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## Summary

Acoustic radiation force (ARF) is a physical phenomenon resulting from the interaction of an acoustic wave with an obstacle placed along its path. Generally, the force exerted on the obstacle is evaluated by integrating the acoustic radiation pressure (due to the presence of the sonic wave) over its time-varying surface.

The magnitude of the force exerted by an acoustic plane wave at any given location can be calculated as:

${\displaystyle |F^{rad}|={\frac {2\alpha I}{c}}}$

where

• ${\displaystyle |F^{rad}|}$ is a force per unit volume, here expressed in kg/(s2cm2);
• ${\displaystyle \alpha }$ is the absorption coefficient in Np/cm (Neper/cm);
• ${\displaystyle I}$ is the temporal average intensity of the acoustic wave at the given location in W/cm2; and
• ${\displaystyle c}$ is the speed of sound in the medium in cm/s.[1][2]

The effect of frequency on acoustic radiation force is taken into account via intensity (higher pressures are more difficult to attain at higher frequencies) and absorption (higher frequencies have a higher absorption rate). As a reference, water has an acoustic absorption of 0.002 dB/(MHz2cm).[3](page number?) Acoustic radiation forces on compressible particles such as bubbles are also known as Bjerknes forces, and are generated through a different mechanism, which does not require sound absorption or reflection.[4]

When a particle is exposed to an acoustic standing wave it will experience a time-averaged force known as the primary acoustic radiation force (${\displaystyle F_{pr}}$).[5] In a rectangular microfluidic channel with coplanar walls which acts as a resonance chamber, the incoming acoustic wave can be approximated as a resonant, standing pressure wave of the form:

${\displaystyle p_{1}=p_{a}cos(kz)}$.

where ${\displaystyle k}$ is the wave number. For a compressible, spherical and micrometre-sized particle (of radius ${\displaystyle a}$) suspended in an inviscid fluid in a rectangular micro-channel with a 1D planar standing ultrasonic wave of wavelength ${\displaystyle \lambda }$, the expression for the primary radiation force (at the far-field region where ${\displaystyle a\ll \lambda }$）becomes then [6][7][8][5]:

${\displaystyle F_{pr}^{1D}=4\pi \Phi ({\tilde {\kappa }},{\tilde {\rho }})a^{3}kE_{ac}sin(2kz)}$

${\displaystyle \Phi ({\tilde {\kappa }},{\tilde {\rho }})={5{\tilde {\rho }}-2 \over 2{\tilde {\rho }}+1}-{\tilde {\kappa }}}$

${\displaystyle E_{ac}={1 \over 4}\kappa _{f}p_{a}^{2}={p_{a}^{2} \over 4\rho _{f}c_{f}^{2}}}$

where

• ${\displaystyle \Phi }$ is the acoustic contrast factor
• ${\displaystyle {\tilde {\kappa }}}$ is relative compressibility between the particle ${\displaystyle \kappa _{p}}$ and the surrounding fluid ${\displaystyle \kappa _{f}}$: ${\displaystyle {\tilde {\kappa }}=\kappa _{p}/\kappa _{f}}$
• ${\displaystyle {\tilde {\rho }}}$ is relative density between the particle ${\displaystyle \rho _{p}}$ and the surrounding fluid ${\displaystyle \rho _{f}}$: ${\displaystyle {\tilde {\rho }}=\rho _{p}/\rho _{f}}$
• ${\displaystyle E_{ac}}$ is the acoustic energy density
• The factor ${\displaystyle sin(2kz)}$ makes the radiation force period doubled and phase shifted relative to the pressure wave ${\displaystyle p_{a}cos(kz)}$
• ${\displaystyle c_{f}}$ is the speed of sound in the fluid

## References

1. ^ Palmeri, Mark; Sharma, Amy; Bouchard, Richard; Nightingale, Roger; Nightingale, Kathryn (October 2005). "A finite-element method model of soft tissue response to impulsive acoustic radiation force". IEEE Trans Ultrason Ferroelectr Freq Control. 52 (10): 1699–1712. doi:10.1109/tuffc.2005.1561624. PMC 2818996. PMID 16382621.
2. ^ McAleavey, S. A.; Nightingale, K. R.; Trahey, G. E. (June 2003). "Estimates of echo correlation and measurement bias in acoustic radiation force impulse imaging". IEEE Trans Ultrason Ferroelectr Freq Control. 50 (6): 631–641. doi:10.1109/tuffc.2003.1209550. PMID 12839175. S2CID 12815598. (subscription required)
3. ^ Szabo, Thomas L. (2013). Diagnostic ultrasound imaging: inside out (2nd ed.). Academic press. ISBN 9780126801453.
4. ^ Leighton, T.G.; Walton, A.J.; Pickworth, M.J.W. (1990). "Primary Bjerknes forces". European Journal of Physics. 11 (1): 47. doi:10.1088/0143-0807/11/1/009.
5. ^ a b Saeidi, Davood; Saghafian, Mohsen; Haghjooy Javanmard, Shaghayegh; Hammarström, Björn; Wiklund, Martin (2019). "Acoustic dipole and monopole effects in solid particle interaction dynamics during acoustophoresis". The Journal of the Acoustical Society of America. 145 (6): 3311–3319. Bibcode:2019ASAJ..145.3311S. doi:10.1121/1.5110303. ISSN 0001-4966. PMID 31255151.
6. ^ Gor'kov, Lev Petrovich (1961). Forces acting on a small particle in an acoustic field within an ideal fluid (in Russian). Dokl. Akad. Nauk SSSR. pp. 140:1, 88–91.
7. ^ Yosioka, K.; Kawasima, Y. (1955-01-01). "Acoustic radiation pressure on a compressible sphere". Acta Acustica United with Acustica. 5 (3): 167–173.
8. ^ Settnes, Mikkel; Bruus, Henrik (2012-01-30). "Forces acting on a small particle in an acoustical field in a viscous fluid". Physical Review E. 85 (1): 016327. arXiv:1110.6037. Bibcode:2012PhRvE..85a6327S. doi:10.1103/PhysRevE.85.016327. ISSN 1539-3755. PMID 22400677. S2CID 35088059.