Sharp-edge-based acoustofluidic chip for programmable pumping, mixing, cell
focusing and trapping†

Alen Pavlic,a‡ Cooper Lars Harshbarger,a,b,c, Luca Rosenthaler,a Jess Gerrit
Snedeker,b,c and Jürg Duala

Precise manipulation of fluids and objects on the micro scale is seldom a
simple task, but nevertheless crucial for many applications in life sciences
and chemical engineering. We present a microfluidic chip fabricated in
silicon-glass, featuring one or several pairs of acoustically excited sharp
edges at side channels that drive a pumping flow throughout the chip and
produce a strong mixing flow in their vicinity. The chip is simultaneously
capable of focusing cells and microparticles that are suspended in the flow.
The multifunctional micropump provides a continuous flow across a wide range
of excitation frequencies
($$80\text{\,}\mathrm{kHz}$-$2\text{\,}\mathrm{MHz}$$), with flow rates
ranging from $\text{\,}\mathrm{nL}\text{\,}{\mathrm{min}}^{-1}$ to
$\text{\,}\mathrm{\SIUnitSymbolMicro L}\text{\,}{\mathrm{min}}^{-1}$,
depending on the excitation parameters. In the low-voltage regime, the flow
rate depends quadratically on the voltage applied to the piezoelectric
transducer, making the pump programmable. The behaviour in the system is
elucidated with finite element method simulations, which are in good agreement
with experimentally observed behaviour. The acoustic radiation force arising
due to a fluidic channel resonance is responsible for the focusing of cells
and microparticles, while the streaming produced by the pair of sharp edges
generates the pumping and the mixing flow. If cell focusing is detrimental for
a certain application, it can also be avoided by exciting the system away from
the resonance frequency of the fluidic channel. The device, with its unique
bundle of functionalities, displays great potential for various bio-chemical
applications.

##

††footnotetext: a Institute for Mechanical Systems, Swiss Federal Institute of
Technology Zurich, Zurich, Switzerland.††footnotetext: b Department of
Orthopedics, Balgrist University Hospital Zurich, University of Zurich,
Zurich, Switzerland.††footnotetext: c Institute for Biomechanics, Swiss
Federal Institute of Technology Zurich, Zurich, Switzerland.††footnotetext: †
Supplemental material available: Experimental videos; details on the numerical
model; admittance analysis; details on measuring the acoustic pressure
amplitude.††footnotetext: ‡ E-mail<EMAIL_ADDRESS>
## Introduction

Biomedical and chemical processes often require small amounts of bio-chemical
substances, including cell and microparticle suspensions, to be moved around
at the micro scale with great precision.1, 2 There are various mechanisms that
have been developed towards this objective, sometimes employing physical
principles of piezoelectricity in combination with a diaphragm,3
electrostatic4 and electromagnetic5 forces, thermal actuation,6 bubble
expansion and collapse,7 electrowetting,8 and acoustics,9, 10, 11, 12, 13, 14,
15 among others.

One particular way of enforcing the fluid motion in microfluidic systems is
through a phenomenon called acoustic streaming.16 This streaming is a steady
fluid motion due to the attenuation of acoustic oscillations near a
boundary17, 18 or in the bulk of the fluid,19 but can also stem from a dynamic
geometric nonlinearity.20 A special form of the boundary-driven streaming is
the streaming near sharp edges, which can be traced back to the acoustic
needle experiments of Hughes and Nyborg 21 in 1962, for the disruption of
cells. Since then, acoustically excited sharp edges have proven to be a
promising technology for streaming-based mixing22, 23, 24, 25, 26, 27, 28 and
pumping9 of fluids, and even for acoustic-radiation-force-based trapping of
particles and cells.29, 30 A series of recent studies revealed a general
streaming flow pattern around a single sharp edge that features an outflowing
vortex at the sharp tip, as shown in Fig. 1(a).31, 28, 24, 22, 23

Fig. 1: Visualization of (a) the typical acoustic streaming flow around a
single sharp edge that oscillates relative to the liquid, and of (b) the
pumping mechanism driven by a pair of sharp edges at a side channel.

In addition, the acoustic radiation force (ARF) is used for particle and cell
trapping and focusing,32 as well as for the separation of particles, cells,
and droplets.33, 34 A suitable ARF for such applications can be achieved in
different systems, one of which is the bulk acoustic wave (BAW) device.35 When
a BAW device is excited, the acoustic waves propagate throughout the whole
device, forming standing waves when the frequency of excitation corresponds to
a certain resonance mode of the fluidic channel or the whole device.36, 37

Devices featuring such acoustic phenomena are recognized, for example, as a
superior alternative to passive microreactors for chemical engineering.38
However, one of the main challenges for the improvement of the technology was
defined as constructing the acoustic microreactors from materials other than
polydimethylsiloxane (PDMS) and lithium niobate, which would reduce the cost
and production complexity while increasing the resistance to extreme reaction
conditions.

