# Acoustic Lateral Recoil Force and Stable Lift of Anisotropic Particles

Mikhail Smagin  School of Physics and Engineering, ITMO University, St.
Petersburg 197101, Russia  Ivan Toftul  School of Physics and Engineering,
ITMO University, St. Petersburg 197101, Russia  Nonlinear Physics Centre,
Research School of Physics, The Australian National University, Canberra, ACT
2601, Australia  Konstantin Y. Bliokh  Theoretical Quantum Physics Laboratory,
Cluster for Pioneering Research, RIKEN, Wako-shi, Saitama 351-0198, Japan
Centre of Excellence ENSEMBLE3 Sp. z o.o., 01-919 Warsaw, Poland  Donostia
International Physics Center (DIPC), Donostia-San Sebastián 20018, Spain
Mihail Petrov  School of Physics and Engineering, ITMO University, St.
Petersburg 197101, Russia<EMAIL_ADDRESS>
###### Abstract

Acoustic forces and torques are of immense importance for manipulation of
particles, in particular in biomedical applications. While such forces and
torques are well understood for small spherical particles with lowest-order
monopole and dipole responses, the higher-order effects for larger anisotropic
particles have not been properly investigated. Here we examine the acoustic
force and torque on an anisotropic (ellipsoid) particle and reveal two novel
phenomena. First, we describe the lateral recoil force, orthogonal to the
direction of the incident wave and determined by the tilted orientation of the
particle. Second, we find conditions for the stable acoustic lift, where the
balanced torque and force produce a stable lateral drift of the tilted
particle. We argue that these phenomena can bring about new functionalities in
acoustic manipulation and sorting of anisotropic particles including
biological objects such as blood cells.

##  I Introduction

Manipulation of objects by means of acoustic waves recently showed significant
progress, demonstrating great potential for biomedical applications, sorting
of cells, acoustic tweezers, controlled levitation, and volumetric displays
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. While acoustic tweezers are
typically used to trap isotropic spherical particles [14, 15, 16, 11], many
systems involve particles with complex shapes [17, 18, 19, 20, 21, 22, 23]
including biological objects [24]. Therefore, theoretical [25, 26, 27, 28, 29]
and experimental [29, 30] studies of acoustic field interactions with
anisotropic particles is currently in high demand.

Acoustic manipulation of complex-shape objects can be challenging, yet it
opens ways for novel functionalities and additional degrees of control. Still,
existing theoretical approaches to acoustic forces lack rigorous analysis of
nontrivial multipole interference in the scattered field and related
acoustomechanical effects. Furthermore, very few studies have properly
examined acoustic torques.

In this work, we examine acoustic forces and torques on anisotropic particles,
such as spheroid (rotational ellipsoid) particles. We employ rigorous
multipole analysis which has shown its efficiency in nanophotonics [31, 32,
33] and acoustics [34, 35, 36].

Importantly, we reveal a lateral recoil force directed orthogonally to the
incident-wave direction and originating from the interference of the monopole
and dipole contributions to the scattered field, Fig. 1. (This effect is an
acoustic counterpart of the optical recoil force originating from the
interference of electric and magnetic dipole contributions [37, 38, 39, 40,
41, 42, 43, 44].) Moreover, we find that the lateral force can be accompanied
by a balanced torque, thereby realizing stable acoustic lift, Fig. 1. In this
case, the tilted particle moves perpendicular to the incident wave without
rotations, akin to the optical lift [45, 46].

Figure 1:  Schematics of the problem. (a) An anisotropic particle in an
incident acoustic plane-wave field exhibits an asymmetric scattering response
resulting in the longitudinal and lateral acoustic force components. (b)
Stable acoustic lift appears when the particle drifts in the lateral direction
without rotation (acoustic torque is balanced).

The paper is organized as follows. First, we derive the recoil (self-action)
corrections to the acoustic force and torque in the general form involving the
wave-induced monopole and dipole moments of the particle. Second, we
investigate the lateral-force effect for small spheroid particles. Third, we
analyse the accompanying torque and find conditions for the stable acoustic
lift (involving higher-order corrections). Finally, we briefly discuss
potential application of this effect for sorting of blood cells or other
anisotropic particles and summarize our findings.

##  II Acoustic Force and Torque with recoil contributions

The most universal method to calculate the radiation force and torque from a
linear acoustic wavefield on an arbitrary scattering body is the integration
of the acoustic momentum and angular momentum flux tensors over a surface $S$
enclosing the scatterer [47, 48]. In this manner, the time-averaged acoustic
force and torque in a monochromatic sound wavefield read [49, 50, 48]:

${\mathbf{F}}=-\oint\limits_{S}\boldsymbol{\mathbf{\hat{\mathcal{T}}}}\cdot\differential\mathbf{S},\qquad{\mathbf{T}}=-\oint\limits_{S}\boldsymbol{\mathbf{\hat{\mathcal{M}}}}\cdot\differential\mathbf{S},$
(1)

