Download Revised APL manuscript suplementary 23-06-2016

Supplementary Material for "Role of shear induced diffusion in acoustophoretic focusing of
dense suspensions"
S. Karthick, A. K. Sen*
Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai-600036, India
*
Author to whom correspondence should be addressed. Email: [email protected]
In this supplementary material we show that the acoustic secondary radiation forces are negligible compare to acoustic
primary radiation forces.
Secondary radiation force.-The secondary radiation force 1 acting on small particles can be expressed as,
(𝜌𝑜 −𝜌𝑝 )2 (3𝑐𝑜𝑠 2 𝜃−1)
𝐹𝑎𝑐𝑠 = 4𝜋𝑎6 [
6𝜌𝑜 𝑑 4
𝑣 2 (𝑧) −
(2𝜋𝑓)2 𝜌0 (𝛽0 −𝛽𝑝 )2
9𝑑 2
𝑝2 (𝑧)]
(S1)
where d is the inter-particle distance, 𝑓 is frequency of the wave, θ is the angle between the centerline of the particle pair
and the direction of the wave propagation, 𝛽𝑝 = 1/𝜌𝑝 𝑐𝑝2 is the compressibility of particle, 𝛽𝑜 = 1/𝜌𝑜 𝑐𝑜2 is the
compressibility of medium, v(z) is the particle velocity amplitude, and p(z) is the acoustic pressure amplitude, v(z) is
maximum at pressure node (centre of the channel) and zero at pressure antinode (walls). Maximum velocity amplitude is
equal to √4𝐸𝑎𝑐 ⁄𝜌𝑜 2. For highly concentrated RBC suspensions, the distance between two cells d approaches diameter of
cell 2𝑎 so the velocity dependent term (1st term in eqn. 3) dominates the pressure dependent term (2nd term in eqn. 3), so
the later can be neglected. Now, by using eqn. 3, we estimate that the maximum secondary radiation force acting between
two RBCs which are in contact is Eac= 20.1 J/m3 is 0.88 pN (repulsive) along the wave direction and -0.44 pN (attractive)
perpendicular to the wave direction. By using eqn. 1, the maximum primary acoustic radiation force on the RBC is
estimated to be 4.0 pN at 20.1 J/m3. In order to estimate the primary and secondary forces, the RBC and plasma properties
are taken as follows3: 𝜌𝑝 = 1100 𝑘𝑔/𝑚3 , 𝛽𝑝 = 3.31 × 10−10 𝑃𝑎 −1 , 𝜌𝑝 = 1025 𝑘𝑔/𝑚3 and co = 1530 𝑚/𝑠. The mean
radius of the RBCs is taken as 𝑎 = 2.8 𝜇𝑚 from the average volume of erythrocytes4 90 fL. In suspensions, only particles
in the neighbourhood mainly contribute towards the secondary force on an individual particle (since 𝐹𝑎𝑐𝑠 ~1⁄𝑑 4 ). If a
particle is located well within a dense suspension, the repulsive or attractive forces due to the neighbouring particles on
all sides of the particles cancel each other so asymmetry is required to create non-zero secondary force on the particles
present in a suspension5. Since only the particles present at the boundary of the suspensions will experience a non-zero
force, and the secondary force is estimated to be a small fraction (~10%) of the primary acoustic radiation force, we
neglect secondary radiation force in our analysis.
The material properties of polystyrene particles and 22.5% aqueous glycerol solution are only used to calculate the energy
density, which are given as follows: radius of the polystyrene beads 𝑎 = 5 𝜇𝑚, density of polystyrene beads 𝜌𝑝 =
1050 𝑘𝑔/𝑚3 , density of 22.5% aqueous glycerol solution 𝜌0 = 1050 𝑘𝑔/𝑚3 , compressibility of polystyrene beads
1/𝜌𝑝 𝑐𝑝2 = 2.16 × 10−10 𝑃𝑎 −1 , velocity of sound in 22.5% aqueous glycerol solution 𝑐𝑜 = 1590 𝑚/𝑠 and viscosity of
22.5% aqueous glycerol solution 𝜂𝑓 = 0.0017 𝑃𝑎𝑠3,6.
References:
1
M. Groschl, ACUSTICA 84, 432 (1998).
2
P.B. Muller, R. Barnkob, P. Augustsson, T. Laurell, M. Rossi, and M. A.G., Phys. Rev. E 023006, 1 (2013).
3
D. Hartono, Y. Liu, L. Tan, Y. Sherlene, and L. Lanry, Lab Chip 4072 (2011).
4
C.E. Mclaren, G.M. Brittenham, and V. Hasselblad, Am. J. Physiol. Circ. Physiol. 252, H857 (1987).
5
G.T. Silva and H. Bruus, Phys. Rev. E 063007, 1 (2014).
6
J.B. Segur and H.E. Oberstar, Ind. Eng. Chem. 43, 2117 (1951).