Here, we present a silicon-glass chip which is driven by a pair of
acoustically excited sharp edges and is simultaneously capable of fluid
pumping and mixing, and of focusing and trapping of cells and microparticles
in a suspension. The robust device provides a continuous flow across a wide
range of excitation frequencies
($$80\text{\,}\mathrm{kHz}$-$2\text{\,}\mathrm{MHz}$$), and programmable flow
rates in the $\text{\,}\mathrm{nL}\text{\,}{\mathrm{min}}^{-1}$ to
$\text{\,}\mathrm{\SIUnitSymbolMicro L}\text{\,}{\mathrm{min}}^{-1}$ range.
The novel pumping mechanism featured in our device is based on a pair of sharp
edges located at a side channel, pumping the liquid out of the side channel,
as depicted in Fig. 1(b). Furthermore, in contrast to a low-frequency sharp-
edge-based pump made of polydimethylsiloxane (PDMS) by Huang _et al._ 9, where
the solid edges oscillate and drive the flow, our device operates at higher
frequencies that support resonances in the solid-liquid system. The resonances
can boost the relative oscillatory velocity between sharp edges and the
liquid, which is known to drive the streaming,28 and consequently the pumping.
The behaviour in the experimental system is explained through finite element
method simulations. It is shown that the streaming around sharp edges drives
the pumping and the mixing, while using a specific frequency can excite a
standing wave in a part of the microfluidic device, leading to a simultaneous
ARF-based focusing or trapping of cells and microparticles. Exciting the
system at an off-resonance frequency of the fluidic channel, allows for the
suppression of the focusing, thus allowing for the particles to remain
dispersed in the pumping flow if desired.

The silicon-glass composition makes our device chemically resistant, solving
one of the challenges of acoustic microreactors for chemical engineering.38
Furthermore, our device operates in a relatively broad frequency range, from
$80\text{\,}\mathrm{kHz}$ to $2\text{\,}\mathrm{MHz}$, enabling the control of
acoustic cavitation, which is interesting for biomedical applications such as
cell lysis.39 The additional functionalities of our device —mixing, focusing,
and trapping —make it attractive for various applications, such as boosting
the sensitivity of bead-based immunoassays or bioassays,40 trapping cells for
medium exchange,41 or improving cell isolation.15

## Theoretical background

The motion of a viscous fluid is governed by the compressible Navier-Stokes
equations

$\rho\left[\frac{\partial\bm{v}}{\partial
t}+(\bm{v}\bm{\cdot}\bm{\nabla})\bm{v}\right]=-\bm{\nabla}p+\eta\nabla^{2}\bm{v}+\left(\eta_{\mathrm{B}}+\frac{\eta}{3}\right)\bm{\nabla}\left(\bm{\nabla}\bm{\cdot}\bm{v}\right),$
(1)

and the continuity equation

$\frac{\partial\rho}{\partial
t}=-\bm{\nabla}\bm{\cdot}\left(\rho\bm{v}\right),$ (2)

with the dynamic viscosity $\eta$ and the bulk viscosity $\eta_{\mathrm{B}}$.
The density $\rho$ is assumed to only be a function of the pressure $p$,
namely

$\rho=\rho(p).$ (3)

The equations are linearized using the regular perturbation approach.42
Accordingly, the physical fields are expanded in a series,
$\square=\square_{0}+\square_{1}+\square_{2}+\dots$, with $\square$
representing the field and the subscript representing the respective order.
The first- and second-order problems, for computing the acoustic fields and
streaming fields, respectively, are described in detail in the supplemental
material.† The second-order problem becomes relevant only when time scales
longer than the period of oscillation $T=1/f$ are considered. Therefore, the
time average $\langle\square_{2}\rangle=1/T\int_{T}\square_{2}\,\mathrm{d}t$
is applied to the second order variables and terms.

### Acoustic radiation force (ARF)

When an acoustic wave inside an acoustofluidic device scatters at an object
such as a microparticle or a cell, it gives rise to the acoustic radiation
force (ARF). The ARF results from the interactions between the scattered and
the background acoustic waves. When an inviscid fluid and a spherical object
can be assumed, which is often the case, the ARF can be approximated as the
negative gradient of the Gor’kov potential,43 namely

$\bm{F}_{\mathrm{rad}}=-\bm{\nabla}U,$ (4)

where the Gor’kov potential can be written as

$U=\frac{4}{3}\pi a^{3}\left(\frac{1}{2}\frac{f_{1}}{c_{0}^{2}\rho_{0}}\langle
p_{1}^{\mathrm{bg}}p_{1}^{\mathrm{bg}}\rangle-\frac{3}{4}\rho_{0}f_{2}\langle\bm{v}_{1}^{\mathrm{bg}}\bm{\cdot}\bm{v}_{1}^{\mathrm{bg}}\rangle\right),$
(5)

with the object’s radius being $a$, the speed of sound in the fluid $c_{0}$,
the equilibrium density $\rho_{0}$, the background acoustic pressure
$p_{1}^{\mathrm{bg}}$, the background acoustic velocity
$\bm{v}_{1}^{\mathrm{bg}}$, and the monopole and the dipole scattering
coefficients $f_{1}$ and $f_{2}$, respectively. This formulation is applicable
under the assumption that the object is small compared to the acoustic
wavelength and far away from the walls. However, a recent study44 showed that
the Gor’kov potential is in most cases valid even in close proximity to a
wall.