where
$\boldsymbol{\mathbf{\hat{\mathcal{T}}}}=\frac{1}{4}\boldsymbol{\mathbf{\hat{I}}}\left[\beta\absolutevalue{p_{\text{tot}}}^{2}-\rho\absolutevalue{\mathbf{v}_{\text{tot}}}^{2}\right]+\frac{1}{2}\rho\real\left(\mathbf{v}_{\text{tot}}^{*}\mathbf{v}_{\text{tot}}\right)$
is the cycle-averaged momentum flux density,
$\boldsymbol{\mathbf{\hat{\mathcal{M}}}}=\mathbf{r}\times\boldsymbol{\mathbf{\hat{\mathcal{T}}}}$
is the corresponding angular momentum flux density;
$p_{\text{tot}}(\mathbf{r}),\mathbf{v}_{\text{tot}}(\mathbf{r})$ are the
complex amplitudes of the total (i.e., incident + scattered) pressure and
velocity fields, $\mathbf{v}^{*}\mathbf{v}$ should be understood as the outer
product of two vectors, $\\{v^{*}_{i}v_{j}\\}$, and $\hat{I}_{ij}=\delta_{ij}$
is the unit dyadic tensor. The real time-dependent pressure and velocity
fields are given by $\bar{p}(\mathbf{r},t)=\real[p(\mathbf{r})\mathrm{e}\mkern
1.0mu^{-{i}\mkern 1.0mu\omega t}]$ and $\bar{{\bf v}}(\mathbf{r},t)=\real[{\bf
v}(\mathbf{r})\mathrm{e}\mkern 1.0mu^{-{i}\mkern 1.0mu\omega t}]$, where
$\omega$ is the wave frequency.

Figure 2: (a) Acoustic Kerker effect and its mechanical manifestation.
Directional scattering gives rise to a recoil force in the direction opposite
to the maximum of the scattered energy flow. (b) The geometry of the problem
with a tilted spheroid particle in an incident plane-wave field.

Expressions (1) are valid for particles of any size and shape, and they
require solution of the full scattering problem to obtain the total wave field
around the particle. In most cases, this can only be done via numerical
methods. However, it is often useful to solve this problem in some
approximation to obtain simplified analytical expressions for the radiation
force and torque in terms of the incident field and properties of the
particle.

We decompose the total field outside the particle as the sum of incident and
scattered fields:
$(p_{\text{tot}},\mathbf{v}_{\text{tot}})=(p_{\text{inc}},\mathbf{v}_{\text{inc}})+(p_{\text{sc}},\mathbf{v}_{\text{sc}})$.
Substituting this decomposition into Eq. (1) yields three terms in the
decomposition of the momentum-flux tensor:
$\mathbf{F}=\oint\left[\boldsymbol{\mathbf{\hat{\mathcal{T}}}}_{\text{inc}}+\boldsymbol{\mathbf{\hat{\mathcal{T}}}}_{\text{mix}}+\boldsymbol{\mathbf{\hat{\mathcal{T}}}}_{\mathrm{recoil}}\right]\cdot\differential\mathbf{S}$.
Here $\boldsymbol{\mathbf{\hat{\mathcal{T}}}}_{\mathrm{inc}}$ includes only
the incident field and is integrated to zero because there is no change of the
momentum due to the free-space propagation;
$\boldsymbol{\mathbf{\hat{\mathcal{T}}}}_{\mathrm{mix}}$ includes terms which
involve both the incident and scattered fields; and
$\boldsymbol{\mathbf{\hat{\mathcal{T}}}}_{\text{recoil}}$ contains only the
scattered fields and thus describes self-interaction.

In general, the scattered field can be decomposed into a sum of multipole
(monopole, dipole, quadrupole, etc.) contributions [51, 52]. For particles
small compared to the wavelength, $ka\ll 1$ ($k$ is the wavenumber, $a$ is the
chracteristic size of the particle), it is sufficient to consider the monopole
$M$ and dipole $\mathbf{D}$ moments which are proportional to the particle
volume, $\propto(ka)^{3}$ [48, 49, 53]. Higher-order multipole contributions
scale as $\propto(ka)^{2n+1}$, where $n$ is the multipole order, and can be
neglected [49, see Table I in SM].

Here we derive the force and torque on a small acoustic particle which
exhibits only the monopole and dipole moments. The mixed term
$\boldsymbol{\mathbf{\hat{\mathcal{T}}}}_{\mathrm{mix}}$ in the force can be
integrated in the near-field zone using the gradient expansion of the incident
field near the particle center. The resulting expression is known in the
literature as a sum of pure monopole and dipole contributions [48, 26, 49]. In
contrast, the self-action term
$\boldsymbol{\mathbf{\hat{\mathcal{T}}}}_{\mathrm{recoil}}$ originates from
the interference of the monopole and dipole contributions and is usually
neglected [48, 26] due to the fact that it has characteristic size dependence
$\propto(ka)^{6}$ in the Rayleigh-particle approximation. Calculating the
radiation force from the mixed and recoil parts of the momentum-flux tensor,
we obtain (see Supplemental Material):

${\mathbf{F}=\underbrace{\frac{1}{2}\real\left[M^{*}\gradient
p+\rho\mathbf{D}^{*}\cdot(\gradient)\mathbf{v}\right]}_{\mathbf{F}_{\text{M}}+\mathbf{F}_{\text{D}}}-\underbrace{\frac{k^{4}}{12\pi}\sqrt{\frac{\rho}{\beta}}\real\left[M^{*}\mathbf{D}\right]}_{\mathbf{F}_{\text{recoil}}}.}$
(2)

Here we omit the index “inc” for brevity: $p\equiv p_{\text{inc}}$ and
$\mathbf{v}\equiv\mathbf{v}_{\text{inc}}$. Note that we define the monopole
moment in a slightly different way, which is related to the conventional
monopole definition as $M=\differential Q/\differential t=-{i}\mkern
1.0mu\omega Q$ [49, 51].

The new recoil term in Eq. (2) is an acoustic counterpart of the similar term
in the optical radiation force, which is produced by the interference of the
electric and magnetic dipole moments [37, 38, 39, 40, 41, 42, 43, 44]. The
interference of the monopole and dipole radiation can result in strongly
directional scattering of the field giving rise to the acoustic recoil force.
For example, Fig. 2 shows the in-phase interference between the monopole and
dipole contributions resulting in the zero backscattering. This effect is
analogous to the Kerker effect in optics [54], and it was recently explored in
acoustics [55, 34, 56]. In this case, the force $\mathbf{F}_{\mathrm{recoil}}$
will be directed oppositely to the direction of the $k$-vector and have purely
pulling nature [39, 43]. In the approximation under consideration, the total
force will be still pushing the particle along the $k$-vector due to the main
$\mathbf{F}_{\text{M}}$ and $\mathbf{F}_{\text{D}}$ terms. However, proper
engineering of the incident and scattered fields beyond the considered limit
can lead to the acoustic pulling [57, 58, 59, 60].

