Microparticles introduced in an acoustic field act as scatterers. The incident and scattered acoustic fields interact, and result in a second-order time-averaged primary acoustic radiation force. The analysis of the acoustic radiation force dates back to the work of King in the 1930’s. Treatment of both standing and travelling acoustic fields were carried out on incompressible spheres, much smaller in size than the wavelength of the field, at the Rayleigh scattering limit (size << wavelength). Yosioka and Kawasima extended this discussion by introducing compressibility of the spheres. These results were summarized and reformulated by Gorkov and a compact equation for the acoustic radiation force in standing wave fields was provided as a gradient of the acoustic potential. Recently, more complex multiphysical phenomena, such as thermoviscous effects or multi-phase fluids were discussed. See References at the end of this blog post for details.

### Radiation force in a standing wave field

The simplest example to consider is the acoustic radiation force on spherical particles in compressible fluid in a standing wave. A travelling acoustic wave can be generated by sinusoidal excitation of an acoustic transducer:

Adding a second transducer, the opposing travelling waves will combine and create a standing wave:

Where the resulting acoustic radiation force is plotted as well. A simplified static image can be given:

The acoustic pressure is denoted in orange. (If total pressure is *p*, the atmospheric pressure is *p*_{0}, then acoustic pressure is *p*_{ac} = *p* – *p*_{0}.) As time elapses, this oscillates, with maximum amplitude at the antinodes, and minimum (ideally zero) amplitude at the nodes. A standing wave that is formed from two head-on travelling waves, has half wavelength distance between acoustic pressure nodes or antinodes.

Now, for most particles and cellular components (main exceptions are lipids or air bubbles), the so-called acoustic contrast factor is positive, meaning that the particles tend to move away from the pressure antinodes towards the pressure nodes. The exact force distribution is sinusoidal, as plotted in red in Fig. 3. Note the period is half compared to the pressure distribution, i.e. the force is proportional to sin(2*ky*), where *k* is the wave number. Blue arrows emphasize the force direction for positive contrast. For negative contrast, the red curve and blue arrows reverse.

For spherical particles in a compressible fluid, the equation has the form:

_{}

where *a*_{p} particle radius, *?* (rho) is density, *?* (kappa) compressibility (inverse of bulk modulus), subscript p is a particle, subscript 0 a fluid property. The angled brackets are time averages, *p*_{in} acoustic pressure, *u*_{in} fluid particle velocity (NOT the same as bulk fluid velocity).

### Simulation results for in-phase plane travelling wave sources

As a very simple approach, we can create a 1.5D model (two elements wide, symmetry on the sides), apply impedance match to the top and bottom, and launch pressure waves here, so they combine into a standing wave. The whole model is a fully water domain.

The acoustic pressure and velocity values are recorded along the centerline, starting at the middle of the model, up to a distance of half the wavelength. Then the above equation (Figure 4) can be directly used to calculate acoustic radiation force. It is averaged over a period, in a moving window approach.

A video of wave propagation (left plot acoustic pressure, right plot *y* velocity):

Here note:

- The middle of the model is an acoustic pressure antinode. This is expected, when two travelling waves are launched head-on, at the same distance from the midpoint
- The acoustic radiation force should move particles away from this point. This can be a first principle check when radiation force results are obtained
- The particle velocity is out-of-phase both spatially and temporally. Spatially there is a velocity node at the midpoint. Temporally there is a 90° phase difference, i.e. one can be described as a ‘cosine’ the other a ‘sine’, as expected from the Euler equation (shown at Eq. 3.15 in Ref. 1, or in Ref. 7 or similar)

Now, the acoustic radiation force can be calculated for a 100 kPa field (this is common, and was used in the simulation), and also using the above equation and the numerical results from the simulation:

There is excellent agreement between theoretical and simulation results. They also pass the first principle test: the 0 normalised position corresponds to the midpoint of the model (pressure antinode), and as we go away (upwards) the force is positive (upwards) as the figure illustrates, forcing particles away from the pressure antinode. At 0.25 normalised position (pressure node) the force below is positive (towards) and above negative (still towards), so particles would indeed move away from pressure antinodes and towards pressure nodes. Note that some literature refers to velocity nodes/antinodes instead and can be confusing, in this case the force moves particles towards velocity antinodes and away from velocity nodes.

Now do the following analysis: instead of averaging on a cycle and plotting that single result, move the averaging window over the results, and plot the resulting radiation force in a spatio-temporal 3D plot:

The first few cycles are omitted where the standing wave is not yet stabilised. Otherwise, as expected, the force does not change temporally, only spatially. Any slice of this graph at a given time instant is equivalent to the 2D graph (Fig. 5).

### Simulation results for out-of-phase plane travelling wave sources

When the phase of the two transducers are not the same, the standing wave pattern will move away from the larger phase transducer. The acoustic radiation force pattern will follow. Investigating the above example with the bottom source having +90° phase compared to the top plane wave source:

Observe that after the initial transient period, the pattern indeed moves up towards +y as expected. The spatial and temporal difference between the pressure and velocity stays the same as before.

The acoustic radiation force moves the same way as the pattern, which manifests itself in a negative argument: *F* = *F*_{0}*sin(2*ky* – 90°), where *F*_{0} is a constant force magnitude:

As expected, the force pattern is temporally the same as the phase modulation is a constant value:

Note here that as previously, any slice of this plot at a given *t* gives Fig. 7.

In this blog post, we reviewed the acoustic radiation force on small spherical particles in acoustic standing wave fields. Stay tuned for our next blog post on the topic discussing the use of transducers to generate pressure fields and investigating modulated fields for more dexterous particle manipulation.

*References*

*Simon, “Modulated Ultrasound-enabled Particle and Cell Separation in Surface Acoustic Wave Microfluidic Devices”, PhD Thesis, Heriot-Watt University, United Kingdom, 2019.**V. King, “On the acoustic radiation pressure on spheres”, Proceedings of the Royal Society of London. Series A – Mathematical and Physical Sciences, vol. 147, no. 861, pp. 212-240, 1934.**Yosioka and Y. Kawasima, “Acoustic radiation pressure on a compressible sphere”, Acta Acustica united with Acustica, vol. 5, no. 3, pp. 167-173, 1955.**P. Gorkov, “On the forces acting on a small particle in an acoustic field in an ideal fluid”, Soviet Physics Doklady, vol. 6, no. 9, pp. 773-775, 1962.**Bruus, “Acoustofluidics 7: The acoustic radiation force on small particles”, Lab Chip, vol. 12, no. 6, pp. 1014-21, 2012.**Settnes and H. Bruus, “Forces acting on a small particle in an acoustical field in a viscous fluid”, Phys Rev E Stat Nonlin Soft Matter Phys, vol. 85, no. 1 Pt 2, p. 016327, 2012.**E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders, Fundamentals of Acoustics. Wiley, 1999.*