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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, (ππ βππ )2 (3πππ 2 πβ1) πΉπππ = 4ππ6 [ 6ππ π 4 π£ 2 (π§) β (2ππ)2 π0 (π½0 βπ½π )2 9π 2 π2 (π§)] (S1) where d is the inter-particle distance, π is frequency of the wave, ΞΈ is the angle between the centerline of the particle pair and the direction of the wave propagation, π½π = 1/ππ ππ2 is the compressibility of particle, π½π = 1/ππ ππ2 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 β4πΈππ βππ 2. For highly concentrated RBC suspensions, the distance between two cells d approaches diameter of cell 2π 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: ππ = 1100 ππ/π3 , π½π = 3.31 × 10β10 ππ β1 , ππ = 1025 ππ/π3 and co = 1530 π/π . The mean radius of the RBCs is taken as π = 2.8 ππ 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 πΉπππ ~1βπ 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 π = 5 ππ, density of polystyrene beads ππ = 1050 ππ/π3 , density of 22.5% aqueous glycerol solution π0 = 1050 ππ/π3 , compressibility of polystyrene beads 1/ππ ππ2 = 2.16 × 10β10 ππ β1 , velocity of sound in 22.5% aqueous glycerol solution ππ = 1590 π/π and viscosity of 22.5% aqueous glycerol solution ππ = 0.0017 πππ 3,6. 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).