ULTRASOUND-INDUCED ACOUSTOPHORETIC MOTION OF . . . PHYSICAL REVIEW E 88, 023006 (2013)
after correcting for a wall-enhanced drag coefficient of 1.032
at the horizontal center plane (see Refs. 27,45,49,50), we extracted
5 μm
ac = (20.6 ± 0.7) J/m3,
the acoustic energy densityE
where the 1σ standard error of the estimated value is stated.
Since both the wall-enhanced drag coefficient and the drag
force from the acoustic streaming fluid velocity are height dependent,
only five trajectories of 5-μm-diameter particles close
to the horizontal center line (z = 0) qualified for use in the fit,
based on a criterion of |z0| 6 μm. The starting positions
(x0,y0,z0) of the five tracks were (34 μm,−115 μm, 6 μm),
(310 μm, −66 μm, −6 μm), (482 μm, −35 μm, −5 μm),
(74 μm, 115 μm, 2 μm), and (350 μm, 128 μm, 0 μm), and
they all reached the vertical center plane y = 0.
Second, the acoustic energy density E
0.5μm
ac for the experiment
with the 0.5-μm-diameter particles was determined,
using the result for E
5μm
ac combined with the fact that Eac
scales as the square of the applied voltage Upp 26. The
measured voltages for the two experiments are U
0.5μm
pp =
(1.62 ± 0.01) V and U
5μm
pp = (0.91 ± 0.01) V, where the
stated error corresponds to the standard deviation of a series of
voltage measurements, with the power turned off in-between
each measurement. The derived value for E
0.5μm
ac , taking into
0.5μm
pp , U
account the errors of U
5μm
pp , and E
5 μm
ac , becomes
E
0.5μm
ac = (U
0.5μm
pp /U
5μm
pp )2E
5μm
ac = (65 ± 2) J/m3, with the
contribution from the error of the measured voltages being
negligible.
Third, based on Eq. (10), the derived value for the energy
0.5μm
ac is used in the analytical expression for the
density E
particle velocities (32). The radiation force is given by Eq. (33)
and the acoustic streaming velocity is given by Eqs. (25)
and (31), using the thermoviscous-corrected amplitude vT
str
Eq. (13). The contribution from the acoustic radiation
force to the 0.5-μm-diameter particle velocity was small
and constituted only 12% of the total particle velocity in
the horizontal center plane z = 0. The contribution from the
radiation force to the 0.5-μm-diameter-particle velocity was
not corrected for the wall-enhanced drag coefficient since this
was minute for these small particles.
To compare the experimental results and the analytical prediction,
we consider the 0.5-μm-diameter particle velocities in
the vertical cross section, yz plane, of the channel as in Figs. 1,
2, and 3. In Fig. 5 are shown color plots of (a) the experimentally
measured acoustophoretic velocities for the 0.5-μmdiameter
particles, (b) the analytical prediction of the same,
and (c) the numerical validation of the analytical result using
the methods of Muller et al. 22. The three data sets are shown
on the same 37 × 15 bin array and with the same color scale.
The experimental and the analytical velocities agree well both
qualitatively and quantitatively, although the experimental
velocities are approximately 20% higher on average. The
experimental results for the particle velocities Fig. 5(a) are
found as the mean of several measurements of the particle
velocity in each bin. The number of measurements performed
in each bin is shown in Fig. 6(a), while the standard error
of the mean (SEM) particle velocity is shown in Fig. 6(b).
These plots show that we typically have between 20 and
70 measurements in each bin and the experimental error is
on average only 1 μm/s, while the relative experimental error
FIG. 5. (Color online) Comparison between experimental, analytical,
and numerical studies of the acoustophoretic particle
velocities up of 0.5-μm-diameter polystyrene particles in water. The
particle velocities up (vectors) and their magnitude color plot ranging
from 0 μm/s (black) to 63 μm/s (white) in all three plots are shown
in the vertical cross section of the microchannel, divided into a pixel
array consisting of 37 × 15 square bins of side length 10 μm. The
axes of the plot coincide with the position of the channel walls.
(a) The APTV measurements of the 0.5-μm-diameter particles,
shown in Fig. 4(b), projected onto the vertical cross section. The
maximum velocity is 63 μm/s. Close to the side walls, experimental
data could not be obtained, which is represented by hatched bins.
(b) Analytical prediction of up based on Eq. (32), taking both the
radiation force and the streaming-induced drag force into account.
The first 20 terms of the Fourier series for v2 Eq. (25) have been
included in the calculation. The maximum velocity is 59 μm/s. There
are no free parameters in this prediction as the acoustic energy density
was calibrated in situ based on measurements of large 5-μm-diameter
particles, shown in Fig. 4(a). (c) Numerical validation of the analytical
result for up using the method described in Muller et al. 22. The
numerical solution has been scaled by the thermoviscous prefactor to
the streaming amplitude (13). The maximum velocity is 59 μm/s.
is on average 4%. The error of the theoretical prediction is
given by the relative error of 4% on the estimated value for the
energy density E
0.5 μm
ac .
The quantitative differences between the experimental
particle velocities Fig. 5(a) and the analytical prediction
Fig. 5(b) are emphasized in Fig. 7, showing the difference
up between the experimental and analytical acoustophoretic
particle speeds
up =
up
exp
−
up
anl
. (38)
We have chosen to consider the difference of the absolute
velocity values |up
exp| − |up
anl
| instead of the absolute value of
the difference |up
exp − up
anl
| because the former allows us to
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