When considering a one-dimensional plane standing wave along a $z$-axis, the
ARF from eq. 4 further simplifies to

$\bm{F}_{\mathrm{rad}}^{\mathrm{1D}}=4\pi a^{3}\Phi
kE_{\mathrm{ac}}\sin\left(2kz\right)\bm{e}_{z},$ (6)

with the acoustic energy density
$E_{\mathrm{ac}}=\frac{p_{\mathrm{a}}^{2}}{4\rho_{0}c_{0}^{2}}$, the pressure
amplitude $p_{\mathrm{a}}$, the ideal wave number $k=\frac{\omega}{c_{0}}$,
and the unit vector $\bm{e}_{z}$ oriented along the $z$-axis. Equation (6) is
attributed to Yosioka and Kawasima 45. The acoustic properties of the object
relative to the fluid are condensed into the acoustic contrast factor

$\Phi=\frac{1}{3}f_{1}+\frac{1}{2}f_{2}=\frac{1}{3}\left[\frac{5\tilde{\rho}-2}{2\tilde{\rho}+1}-\tilde{\kappa}\right],$
(7)

in which the relative compressibility
$\tilde{\kappa}=\frac{\kappa_{\mathrm{obj}}}{\kappa_{0}}$ and density
$\tilde{\rho}=\frac{\rho_{\mathrm{obj}}}{\rho_{0}}$ reflect the ratios between
the properties of the object $\square_{\mathrm{obj}}$ and the fluid
$\square_{0}$.

Fig. 2: (a) Experimental device, featuring a silicon-glass chip, piezoelectric
transducer (PZT), holder from acrylic glass (PMMA), and connections made out
of fused silica capillaries. (b) Three different device designs that are
featured in the manuscript next to a one Swiss franc coin for scale. (c) The
schematic cross-section of a device, explaining individual layers. The size of
an individual silicon-glass chip without supporting elements is $1\times
12\times$24\text{\,}\mathrm{mm}$$. Fig. 3: (a) Geometry and mesh used in the
numerical simulations, with the description of geometrical parameters and
boundary conditions applied in the numerical model. The blue region
corresponds to the modeled fluid domain, whereas the grey area represents the
solid structure of the device, which we assume to be rigid. (b) Detailed view
of the geometry of sharp edges, with a refined mesh at the tips of sharp edges
and at boundaries. The parameters that define the geometry of the sharp edge
pair are the height of the sharp edges $h$, the angle of each sharp edge
$\alpha$, and the rounding radius of the edges $r$. (c) Region of interest on
an actual device that corresponds to the numerical model. The size of the chip
in (c) is $12\times$24\text{\,}\mathrm{mm}$$.

## Materials and methods

### Devices and fabrication

The lab-on-a-chip devices shown in Fig. 2 were produced through cleanroom
processing and feature a two-layered structure made of silicon and glass. The
channel designs were first patterned on a silicon wafer
($500\pm$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ thickness) through
photolithography (resist: S1828/S1818, Shipley, $4000\,\mathrm{rpm}$;
developer: AZ351B, Microchemicals) and then etched with an inductively coupled
plasma deep reactive ion etching (ICP-DRIE) machine (Estrellas, Oxford
instruments) to a depth of $\sim$184\text{\,}\mathrm{\SIUnitSymbolMicro m}$$.

Afterwards, a glass wafer ($500\text{\,}\mathrm{\SIUnitSymbolMicro m}$
thickness) was anodically bonded onto the silicon wafer. The wafer was then
diced into individual $12\times$24\text{\,}\mathrm{mm}$$ chips with a wafer
saw (DAD3221, Disco corporation). Fused silica capillaries
($164\pm$6\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ outer diameter,
$100\pm$6\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ inner diameter, Molex) were
inserted into the inlets and outlets of the chips and fixed with a two-
component glue (5 Minute Epoxy, Devcon). A piezoelectric transducer (PZT)
(length$\times$width$\times$thickness $=$ $10\times
2\times$1\text{\,}\mathrm{mm}$$ or $16.75\times
8.74\times$17.4\text{\,}\mathrm{mm}$$, Pz26, Meggitt Ferroperm) was glued to
the backside of each device with a conductive epoxy (H20E, EPO-TEK). Copper
cables ($0.15\text{\,}\mathrm{mm}$ diameter) were connected to the PZT using a
conductive silver paste and glued to a device with instant glue for mechanical
stability.

To fix the devices under the microscope, a chip-holder was designed and laser-
cut from acrylic glass (PMMA), with top and bottom parts being held together
by two bolts, as depicted in Fig. 2(a).