It should be noted that contributions of higher-order multipoles, neglected in
the monopole-dipole approximation, can compete with the recoil force. For
example, the quadrupole force contribution can have the $\propto(ka)^{5}$
dependence, since the quadrupole polarizability is connected to the second Mie
scattering coefficient $a_{2}$ which has a known scaling dependence —
$\real(a_{2})\propto(ka)^{10}$ and $\imaginary(a_{2})\propto(ka)^{5}$ [49, see
Table 1 in SM] and Ref. [53]. In optics the higher order quadrupole force or
torque can be found in Refs. [61, 62], as well as the connection between Mie
coefficients and polarizabilities [63, 64]

Using the same approach, but now integrating the angular momentum flux
$\boldsymbol{\mathbf{\hat{\mathcal{M}}}}$ in Eq. (1), we derive expression for
the acoustic torque in the monopole-dipole approximation with the self-action
taken into account (see Supplemental Material):

${\mathbf{T}=\underbrace{\frac{\rho}{2}\real\left(\mathbf{D}^{*}\crossproduct\mathbf{v}\right)}_{\mathbf{T}_{\text{D}}}-\underbrace{\frac{\rho
k^{3}}{24\pi}\imaginary\left(\mathbf{D}^{*}\crossproduct\mathbf{D}\right)}_{\mathbf{T}_{\text{recoil}}}.}$
(3)

The monopole radiation does not contribute to the torque since it carries no
angular momentum. The main dipole term $\mathbf{T}_{\text{D}}$ has been known
before [49], while the recoil term $\mathbf{T}_{\text{recoil}}$ is novel. This
term also has an optical counterpart [65, 66], and it is crucial for the
angular momentum conservation. In particular, the radiation torque on an
isotropic lossless particle must vanish, and this is ensured by the balance of
pure dipole and recoil contributions [49, 67, 68, 69, 53].

Equation (2) and (3) are the key general results of this work.

Figure 3: Acoustic force on a Rayleigh prolate spheroid particle with
$ka=0.02$, $kb=0.1$ oriented at the angle $\theta=\pi/4$ with respect to the
incident plane wave vs. relative density and compressibility of the particle.
The black dots indicate examples of specific particle materials, assuming
water as the host material with $c=1500$ m/s, $\rho=997~{}\mathrm{kg/m}^{3}$.
(a) The absolute value of total acoustic force $F=|{\bf F}|$ normalized
normalized by the pressure force on a spheroid particle $F_{0}$ (see
explanation in the text). (b) The ratio of the lateral and longitudinal force
components, $F_{\perp}/F_{\parallel}$. Since the longitudinal force is always
positive, there are areas of negative and positive lateral force. The dashed
curves show an approximate analytical solution for the extremum values of the
particles’ parameters, Eq. (12). (c) Examples of the radiation diagrams for
the scattered acoustic intensity for particles made of specific materials in
the water environment. The direction of the recoil force, opposite to the
scattered momentum, depends on the relative phase of the excited monopole and
dipole moments.

##  III Acoustic Force on a Small Spheroid Particle

###  III.1 General equations

We now analyse acoustic forces acting on the simplest example of anisotropic
particle: a spheroid (rotational ellipsoid) with two independent semi-axes
$a>b=c$, Fig. 2(b). The material parameters of the particles are the density
$\rho_{\text{p}}$, compressibility $\beta_{\text{p}}$, and the speed of sound
$c_{\text{p}}=1/\sqrt{\rho_{\text{p}}\beta_{\text{p}}}$, whereas the
corresponding parameters of the surrounding medium are $\rho$, $\beta$,
$c=1/\sqrt{\rho\beta}$. We assume the incident field to be an $x$-propagating
plane wave:

$p=p_{0}\mathrm{e}\mkern 1.0mu^{{i}\mkern
1.0mukx}\,\quad\mathbf{v}=v_{0}\,\bar{\bf x}\,\mathrm{e}\mkern
1.0mu^{{i}\mkern 1.0mukx}\,.$ (4)

Here $p_{0}$ and $v_{0}=p_{0}\sqrt{{\beta}/{\rho}}$ are the pressure and
velocity field amplitudes, whereas the overbar denotes the unit vector of the
corresponding axis.

For any small anisotropic particle, one can express the monopole and dipole
moments via the incident-wave fields as [49, 70]

${M=\beta\alpha_{\text{M}}p,\qquad\mathbf{D}=\boldsymbol{\mathbf{\hat{\alpha}}}_{\text{D}}\mathbf{v},}$
(5)

where $\alpha_{\text{M}}$ is a scalar monopole polarizability, and
$\hat{\boldsymbol{\alpha}}_{\text{D}}$ is the dipole polarizability dyadic
tensor. For an isotropic spherical particle,
$\boldsymbol{\mathbf{\hat{\alpha}}}_{\text{D}}=\alpha_{\text{D}}\boldsymbol{\mathbf{\hat{I}}}$.
For a spheroid particle whose longer semi-axis $a$ is aligned with the
$y$-axis, the dipole polarizability is diagonal:
$\boldsymbol{\mathbf{\hat{\alpha}}}_{\text{D}}=\mathrm{diag}\\{\alpha^{\text{D}}_{s},\alpha^{\text{D}}_{l},\alpha^{\text{D}}_{s}\\}$.
Here $\alpha^{\text{D}}_{l}$ and $\alpha^{\text{D}}_{s}$ correspond to the
dipolar responses for the velocity field aligned with the longer and shorter
semi-axes of the spheroid.

