Acoustic radiation force (ARF) is a physical phenomenon resulting from the interaction of anacoustic wave with an obstacle placed along its path. Generally, the force exerted on the obstacle is evaluated by integrating theacoustic 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:

${\displaystyle |F^{\mathrm{r}\mathrm{a}\mathrm{d}}|=\frac{2\alpha I}{c}}$

where

• ${\displaystyle |F^{\mathrm{r}\mathrm{a}\mathrm{d}}|}$ is a force per unit volume, here expressed in kg/(s²cm²);
• ${\displaystyle \alpha }$ is theabsorption coefficient inNp/cm (nepers per cm);
• ${\displaystyle I}$ is the temporal averageintensity of the acoustic wave at the given location inW/cm²; and
• ${\displaystyle c}$ is thespeed of sound in the medium in cm/s.^[1][2]

The effect offrequency 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 0.002dB/(MHz²cm).^{[3](page number?)} Acoustic radiation forces on compressible particles such asbubbles are also known asBjerknes forces, and are generated through a different mechanism, which does not require soundabsorption orreflection.^[4] Acoustic radiation forces can also be controlled through sub-wavelength patterning of the surface of the object.^[5]

When a particle is exposed to an acoustic standing wave it will experience a time-averaged force known as the primary acoustic radiation force (${\displaystyle F_{pr}}$)_.^[6] In a rectangular microfluidic channel with coplanar walls which acts as aresonance chamber, the incoming acoustic wave can be approximated as aresonant,standing pressure wave of the form:

${\displaystyle p_1=p_a\cos kz}$.

where${\displaystyle k}$ is thewave number.For acompressible, spherical andmicrometre-sized particle (of radius${\displaystyle a}$) suspended in aninviscid 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^[7][8][9][6]:

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

${\displaystyle \mathrm{\Phi }(\tilde{\kappa },\tilde{\rho })=\frac{1}{3}\left[\frac{5\tilde{\rho }-2}{2\tilde{\rho }+1}-\tilde{\kappa }\right]}$

${\displaystyle E_{\mathrm{a}\mathrm{c}}=\frac{1}{4}{\kappa }_f{p}_a^2=\frac{p_a^2}{4{\rho }_f{c}_f^2}}$

where

• ${\displaystyle \mathrm{\Phi }}$ is theacoustic contrast factor
• ${\displaystyle \tilde{\kappa }}$ is relativecompressibility between the particle${\displaystyle {\kappa }_p}$ and the surrounding fluid${\displaystyle {\kappa }_f}$:${\displaystyle \tilde{\kappa }={\kappa }_p/{\kappa }_f}$
• ${\displaystyle \tilde{\rho }}$ is relativedensity between the particle${\displaystyle {\rho }_p}$ and the surrounding fluid${\displaystyle {\rho }_f}$:${\displaystyle \tilde{\rho }={\rho }_p/{\rho }_f}$
• ${\displaystyle E_{\mathrm{a}\mathrm{c}}}$ is the acousticenergy density
• The factor${\displaystyle \sin 2kz}$ makes the radiation force period doubled andphase shifted relative to the pressure wave${\displaystyle p_a\cos kz}$
• ${\displaystyle c_f}$ is thespeed of sound in the fluid

• Acoustic tweezers
• Radiation pressure

• Acoustics

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