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Three-dimensional numerical simulation of particle acoustophoresis: COMSOL implementation and case studies

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Abstract

As a microfluidic-based noncontact manipulation and separation technique, acoustophoresis is the motion of the particles or cells within a host fluid under the effect of acoustic waves. Wave scattering off the particle surface generates both acoustic radiation forces and interparticle interaction forces that move the particles relative to the host fluid and each other, respectively. In this paper, an implementation in the commercial finite element software COMSOL Multiphysics for three-dimensional simulations of particle acoustophoresis is presented. Case studies with various particle types including rigid, compressible, elastic, core–shell and different particle geometries including spherical and spheroidal particles as well as pair-particle are simulated. The simulation results are verified by comparing the scattered velocity potential around the particle and the acoustic forces on the particle, against the existing analytical solutions. Furthermore, the COMSOL Livelink for MATLAB is utilized to implement an incremental simulation algorithm for the particle migration under the effect of a standing wave in a microchannel, based on the Stokes drag formula. The acoustophoretic motion simulations are performed for single particle and pair-particle case studies to demonstrate the presented implementation capabilities in modeling multi-particle tracking in real applications.

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source particle distance from the wave node, for different interparticle distances of a , b, and c

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source and probe particles, over simulation time

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Abbreviations

\(A\) :

Particle cross-section (μm2)

\(a:\) :

Particle outer radius (μm)

\(b:\) :

Inner radius of core–shell particle (μm)

\({c}_{f}:\) :

Sound wave speed in fluid (m/s)

\({c}_{c}:\) :

Compressional wave speed in elastic particle material (m/s)

\({c}_{s}:\) :

Shear wave speed in elastic particle material (m/s)

\(E:\) :

Young’s modulus (MPa)

\({E}_{ac}:\) :

Acoustic energy density (N/m2)

\({F}^{rad}:\) :

Acoustic radiation force (N)

\(f:\) :

Stimulation frequency (MHz)

\({f}_{m}:\) :

Monopole scattering coefficient

\({f}_{d}:\) :

Dipole scattering coefficient

\(H:\) :

Channel height (μm)

\(h:\) :

Particle distance from anti-node (μm)

\({h}_{s}:\) :

Distance between the source particle and the first wave node (μm)

\(i:\) :

Complex variable equal to \(\sqrt{-1}\)

\({\varvec{I}}\) :

Second-order identity tensor

\(k:\) :

Wave number (1/m)

\({k}_{\perp }:\) :

Added mass coefficient of spheroid in the perpendicular direction

\({k}_{\parallel }:\) :

Added mass coefficient of spheroid in the parallel direction

\(L:\) :

Inter-particle distance (μm)

\({\varvec{n}}:\) :

Surface normal outgoing vector

\(p:\) :

Fluid pressure (Pa)

\({p}_{in}:\) :

Incident wave pressure (Pa)

\({P}_{sc}:\) :

Scattered wave pressure (Pa)

\({\varvec{r}}:\) :

Surface element vector from the center of particle

\(r:\) :

Position relative to the center of particle (μm)

\({r}_{a}:\) :

Wave-perpendicular spheroidal radius (μm)

\({r}_{b}:\) :

Wave-parallel spheroidal radius (μm)

\({r}_{p}:\) :

Position relative to the center of probe particle (μm)

\({r}_{s}:\) :

Position relative to the center of source particle (μm)

\({r}_{sp}:\) :

Distance between source and probe particles (μm)

\(S:\) :

Surface area (μm2)

\({T}_{rad}:\) :

Radiation torque (Nm)

\(U\left({r}_{p}|{r}_{s}\right):\) :

Potential energy of the pair interaction (μJ)

\(V\) :

Particle volume (μm3)

\({v}_{p}:\) :

Particle speed (m/s)

\({\varvec{v}}:\) :

Fluid velocity (m/s)

\({v}_{in}:\) :

First-order incident velocity (m/s)

\({v}_{sc}:\) :

Scattered wave velocity (m/s)

\({Y}_{st}:\) :

Acoustic radiation force function

\(\delta :\) :

