Generalized potential theory for close-range acoustic interactions in the Rayleigh limit

Under an external acoustic field, particles experience radiation forces that bring them to certain trapping locations, such as pressure or velocity nodes for the case of plane standing wave. Due to acoustical interactions, particles form clusters on reaching those trapping locations. In this work, by using the far-field evaluation of scattered fields, a generalized force potential is formulated that gives both the primary and interaction forces for particles with size much smaller than the wavelength (Rayleigh limit). The generalized potential for the primary force is the same as the Gorkov's potential. The interaction potential and forces between a pair of particles at the zero-primary-force locations are studied for the two cases of planar and nonplanar (Bessel) standing waves. It was found that the interaction forces are predominantly dependent on the product of the external acoustic field and the scattered fields from the adjacent particles. Besides the line formation, other cluster shapes are shown to be plausible for three solid particles agglomerating under a plane standing wave. The mutual interaction force between particles of different material properties was found to be not equal and opposite in general, suggesting that they do not form an action and reaction pair. From the interaction patterns due to the nonplanar field of a Bessel standing wave, it is inferred that many cluster configurations are possible since particles near the stable trapping locations attract each other from more than one direction. The advantage of using the generalized force potential is that it provides physical insight for the acoustical manipulation of small particles in any external field with arbitrary wave front, such as those used in acoustic holography.

Figure 1

Ensemble of $N$ particles in an external acoustic field and the illustration of near-field and far-field regions.

Figure 2

The normalized interaction potential $g_{\mathrm{in}}$ in the $x\text{−}z$ plane for a sphere pair with the same size and the source sphere located at the origin of the coordinate system on the pressure nodal plane, denoted by P.N.

Figure 3

Projection of the normalized interaction potential $g_{\mathrm{in}}$ for the probe sphere on the $x\text{−}z$ plane for a pair of spheres under a plane standing wave. Panels (a) and (c) show the contribution from the product of incident field and the scattered field of the source particle, denoted by $[0,s]$, in Eq. (21). The contributions from the self-product term, denoted by $[s,s]$, are shown in panels (b) and (d).

Figure 4

Magnitude and direction angle of the interaction forces acting on the probe and source spheres, placed in the $x\text{−}z$ plane. Panels (a) and (b) show the forces for the case of two solid spheres, while (c) and (d) show the case of solid-bubble interactions. The probe sphere is shifted along a path parallel to the nodal plane at a distance of $3a$ from the plane.

Figure 5

Projection of the normalized interaction potential $g_{\mathrm{in}}$ for the probe sphere on the $x\text{−}z$ plane for three spheres with the same size under a plane standing wave. The panels in the last column show the contribution from the cross products of the scattering fields from the two source spheres, denoted by $[s_1,s_2]$ and $[s_2,s_1]$, to the partial interaction potential due to each source sphere, as expressed in Eq. (21).

Figure 6

Contrast factor of the interaction force ${\mathbf{S}}_{\mathrm{in}}$ acting on a target sphere from the two touching source spheres on the pressure nodal plane.

Figure 7

Contrast factor of the primary force $S_{\mathrm{pr}}$ with respect to the material properties of particles at $(kz=\pi /4,R=0)$, where the force is nonzero and acts only in the wave direction ($z$ axis). The results for PSW and BSW with $\gamma =\pi /3$ are shown in the left and right panels, respectively.

Figure 8

The $[0,s]$ and $[s,s]$ contributions to the interaction potential under a BSW for a pair of spheres with the source sphere being located at the intersection of the pressure nodal plane at $kz=0$ and $kR=0$, 2.78, and 4.42.

Figure 9

The $x\text{−}y$ and $x\text{−}z$ projections of the contrast factor of the interaction force under a BSW for a pair of spheres with the source sphere being located at the intersection of the pressure nodal plane at $kz=0$ and $kR=0$, 2.78, and 4.42. The vectors length are normalized and their magnitude is shown by the background contours.

Figure 10

Contrast factor of the total force for a pair of solid spheres with different locations for the source sphere at $kR=0,2.78,$ and 4.42 on the pressure nodal plane at $kz=0$. The force vectors are normalized while the background colors show their magnitude.

Figure 11

Comparison of the theoretical estimation of acoustic interaction force for a pair of PS beads in water from the experiments in Ref. [23]. The shaded band shows the area of experimental data points. The results of the present model are shown with a blue dashed line with square markers.