ULTRASOUND-INDUCED ACOUSTOPHORETIC MOTION OF . . . PHYSICAL REVIEW E 88, 023006 (2013)
up between Figs. 8(a) and 8(b) is due to differences in the
resonance modes, i.e., Eac is not the same even though Upp is.
V. DISCUSSION
The comparison shows good agreement between the experimental
measurements and the analytical prediction of
the streaming-induced particle velocities. The qualitative
agreement is seen in Fig. 5 for the two-dimensional topology
of the particle motion, and in Fig. 8 for the effect of the side
walls. Quantitatively, the experimental and analytical results
agree within a mean relative difference of approximately 20%,
a low deviation given state-of-the-art in the field. However, as
illustrated by the statistical analysis in Fig. 7, the differences
up are larger than the estimated 1σ errors. This could indicate
a minor systematic error in the experimental procedure or in
the theoretical model, or be due to underestimation of the
experimental error involved in the analytical prediction.
In the 5-μm-diameter-particle experiment, the acoustic
energy density is determined using only five particle trajectories
close to the channel center z = 0. This is reasonable
as the error of the calculated energy density is relatively
low (3%), however, a calculation based on more particle
trajectories would be desirable. This can be realized through
more experimental repetitions or through implementation of
the 2D dependence of the wall-enhanced drag force, allowing
for use of off-center particle trajectories. One source of error
that has not yet been discussed arises from the assumption
made that the acoustic energy density, and thus the acoustic
forces, does not depend on the x position in the investigated
field of view. In the same setup, Augustsson et al. 28 observed
negligible field gradients in the x direction in some field of
views and significant ones in others. This inhomogeneity was
considered here when making the measurements: we made
sure to check that the five 5-μm-diameter-particle trajectories
sample the x range reasonably well and exhibit only negligible
variations in the acoustic energy density as a function of the
x position.
In the 0.5-μm-diameter-particle experiment, the statistics
and sampling of the x range are good, however, they could
still be improved to achieve better statistics close to the walls.
The relative positions of the 0.5-μm-diameter particles are
accurately determined by use of the APTV technique, whereas
the absolute position in the channel, which was used to
compare with theory, is difficult to determine precisely and
might also be improved. Furthermore, accurate measurements
of the channel dimensions are also important, as these are key
parameters in the theoretical model.
The analytical model could be improved in several ways.
The treatment of the liquid could be extended by including
thermal dependence of more material parameters such as
the specific heat capacity ratio γ , thermal expansion αp,
compressibility κs , and speed of sound c0. The influence
of the surrounding chip material could be included, thereby
relaxing the assumptions of infinite acoustic impedance (ideal
reflection) and infinite thermal conduction (ideal heat sink) of
the channel walls. Solving the full elastic wave problem in
the whole chip is beyond analytical solutions, but is, however,
possible with numerical models. This might be necessary to
achieve accurate quantitative agreement between theoretical
predictions and experiments. Furthermore, the analytical and
numerical models assume an ideal rectangular channel cross
section, which is crucial since the generating mechanism for
the acoustic streaming takes place within the μm-thin acoustic
boundary layer. Even small defects, such as uneven surfaces on
the μm scale, might lead to changes in the acoustic streaming
velocity field.
VI. CONCLUSIONS
In this work, we have derived an analytical expression
for the acoustophoretic velocity of microparticles resulting
from the acoustic radiation force and the acoustic streaminginduced
drag force in a rectangular microchannel, and we have
successfully compared it with a direct numerical solution of
the governing equations.We have also accurately measured 3D
trajectories of 0.5-μm-diameter and 5-μm-diameter particles
in an acoustically actuated microchannel, with an average
relative experimental error of 4% for the 0.5-μm-particle velocities.
This allowed us to perform a quantitative comparison
in 3D between theory and experiments of streaming-induced
particle velocities in a rectangular channel. The analytical
derivation successfully predicted the measured streaminginduced
0.5-μm-diameter-particle velocities, with qualitative
agreement and quantitative differences around 20%, a low
deviation given state-of-the-art in the field. This shows that
the time-averaged second-order perturbation model of the
governing equations yields an adequate description of the
acoustophoretic particle motion.
The differences between the theoretical prediction and the
experimental results emphasize the need for further extensions
of the analytical model, along with improved numerical
simulations 22. Aiming for more detailed quantitative studies
of acoustophoresis, the results also stress the need for improved
accuracy of the measurements of the channel dimensions and
the absolute positions of the particles in the microchannel. The
trinity of analytical, numerical, and experimental studies of the
acoustophoretic particle motion enhances the understanding of
acoustophoresis and opens up for more elaborate and broader
applications.
ACKNOWLEDGMENTS
This work was supported by the Danish Council for
Independent Research, Technology and Production Sciences,
Grants No. 274-09-0342 and No. 11-107021, the German
Research Foundation (DFG), under the individual grants
program KA 1808/12-1, the Swedish Governmental Agency
for Innovation Systems, VINNOVA, the program Innovations
for Future Health, Cell CARE, Grant No. 2009-00236, and the
Swedish Research Council, Grant No. 621-2010-4389.
1 H. Bruus, J. Dual, J. Hawkes, M. Hill, T. Laurell, J. Nilsson, S.
Radel, S. Sadhal, and M. Wiklund, Lab Chip 11, 3579 (2011).
2 T. Laurell, F. Petersson, and A. Nilsson, Chem. Soc. Rev. 36,
492 (2007).
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