Without loss of generality, we can assume that the particle is rotated by the
angle $\theta$ with respect to the incident wave in the $(x,y)$ plane, as
shown in Fig. 2(b). Then, the dipole polarizability tensor is no longer
diagonal and it becomes

${\boldsymbol{\mathbf{\hat{\alpha}}}_{\text{D}}=\begin{pmatrix}\alpha^{\text{D}}_{s}\cos^{2}\\!\theta+\alpha^{\text{D}}_{l}\sin^{2}\\!\theta&(\alpha^{\text{D}}_{s}-\alpha^{\text{D}}_{l})\sin\theta\cos\theta&0\\\
(\alpha^{\text{D}}_{s}-\alpha^{\text{D}}_{l})\sin\theta\cos\theta&\alpha^{\text{D}}_{l}\cos^{2}\\!\theta+\alpha^{\text{D}}_{s}\sin^{2}\\!\theta&0\\\
0&0&\alpha^{\text{D}}_{s}\end{pmatrix}\\!.}$ (6)

Substituting this into Eqs. (4) and (5) yields the dipole moments longitudinal
and transverse with respect to the incident velocity field:

$\displaystyle{\bf
D}_{\parallel}=v_{0}\\!\left[\alpha^{\text{D}}_{l}\sin^{2}\\!\theta+\alpha^{\text{D}}_{s}\cos^{2}\\!\theta\right]\bar{\bf
x},$ $\displaystyle{\bf
D}_{\perp}=v_{0}\\!\left(\alpha^{\text{D}}_{s}-\alpha^{\text{D}}_{l}\right)\sin\theta\cos\theta\,\bar{\bf
y}.$ (7)

Substituting Eqs. (4)–(7) into Eq. (2), we find the monopole, dipole, and
recoil parts of the radiation force on the spheroid:

$\displaystyle\mathbf{F}_{\mathrm{M}}$
$\displaystyle=\frac{1}{2}k\beta|p_{0}|^{2}\,\mathrm{Im}\\!\left(\alpha_{\text{M}}\right)\bar{\bf
x}\,,\qquad\mathbf{F}_{\mathrm{D}}=\frac{1}{2}k\beta\absolutevalue{p_{0}}^{2}\left[\imaginary\\!\left(\alpha^{\text{D}}_{l}\right)\sin^{2}\\!\theta+\imaginary\\!\left(\alpha^{\text{D}}_{s}\right)\cos^{2}\\!\theta\right]\bar{\bf
x}\,,$ (8) $\displaystyle\mathbf{F}_{\mathrm{recoil}}$
$\displaystyle=-\frac{\beta
k^{4}|p_{0}|^{2}}{12\pi}\,\real\\!\left[\alpha_{\text{M}}^{*}(\alpha^{\text{D}}_{l}\sin^{2}\\!\theta+\alpha^{\text{D}}_{s}\cos^{2}\\!\theta)\right]\bar{\bf
x}-\frac{\beta
k^{4}|p_{0}|^{2}}{12\pi}\sin\theta\cos\theta\,\real\\!\left[\alpha_{\text{M}}^{*}(\alpha^{\text{D}}_{s}-\alpha^{\text{D}}_{l})\right]\bar{\bf
y}\,.$ (9)

These equations show that the monopole and dipole forces are directed along
the incident plane wave, while the recoil force generally has a nonzero
lateral force component. This lateral force depends on the orientation of the
particle, as well as on the relative phases and amplitudes of the monopole and
dipole polarizabilities.

In the Supplemental Material, we derive the monopole and dipole
polarizabilities of a small ellipsoid particle:

$\displaystyle({\alpha_{\text{M}}})^{-1}=({\alpha_{\text{M}}^{\text{st}}})^{-1}-\dfrac{{i}\mkern
1.0muk^{3}}{4\pi},~{}~{}\left(\boldsymbol{\mathbf{\hat{\alpha}}}_{\text{D}}\right)^{-1}=\left(\boldsymbol{\mathbf{\hat{\alpha}}}_{\text{D}}^{{}_{\text{st}}}\right)^{-1}-\dfrac{{i}\mkern
1.0muk^{3}}{12\pi}\boldsymbol{\mathbf{\hat{I}}}.$ (10)

Here

${\alpha_{\text{M}}^{\text{st}}=V_{\text{e}}\
(\bar{\beta}-1),\quad\alpha^{\text{st}}_{\text{D},i}=V_{\text{e}}\frac{\bar{\rho}-1}{\bar{\rho}+L_{i}\left[1-\bar{\rho}\right]}}$
(11)

are the static polarizabilities, where $\bar{\beta}=\beta_{\text{p}}/\beta$
and $\bar{\rho}=\rho_{\text{p}}/\rho$ are the relative compressibility and
density of the particle, $V_{\text{e}}=4\pi\,abc/3$ is the ellipsoid volume,
$L_{\\{1,2,3\\}}=(abc/2)\int\limits_{0}^{\infty}\left[\left(\\{c,a,b\\}^{2}+q\right)\\!f(q)\right]^{-1}\differential
q$, $f(q)=\sqrt{(q+a^{2})(q+b^{2})(q+c^{2})}$, and
$\alpha^{\text{st}}_{\text{D},i}$ are the diagonal components of
$\boldsymbol{\mathbf{\hat{\alpha}}}_{\text{D}}^{{}_{\text{st}}}$ for the
coordinate axes aligned with the principal axes of the ellipsoid. For
spherical particle, the geometric factors $L_{1}=L_{2}=L_{3}=1/3$, and Eqs.
(11) agree with previously known results [35, 49].

