Fig. 9 Acoustophoresis in a high-aspect-ratio channel. The setup is
identical to Fig. 4(a) except that for the fixed width w = 0.38 mm the
channel height h has been increased from 0.16 mm (aspect ratio h/w =
0.42) to 0.76 mm (aspect ratio h/w = 2). (a) Vector plot (white arrows),
similar to Fig. 7(b), of the time-averaged second-order streaming velocity
Sv2T and color plot from 0 mm s21 (dark blue) to 4.2 mm s21 (dark red)
of its magnitude. (b) Particle tracing plot for 1-mm-diameter polystyrene
particles corresponding to Fig. 8(c) but for time t = 100 s, aspect ratio
h/w = 2, and velocity ranging from 0 mm s21 (dark blue) to 3.3 mm s21
(dark red). In this high aspect-ratio geometry the acoustic streaming flow
rolls are located near the top and bottom walls leaving the center region
nearly streaming free. Fig. 10 Acoustophoresis in a 50% glycerol-in-water mixture. The setup
theory in a rectangular channel. This is perhaps not a surprise, as
this theory is derived for an infinite parallel-plate channel.
In Fig. 10(c) is shown that the viscous boundary-layer
thickness in the glycerol-in-water mixture at 2.27 MHz is d =
0.79 mm, a factor 2.1 larger than the value d = 0.38 mm in water
at 1.97 MHz shown in Fig. 7(c). As the two resonance
frequencies only differ by 10%, the change in the boundarylayer
thickness is mainly due to the viscosity ratio, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
5 mPa s
p
=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 mPa s
p
&2:2.
Finally, fromeqn (10) usingY=3/8 and W = 0.031, we calculated
the critical particle diameter to be 2ac = 9.5 mm for the cross-over
from radiation-dominated to streaming-dominated acoustophoretic
motion in the glycerol-in-water system. This value explains why
the particle trajectories for the 5-mm-diameter polystyrene particles
in Fig. 10(d) appear to be much more influenced by the acoustic
streaming rolls, compared to the same-sized particles in water,
Fig. 8(f). Instead, Fig. 10(d) resembles more the motion of the
1-mm-diameter particles in water, Fig. 8(c). This resemblance can be
quantified by the ratio a/ac: for 5-mm-diameter particles in the
glycerol-in-water mixture it is 0.52, while for 1-mm-diameter
particles in pure water it is 0.50, only 4% lower. Note that because
of the reduction in streaming velocity by the above-mentioned
factor of 15, we have chosen to follow the particles in the glycerolin
water mixture for 150 s and in water only for 10 s.
V Concluding discussion
The finite element method was successfully used to model the
acoustophoretic motion of microparticles inside a microchannel
is identical to Fig. 4(a) except that the resonance frequency is increased to
f = c0/(2w) = 2.27 MHz. (a) Color plot of the pressure p1 showing the
horizontal half-wave resonance. (b) Vector plot (white arrows) of the
time-averaged second-order streaming velocity Sv2T and color plot of its
magnitude corresponding to Fig. 7(b). (c) Zoom-in on the 0.4-mm-thick
boundary layer near the bottom wall corresponding to Fig. 7(c). (d)
Particle tracing plot for 5-mm-diameter polystyrene particles corresponding
to Fig. 8(f) but for 150 s. Contrary to the water-filled channel in
Fig. 8(f), the motion of the 5-mm-particles in the more viscous glycerol-inwater
mixture are dominated by the streaming-induced drag, whereby the
particle trajectories end up looking more like those of the smaller 1-mmdiameter
particles Fig. 8(c).
subject to a transverse horizontal half-wave ultrasound resonance.
The motion is due to the combined effect of Stokes drag from the
time-averaged second-order streaming flow and the acoustic
radiation forces. To achieve this, the first-order acoustic field of a
standing wave was determined inside a microchannel cavity by
solving the linearized compressional Navier–Stokes equation, the
continuity equation, and the heat equation, while resolving the
boundary layers near rigid walls. The first-order field was then
used to determine the streaming flow and the acoustic radiation
forces, and from this the time-dependent trajectories of an
ensemble of non-interacting microparticles was calculated.
A main result is the characterization of the cross over from
streaming-dominated to radiation-dominated acoustophoretic
microparticle motion as a function of particle diameter,
geometry, and viscosity. Using a water-filled shallow microchannel
as the base example, we demonstrated how to get rid of
streaming effects in the center region of a microchannel by
4626 | Lab Chip, 2012, 12, 4617–4627 This journal is The Royal Society of Chemistry 2012
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Published on 23 July 2012 on http://pubs.rsc.org | doi:10.1039/C2LC40612H
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