PHYSICAL REVIEW E 88, 023006 (2013)
Ultrasound-induced acoustophoretic motion of microparticles in three dimensions
P. B. Muller,1 M. Rossi,2 A´ . G. Mar´ın,2 R. Barnkob,1 P. Augustsson,3 T. Laurell,3,4 C. J. Ka¨hler,2 and H. Bruus1,*
1Department of Physics, Technical University of Denmark, DTU Physics Building 309, DK-2800 Kongens Lyngby, Denmark
2Universit¨at der Bundeswehr M¨unchen, Werner-Heisenberg-Weg 39, 85579 Neubiberg, Germany
3Department of Measurement Technology and Industrial Electrical Engineering, Lund University, PO-Box 118, S-221 00 Lund, Sweden
4Department of Biomedical Engineering, Dongguk University, Seoul, South Korea
(Received 1 March 2013; published 8 August 2013)
We derive analytical expressions for the three-dimensional (3D) acoustophoretic motion of spherical
microparticles in rectangular microchannels. The motion is generated by the acoustic radiation force and the
acoustic streaming-induced drag force. In contrast to the classical theory of Rayleigh streaming in shallow, infinite,
parallel-plate channels, our theory does include the effect of the microchannel sidewalls. The resulting predictions
agree well with numerics and experimental measurements of the acoustophoretic motion of polystyrene spheres
with nominal diameters of 0.537 and 5.33 μm. The 3D particle motion was recorded using astigmatism
particle tracking velocimetry under controlled thermal and acoustic conditions in a long, straight, rectangular
microchannel actuated in one of its transverse standing ultrasound-wave resonance modes with one or two
half-wavelengths. The acoustic energy density is calibrated in situ based on measurements of the radiation
dominated motion of large 5-μm-diameter particles, allowing for quantitative comparison between theoretical
predictions and measurements of the streaming-induced motion of small 0.5-μm-diameter particles.
DOI: 10.1103/PhysRevE.88.023006 PACS number(s): 47.15.−x, 43.25.Nm, 43.25.Qp, 43.20.Ks
I. INTRODUCTION
Acoustofluidics is gaining increasing interest in lab-ona
chip and microfluidics applications. Techniques based on
acoustofluidic forces permit us to perform a large variety of
different tasks such as trapping, separation and sorting of cells,
particle manipulation, and generation of fluid motion in a nonintrusive
way 1,2. Acoustic forces allow for nondestructive
and label-free particle handling based on size, density, and
compressibility. Experimentally, the acoustophoretic motion
of particles is driven by an ultrasonic standing wave that
generates acoustic radiation forces on the particles and acoustic
streaming in the fluid, which exerts a Stokes drag force on
the particles. Theoretically, such phenomena are described
by complex, nonlinear governing equations sensitive to the
boundary conditions and are thereby difficult to predict. Therefore,
the development of analytical and numerical methods
that are able to accurately predict the acoustophoretic motion
of different particle or cell types is currently a major challenge
in the design of acoustofluidic systems.
Regarding the acoustic radiation force on microparticles,
recent theoretical studies by Doinikov 3, Danilov and
Mironov 4, as well as Settnes and Bruus 5 have advanced
the theoretical treatment, beyond the seminal contributions by
King 6, Yosioka and Kawasima 7, and Gorkov 8. We use
these models without any new contributions. However, so far
theoretical treatments of the acoustic streaming generated by
standing acoustic waves have not included an analysis of the
effect of the vertical sidewall in rectangular microchannels,
instead the focus has been on the idealized parallel-plate
geometry 9–14 or single-wall systems 15–18. Also, in most
theoretical work either the radiation force or the streaming
effects have been studied separately, but not combined with
wall effects to obtain a complete description of microparticle
*bruus@fysik.dtu.dk
acoustophoresis. Recently, a number of numerical studies of
acoustic streaming 19–21 and acoustophoresis 22,23 have
appeared in the literature. In this work, we present a theoretical
analysis of acoustic streaming, taking the effect of the vertical
sidewalls into account, and apply it to a theoretical study of
microparticle acoustophoresis in rectangular microchannels.
Our results (both with and without vertical sidewalls) are valid
for channel heights and acoustic wavelengths much larger than
the acoustic boundary layer thickness, thus extending previous
results for parallel-plate systems that are only valid for heights
much smaller than the acoustic wavelength 9,10.
To guide and control the theoretical developments, precise
experimental measurements of the acoustophoretic motion of
microparticles are necessary, and particle-based velocimetry
techniques are among the best methods available. The work of
Hags¨ater et al. 24 was one of the first to use microparticle
image velocimetry (μPIV) in resonant microfluidic chips.
In their case, the measurements were employed to visualize
the resonance modes in the microchip, using the radiationdominated
horizontal motion of 5-μm-diameter particles and
the associated horizontal acoustic streaming pattern using
1-μm-diameter particles. Using a similar μPIV technique,
Manneberg et al. 25 characterized multiple localized ultrasonic
manipulation functions in a single microchip. Barnkob
et al. 26 and Koklu et al. 27 also studied acoustophoretic
motion of large particles (5- and 4-μm-diameter, respectively),
but instead used particle tracking velocimetry (PTV) to
obtain particle paths, which were compared with theoretical
results. Later, Augustsson et al. 28 employed both PTV and
μPIV to make high-accuracy measurements of the acoustic
energy density as well as the temperature and frequency
dependence of acoustic resonances in microchannels filled
with 5-μm-diameter particles dominated by the radiation
force. Such approaches have successfully been applied to the
two-dimensional (2D) motion of particles in the optical focal
plane in simple geometries and resonances. Recently, Dron
et al. 29 used defocusing of particle images to measure
1539-3755/2013/88(2)/023006(12) 023006-1 ©2013 American Physical Society