The second terms in the right-hand sides of Eqs. (10) are the radiation
corrections due to rescattering of the field, which are well known in optics
and are always present due to the optical theorem requirements [71, 72, 73,
74, 75, 76]. These corrections describe non-zero radiation force on small
lossless particles $\propto\imaginary(\alpha_{\text{M},\text{D}})$ [77]. In
acoustics, the radiation corrections to the polarizabilities were previously
derived for spheres via Taylor expansion of the Mie coefficients in the
Generalized Lorenz-Mie Theory [35, 53].

The acoustic dipole polarizability tensor for an ellipsoid particle, Eqs. (6),
(10), (11), and the corresponding radiation forces (8), (9) are the main
results of this section.

###  III.2 Lateral force

The most interesting outcome of the expressions for the acoustic force on a
small anisotropic particle is the lateral recoil force component
$F_{\perp}\equiv F_{y}$ in Eq. (9). Here we analyse this qualitatively novel
force contribution. First, since $F_{\perp}(\theta)\propto\sin 2\theta$, the
lateral force is maximum at the orientation angle $\theta=\pi/4$.

Next, Figure 3(a) shows the absolute value of the total acoustic force,
$F=|{\bf F}|$, Eqs. (2), (8), and (9), as dependent on the relative density
$\bar{\rho}$ and compressibility $\bar{\beta}$ for $\theta=\pi/4$. The plot is
divided into four quadrants $\bar{\rho}\lessgtr 1$, $\bar{\beta}\lessgtr 1$
corresponding to different types of acoustic materials. The total force in
this and all subsequent figures is normalized by
$F_{0}=\sigma_{\text{geom}}\beta\absolutevalue{p_{0}}^{2}/2$, which is the
plane-wave radiation pressure force acting on a particle with the extinction
cross-section equal to its geometrical cross-section $\sigma_{\text{geom}}=\pi
ab$.

The effectiveness of the lateral force can be characterized by its ratio to
the longitudinal force, $F_{\perp}/F_{\parallel}\equiv F_{y}/F_{x}$. The
dependence of this ratio on $\bar{\rho}$ and $\bar{\beta}$ is shown in Fig.
3(b). While the longitudinal force is always positive, the transverse force
changes its sign across the line $\bar{\beta}=1$. Moreover, one can see that
the effectiveness $F_{\perp}/F_{\parallel}$ has pronounced extrema and can be
considerably enhanced for specific parameter values [the red and blue zones in
Fig. 3(b)].

Analyzing Eqs. (8) and (9) in the Rayleigh limit $ka\ll 1$ and for $\theta\ll
1$, we find the condition for the maximum of
$\absolutevalue{F_{\perp}/F_{\parallel}}$, which yields to a compact formula

$\bar{\beta}^{\text{cr}}_{1,2}=1\pm\frac{\absolutevalue{\bar{\rho}^{\text{cr}}-1}}{{\sqrt{3}}\absolutevalue{L_{s}(1-\bar{\rho}^{\text{cr}})+\bar{\rho}^{\text{cr}}}}.$
(12)

These curves, indicating the maximum effectiveness of the lateral force are
shown in Fig. 3(b). Even though the color map is for $\theta=\pi/4$, the
curves provide a reasonable approximation for the areas of maximum lateral
force ratio.

Finally, to explain the physical mechanism underlying the effective lateral
force, in Fig. 3(c) we plot scattered-field diagrams and the recoil force
direction for spheroid particles made of different materials, which are also
marked in Figs. 3(a,b). One can see that these diagrams differ drastically
from each other because of the difference in the relative phases and
magnitudes of the excited dipole and monopole moments. The most effective
lateral force appears for the parameters corresponding to the negative
longitudinal recoil force $F_{{\rm recoil},x}<0$, suppressing the main
radiation force $F_{{\rm M},x}+F_{{\rm D},x}>0$ (the acoustic Kerker effect),
and high-magnitude lateral force $F_{{\rm recoil},y}$ (see the polypropelene
case).

##  IV Stable Acoustic Lift

The acoustomechanical behaviour of a particle is determined by both the
acoustic radiation force and torque. In particular, to produce a stable
lateral drift (lift) of the particle, there should be a nonzero lateral force
(which requires a tilted orientation of the spheroid particle), as well as
vanishing torque to avoid rotation of the particle. Moreover, dynamical
stability of the system can be achieved only when minor rotational
perturbations lead to the emergence of a restorative torque in the opposite
direction, as depicted in Fig. 4(a). For the orientation angle $\theta$ under
consideration and the corresponding $z$-directed torque, this means $T_{z}=0$
and $\partial T_{z}/\partial\theta<0$.

Figure 4:  (a) Schematics of rotationally stable and unstable particle
configurations. The stable configuration is achieved when the torque vanishes
and small rotational perturbations induce restoring torques. (b,c) Acoustic
torque and lateral force on a prolate Rayleigh spheroid particle vs. the
orientation angle $\theta$. The only stable point $\theta=0$ is trivial, and
the lateral force vanishes there.

###  IV.1 The Rayleigh approximation

In the Rayleigh approximation $\\{ka,kb\\}\ll 1$, for a spheroid particle
illuminated by an $x$-propagating plane wave, Eq. (3) yields
$\mathbf{T}=T_{z}\bar{\mathbf{z}}$,

$T_{z}=-\frac{\beta\absolutevalue{p_{0}}^{2}}{4}\sin{2\theta}\left(\alpha^{\text{st}}_{\text{D},s}-\alpha^{\text{st}}_{\text{D},l}\right)\left(1+\frac{k^{6}}{72\pi^{2}}\alpha^{\text{st}}_{\text{D},s}\alpha^{\text{st}}_{\text{D},l}\right)$
(13)

Thus, the torque has the same $\propto\sin 2\theta$ dependence as the lateral
force $F_{y}$, and a nontrivial tilted-particle configuration with vanishing
torque can be achieved only for
$\alpha^{\text{st}}_{\text{D},s}\alpha^{\text{st}}_{\text{D},l}=-72\pi^{2}/k^{6}$.
However, it follows from Eq. (11) and the fact that $L_{i}<1$ that for a
particle with isotropic effective mass density the elements of polarizability
tensor always have the same sign and this condition cannot be satisfied, and
hence there is no stable acoustic lift for a spheroid particle in the Rayleigh
regime.

