Acoustic interaction forces between small particles in an ideal fluid

We present a theoretical expression for the acoustic interaction force between small spherical particles suspended in an ideal fluid exposed to an external acoustic wave. The acoustic interaction force is the part of the acoustic radiation force on one given particle involving the scattered waves from the other particles. The particles, either compressible liquid droplets or elastic microspheres, are considered to be much smaller than the acoustic wavelength. In this so-called Rayleigh limit, the acoustic interaction forces between the particles are well approximated by gradients of pair-interaction potentials with no restriction on the interparticle distance. The theory is applied to studies of the acoustic interaction force on a particle suspension in either standing or traveling plane waves. The results show aggregation regions along the wave propagation direction, while particles may attract or repel each other in the transverse direction. In addition, a mean-field approximation is developed to describe the acoustic interaction force in an emulsion of oil droplets in water.

Figure 1

Sketch of the external incident wave ${\varphi }_{\mathrm{in}}\left(\mathit{r}\right)$ (red straight lines) scattered by a suspension of small spherical particles with radii $a_s\ll \lambda$. The scattered wave ${\varphi }_{\mathrm{sc}}\left({\mathit{r}}_p\right|{\mathit{r}}_s)$ (green dashed curves) from a source particle located at ${\mathit{r}}_s$ (black sphere) is probed at the position ${\mathit{r}}_p$.

Figure 2

The acoustic interaction pair potential $U\left({\mathit{r}}_p\right|\mathbf{0})$ [Eq. (14), contours] and force ${\mathit{F}}_{\mathrm{int}}^{\mathrm{rad}}\left({\mathit{r}}_p\right|\mathbf{0})=-\nabla U$ (arrows) between a pair of identical 12-$\mu \mathrm{m}$ particles induced by the traveling plane wave (TPW) Eq. (13), with the source particle located at the origin, ${\mathit{r}}_{\text{s}}=\mathbf{0}$, for $kr>0.2$. (a) Silicone oil droplets (oil-oil), with the probe ${\mathit{r}}_p=(x,y,0)$ in the transverse $xy$ plane. (b) Same as (a) but for polystyrene microparticles (ps-ps). (c) Same as (a), but with the probe ${\mathit{r}}_p=(x,0,z)$ in the parallel $xz$ plane (oil-oil). (d) Same as (b), but with the probe ${\mathit{r}}_p=(x,0,z)$ in the parallel $xz$ plane (ps-ps).

Figure 3

The acoustic interaction pair potential $U\left({\mathit{r}}_p\right|\mathbf{0})$ [Eq. (19), contours] and force ${\mathit{F}}_{\mathrm{int}}^{\mathrm{rad}}\left({\mathit{r}}_p\right|\mathbf{0})=-\nabla U$ (arrows) between a pair of identical 12-$\mu \mathrm{m}$ particles induced by the standing plane wave (SPW) Eq. (18), with $kh=\pi /2$ and the source particle located at the origin, ${\mathit{r}}_{\text{s}}=\mathbf{0}$ for $kr>0.2$. (a) Silicone oil droplets (oil-oil), with the probe ${\mathit{r}}_p=(x,y,0)$ in the transverse $xy$ plane. (b) Same as (a) but for polystyrene microparticles (ps-ps). (c) Same as (a), but with the probe ${\mathit{r}}_p=(x,0,z)$ in the parallel $xz$ plane (oil-oil). (d) Same as (b), but with the probe ${\mathit{r}}_p=(x,0,z)$ in the parallel $xz$ plane (ps-ps).

Figure 4

The acoustic interaction pair potential $U\left({\mathit{r}}_p\right|\mathbf{0})$ [Eq. (19), contours] and force ${\mathit{F}}_{\mathrm{int}}^{\mathrm{rad}}\left({\mathit{r}}_p\right|\mathbf{0})=-\nabla U$ (arrows) between a pair of identical 12-$\mu \mathrm{m}$ particles induced by the standing plane wave (SPW) Eq. (18), with the source particle located at the origin, ${\mathit{r}}_{\text{s}}=\mathbf{0}$ for $kr>0.2$. (a) Rigid microparticles (rigid-rigid), with $f_{0,p}=f_{0,s}=f_{1,p}=f_{1,s}=1$, $kh=\pi /2$ (antinode), and the probe ${\mathit{r}}_p=(x,y,0)$ in the transverse $xy$ plane. (b) Same as (a) but for microbubbles (bubble-bubble) with $f_{0,p}=f_{0,s}\approx -{10}^5$, $f_{1,p}=f_{1,s}\approx -2$, and $kh=0$ (node). (c) Same as (a), but with the probe ${\mathit{r}}_p=(x,0,z)$ in the parallel $xz$ plane (rigid-rigid). (d) Same as (b), but with the probe ${\mathit{r}}_p=(x,0,z)$ in the parallel $xz$ plane (bubble-bubble).

Figure 5

Same as Figs. 3 and 3, but with two different particles (oil-ps): a 12-$\mu \mathrm{m}$ silicone oil droplet as the probe particle at ${\mathit{r}}_p=\mathit{r}$ and a 12-$\mu \mathrm{m}$ polystyrene particle as the source particle at $\mathit{r}=\mathbf{0}$.

Figure 6

The normalized interaction potential $\mathcal{U}/{\mathcal{U}}_0$ (contour plot) of the acoustic interaction force ${\mathit{F}}_{\mathrm{int}}^{\mathrm{rad}}$ (arrows) in an aqueous emulsion of soybean oil droplets of radius $a_s=12$ $\mu \mathrm{m}$ in the external standing plane wave (SPW) defined in Eq. (18). The emulsion consists of $N\approx 3800$ droplets in a cylindrical disk region of radius $R=5$ mm and thickness $2a_s$ at the nodal $xy$ plane. Here ${\mathcal{U}}_0=0.2$ fJ.