Paper Lab on a Chip
samples including blood,20–23 raw milk,24 circulating tumor
cells,25,26 and yeast.27
For bulk acoustic waves (BAW) microchannel acoustophoresis
is usually carried out in the 1–10 MHz frequency
range and particles are focused along a single dimension. For
larger particles, the acoustically induced particle motion is
dominated by the primary acoustic radiation force, whereas
the motion of smaller particles is instead dominated by the
acoustic streaming-induced drag force of the suspending
liquid.28,29 Attempts have been made to address the need for
bacterial or other sub-micrometer particle manipulation and
enrichment in acoustic standing-wave systems. Bacteria have
been processed with some success in batch mode using ultrasound
to agglomerate them,30,31 and a quarter-wavelength
acoustic device was used to concentrate 1 μm particles in
continuous flow.32 However, no systems have yet emerged
that enable continuous flow-based focusing of bacteria or
other sub-micrometer particles at recovery rates above 90%,
relevant when handling highly dilute suspensions.
This paper presents continuous flow-based sub-micrometer
particle focusing using two-dimensional BAW-acoustophoresis.
The use of two-dimensional focusing has previously only been
explored for particles larger than 5 μm in diameter.25,26,33–35
In contrast to the case with one-dimensional standing acoustic
waves, the simultaneous excitation of two orthogonal resonances
generates an acoustic streaming velocity field that
does not counteract the primary radiation force. A numerical
model that predicts a streaming field with essentially a single
large vortex centered in the cross section of the channel, in
agreement with experimental data, is also presented.
Theory
Particles in a standing-wave acoustophoresis system are primarily
affected by two forces: the acoustic radiation force
from scattering of the acoustic wave on the particles, and the
drag force from the acoustic streaming velocity field of the
fluid generated by viscous stresses in the acoustic boundary
layers. The interplay between these two forces and the
regimes in which they each dominate the particle motion in
acoustophoresis systems have been studied extensively by
Barnkob et al.29 Through theoretical derivation and experimental
verification, these authors have described how the
motion of large particles is dominated by the acoustic radiation
force while the motion of small particles is dominated
by the drag force from the acoustic streaming.
To theoretically determine the critical particle diameter
2ac, where the crossover from radiation force-dominated
particle motion to acoustic streaming-induced drag forcedominated
particle motion occurs, the magnitudes of the two
forces are equated, resulting in the following equation valid
for single-particle motion in a half-wavelength resonance:29
2a 12s v 1 6
m,
c
f .
(1)
where s is a factor related to the channel geometry, v is the
kinematic viscosity of the medium, Φ is the acoustic contrast
factor, and f is the frequency of the acoustic field. The
numerical value is calculated for a polystyrene particle in
water and a frequency of f = 3.19 MHz. The geometrical value
used is s = 0.47 for a particle near the top or bottom walls,
and includes thermal effects.29 The critical particle size is
independent of the applied peak-to-peak voltage Upp driving
the piezo-ceramic ultrasound transducer, because both the
radiation force and the streaming depend linearly on the
energy density of the standing acoustic wave. In contrast, it
can be seen in eqn (1) that the critical particle size does
depend on the material parameters v of the fluid and Φ of
the fluid and particles, and on the actuation frequency f.
Increasing the frequency to achieve radiation forcedominated
motion of smaller particles is a relatively straightforward
solution, but such an increase often necessitates
reduced channel dimensions, which drastically reduces the
throughput of the device. In this paper we propose another
solution, namely to change the whole acoustic resonance
such that the acoustic radiation force and the acoustic
streaming-induced drag work together in focusing the
particles.
The acoustic streaming and acoustophoretic particle
motion in a microchannel cross section have been studied
numerically by Muller et al.28 The method is valid for long,
straight microchannels of constant rectangular cross section
and employs a pertubation approach to the pressure, temperature
and velocity fields. Briefly, the numerical scheme is as
follows. The first-order acoustic fields are solved in the frequency
domain for an oscillating velocity boundary condition
on the walls of the rectangular channel domain. From the
first-order fields, the acoustic radiation force is calculated
from the expression given by Settnes and Bruus,36 while the
steady acoustic streaming velocity field is calculated numerically
by solving the time-averaged second-order Navier–Stokes
equation and continuity equation.27 This method only considers
actuation at a single frequency, but can readily be
extended to consider actuation with two frequencies by
superposition of the second-order streaming flows. For this
superposition to be valid the separation of the two frequencies
should be much larger than the width of the resonances,
which is typically on the order of 10 kHz, such that the firstorder
fields of the two resonances do not couple in the timeaveraged
second-order source terms for the streaming velocity
field.28 However, if the two resonance frequencies are
closely spaced, resulting in overlapping resonance curves, the
two resonances can be excited simultaneously at a single frequency.
In this case, the first-order fields of the two resonances
couple in the time-averaged second-order source
terms, and consequently the streaming velocity field cannot
be calculated by superposition of second-order streaming
flows. This case of close-lying overlapping resonances is the
subject of the numerical analysis in the following section,
where we show that the relative phases of the wall oscilations
control the structure of the streaming flow, and that specific
2792 | Lab Chip, 2014, 14, 2791–2799 This journal is © The Royal Society of Chemistry 2014