We performed numerical calculations of the acoustic force and torque on a
spheroid silicone rubber particle with parameters $kb=0.01,ka=0.002$ using
Comsol Multiphysics. The results are shown in Figs. 4(c,d). One can see
perfect agreement with the analytical expressions in the Rayleigh regime. The
only stable point with $T_{z}=0$ and $\partial T_{z}/\partial\theta<0$ is
$\theta=0$, when the spheroid is oriented perpendicular to the incident field,
and the lateral force vanishes. The acoustic radiation torque in this and all
subsequent plots is normalized by
$T_{0}=F_{0}/k=\sigma_{\text{geom}}\beta\absolutevalue{p_{0}}^{2}/2k$.

###  IV.2 Beyond the Rayleigh approximation

Since the stable optical lift was achieved for structures much larger than the
wavelength [45, 46], one can expect that acoustic lift effect may also be
discovered beyond the Rayleigh limit. In this section, we extend our analysis
to the Mie-resonant scattering. For numerical calculations, we choose a highly
resonant prolate spheroid with $a/b=0.2$, material parameters $\bar{\beta}=3$,
$\bar{\rho}=7$, and variable parameter $ka$. The Rayleigh-limit expression for
the torque, Eq. (13), is not valid for larger particles because of: (i) higher
multipole contributions and (ii) nonlocal effects for each spherical multipole
since spheroid has lower symmetry compared to sphere [36].

To address this problem we employ the well-established partial-wave expansion
[52, 51]. First, we decompose the incident and scattered fields into spherical
multipoles:

$p_{{}^{\text{inc}}_{\text{sc}}}(r,\varphi,\vartheta)=p_{0}\sum_{n=0}^{\infty}\sum\limits_{m=-n}^{n}\>{}^{A_{mn}}_{S_{mn}}R_{n}(kr)Y_{n}^{m}(\varphi,\vartheta),$
(14)

where $A_{n}^{m}$ and $S_{n}^{m}$ are the expansion coefficients of the
incident and scattered fields, $R_{n}$ are the radial spherical functions
containing Hankel and Bessel functions for the scattered and incident fields
respectively, and $Y_{n}^{m}$ are the spherical harmonics of order $n$ and
with the azimuthal number $m$ correspondent to the angular momentum about the
$z$-axis. The torque can be expressed from the exact equation (1) taking all
excited multipoles into account [78]:

$T_{z}=\sum\limits_{nm}T_{z}^{(nm)}=\frac{\beta\absolutevalue{p_{0}}^{2}}{2k^{3}}\real\sum_{n=1}^{\infty}\sum_{m=-n}^{n}\\!m\\!\left(A_{n}^{m*}+S_{n}^{m*}\right)\\!S_{n}^{m}.$
(15)

This equation is valid for arbitrary particle and arbitrary incident field.
All information about the particle and the incident field is encoded in the
the coefficients $S_{n}^{m}$ and $A_{n}^{m}$, respectively.

The total dipole contribution to the torque (including nonlocal effects) is
given by the $n=1$ terms in Eq. (15):
$T_{z,\text{D}}=\sum\limits_{m}T_{z}^{(1m)}$. This dipole torque can still be
expressed via Eq. (3) using the total induced dipole moment $\mathbf{D}$.
However, due to the nonlocal effects, the excited dipole moment can no longer
be expressed through the local velocity ${\bf v}$, i.e., Eq. (5) is no longer
valid. Instead, the general nonlocal relation between the induced dipole
moment and the incident acoustic field takes on the form [79]:

$\mathbf{D}={\boldsymbol{\chi}}^{\text{D}}p+\boldsymbol{\mathbf{\hat{\alpha}}}^{\text{D}}\mathbf{v}+{\boldsymbol{\mathbf{\hat{\beta}}}^{\text{D}}}\gradient\mathbf{v}+\boldsymbol{\mathbf{\hat{\gamma}}}^{\text{D}}\gradient\gradient\mathbf{v}+\dots\,,$
(16)

where {${\boldsymbol{\chi}}^{\text{D}}$,
${\boldsymbol{\mathbf{\hat{\alpha}}}^{\text{D}}}$,
${\boldsymbol{\mathbf{\hat{\beta}}}^{\text{D}}}$,
${\boldsymbol{\mathbf{\hat{\gamma}}}^{\text{D}}}$} are tensors of the first,
second, third and fourth rank, respectively, which depend on the particle’s
parameters and orientation. These tensors describe the coupling between the
{monopole, dipole, quadrupole, octupole} modes in the incident field and the
dipole mode of the particle. For instance, the first term in Eq. (16) is known
as the Willis coupling [80, 81, 82, 27, 70, 83, 84]. Depending on the
particle’s symmetry some of the coefficients in the decomposition (16) may
vanish. In particular, from the group-theory analysis of the scattered
multipolar contributions one can conclude that
${\boldsymbol{\chi}}^{\text{D}}\equiv{\bf 0}$ and
${\boldsymbol{\mathbf{\hat{\beta}}}^{\text{D}}}\equiv{\bf 0}$ for a spheroid
and any particle possessing inversion symmetry [85, 36]. More details on the
decomposition (16) and individual contributions can be found in the
Supplemental Material.