Viscous boundary-layer thickness (μm)

\(\varepsilon\) :

Ellipsoidal eccentricity

\({\epsilon }_{p}\) :

Dimensionless scattering parameter

\(\theta\) :

Angular distraction of ellipsoidal particle relative to wave propagation direction (rad)

\({\kappa }_{f}\) :

Fluid compressibility (Pa−1)

\(\stackrel{\sim }{\kappa }\) :

Particle relative compressibility

\({\kappa }_{p}\) :

Particle compressibility (Pa−1)

\(\lambda\) :

Wave length (μm)

\(\mu\) :

Fluid viscosity (Pa s)

\(\upsilon\) :

Poisson’s ratio

\(\rho\) :

Fluid density field (Kg/m3)

\({\rho }_{f}\) :

Fluid unperturbed density (Kg/m3)

\(\stackrel{\sim }{\rho }\) :

Particle relative density

\({\rho }_{p}\) :

Particle density (Kg/m3)

\({\rho }_{in}\) :

First-order incident density (kg/m3)

\(\sigma\) :

Stress (N/m2)

\(\Psi\) :

Acoustic contrast factor

\(\phi\) :

Fluid velocity potential (m2/s)

\({\phi }_{in}\) :

Incident wave velocity potential (m2/s)

\({\phi }_{sc}\) :

Scattered wave potential (m2/s)

\({\phi }_{ext}\) :

External incident wave potential (m2/s)

\({\phi }_{sc}(\left({r}_{p}|{r}_{s}\right))\) :

Scattered potential of the source particle at the probe position (m2/s)

\(\omega\) :

Angular frequency of the acoustic field (rad/s)

0:

Unperturbed state

1:

First-order perturbed state

2:

Second-order perturbed state

References

  1. Ankrett DN, Carugo D, Lei J, Glynne-Jones P, Townsend PA, Zhang X, Hill M (2013) The effect of ultrasound-related stimuli on cell viability in microfluidic channels. J Nanobiotechnol 11:1–5. https://doi.org/10.1186/1477-3155-11-20

    Article  Google Scholar 

  2. Laurell T, Petersson F, Nilsson A (2007) Chip integrated strategies for acoustic separation and manipulation of cells and particles. Chem Soc Rev 36:492–506

    Article  Google Scholar 

  3. Lenshof A, Laurell T (2010) Continuous separation of cells and particles in microfluidic systems. Chem Soc Rev 39:1203–1217

    Article  Google Scholar 

  4. Petersson F, Nilsson A, Holm C, Jönsson H, Laurell T (2004) Separation of lipids from blood utilizing ultrasonic standing waves in microfluidic channels. Analyst 129:938–943

    Article  Google Scholar 

  5. Petersson F, Nilsson A, Holm C, Jönsson H, Laurell T (2005) Continuous separation of lipid particles from erythrocytes by means of laminar flow and acoustic standing wave forces. Lab Chip 5:20–22

    Article  Google Scholar 

  6. Nilsson A, Petersson F, Jönsson H, Laurell T (2004) Acoustic control of suspended particles in micro fluidic chips. Lab Chip 4:131–135

    Article  Google Scholar 

  7. Lenshof A, Magnusson C, Laurell T (2012) Acoustofluidics 8: applications of acoustophoresis in continuous flow microsystems. Lab Chip 12:1210–1223. https://doi.org/10.1039/c2lc21256k

    Article  Google Scholar 

  8. Evander M, Nilsson J (2012) Acoustofluidics 20: applications in acoustic trapping. Lab Chip 12:4667–4676. https://doi.org/10.1039/c2lc40999b

    Article  Google Scholar 

  9. Doinikov AA (1994) Acoustic radiation pressure on a rigid sphere in a viscous fluid. Proc R Soc Lond Ser A Math Phys Sci 447:447–466