The devices were designed to have a $1\text{\,}\mathrm{mm}$ wide main channel
across the length of the device, to which one or two loops of side channels
are connected, as visible in Fig. 2(b). An example of a single side channel
loop device is shown in Fig. 3. The default width of side channels is
$500\text{\,}\mathrm{\SIUnitSymbolMicro m}$, except for side channel segments
that start with a pair of sharp edges, for which the width varies between
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
$500\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The width of this side channel
segment is equal to the spacing between the pair of sharp edges, and is
specified for each device alongside the corresponding results. All sharp edges
in the study feature a $10\text{\,}\mathrm{\SIUnitSymbolDegree}$ apex angle,
and are oriented such that one of the sides of the edge is aligned with a wall
of the adjacent side channel. Measurements of the geometry of devices at the
end of the production revealed the rounding radius of the sharp edges of
$\sim$0.9\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, the apex angle of a sharp
edge of $\sim$10.3\text{\,}\mathrm{\SIUnitSymbolDegree}$$, the length of the
protrusion of sharp edges into the main channel of
$\sim$275\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, and the deviation of
channel widths in the order of $<$1\text{\,}\mathrm{\char 37\relax}$$.

### Experimental setup

For visual observation of phenomena within the channels of the lab-on-a-chip
devices we used a microscope (Axioscope, Zeiss) with a blue LED (COP4-A,
Thorlabs) and a high-speed camera (HiSpec1 2G Mono, Fastec Imaging). To
visualize the flow field and demonstrate the acoustic forces, yeast cells
(Coop Supermarket) and polystyrene (PS) beads
($5.19\pm$0.14\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, microParticles GmbH)
were used. Pressure pumps (Flow EZ, Fluigent) were connected to the inlet
capillaries of a device, for supplying the particles/cells dispersed in water,
as well as for re-dispersing the particles/cells within the chip in-between
experiments. A function generator (Model DS345, Stanford Research Systems) and
an amplifier (HSA 4101, NF Corporation) were used to drive the PZT, and
monitored through an oscilloscope (Model 9410, LeCroy).

Devices were characterized by measuring the admittance of the PZT with an
impedance analyzer (Sine Phase Z-Check 16777k).†

The pumping velocity was estimated through manual particle tracking
velocimetry (Fiji46) of the PS particles. The flow rate was computed by
assuming that the flow profile corresponds to the Poiseuille flow through a
rectangular cross-section,47 and that the velocity measured in the middle of
the channel corresponds to the maximal velocity of the flow profile. The
trajectories of PS particles that were used for estimation of the acoustic
pressure amplitude ($p_{\mathrm{a}}$) were obtained through a particle
tracking velocimetry plugin TrackMate48 in Fiji,46 and post-processed with a
custom Matlab49 code (described in the supplemental material†).

The time-averaged paths of PS particles through the system were obtained by
applying the variance mode to a stack of several video frames in Adobe
Photoshop.

### Numerical model

The numerical model resembling the region of interest (ROI) of one of the
devices, as indicated in Fig. 3(c), was based on a finite element method
framework of COMSOL Multiphysics v. 5.6.50 The geometry with all the relevant
dimensions, and the applied boundary conditions are shown in Fig. 3(a),(b). We
only modeled the fluid domain in 2D, represented by the blue colored region in
Fig. 3(a),(b), taking the walls as rigid boundaries. The material properties
used for the modeling are given in Table 1.

| H2O | Unit |  | PS | Unit
---|---|---|---|---|---
$\rho_{0}$ | $1000$ | $\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}$ | $\rho_{\mathrm{obj}}$ | $1050$ | $\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{-3}$
$c_{0}$ | $1481$ | $\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ | $c_{\mathrm{obj}}^{\mathrm{P}}$ | $2400$ | $\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$
$\eta$ | $1.002$ | $\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$ | $c_{\mathrm{obj}}^{\mathrm{S}}$ | $1150$ | $\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$
$\eta_{\mathrm{B}}$ | $3.09$ | $\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$ | $\Phi$ | $0.175$ | -
Table 1: The material parameters for water (H2O) and polystyrene51 (PS). The
solid elastic polystyrene is defined through the speed of primary and
secondary waves, $c_{\mathrm{obj}}^{\mathrm{P}}$ and
$c_{\mathrm{obj}}^{\mathrm{S}}$, respectively. The acoustic contrast factor
$\Phi$ is computed using
$\kappa_{\mathrm{obj}}=1/(\rho_{\mathrm{P}}[(c_{\mathrm{obj}}^{\mathrm{P}})^{2}-\frac{4}{3}(c_{\mathrm{obj}}^{\mathrm{S}})^{2}])$.

The perturbation approach presented in the Theoretical background section and
in the supplemental material† was used for the numerical modeling, similarly
to previous studies.52 The fluid domain is excited through boundary conditions
specified in Fig. 3(a), by applying a zero acoustic pressure at the left-hand
side boundary of the main channel, and an oscillatory velocity with an
amplitude of $v_{\mathrm{a}}$ along the $x$-direction at the right-hand side
boundary of the main channel. The rounding of the sharp edges is defined
through the viscous boundary layer as $r=\delta/10$, which ensures that the
resulting streaming is independent of $r$, as demonstrated by Zhang _et al._
24. The detailed description of the numerical model is given in the
supplemental material.†

## Results

The investigated device features various functionalities that are graphically
summarized in Fig. 4. The two main being the pumping flow due to a single pair
of sharp edges and the focusing of cells and microparticles due to the ARF in
a standing acoustic wave. Other functionalities that are demonstrated in the
manuscript are: trapping of cells with the localized pressure nodes in the
side channels, and fluid mixing due to the strong acoustic streaming field
near the two sharp edges. The results that demonstrate and characterize these
functionalities are further discussed in the following sections.

