Theoretical framework to predict the acoustophoresis of axisymmetric irregular objects above an ultrasound transducer array

Acoustic radiation force and torque arising from wave scattering are able to translate and rotate matter without contact. However, the existing research mainly focused on manipulating simple symmetrical geometries, neglecting the significance of geometric features. For the nonspherical geometries, the shape of the object strongly affects its scattering properties, and thus the radiation force and torque as well as the acoustophoretic process. Here, we develop an analytical framework to calculate the radiation force and torque exerted on the sound-hard or the sound-soft, axisymmetric particles excited by a user-customized transducer array based on a conformal transformation approach, capturing the significance of the geometric features. The derivation framework is established under the computation coordinate system (CCS), whereas the particle is assumed to be static. For the dynamic processes, the rotation of particle is converted as the opposite rotation of transducer array, achieved by employing a rotation transformation to tune the incident driving field in the CCS. Later, the obtained radiation force and torque in the CCS should be transformed back to the observation coordinate system for force and torque analysis. The radiation force and torque exerted on particles with different orientations are validated by comparing the full three-dimensional numerical solution in different phase distributions. It is found that the proposed method presents superior computational accuracy, high geometric adaptivity, and good robustness to various geometric features, while the computational efficiency is more than 100 times higher than that of the full numerical method. Furthermore, it is found that the dynamic trajectories of particles with different geometric features are completely different, indicating that the geometric features can be a potential degree of freedom to tune acoustophoretic process. The ability to predict the acoustophoretic process of nonspherical particles above a user-customized transducer array has improved our understanding of the effect of shape asymmetry, which can also be used to verify the effectiveness of acoustic tweezers in manipulating nonspherical objects.

Figure 1

Rotation transformation between the computation coordinate system $\left(x,y,z\right)$ and the observation coordinate system $\left(x^{\prime },y^{\prime },z^{\prime }\right)$.

Figure 2

Geometric description of the position relationship of transducers in the OCS and the CCS. The probe transducer (marked as $q$) and the source transducers (marked as $j$) can be linked by a relative position vector ${\stackrel{\rightharpoonup }{r}}^{\left(jq\right)}$ under the CCS, in which the potential field from the source transducer can be expressed in the probe transducer system $\left(x_{\mathrm{t},q},y_{\mathrm{t},q},z_{\mathrm{t},q}\right)$ with the help of the translation addition theorem [Eq. (9)]. Between the OCS and the CCS, a rotation transformation of Eq. (7) is needed to transform the relative position vector from ${\stackrel{\rightharpoonup }{r}}^{{}^{\prime }\left(jq\right)}$ to ${\stackrel{\rightharpoonup }{r}}^{\left(jq\right)}$.

Figure 3

Conformal transformation mapping of an axisymmetric particle in physical space to a sphere in mapping space. The particle is symmetric about the $z$ axis. The particle surface in the mapping coordinate $\left(u,w,v\right)$ system can be described by the radial coordinate $u=u_0$, independent with the polar angular coordinate, $w$, and the azimuthal angular coordinate, $v$. On the $zOf$ slice plane, defined as arbitrary cross-sectional plane along the symmetric axis ($z$ axis), the radial coordinate satisfies $r_{\mathrm{s}}\left(u_0,w\right)=\sqrt{f^2\left(u_0,w\right)+g^2\left(u_0,w\right)}$ in the real system, while $r_{\mathrm{s}}\left(\theta \right)e^{i\theta }=g\left(u_0,w\right)+f\left(u_0,w\right)i$ in the complex system (the azimuthal angular coordinates, $\varphi$ or $v$, are not involved for the axisymmetric reason). The mapping functions $g\left(u_0,w\right)$ and $f\left(u_0,w\right)$ are introduced to connect the physical space and the mapping space.

Figure 4

RMSE of the normalized radial intensity (${\overline{I}}_{\mathrm{r}}$) between the modal expansion approximations and the theoretical results along the probe arc ($\theta \in \left[-{\theta }_{\mathrm{apx}},{\theta }_{\mathrm{apx}}\right]$). (a) The geometric relationship among the approximated space, the probe circle, and the probe arc. The approximated space is a spherical domain with a radius of $R,$ and its center is consistent with the mass center of the contained scatterer. The radius of the probe circle is $d_{\mathrm{t}}$ and its center is located at the center of the transducer surface. The intersection of the approximated space and the probe circle is defined as the probe arc. (b)–(g) Visualization of the differences of the normalized radial intensity along the probe arc. The solid and dashed red curves denote the results based on the modal expansion series and theoretical solutions, respectively. (h) RMSE of ${\overline{I}}_r$ along the probe arc as a function of $R$ and $d_{\mathrm{t}}$. The brown dots represent the cases given in (b)–(g).

Figure 5

Comparisons of the radiation force ${\stackrel{\rightharpoonup }{F}}_{\mathrm{rad}}$ and torque ${\stackrel{\rightharpoonup }{T}}_{\mathrm{rad}}$ acting on the particles with different geometric features (average radius of $a=2\mathrm{mm}$) based on the analytical expansion series [Eqs. (25) and (26)] and the FEM results in a five-transducer system. The radiation force and torque are plotted as a function of the rotation angle along the $x^{\prime }$ axis, ${\theta }_{\mathrm{rot},x^{\prime }}$, under different phase distributions. Subfigures (a), (d), and (g) display the predictions of the radiation force, while (b), (e), and (h) are of the radiation torque for (c) an ellipsoid, (f) a cone, and (i) a diamond, respectively. The circle marks represent the results based on the full three-dimensional FEM. In contrast, the dotted, dashed, and solid curves mean the data collected based on the analytical method along the $x^{\prime }$-, the $y^{\prime }$-, and the $z^{\prime }$ axes, respectively. The radial velocity of the transducers are all set to ${\widehat{v}}_0=1.5\mathrm{m}/\mathrm{s}$, while four groups of phase distributions are applied to the transducer array.

Figure 6

Translational and rotational dynamics of different particles with the same averaged radius of 0.5 mm. The translational trajectories of different particles at different moments along (a) $x^{\prime }$-, (b) $y^{\prime }$-, and (c) $z^{\prime }$ axes. The rotational angles of different particles at different moments along (d) $x^{\prime }$-, (e) $y^{\prime }$-, and (f) $z^{\prime }$ axes. If necessary, the gray regions are zoomed in for more details of the corresponding figures. (g) Three-dimensional translational trajectory (solid line) and particle orientation (arrow). The arrows represent the symmetric axis of the particle, while the color of the arrow is used to represent the increase of time. The time intervals represented by any adjacent arrows are the same. These are 20 arrows showing the position and orientation of the particles from the start of the calculation (red arrow) to the end of the calculation (yellow arrow). Note that only the trajectory and orientation of the ellipsoidal particle are visualized, as the trajectory and orientation for other particles are not significantly different.

Figure 7

The same as in Fig. 6, but increasing the averaged radius to 1 mm. Note that only the trajectory and orientation of the ellipsoidal particle are visualized in (g), as the trajectory and orientation for other particles are not significantly different.

Figure 8

The same as in Fig. 6, but increasing the averaged radius to 2 mm.

Figure 9

The same as in Fig. 6, but increasing the averaged radius to 3 mm.