Interparticle attraction along the direction of the pressure gradient in an acoustic standing wave

Scattering of an acoustic wave by particles gives rise to microstreaming, as well as to acoustic radiation and interaction forces on the particles. We numerically study these steady, nonlinear phenomena for a case of two elastic spheres in a standing wave. We show that if one or both spheres are smaller or comparable to the viscous boundary layer, the microstreaming close to the pressure node can lead to an interparticle attraction along the direction of the pressure gradient of the wave. Similar behavior is observed when, instead of size, density of one of the spheres is sufficiently larger relative to the other sphere. These findings could promote the acoustic manipulation of nanoparticles and bacteria.

Figure 1

The comsol model of two particles is parametrized in the cylindrical coordinate system $(r,\theta ,z)$. The model is symmetric with respect to the $z$ axis, which is parallel to the direction of the pressure gradient of the one-dimensional incident standing wave. Interparticle distance is defined as the surface-to-surface distance between the two particles along the $z$ axis. Perfectly matched layer (PML) surrounds the fluid domain.

Figure 2

Acoustic forces on Particle 2 approaching Particle 1 that is centered at the pressure node. Particle 2 is located in the positive $z$ direction from Particle 1, along the $z$ axis. Forces with respect to the surface-to-surface interparticle distance ($1\mu \mathrm{m}\to 10\mu \mathrm{m}$) are shown in panels (a)–(g) for different combinations of polystyrene (PS) and copper (Cu) particles of $5\mu \mathrm{m}$ (5), $1\mu \mathrm{m}$ (1), and $0.5\mu \mathrm{m}$ (0.5) radii. In water, at $f=500\mathrm{k}\mathrm{Hz}$ ($\delta =0.74\mu \mathrm{m}$), with the pressure amplitude of $500\mathrm{k}\mathrm{Pa}$. “AIF - comsol” is the AIF on Particle 2 resulting from our model; “AIF - Silva & Bruus” is the AIF from the inviscid theory [25]; “Total - comsol” is the total acoustic force on Particle 2 from the two-particle configuration of our model. The reversal of the AIF on Particle 2 is demonstrated in panels (b), (d), (f), (g). The associated transitions in the Eulerian streaming patterns are shown in panels (h) and (i), for two polystyrene particles, of $5\mu \mathrm{m}$ radius (Particle 1, green) and $0.5\mu \mathrm{m}$ radius (Particle 2, red), respectively. The streaming around two particles of $5\mu \mathrm{m}$ radius, but of different material, namely, copper (Particle 1, green) and polystyrene (Particle 2, red), are given in panels (j) and (k). Plots (i) and (k) depict streaming at $9\mu \mathrm{m}$ surface-to-surface distance, plot (h) at the distance of $1\mu \mathrm{m}$, and plot (j) at the distance of $3\mu \mathrm{m}$. Particle 1 (green) is in all cases centered at the pressure node. Brightness of color indicates the magnitude of the streaming velocity, with the maximal magnitude of $0.59\mathrm{m}\mathrm{m}{\mathrm{s}}^{-1}$ in panels (h), (i), and $3.4\mathrm{m}\mathrm{m}{\mathrm{s}}^{-1}$ in panels (j), (k).

Figure 3

Regions of attraction and repulsion of Particle 2 (radius $a_2$) to/from Particle 1 (radius $a_1$) that is centered at the pressure node. In water, at $f=500\mathrm{k}\mathrm{Hz}$ ($\delta =0.74\mu \mathrm{m}$), with the pressure amplitude of $500\mathrm{k}\mathrm{Pa}$. Particle 2 is located in the positive $z$ direction from Particle 1, along the $z$ axis. The surface-to-surface distance between the particles is $1\mu \mathrm{m}$; $\times$ indicates $a_1=a_2=5\mu \mathrm{m}$; $•$ indicates $a_1=a_2=1\mu \mathrm{m}$. Coloring represents the AIF on Particle 2, normalized by the average surface area of the two particles $S_{\mathrm{avg}}$. Regions on the darker side of the dashed line, identifying $AIF=0$, represent the attraction of Particle 2 towards Particle 1, while the brighter regions indicate the repulsion. Particle 2 is made of polystyrene in panels (a), (d), (e), (f), whereas Particle 1 varies between polystyrene in panel (a), copper in panel (d), titanium in panel (e), and glass in panel (f). The combination of two copper particles is shown in panel (c), and the combination of polystyrene Particle 1 and copper Particle 2 in panel (b)

Figure 4

Acoustic forces on Particle 2 approaching Particle 1 that is centered at the pressure node. Particle 2 is located in the positive $z$ direction from Particle 1, along the $z$ axis. Forces on Particle 2 are plotted with respect to the surface-to-surface interparticle distance ($10\mu \mathrm{m}\to 20\mu \mathrm{m}$) for different combinations of polystyrene (PS) and copper (Cu) particles of $5\mu \mathrm{m}$ (5), $1\mu \mathrm{m}$ (1), and $0.5\mu \mathrm{m}$ (0.5) radii. In water, at $f=500\mathrm{k}\mathrm{Hz}$ ($\delta =0.74\mu \mathrm{m}$), with the pressure amplitude of $500\mathrm{k}\mathrm{Pa}$. “AIF - comsol” is the AIF resulting from our model; “AIF - Silva & Bruus” is the AIF from the inviscid theory [25]; “Total - comsol” is the total force on Particle 2 in the two-particle configuration resulting from our model; “AIF noStr - comsol” is the AIF from our model simplified by using assumptions from Eq. (4). The microstreaming causes the AIF to retain a significant magnitude relative to the total force at large interparticle distances in panels (a)–(c).