Fig. 4: Overview of the four main functionalities of the device. (1.) Fluid
pumping, (2.) cell/microparticle focusing, (3.) fluid mixing, and (4.)
cell/microparticle trapping.

### Pumping and mixing

Fig. 5: (a) Time-averaged paths of $5.19\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ polystyrene particles in the experimental device that is excited at
$f=$802\text{\,}\mathrm{kHz}$$ with $22\text{\,}\mathrm{V}_{pp}$. The
particles reveal the mixing around the two sharp edges, the focusing in the
central region of the main channel, and the pumping through the main and side
channel. Dimensions in (a) and (b) correspond to the measurements on the
experimental device, yielding the spacing between the two edges of
$181.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$. (c) Numerically, we found a
matching streaming field and an acoustic pressure field that yields the
pressure node at the middle of the main channel, at
$f=$746\text{\,}\mathrm{kHz}$$. In the model, the spacing between the two
edges was adjusted to $180\text{\,}\mathrm{\SIUnitSymbolMicro m}$, to match it
to the spacing of the experimental device from (a). To avoid influence of
effects near the ends of the main channel, the numerical model extends past
the region that is shown here that matches the region from (a), observed in
experiments. (d) The Gor’kov potential and the acoustic radiation force
(arrows) further justify the focusing of the polystyrene particles at the
pressure node. The time-averaged paths of PS particles through the system were
obtained by applying the variance mode to a stack of several video frames in
Adobe Photoshop. The direction of the experimentally observed flow near the
sharp edges in (a) matches the direction of the numerically-predicted
streaming in (c).  Fig. 6: Velocity of the pumping flow in the middle of the
side channel parallel to the main channel (equivalent to the evaluation point
$P$ in Fig. 3), in dependence of the applied voltage squared. The device
corresponds to the one in Fig. 5, as well as the frequency, at
$f=$802\text{\,}\mathrm{kHz}$$. The graph is split into regions A:
demonstrating a linear relationship between the streaming velocity and applied
voltage squared, B: demonstrating the same linear relationship as in A but
with a decreased slope, and C: a region where the streaming velocity cannot be
increased by increasing the applied voltage squared, indicating a dominant
influence of an unknown nonlinear physical mechanism. Fig. 7: Time-averaged
paths of $5.19\text{\,}\mathrm{\SIUnitSymbolMicro m}$ polystyrene particles in
devices that feature two side channel loops with four pairs of sharp edges,
with spacing between the two edges of $100\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ or $500\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The width of the main
channel of the two devices is $1\text{\,}\mathrm{mm}$. (a) Pairs of sharp
edges with equal spacing between the two edges are opposing each other
($100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ on the left and
$500\text{\,}\mathrm{\SIUnitSymbolMicro m}$ on the right). The device was
excited at $f=$1.371\text{\,}\mathrm{MHz}$$ with
$V_{\mathrm{pp}}=$30\text{\,}\mathrm{V}$$. (b) Pairs of different spacing
between the two edges are opposing each other
($100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ bottom left and top right,
$500\text{\,}\mathrm{\SIUnitSymbolMicro m}$ top left and bottom right). The
device was excited at $f=$780\text{\,}\mathrm{kHz}$$ with
$V_{\mathrm{pp}}=$6\text{\,}\mathrm{V}$$. The arrows show the direction of the
pumping flow, while the time-averaged paths of polystyrene particles reveal
regions of focusing in (a) and a uniform flow in (b). The weak flow in the
main channel in (b) is a result of a slight imbalance between the pumping
strength of equally spaced sharp edge pairs, which is likely due to the
spatial variability of the oscillatory velocity of the sharp edges relative to
the liquid. The time-averaged paths of PS particles through the system were
obtained by applying the variance mode to a stack of several video frames in
Adobe Photoshop.

The pumping flow is demonstrated in Fig. 5(a) through the time-averaged paths
of $5.19\text{\,}\mathrm{\SIUnitSymbolMicro m}$ diameter PS particles in
water. Near the two sharp edges, the particles follow the streamlines of the
vortices that are typical around sharp edges,28 and that are usable for
mixing25, 26, 27 at the micro scale. The pumping flow in the counterclockwise
direction, which results from the streaming around the two sharp edges,9 moves
the particles from the mixing to the focusing region, and then back to the
mixing region by pumping the particles through the side channel. The sharp
edge angles of $\sim$10\text{\,}\mathrm{\SIUnitSymbolDegree}$$ are employed to
maximize the streaming velocity, since it has been shown by Zhang _et al._ 24
and by Doinikov _et al._ 28 that the sharper the angle, the higher the
streaming velocity at a fixed amplitude of the first-order excitation
velocity. The video corresponding to Fig. 5(a) is available in the
supplemental material (video_1_fig_5).†

