Lab on a Chip Paper
values can lead to structures qualitatively different from the
standard quadrupolar Rayleigh streaming flow observed for
the half-wavelength resonance.37
Numerical analysis
The following numerical analysis is a generic investigation
of the acoustophoretic motion of 0.5 μm-diameter particles
in a nearly-square channel cross section. It is not intended to
be a direct numerical simulation of the actual experiments
presented in this paper; nevertheless, it predicts the existence
of two fundamentally different acoustic streaming patterns
relevant for the interpretation of the experiments. The numerical
analysis employs the method presented by Muller et al.28
and to avoid spurious effects of perfect square symmetry
and to imitate the uncertainty in microchannel fabrication,
the cross-sectional dimensions of the microchannel in the
model was chosen to be 230.5 μm wide and 229.5 μm
high. The parameters used in the model correspond to the
biologically relevant temperature of 37 °C. The first-order
velocity boundary condition applied to the walls was ubc = u0
cos(ωt)ey on the left and right walls and ubc = u0 cos(ωt + ϕ)ez
on the top and bottom walls, where u0 is the amplitude,
ω = 2πf is the angular frequency of the transducer, ϕ is a
constant phase shift, and ey and ez are the unit vectors in
the transverse horizontal and vertical directions, respectively.
Because both the acoustic streaming-induced drag force and
the acoustic radiation force depend non-linearly on the
oscillating velocity boundary conditions, the consequences of
changing the phase shift ϕ between the two wall pairs cannot
easily be deduced analytically.
To characterize the resonances of the nearly-square channel,
the average acoustic energy density,38 denoted Eac, was
calculated numerically for a range of frequencies, shown in
Fig. 1. This was done for several different actuations of the
nearly-square channel. In each panel an inset shows a sketch
of the channel geometry and which walls are actuated and by
which phase factor cos(ωt + ϕ). In Fig. 1(a) the nearly-square
channel was actuated in phase on the left/right walls to
obtain the usual horizontal half-wavelength resonance, showing
up as a Lorentzian peak centered around the resonance
frequency f1 = 3.3032 MHz. In Fig. 1(b) the nearly-square
channel was actuated in phase on the top/bottom walls
resulting in a peak at the slightly higher resonance frequency
f2 = 3.3176 MHz corresponding to the vertical half-wavelength
resonance. f2 is slightly higher than f1 because the height of
the nearly-square channel is slightly smaller than the width.
In Fig. 1(c) the nearly-square channel was actuated in phase on
all four walls (ϕ = 0). Due to the finite width of the two resonance
peaks, this actuation simultaneously excites both
the horizontal and the vertical half-wavelength resonances,
resulting in a resonance curve with two peaks and a plateau
in between, in contrast to the previous single-peak resonance
curves. As a guide to the eye, the single-peak resonance
curves from Fig. 1(a–b) are included in Fig. 1(c) in grey. The
frequency mid-way between the two resonance peaks is
Fig. 1 Resonance curves, obtained by plotting the average acoustic
energy density Eac vs. the frequency f = ω/2π of the wall actuations.
(a) A nearly-square channel, 230.5 μm by 229.5 μm cross section, with
the left/right walls vibrating in phase. (b) The nearly-square channel
with the top/bottom walls vibrating in phase. (c) The nearly-square
channel with all walls vibrating in phase. The resonance curves from
(a) and (b) are shown in grey. (d) Same as (c) except that the top/bottom
wall pair vibrate with a phase shift of ϕ = π/2 relative to the left/right
wall pair. f1 and f2 are the two resonance frequencies corresponding
to the horizontal and vertical half-wavelength resonance, respectively.
fm indicates the middle frequency between the two resonance peaks.
All walls have the same oscillation amplitude u0.
fm = (f1 + f2)/2 = 3.3104 MHz. At this particular frequency, the
amplitudes of the horizontal and the vertical resonances are
the same, however much reduced relative to the two resonance
maxima. In Fig. 1(d) the nearly-square channel was
actuated on all four walls, but the phase of the actuation on
the top/bottom wall pair was shifted relative to the left/right
wall pair by ϕ = π/2. The resulting resonance curve is the
same; however, as we will see below, the second-order steady
acoustic streaming velocity field changes significantly by
introducing this phase shift.
We now study the acoustophoretic motion of 0.5 μmdiameter
particles in the nearly-square channel cross section,
shown in Fig. 2, for each of the four actuations shown in
Fig. 1. This particle motion results from the acoustic radiation
force and the streaming-induced drag force, both
second-order acoustic effects.28 Given the small particle
diameter, the acoustophoretic motion is dominated by the
drag force from the acoustic streaming.29 Fig. 2 contains four
rows (a–d) corresponding to the four cases in Fig. 1. The
actuation frequency was f1 in (a), f2 in (b), and fm in (c–d).
For each case, column 1 shows the first-order acoustic
pressure, column 2 shows the acoustic radiation force
together with streamlines of the steady streaming velocity
field, and column 3 shows the acoustophoretic trajectories
of 0.5 μm-diameter particles.
For the two cases (a–b) the weak radiation force acts to
focus the particles towards a center line, but as the particle
motion is dominated by the streaming-induced drag force,
they follow the quadrupolar streaming flow of the 1D halfwavelength
resonance. For the two cases (c–d) both the horizontal
and the vertical half-wavelength resonances are exited
simultaneously at the single frequency fm, and the streaming
flow is qualitatively different from the usual quadrupolar structure.
For ϕ = 0 (c), the streaming flow consists of two larger
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