Figure 11: Maximizing the lateral distance of two particles with different
radii $a/w=0.2$ and $0.3$ using the same control-force protocol
$f_{\mathrm{ctl}}^{*}(t)$. The particles both enter at the same inlet at
$x_{i}$ and travel an axial distance with optimal value $z_{t}=500w$ during
time $T^{*}$. Left: Trajectories of the two particles for both initial
conditions. Inset: Axial separation over time of the two particles. Right:
Optimal control-force protocols for the two initial lateral positions
$x_{i}=0.2w$ and $0.5w$.

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 Fig. 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 Sect. 3.3 we already formulated the
appropriate cost functional for maximimizing the lateral distance between both
particles. In Fig. 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 either at $x_{i}=0.2w$ or
$x_{i}=0.5w$. Again, we assume 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 Fig. 11, left) is
pushed towards the centerline, while the larger particle (dashed lines) moves
towards the channel wall during the second halves 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.

Figure 12: Instantaneous stable fixed points resulting from the applied forces
in Fig. 11(left).

To develop a better understanding for the optimal control-force protocols of
Fig. 11, right, we show in Fig. 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 Fig. 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. Then 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 Fig. 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 reach it. Nevertheless, this
initial behavior causes the minima in the dashed trajectories of Fig. 11,
left. Then 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, since 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 Fig. 11, left).

##  5 Conclusions

We applied concepts from optimal control theory to a setup from inertial
microfluidics and managed to precisely steer single particles from a
microchannel inlet to an outlet using a time-dependent axial force, which
controls the lateral inertial lift force via the Saffman effect. Our results
show that the optimal control force exploits conventional inertial migration
since in the beginning it is zero so that the particle drifts towards its
lateral equilibrium position. Only then the control force is switched on so
that the particle is pushed towards a target position. Due to this property
steering with an optimized control force is cheaper than a strategy where a
constant axial force is used for steering. Additionally, the optimal-control
strategy can be implemented for different channel lengths, which makes this
approach versatile. We also used the optimal control-force protocol for a
single particle to demonstrate that even a pulse of particles spread along the
channel axis can be steered to a target with only a small spread around the
exact target position. Finally, we showed how a single optimized control-force
protocol can separate particles of similar radii $a_{1}=0.2w$ and
$a_{2}=0.3w$. The lateral distances reached for a channel length of $500w$ and
different initial positions are considerably larger than a passive strategy
could achieve.

It would be interesting to explore different channel geometries in the future,
such as rectangular cross sections with different aspect ratio or triangular
cross sections Kim et al. (2016), because they strongly influence the
locations of the fixed points of the inertial lift force. For triangular
microchannels this would require to determine the lift-force profile in the
cross-sectional plane and not just on one axis.

We consider particle steering by an optimal axial control force as an
innovative method for targeting precise positions at the channel outlets,
which will then have implications for particle separation and filtration. We
hope that our work stimulates future efforts towards an experimental
realization. As we outlined above, to realize the axial control force, we
suggest the use of electric fields in combination with electrophoresis, which
has already been applied in experiments on the Saffman effect and inertial
migration Kim and Yoo (2009); Yuan et al. (2016) and also studied in theory
Choudhary et al. (2019). Typically, micron-sized particles can exhibit
electrophoresis and it is also realized for biological cells Korohoda and Wilk
(2008).

Conceptualization, F.R., C.S., and H.S.; methodology, F.R., C.S., and H.S.;
software, F.R, and C.S..; validation, F.R., C.S., and H.S.; formal analysis,
F.R., C.S., and H.S.; investigation, F.R.; resources, F.R., and C.S.; data
curation, F.R., and C.S.; writing–original draft preparation, F.R, and H.S.;
writing–review and editing, F.R.,C.S., and H.S.; visualization, F.R., and
H.S.; supervision, H.S.; project administration, H.S.; funding acquisition,
H.S. All authors have read and agreed to the published version of the
manuscript.

This research was funded by Deutsche Forschungsgemeinschaft through the
priority program SPP 1726 (grant number STA352/11) and through the
collaborative research center SFB 910.

###### Acknowledgements.

We thank F. Tröltzsch and A. Resing for fruitful discussions. The authors
declare no conflict of interest. The funders had no role in the design of the
study; in the collection, analyses, or interpretation of data; in the writing
of the manuscript, or in the decision to publish the results. yes References

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