The behaviour corresponding to the device from Fig. 5(a) is analyzed with the
help of the numerical model. The resulting acoustic pressure field is shown
through colors in Fig. 5(c), indicating that the pressure nodal plane appears
near the middle of the main channel, at the frequency of
$746\text{\,}\mathrm{kHz}$. This frequency is lower than in an experimental
setting (Fig. 5a) due to simplifying the problem to two dimensions, and
assuming rigid walls, both of which can influence the resonance frequency. The
Gor’kov potential (color) and the ARF (arrows) in the focusing region are
shown in Fig. 5(d), and confirm that the PS particles are being forced to the
middle of the channel. Furthermore, the simulation of the Eulerian acoustic
streaming, the streamlines of which overlay the pressure field in Fig. 5(c),
indicates the pumping flow as well as the mixing flow close to the two edges.
Qualitatively, our simplified two-dimensional numerical rigid-wall model that
is based on the regular perturbation technique42, 28 captures the observed
pumping and mixing phenomena very well. However, the approach and the solution
for the streaming velocity might be questionable in the proximity of the sharp
tips, due to high velocities that can develop at those nearly-singular
points.31 In order to improve the accuracy in that region, more sophisticated
techniques would have to be applied, such as the direct numerical simulations
(DNS)24, 31 or the analytical approaches suitable for the so-called fast
streaming.53 It is important to note that the region of concern near the sharp
tips has a characteristic length in the order of the viscous boundary layer
thickness
$\delta=\sqrt{\eta/\pi\rho_{0}f}=$0.63\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
at $f=$802\text{\,}\mathrm{kHz}$$ in water.

In the device from Fig. 5, we measured the flow rate in dependence of the
applied voltage squared, at $f=$802\text{\,}\mathrm{kHz}$$, as shown in Fig.
6. Specifically, we measured the velocity of the
$5.19\text{\,}\mathrm{\SIUnitSymbolMicro m}$ PS particles in the pumping flow
in the middle of the side channel that is parallel to the main channel, and
transformed this velocity to the flow rate of a Poiseuille flow through the
rectangular cross section of the channel.† At the applied voltages of below
$V_{\mathrm{pp}}=$10\text{\,}\mathrm{V}$$, the good linear fit in the region
“A” of Fig. 6 corresponds to the expected linear relation between the
streaming velocity and the applied voltage squared.† However, between
$V_{\mathrm{pp}}=$10\text{\,}\mathrm{V}$$ and
$V_{\mathrm{pp}}=$22\text{\,}\mathrm{V}$$, in the region “B”, the slope of the
fitted linear curve is steeper, indicating a relative boost of the streaming
velocity and a nonlinear dependence of the streaming velocity on the applied
voltage squared. Interestingly, this is the opposite behaviour to that
commonly observed in the streaming around sharp edges; for a single sharp
edge, the slope of the increase in the streaming velocity with the increase of
the applied voltage squared, would start decreasing after a certain threshold
amplitude is reached.31, 24 There is another regime, at above
$V_{\mathrm{pp}}=$22\text{\,}\mathrm{V}$$ \- the region “C” in Fig. 6, where
the streaming velocity is more or less independent of the applied voltage. The
observed nonlinear behaviour could indicate various physical mechanisms at
play, such as the fluid turbulence, the maximal particle velocity threshold in
the solid,54 or nonlinearities associated with the piezoelectric transducer
and the conductive epoxy layer connecting the transducer to the device.

The source of the deviation from the linear relation between the streaming
velocity magnitude and the applied voltage squared, in case of the sharp edge
streaming, is not generally understood. As pointed out by Ovchinnikov _et al._
31, the nonlinearity could come from the increased influence of the third-
order inertial term
$\left(\left<\bm{v}_{2}\right>\bm{\cdot}\bm{\nabla}\right)\bm{v}_{1}$ that is
neglected in the second-order momentum conservation equation (eq. (S4) in the
supplemental material†); or from the interference of the streaming vortices
with the surrounding geometry, according to Zhang _et al._ 24. Furthermore,
the rounding radius of the sharp edges $r$ was measured to be
$\sim$0.9\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, which is comparable to the
viscous boundary layer thickness in the investigated range
($\delta\approx$0.56\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ at
$f=$1\text{\,}\mathrm{MHz}$$ in water), and could be affecting the streaming
flow. Zhang _et al._ 24 showed that the streaming velocity around a sharp edge
decreases as $r$ increases, at $r>\delta$.

The flow rate in our devices reaches up to
$4.1\text{\,}\mathrm{\SIUnitSymbolMicro L}\text{\,}{\mathrm{min}}^{-1}$ at
$V_{\mathrm{pp}}=$28\text{\,}\mathrm{V}$$ and $f=$802\text{\,}\mathrm{kHz}$$,
which corresponds to the pumping pressure of $\sim$12\text{\,}\mathrm{Pa}$$,
approximated through the Hagen-Poiseuille law47, considering the whole side
channel loop. This is in the same order as the maximal flow rate and the
pumping pressure generated in the previously reported PDMS acoustic sharp-edge
micropump at $V_{\mathrm{pp}}=$50\text{\,}\mathrm{V}$$ and
$f=$6.5\text{\,}\mathrm{kHz}$$, namely
$\sim$8\text{\,}\mathrm{\SIUnitSymbolMicro L}\text{\,}{\mathrm{min}}^{-1}$$
and $\sim$76\text{\,}\mathrm{Pa}$$; however, we use a single pair of sharp
edges, whereas the PDMS device features twenty sharp edges.

