1D = 4pW(k˜ ,r˜ ,d ˜
F rad
)a3qEacsin(2qy), (13a)
W ~k,~r,~d
~
1
3
f1ðk~Þz
1
2
h i
Re f2 ~r,~d
, (13b)
where W is the so-called acoustic contrast factor.
The time-averaged Stokes drag force F drag on a spherical
particle of radius a moving with velocity u in a fluid having the
streaming velocity Sv2T is given by the usual expression
F drag = 6pga(Sv2T 2 u), (14)
valid for particles sufficiently far from the channel walls.28
III Numerical model
In the following we present the idealized numerical model, and how
we implement and solve the governing equations for thismodel using
the finite element software COMSOL Multiphysics 4.2a, see ref. 29.
A Model system and computational domain
Given the detailed measurements of the acoustophoretic motion
and the successful comparison with theoretical predictions
presented in ref. 7 and 8, it is natural to use an idealization of
their straight microchannel of length 35 mm and rectangular cross
section as a model system in our numerical study. We neglect the
chip structure and simply represent the silicon–glass chip as hardwall
boundary conditions. We further neglect any axial dynamics
in the long straight channel, and thus restrict our analysis to the
rectangular cross section V of width w = 0.38 mm and height h =
0.16 mm in the vertical yz-plane, see Fig. 2. Finally, we represent
the ultrasonic piezo transducer by the velocity boundary condition
eqn (4). The particle suspensions are modeled as being monodisperse
and containing non-interacting, spherical polystyrene
particles with diameters of 0.5, 1.0, 2.0, 3.0, or 5.0 mm, respectively.
The model system has a horizontal half-wave resonance across
the width w given by the frequency f = v/(2p) = c0/(2w), equal to
1.97 MHz for water. To excite this resonance, we let all external
acoustic excitations have a harmonic time dependence of
frequency f = 1.97 MHz. All relevant material parameters are
listed in Table 1 .
B Particle tracing model
In order to study the acoustophoretic motion of N particles
suspended in the microchannel, we apply the COMSOL Particle
Tracing Module, which closely mimics experimental particle
2) from eqn (2b).
tracing and velocimetry.7,8 This module provides a Lagrangian
description of the motion of the particles, each of which is
treated as a point particle governed by Newton’s law of motion,
and thus involving one ordinary differential equation (ODE) for
each spatial direction. Consequently, in total 2N ODEs are
solved for the particle suspension. The input are the particle
masses mj and all forces Fi(rj) acting on each particle at position
rj. The ODE for the jth particle with velocity vj = drj/dt is
mj
dvj
dt
~
X
i
Fi(rj): (15)
Neglecting gravitational effects, the forces acting on a particle in
our model are the radiation force F rad, eqn (11) , and the Stokes
drag force F drag, eqn (14). These forces are calculated
numerically as described in the following sub-section.
C Numerical procedure
We have used the following sequential procedure to solve the
problem numerically in COMSOL:
(i) The first-order acoustic fields of eqn (5), subject to the
boundary conditions of eqn (4), are calculated using the
predefined Thermoacoustic Physics Interface.
Fig. 2 (a) End-view sketch of the acoustophoresis microchip with a
fluidic channel of width w = 0.38 mm and height h = 0.16 mm used in
experiments.7,8 It consists of a silicon chip (dark gray), a pyrex lid (light
gray), water (blue), and a piezo transducer (black). (b) The corresponding
two-dimensional computational domain V (blue) surrounded by rigid
walls hV (black) implemented in our numerical model.
Table 1 Model parameters. The parameters are given at temperature
T = 25u and taken from the COMSOL Material Library unless explicit
stated otherwise.
Polystyrene
Density30 rps 1050 kg m23
Speed of sound31 (at 20 uC) cps 2350 m s21
Poisson’s ratio32 sps 0.35
Compressibilitya kps 249 TPa21
Water
Density r0 998 kg m23
Speed of sound c0 1495 m s21
Compressibilityb k0 448 TPa21
Viscosity g 0.893 mPa s
Visc. boundary layer, 1.97 MHz d 0.38 mm
Thermal conductivity kth 0.603 W m21 K21
Specific heat capacity Cp 4183 Jkg21K21
Specific heat capacity ratioc c 1.014
Thermal diffusivityd Dth 1.43 6 1027 m2 s21
Thermal expansion coeff.e a 2.97 6 1024 K21
50% glycerol-in-water mixture
Density34 r0 1129 kg m23
Speed of sound35 c0 1725 m s21
Compressibilityb k0 298 TPa21
Viscosity34 g 5.00 mPa s
Visc. boundary layer, 2.27 MHz d 0.79 mm
Thermal cond.36 (at 20 uC) kth 0.416 W m21 K21
Specific heat cap.37 (at 1.7 uC) Cp 3360 J kg21 K21
Specific heat capacity ratioc c 1.043
Thermal diffusivityd Dth 1.10 6 1027 m2 s21
Thermal expansion coeff.e a 4.03 6 1024 K21
a Calculated as kps~
3(1{sps)
1zsps
1
(rpsc2
ps)
from ref. 33.
b Calculated as k0 = 1/(r0c0
c Calculated from T0a2/(r0Cpk) = c 2 1.
d Calculated as Dth = kth/(r0Cp).
e Calculated from eqn (2c).
4620 | 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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