    MathSciNet  MATH  Google Scholar 

  10. Doinikov AA (1994) Acoustic radiation pressure on a compressible sphere in a viscous fluid. J Fluid Mech 267:1–22

    Article  MathSciNet  MATH  Google Scholar 

  11. King LV (1934) On the acoustic radiation pressure on spheres. Proc R Soc Lond Ser A-Math Phys Sci 147:212–240

    Google Scholar 

  12. Hartono D, Liu Y, Tan PL, Then XYS, Yung L-YL, Lim K-M (2011) On-chip measurements of cell compressibility via acoustic radiation. Lab Chip 11:4072–4080

    Article  Google Scholar 

  13. Weiser MAH, Apfel RE, Neppiras EA (1984) Interparticle forces on red cells in a standing wave field. Acta Acust United Acust 56:114–119

    MATH  Google Scholar 

  14. Zheng X, Apfel RE (1995) Acoustic interaction forces between two fluid spheres in an acoustic field. J Acoust Soc Am 97:2218–2226. https://doi.org/10.1121/1.411947

    Article  Google Scholar 

  15. Garcia-Sabaté A, Castro A, Hoyos M, González-Cinca R (2014) Experimental study on inter-particle acoustic forces. J Acoust Soc Am 135:1056–1063

    Article  Google Scholar 

  16. Barnkob R, Augustsson P, Laurell T, Bruus H (2012) Acoustic radiation- and streaming-induced microparticle velocities determined by microparticle image velocimetry in an ultrasound symmetry plane. Phys Rev E Stat Nonlinear Soft Matter Phys 86:056307. https://doi.org/10.1103/PHYSREVE.86.056307/FIGURES/9/MEDIUM

    Article  Google Scholar 

  17. Marefati S, Ghassemi M, Ghazizadeh V (2022) Investigation of effective parameters on streaming-induced acoustophoretic particle manipulation in a microchannel via three-dimensional numerical simulation. Phys Fluids 34:012008. https://doi.org/10.1063/5.0077392

    Article  Google Scholar 

  18. Lei J, Cheng F, Li K (2020) Numerical simulation of boundary-driven acoustic streaming in microfluidic channels with circular cross-sections. Micromachines 11:240. https://doi.org/10.3390/MI11030240

    Article  Google Scholar 

  19. Namnabat MS, Moghimi Zand M, Houshfar E (2021) 3D numerical simulation of acoustophoretic motion induced by boundary-driven acoustic streaming in standing surface acoustic wave microfluidics. Sci Rep 11:1–16. https://doi.org/10.1038/s41598-021-90825-z

    Article  Google Scholar 

  20. Hsu JC, Chao CL (2020) Full-wave modeling of micro-acoustofluidic devices driven by standing surface acoustic waves for microparticle acoustophoresis. J Appl Phys 128:124502. https://doi.org/10.1063/5.0017933

    Article  Google Scholar 

  21. Chen C, Zhang SP, Mao Z, Nama N, Gu Y, Huang PH, Jing Y, Guo X, Costanzo F, Huang TJ (2018) Three-dimensional numerical simulation and experimental investigation of boundary-driven streaming in surface acoustic wave microfluidics. Lab Chip 18:3645–3654. https://doi.org/10.1039/C8LC00589C

    Article  Google Scholar 

  22. Nama N, Barnkob R, Mao Z, Kähler CJ, Costanzo F, Huang TJ (2015) Numerical study of acoustophoretic motion of particles in a PDMS microchannel driven by surface acoustic waves. Lab Chip 15:2700–2709. https://doi.org/10.1039/C5LC00231A

    Article  Google Scholar 

  23. Skov NR, Sehgal P, Kirby BJ, Bruus H (2019) Three-dimensional numerical modeling of surface-acoustic-wave devices: acoustophoresis of micro-and nanoparticles including streaming. Phys Rev Appl 12:044028. https://doi.org/10.1103/PHYSREVAPPLIED.12.044028/FIGURES/8/MEDIUM

    Article  Google Scholar 

  24. Embleton TFW (1954) Mean force on a sphere in a spherical sound field. I. (Theoretical). J Acoust Soc Am 26:40–45

    Article  MathSciNet  Google Scholar 

  25. Embleton TFW (1954) Mean Force on a Sphere in a Spherical Sound Field. II. (Experimental). J Acoust Soc Am 26:46–50. https://doi.org/10.1121/1.1907287