To demonstrate a wide range of operating frequencies that can be used to
operate our micropumping mechanism, we used a larger PZT ($16.75\times
8.74\times$17.4\text{\,}\mathrm{mm}$$) that has a resonance frequency in the
thickness direction at $\sim$80\text{\,}\mathrm{kHz}$$. The maximal flow rate
observed at $f=$78.26\text{\,}\mathrm{kHz}$$, for the same voltage range as in
Fig. 6, was $4.46\text{\,}\mathrm{\SIUnitSymbolMicro
L}\text{\,}{\mathrm{min}}^{-1}$ at $V_{\mathrm{pp}}=$35\text{\,}\mathrm{V}$$.
The corresponding video, showing the fluid pumping at
$f=$78.26\text{\,}\mathrm{kHz}$$ without the cell/microparticle focusing, is
provided in the supplemental material† (video_2).

Since the rate of the sharp edge pumping flow is controlled independently of
the general flow through the whole device, which we control with the external
pressure pump, we can fine-tune the flow rate in any of the side channel
loops, and even bring it to stagnation, as shown in a video (video_3) in the
supplemental material.† This could potentially help increase the speed of
operations like medium exchange or addition of a new sample, as the reduced
flow in the side channel, while the PZT is excited, could impede the flushing
of the trapped cells due to shear forces. The novelty of the device therefore
lies in, but is not limited to, the programmable multifunctionality of the
device, where each feature can either be used, in an arbitrary order,
individually or simultaneously.

We also observed a dependence of the pumping flow on the spacing between the
two edges. The pumping flows in the devices with two side channel loops in
Fig. 7, with two pairs of sharp edges per loop, indicate that the narrower
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ spacing between the sharp edges
generates a stronger pumping flow than the
$500\text{\,}\mathrm{\SIUnitSymbolMicro m}$ spacing, when other conditions
(e.g. $V_{\mathrm{pp}}$, $f$, PZT placement, glue layer thickness, etc.) are
kept constant. For each loop in Fig. 7, one of the sharp-edge pairs has a
spacing of $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$, while the spacing is
$500\text{\,}\mathrm{\SIUnitSymbolMicro m}$ for the other. Both ends of a side
channel loop should facilitate the pumping, as was verified through two
additional devices featuring a single side channel loop (same design as in
Fig. 5a), with the spacing between the two edges of
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
$500\text{\,}\mathrm{\SIUnitSymbolMicro m}$, respectively. However, based on
the observed flow directions (Fig. 7), the pumping from the sharp-edge pair
with the narrower spacing prevails.

Experimental videos showcasing phenomena from Fig. 7 for PS particles
(video_4_fig_7a and video_5_fig_7b) and for yeast cells (video_6 and video_7)
are available as part of the supplemental material.†

The streaming vortices around the sharp edges that are analogous to ours, and
that are responsible for the fluid mixing, have been investigated
extensively,28, 22, 23, 24, 31 which is why we omit a detailed analysis on the
subject.

### Focusing and trapping of cells and microparticles

Objects of positive acoustic contrast ($\Phi>0$) are, in accordance with eq.
(6), forced towards pressure nodes of a one-dimensional standing wave. This
links the PS particles being focused towards the middle of the main channel in
Fig. 5(a), and the simulated acoustic pressure in Fig. 5(c) indicating the
pressure nodal line in the focusing region. The Gor’kov potential (color) and
the ARF (arrows) in Fig. 5(d), further confirm that the PS particles are
forced towards the middle of the main channel.

The hard silicon walls of our devices provide a strong acoustic impedance
mismatch with respect to water, $19.79\cdot
10^{6}\,$\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{-2}\text{\,}{\mathrm{s}}^{-1}$$
compared to $1.49\cdot
10^{6}\,$\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{-2}\text{\,}{\mathrm{s}}^{-1}$$,
leading to the acoustically hard wall boundary condition for water at the
solid-fluid interface ($\bm{v}_{1}\approx\bm{0}$ at the interface) being a
good model.55 This can result in a quasi one-dimensional standing wave between
the walls of the main channel, featuring a $n\lambda/2$ ultrasound resonance
mode, where $n=1,2,3,...$ represents an individual resonance mode and
$\lambda=c_{0}/f$ the acoustic wavelength.42 For a fixed channel width, the
resonance mode that is excited depends on the frequency of excitation:
$\lambda/2$ mode, with its single pressure nodal plane, is visible at
$f=$802\text{\,}\mathrm{kHz}$$ in the $1\text{\,}\mathrm{mm}$ wide main
channel in Fig. 5(a), and at $f=$1.371\text{\,}\mathrm{MHz}$$ in the
$0.5\text{\,}\mathrm{mm}$ wide side channels in Fig. 7(a). The $\lambda$
resonance mode, featuring two pressure nodal lines, was observed at
$f=$1.371\text{\,}\mathrm{MHz}$$ in the $1\text{\,}\mathrm{mm}$ wide main
channel in Fig. 7(a).