Figure 5:  Acoustic torque and lateral force on a prolate spheroid particle
with eccentricity $e=0.98$ oriented at the angle $\theta=\pi/4$ vs. the
dimensionless particle size $ka$ in the Mie regime. The relative material
parameters are $\bar{\beta}=3$ and $\bar{\rho}=7$. (a) The acoustic torque
$T_{z}$ exhibits two stable zeros $S_{1,2}$ and unstable zero $U$. The lateral
force $F_{\perp}$ is nonzero in these points, enabling acoustic lift (see also
Fig. 6). (b) Multipolar contributions to the net acoustic torque, which shows
stable lift point $S_{\text{D}}$ (corresponding to $S_{2}$) already in the
dipole approximation including higher-order nonlocal contributions to the
dipole moment, Eq. (16). Figure 6:  Color maps of the acoustic torque $T_{z}$
(a) and lateral force $F_{\perp}$ (b) on a prolate spheroid particle in the
Mie regime vs. the dimensionless particle size $ka$ and orientation $\theta$.
The torque is normalized by $T_{\text{max}}$, which is the absolute value of
the maximum torque for each value of $ka$. The rotationally stable
configurations of the particle correspond to the lines with ${T}_{z}=0$ and
$\partial T_{z}/\partial\theta<0$. The rotationally stable configuration in
(a) with nonzero lateral force in (b) provide a stable acoustic lift.

Thus, the total torque in the Mie regime has contributions from all multipole
amplitudes. Figure 5(a) shows the numerically calculated acoustic force and
torque on a Mie-resonant spheroid particle oriented at $\theta=\pi/4$ versus
the $ka$ parameter. In sharp contrast to the Rayleigh regime, the torque
vanishes, $T_{z}=0$, for some values of $ka$, while the lateral force is non-
zero. Further analysis (see Supplemental Material) shows that the points
$S_{1}$ ($ka=1.91$) and $S_{2}$ ($ka=3.12$) correspond to the stable lift
configuration with $\partial T_{z}/\partial\theta<0$.

The multipolar origin of these stable zero-torque points can be recognized by
analysing the individual multipole contributions ($n=1,2,3,...$), shown in
Fig. 5(b). One can see that the first stability point $S_{1}$ appears away
from multipolar resonances due to the mutual cancellation of different
multipole contributions. In turn, the second stability point $S_{2}$ is close
to the multipolar resonance, where not only the sum of mutlipoles contribution
vanishes, but also each multipole contribution vanishes in the vicinity of
this point. Such stability point appears already in the dipole response
($n=1$) taking into account nonlocal effects, Eq. (16) (see Supplemental
Material).

To describe the dependence of the stable acoustic lift on the orientation
angle $\theta$, in Fig. 6 we plot the acoustic torque and lateral force as
dependent on $ka$ (in the range including the stable zero-torque point $S_{1}$
and unstable point $U$) and $\theta$. One can see that these zero-torque
points evolve smoothly with $\theta$, so that the stable acoustic lift can
appear at any orientation of the particle.

##  V Example: Sorting of blood cells

We briefly discuss potential application of the acoustic lateral force and
lift for acoustic sorting of particles with different anisotropies: e.g.,
near-spherical leukocytes (white blood cells) and spheroid erythrocytes (red
blood cells). The separation of red blood cell via standard acoustic methods
is challenging [21], and most methods of acoustic or microfluidic sorting of
particles involve the size and material parameters but not the shapes [4, 5,
6, 7].

Figure 7:  (a) Schematics of the possible application of acoustic lift for
acoustofluidic separation of the white and red blood cells using their shape
anisotropy. (b) Frequency dependences of the acoustic lateral force and torque
exerted by a plane-wave on a red blood cell particle in blood plasma at the
orientation angle $\theta=\pi/4$. The values are normalized by the
corresponding maximum absolute values. There is a point of stable lift at
$ka\simeq 2$ which corresponds to the frequency of about $90$ MHz.

The main idea is depicted in Fig. 7(a). An acoustic wave propagates along the
microchannel containing red and white blood cells. It produces lateral forces
(with different signs for different orientations) on the red blood cells due
to their elongated shapes, while spherical white blood cells experience only
the longitudinal pressure force. The three-channel geometry used in
microfluidic systems implements the sorting, as shown in Fig. 7(b).

We calculate acoustic force and torque exerted by a plane wave incident on a
red blood cell submerged in blood plasma. The red blood cell geometry is
modelled as a biconcave solid disc described by the implicit equation from
[86]. The size and shape parameters are taken from the table in [25]. The
acoustic and mechanical parameters of erythrocytes and blood plasma are taken
from [87, 88]. For simplicity, we neglect thermoviscous effects.

A red blood cell has the same symmetry as a spheroid, so that our previous
analysis can be applied. The results of numerical calculations are shown in
Fig. 7(c). The frequency dependence of the acoustic torque exhibits a stable-
lift point with $\omega/2\pi\simeq 90$ MHz corresponding to $ka\simeq 2$. For
this frequency, the ratio of the lateral-to-longitudinal force components is
$|F_{\perp}/F_{\parallel}|\simeq 0.2$. Assuming the incident pressure wave
amplitude $p_{0}=10$ kPa, the lateral force is $|F_{\perp}|\simeq 8$ fN, which
is significant for the red blood cell weight $m\simeq 30$ picograms [89].

