ULTRASOUND-INDUCED ACOUSTOPHORETIC MOTION OF . . . PHYSICAL REVIEW E 88, 023006 (2013)
the viscosity-dependent time-averaged radiation force Frad in
the experimentally relevant limit of the wavelength λ being
much larger than both the particle radius a and the boundary
layer thickness δ was given recently by Settnes and Bruus 5.
For the case of a 1D transverse pressure resonance Eq. (7b),
the viscosity-dependent acoustic radiation force on a particle
reduces to the x- and z-independent expression
Frad(˜y
) = 4πa3knEac ( ˜κ, ˜ ρ, ˜ δ) sinnπ(˜y
+ 1)ey . (33)
The acoustic contrast factor is given in terms of the material
parameters as
(κ˜ ,ρ˜,δ˜) = 1
3
f1( ˜κ) + 1
2
Ref2( ˜ ρ,˜ δ), (34a)
f1( ˜κ) = 1 − ˜κ, (34b)
f2( ˜ ρ,˜ δ) = 21−(δ˜)(ρ˜ − 1)
2 ˜ ρ + 1 − 3( ˜ δ)
, (34c)
( ˜ δ) = −3
2
1 + i(1 + ˜ δ) ˜ δ, (34d)
where ˜κ = κp/κs, ˜ ρ = ρp/ρ0, and ˜ δ = δ/a. Using Eq. (33) for
the transverse resonance, urad only has a horizontal component
urad
y :
urad
y
= u0
a2
a
2
0
sinnπ(˜y
+ 1), n= 1,2,3, . . . , (35a)
where the characteristic particle radius a0 is given by
a0 = δ
3
, (35b)
with δ given by Eq. (6). The acoustophoretic particle velocity
up will in general have nonzero z components, due to the
contribution from the acoustic streaming v2. However, for
the special case of particles in the horizontal center plane
= 0 of a parallel-plate or rectangular channel, the vertical
streaming velocity component vanishes, v2z( ˜ y,0) = 0. From
Eqs. (19a) and (35a), we find that the horizontal particle
velocity component u
˜z
p
y ( ˜ y,0) in a parallel-plate channel is given
by the sinusoidal expression
up
y ( ˜ y,0) = u0
a2
a
2
0
(nα,0)
− KT A
sinnπ(˜y
+ 1). (36)
Since by Eq. (20a) A
(nα,0) is always negative, it follows that
the streaming-induced drag and the radiation force have the
same direction in the horizontal center plane of the channel.
For the rectangular channel using Eq. (25a), the expression for
u
p
y ( ˜ y,0) becomes
up
y ( ˜ y,0) = u0
a2
a
2
0
sinnπ(˜y
+ 1)
+KT
∞
am sin(mπ˜y
m=1
) A
(mα,0)
⊥(mα
+bmA
−1,˜y
)
, (37)
which is not sinusoidal in ˜y
but still proportional to u0. This
particular motion in the ultrasound symmetry plane is studied
in detail in Ref. 45.
III. EXPERIMENTS
We have validated experimentally the analytical expressions
derived above by measuring trajectories of micrometersized
polystyrene particles displaced by acoustophoresis in
a long, straight silicon-glass microchannel with rectangular
cross section. A fully three-dimensional evaluation of the
particle trajectories and velocities was performed by means
of the astigmatism particle tracking velocimetry (APTV)
technique 33,34 coupled to the temperature-controlled and
automated setup presented in Ref. 28. APTV is a singlecamera
particle tracking method in which an astigmatic
aberration is introduced in the optical system by means
of a cylindrical lens placed in front of the camera sensor.
Consequently, an image of a spherical particle obtained in such
a system shows a characteristic elliptical shape unequivocally
related to its depth position z. More details about calibration
and uncertainty of this technique, as well as comparison with
other whole-field velocimetry methods for microflows, can be
found in Refs. 34,35.
A. Acoustophoresis microchip
The acoustophoresis microchip used for the experimentwas
the one previously presented in Refs. 26,28,45. In Ref. 28
the microchip and the experimental setup are described in
details; here, we give a brief description. A rectangular cross
section channel (L = 35 mm,w = 377 μm, and h = 157 μm)
was etched in silicon.APyrex lidwas anodically bonded to seal
the channel and provided the optical access for the microscope.
The outer dimensions of the chip are L = 35 mm, W =
2.52 mm, and H = 1.48 mm. Horizontal fluidic connections
were made at the ends of the microchip. From top and down,
glued together using ethyl-2-cyanoacrylat (ExpressLim, Akzo
Nobel Bygglim AB, Sweden), the chip was placed on top of
a piezoceramic transducer (piezo) (35 mm×5 mm×1 mm,
PZT26, Ferroperm Piezo-ceramics, Denmark), an aluminum
slab to distribute heat evenly along the piezo, and a Peltier
element (standard 40mm×40 mm, Supercool AB, Sweden) to
enable temperature control. The temperaturewas kept constant
at 25 ◦C, based on readings from a temperature sensor placed
near the chip on top of the piezo. This chip stack was mounted
on a computer-controlled xyz stage. Ultrasound vibrations
propagating in the microchip were generated in the piezo
by applying an amplified sinusoidal voltage from a function
generator, and the resulting piezo voltage Upp was monitored
using an oscilloscope.
B. APTV setup and method
The images of the particles in the microfluidic chip were
taken using an epifluorescent microscope (DM2500 M, Leica
Microsystems CMS GmbH, Wetzlar, Germany) in combination
with a 12-bit, 1376×1040 pixels, interline transfer CCD
camera (Sensicam QE, PCO GmbH). The optical arrangement
consisted of a principal objective lens with 20× magnification
and 0.4 numerical aperture and a cylindrical lens with focal
length fcyl = 150 mm placed in front of the CCD sensor of the
camera. This configuration provided a measurement volume
of 900 × 600 × 120 μm3 with an estimated uncertainty in the
particle position determination of ±1 μm in the z direction
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