Fig. 8: Trapping and focusing of yeast cells in water, demonstrated using a
device featuring a $260\text{\,}\mathrm{\SIUnitSymbolMicro m}$ spacing between
the two sharp edges, and is otherwise equal to the device from Fig. 5, with
$1\text{\,}\mathrm{mm}$ wide main channel. The image is a mosaic of several
snapshots of channel segments, taken while the device was being excited at
$f=$714\text{\,}\mathrm{kHz}$$ with
$V_{\mathrm{RMS}}=$30\text{\,}\mathrm{V}$$. The dark spots in the channel are
the trapped clusters of yeast. The direction of the pumping flow is indicated
by the arrow. The illustrated quasi-one-dimensional pressure fields along the
side channels indicate the corresponding $n=4$ and $n=8$ resonance modes that
focus and trap the yeast cells.

The acoustic pressure amplitude was measured in the region of the main channel
outside the loop, where the mixing and pumping streaming flows were not
evident (see the supplemental material† for details). The resulting average
pressure amplitude $p_{\mathrm{a}}=$1.2\text{\,}\mathrm{MPa}$$ corresponds to
$E_{\mathrm{ac}}=$160\text{\,}\mathrm{J}\text{\,}{\mathrm{m}}^{-3}$$, which is
in the typical range for silicon-glass chips,56 considering the
$V_{\mathrm{pp}}=$22\text{\,}\mathrm{V}$$ excitation.

In addition to the one-dimensional standing waves perpendicular to the channel
mentioned so far, other resonance modes can be excited inside a device,
yielding interesting trapping regions. Figure 8 shows clusters of trapped
yeast cells in the side channel loop, showcasing the $n=4$ resonance mode in
the two channels perpendicular to the main channel and the $n=8$ mode along
the channel parallel to the main channel, while also yielding the $n=1$ mode
across the main channel, with a single pressure nodal line. The trapped cells
in Fig. 8 are disconnected across the channel width in the side channel loop
due to the pumping flow that features the highest velocity in the middle of
the channel cross-section and obstructs the trapping.

Such trapping capabilities, in combination with the flow control through the
side channel, could improve the efficiency of the cell isolation process15 or
the medium exchange.41 The acoustic radiation force that is the mechanism
behind the trapping and focusing in our device, scales with the volume of a
cell, whereas the drag force from the acoustic streaming scales with the
radius of the cell. Consequently, fine-tuning the flow rate through the side
channel loop and the strength of the trapping, by modifying the voltage
applied to the PZT and the external pressure difference applied to the main
channel, could lead to size-selective separation of cells, with a programmable
size threshold.

## Conclusions

The presented microfluidic chip offers a platform that is especially relevant
for applications where multiple functions are needed within the same chip,
particularly pumping, mixing, and focusing or trapping of cells and
microparticles. We demonstrated a continuous pumping in the range of
excitation frequencies from $80\text{\,}\mathrm{kHz}$ to above
$1\text{\,}\mathrm{MHz}$, with the corresponding programmable flow rates of up
to $\sim$4\text{\,}\mathrm{\SIUnitSymbolMicro L}\text{\,}{\mathrm{min}}^{-1}$$
($\sim$12\text{\,}\mathrm{Pa}$$ pumping pressure). The pumping is always
paired with the mixing vortices near the sharp edges that drive the pumping
flow. We also demonstrated the focusing and trapping capabilities of our
device by predicting and observing the behavior of polystyrene microparticles
and yeast cells in combination with different resonance modes of the fluidic
channels.

Our silicon-glass device is chemically resistant, solving one of the main
problems of acoustic microreactors for chemical engineering.38 The pumping and
strong mixing at the sharp edges could provide a tool for cell disruption,
similar to what Hughes and Nyborg 21 demonstrated, but with the ability to
disrupt larger samples in a very controlled manner. The unique precisely
programmable multifunctionality that the device can provide, could also be
beneficial in studying the dynamics of active matter, such as bacteria57 or
Janus particles.58 Furthermore, it has been previously demonstrated that
imposing a standing acoustic wave can significantly improve the sensitivity of
particle-based immunoassays.40 The combination of mixing, focusing, and
pumping that our device provides, could lead to an even higher sensitivity of
these methods.

In the future, the device design will be further optimized based on the needs
of specific applications. The process will be aided by the presented numerical
model that is capable of predicting the acoustofluidic phenomena inside such
lab-on-a-chip devices. Furthermore, the thin film bulk acoustic wave (TFBAW)
technology59 could be applied to mitigate the variability in production, and
to potentially improve the efficiency of the devices. This would also allow
for a more in-depth experimental analysis of the influence of various
geometrical and excitation parameters on the observed phenomena, further
aiding the optimization of the device design.

## Conflicts of interest

There are no conflicts to declare.

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