    Article  MathSciNet  Google Scholar 

  26. Embleton TFW (1956) The radiation force on a spherical obstacle in a cylindrical sound field. Can J Phys 34:276–287

    Article  MathSciNet  MATH  Google Scholar 

  27. Yosioka K, Kawasima Y (1955) Acoustic radiation pressure on a compressible sphere. Acta Acust United Acust 5:167–173

    Google Scholar 

  28. Gor’kov LP (1962) On the forces acting on a small particle in an acoustical field in an ideal fluid, in: Sov. Phys. Dokl, pp 773–775

  29. Hasegawa T (1979) Acoustic radiation force on a sphere in a quasistationary wave field—experiment. J Acoust Soc Am 65:41–44

    Article  Google Scholar 

  30. Mitri FG (2005) Acoustic radiation force acting on elastic and viscoelastic spherical shells placed in a plane standing wave field. Ultrasonics 43:681–691

    Article  Google Scholar 

  31. Doinikov AA (1997) Acoustic radiation force on a spherical particle in a viscous heat-conducting fluid. I. General formula. J Acoust Soc Am 101:713–721

    Article  Google Scholar 

  32. Doinikov AA (1997) heat-conducting fluid. I. General formula 101:713–721

  33. Danilov SD, Mironov MA (2000) Mean force on a small sphere in a sound field in a viscous fluid. J Acoust Soc Am 107:143–153. https://doi.org/10.1121/1.428346

    Article  Google Scholar 

  34. Karlsen JT, Bruus H (2015) Forces acting on a small particle in an acoustical field in a thermoviscous fluid. Phys Rev E Stat Nonlinear Soft Matter Phys. https://doi.org/10.1103/PhysRevE.92.043010

    Article  Google Scholar 

  35. Embleton TFW (1962) Mutual interaction between two spheres in a plane sound field. J Acoust Soc Am 34:1714–1720

    Article  MathSciNet  Google Scholar 

  36. Crum LA (1975) Bjerknes forces on bubbles in a stationary sound field. J Acoust Soc Am 57:1363–1370

    Article  Google Scholar 

  37. Doinikov AA, Zavtrak ST (1997) Radiation forces between two bubbles in a compressible liquid. J Acoust Soc Am 102:1424–1431

    Article  Google Scholar 

  38. Bjerknes V (1906) Fields of Force: a course of lectures in mathematical physics delivered December 1 to 23, 1905, Columbia University Press

  39. Silva GT, Bruus H (2014) Acoustic interaction forces between small particles in an ideal fluid. Phys Rev E 90:63007

    Article  Google Scholar 

  40. Sepehrirahnama S, Chau FS, Lim K-M (2015) Numerical calculation of acoustic radiation forces acting on a sphere in a viscous fluid. Phys Rev E 92:63309

    Article  MathSciNet  Google Scholar 

  41. Sepehrirahnama S, Lim K-M, Chau FS (2015) Numerical study of interparticle radiation force acting on rigid spheres in a standing wave. J Acoust Soc Am 137:2614–2622. https://doi.org/10.1121/1.4916968

    Article  Google Scholar 

  42. Sepehrirahnama S, Chau FS, Lim KM (2016) Effects of viscosity and acoustic streaming on the interparticle radiation force between rigid spheres in a standing wave. Phys Rev E. https://doi.org/10.1103/PhysRevE.93.023307

    Article  Google Scholar 

  43. Wang J, Dual J (2009) Numerical simulations for the time-averaged acoustic forces acting on rigid cylinders in ideal and viscous fluids. J Phys A Math Theor 42:285502

    Article  MathSciNet  MATH  Google Scholar 

  44. Haydock D (2005) Lattice Boltzmann simulations of the time-averaged forces on a cylinder in a sound field. J Phys A Math Gen 38:3265

    Article  MathSciNet  MATH  Google Scholar 

  45. Wijaya FB, Lim K-M (2015) Numerical calculation of acoustic radiation force and torque acting on rigid non-spherical particles. Acta Acust United Acust 101:531–542

