Fig. 8 (a) The starting positions (dots) of 144 evenly distributed particles at t = 0 s in the computational domain at the onset of the horizontal halfwave
1.97-MHz resonance shown in Fig. 5 and 7. The following five panels show the trajectories (colored lines) and positions (dots) that the particles
have reached by acoustophoresis at t = 10 s for five different particle diameters: (b) 0.5 mm, (c) 1 mm, (d) 2 mm, (e) 3 mm, and (f) 5 mm. The colors
indicate the instantaneous particle velocity u ranging from 0 mm s21 (dark blue) to 44 mm s21 (dark red). The lengths of the trajectories indicate the
distance covered by the particles in 10 s. Streaming-induced drag dominates the motion of the smallest particles, which consequently are being advected
along the acoustic streaming rolls of Fig. 7(b). In contrast, the acoustic radiation force dominates the motion of the larger particles, which therefore are
forced to the vertical nodal plane at y = 0 of the first-order pressure p1 shown in Fig. 5(a).
As a consequence, the acoustophoretic motion of particles in the
center region is controlled by the radiation force. This is illustrated
in Fig. 9(b), where trajectories of small 1-mm-diameter particles are
shown. For 2h/4 , z , h/4 their motion is similar to the radiationforce
dominated motion of the larger 5-mm-diameter particles
moving in the shallow channel with h/w = 0.42 as shown in Fig. 8(f).
Near the top and bottom walls, the 1 mm diameter particles exhibit
the usual small-particle streaming-induced motion.
Clearly, geometry can be used to obtain more control of the
acoustophoretic motion of suspended particles in microchannels.
F Streaming in a high-viscosity buffer
According to eqn (20), the critical particle diameter for crossover
between radiation-dominated and streaming-dominated
acoustophoretic motion is proportional to the boundary layer
thickness d~
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2g=(r0v)
q
. Obviously, viscosity can also be used to
control acoustophoresis.We therefore studied the effects of replacing
water (g = 1 mPa s) with glycerol mixtures, in particular the 50%
glycerol-in-water mixture (g = 5 mPa s), for which the relevant
material parameters are listed in Table 1.
First, to ensure comparable conditions, we wanted to excite
the horizontal half-wave resonance in the glycerol-in-water
system. As the speed of sound of the glycerol mixture is 15%
larger than that of water, we found the resonance frequency to be
f = c0/(2w) = 2.27 MHz. This frequency was used in the velocity
boundary condition eqn (17a) to calculate the results shown in
Fig. 10 for the first-order pressure field, the time-averaged
second-order streaming velocity field, and particle trajectories
for 5-mm-diameter polystyrene particles.
The glycerol-in-water and the water system are actuated with
the same boundary velocity given in eqn (18) , but the difference
in viscosity of the two liquids leads to different acoustic
responses. The amplitude of the induced first-order resonance
pressure is reduced by a factor of 2.6 from 0.243 MPa in the lowviscosity
water of Fig. 5(a) to 0.094 MPa in the high-viscosity
glycerol mixture of Fig. 10(a). Likewise, the induced streaming
velocity Svbnd
2y T near the boundary is reduced by a factor of 15
from 6.42 mm s21 in water, Fig. 7(b), to 0.43 mm s21 in glycerolin
water, Fig. 10(b). In contrast, given the validity of Rayleigh’s
streaming theory, the velocity ratio Y = c0Svbnd
2 should be
2y T/U1
independent of viscosity. For the glycerol-in-water mixture it is
0.336 deviating 8% from the value in water, see Section IV C, and
10% from the Rayleigh value 3/8 of eqn (10) . The significant
difference in the numerically determined values of Y for water
and glycerol-in-water points to the inadequacy of the Rayleigh
This journal is The Royal Society of Chemistry 2012 Lab Chip, 2012, 12, 4617–4627 | 4625
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Published on 23 July 2012 on http://pubs.rsc.org | doi:10.1039/C2LC40612H
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