    Article  Google Scholar 

  46. Wijaya FB, Sepehrirahnama S, Lim K-M (2018) Interparticle force and torque on rigid spheroidal particles in acoustophoresis. Wave Motion 81:28–45

    Article  MathSciNet  MATH  Google Scholar 

  47. Fisher KA, Miles R (2008) Modeling the acoustic radiation force in microfluidic chambers. J Acoust Soc Am 123:1862–1865

    Article  Google Scholar 

  48. Glynne-Jones P, Mishra PP, Boltryk RJ, Hill M (2013) Efficient finite element modeling of radiation forces on elastic particles of arbitrary size and geometry. J Acoust Soc Am 133:1885–1893

    Article  Google Scholar 

  49. Garbin A, Leibacher I, Hahn P, Le Ferrand H, Studart A, Dual J (2015) Acoustophoresis of disk-shaped microparticles: a numerical and experimental study of acoustic radiation forces and torques. J Acoust Soc Am 138:2759–2769

    Article  Google Scholar 

  50. Bruus H (2012) Acoustofluidics 2: perturbation theory and ultrasound resonance modes. Lab Chip 12:20–28

    Article  Google Scholar 

  51. Zhou S, Rabczuk T, Zhuang X (2018) Phase field modeling of quasi-static and dynamic crack propagation: COMSOL implementation and case studies. Adv Eng Softw 122:31–49. https://doi.org/10.1016/J.ADVENGSOFT.2018.03.012

    Article  Google Scholar 

  52. Emami MM, Jamshidian M, Rosen DW (2021) Multiphysics modeling and experiments of grayscale photopolymerization with application to microlens fabrication. J Manuf Sci Eng Trans ASME. https://doi.org/10.1115/1.4050549/1104368

    Article  Google Scholar 

  53. Samaniego E, Anitescu C, Goswami S, Nguyen-Thanh VM, Guo H, Hamdia K, Zhuang X, Rabczuk T (2020) An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Comput Methods Appl Mech Eng 362:112790. https://doi.org/10.1016/J.CMA.2019.112790

    Article  MathSciNet  MATH  Google Scholar 

  54. Settnes M, Bruus H (2012) Forces acting on a small particle in an acoustical field in a viscous fluid. Phys Rev E Stat Nonlinear Soft Matter Phys 85:1–12. https://doi.org/10.1103/PhysRevE.85.016327

    Article  Google Scholar 

  55. Bruus H (2012) Acoustofluidics 7: The acoustic radiation force on small particles. Lab Chip 12:1014–1021. https://doi.org/10.1039/c2lc21068a

    Article  Google Scholar 

  56. Leibacher I, Dietze W, Hahn P, Wang J, Schmitt S, Dual J (2014) Acoustophoresis of hollow and core-shell particles in two-dimensional resonance modes. Microfluid Nanofluidics 16:513–524

    Article  Google Scholar 

  57. Marston PL, Wei W, Thiessen DB (2006) Acoustic radiation force on elliptical cylinders and spheroidal objects in low frequency standing waves. AIP Conf Proc 838:495–499. https://doi.org/10.1063/1.2210403

    Article  Google Scholar 

  58. Fan Z, Mei D, Yang K, Chen Z (2008) Acoustic radiation torque on an irregularly shaped scatterer in an arbitrary sound field. J Acoust Soc Am 124:2727. https://doi.org/10.1121/1.2977733

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, 84156-83111, Iran

    S. M. Zareei & S. Ziaei-Rad

  2. National University of Singapore, 9 Engineering Drive 1, Singapore, 117575, Singapore

    S. Sepehrirahnama

  3. Institute of High Performance Computing, A*STAR, 1 Fusionopolis Way, #16-16 Connexis, Singapore, 138632, Singapore

    M. Jamshidian

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Zareei, S.M., Sepehrirahnama, S., Jamshidian, M. et al. Three-dimensional numerical simulation of particle acoustophoresis: COMSOL implementation and case studies. Engineering with Computers 39, 735–750 (2023). https://doi.org/10.1007/s00366-022-01663-0

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