##  VI Conclusions

We have examined acoustic forces and torque on an anisotropic (spheroid)
particle in an incident plane-wave field. Within the monopole-dipole
approximation, we have derived exact analytical expressions, including
previously unknown recoil contributions. Similar recoil terms are well studied
in optical forces and torques [37, 38, 39, 40, 41, 42, 43, 44]. Importantly,
the acoustic recoil force on an anisotropic particle can have lateral
component, orthogonal to the incident-wave direction. The sign of this lateral
force is associated with the relative phases between the induced monopole and
dipole and can be controlled by physical and geometric parameters of the
particle.

We have also examined acoustic torque on a spheroid particle in the presence
of the lateral force. In particular, we have found conditions for the
rotationally stable lateral drift of the particle, i.e., stable acoustic lift.
The optical counterpart of this effect has been described in [45, 46].
Notably, the stable acoustic lift does not appear in the Rayleigh limit and
requires larger Mie particles involving higher-order terms in the scattering
problem.

Finally, we have proposed a possible application of the acoustic-lift effect
for sorting of red and white blood cells, utilizing their different
anisotropic shapes. We have shown that there exists a set of parameters where
such sorting can be implemented in a simple acoustofluidic setup. Of course,
this is only a proof-of-principle consideration, and a more detailed
consideration including thermoviscous and acoustic streaming effects [90] is
required for practical implementation. Furthermore, for dense solutions the
multiple rescattering should be taken into account [91].

Thus, our results reveal several new acoustomechanical phenomena, which can
provide novel functionalities for acoustic and microfluidic manipulation of
particles. To further explore these phenomena, it would be important to
examine recoil forces and torques in structured incident fields [92], such as
evanescent waves [49, 55, 56] or vortex beams [93, 94, 95, 96], as well as for
structured particles made of nontrivial inhomogeneous or elastic materials
[97].

###### Acknowledgements.

The authors are grateful to Kristina Frizyuk for valuable discussions. This
work was supported by Russian Academic Leadership Programm Priority 2030. The
numerical simulations were supported by Russian Science Foundation (grant No.
20-72-10141). K.Y.B. thanks ENSEMBLE3 Project (MAB/2020/14) which is carried
out within the International Research Agendas Programme (IRAP) of the
Foundation for Polish Science co-financed by the European Union under the
European Regional Development Fund and Teaming Horizon 2020 programme of the
European Commission; the TEAM/2016-3/29 Grant within the TEAM program of the
Foundation for Polish Science co-financed by the European Union under the
European Regional Development Fund.

## References

  * Ozcelik  _et al._ [2018] A. Ozcelik, J. Rufo, F. Guo, Y. Gu, P. Li, J. Lata, and T. J. Huang, Acoustic tweezers for the life sciences, Nat. Methods 15, 1021 (2018).
  * Meng  _et al._ [2019] L. Meng, F. Cai, F. Li, W. Zhou, L. Niu, and H. Zheng, Acoustic tweezers, J. Phys. D: Appl. Phys. 52, 273001 (2019).
  * Baudoin and Thomas [2020] M. Baudoin and J.-L. Thomas, Acoustic Tweezers for Particle and Fluid Micromanipulation, Annu. Rev. Fluid Mech. 52, 205 (2020).
  * Fan  _et al._ [2022] Y. Fan, X. Wang, J. Ren, F. Lin, and J. Wu, Recent advances in acoustofluidic separation technology in biology, Microsyst. Nanoeng. 8, 1 (2022).
  * Rufo  _et al._ [2022] J. Rufo, F. Cai, J. Friend, M. Wiklund, and T. J. Huang, Acoustofluidics for biomedical applications, Nat. Rev. Methods Primers 2, 1 (2022).
  * Wu  _et al._ [2019] M. Wu, A. Ozcelik, J. Rufo, Z. Wang, R. Fang, and T. Jun Huang, Acoustofluidic separation of cells and particles, Microsyst. Nanoeng. 5, 1 (2019).
  * Olofsson  _et al._ [2020] K. Olofsson, B. Hammarström, and M. Wiklund, Acoustic separation of living and dead cells using high density medium, Lab Chip 20, 1981 (2020).
  * Mura  _et al._ [2013] S. Mura, J. Nicolas, and P. Couvreur, Stimuli-responsive nanocarriers for drug delivery, Nat. Mater. 12, 991 (2013).
  * Maresca  _et al._ [2018] D. Maresca, A. Lakshmanan, M. Abedi, A. Bar-Zion, A. Farhadi, G. J. Lu, J. O. Szablowski, D. Wu, S. Yoo, and M. G. Shapiro, Biomolecular Ultrasound and Sonogenetics, Annu. Rev. Chem. Biomol. Eng. 9, 229 (2018).
  * Wu  _et al._ [2023] D. Wu, D. Baresch, C. Cook, Z. Ma, M. Duan, D. Malounda, D. Maresca, M. P. Abundo, J. Lee, S. Shivaei, D. R. Mittelstein, T. Qiu, P. Fischer, and M. G. Shapiro, Biomolecular actuators for genetically selective acoustic manipulation of cells, Sci. Adv. 9, eadd9186 (2023).
  * Hirayama  _et al._ [2019] R. Hirayama, D. Martinez Plasencia, N. Masuda, and S. Subramanian, A volumetric display for visual, tactile and audio presentation using acoustic trapping, Nature 575, 320 (2019).
  * Fushimi  _et al._ [2019] T. Fushimi, A. Marzo, B. W. Drinkwater, and T. L. Hill, Acoustophoretic volumetric displays using a fast-moving levitated particle, Appl. Phys. Lett. 115, 10.1063/1.5113467 (2019).
  * Hossein and Angeli [2023] F. Hossein and P. Angeli, A review of acoustofluidic separation of bioparticles, Biophys. Rev. , 1 (2023).
  * Wixforth  _et al._ [2004] A. Wixforth, C. Strobl, Ch. Gauer, A. Toegl, J. Scriba, and Z. V. Guttenberg, Acoustic manipulation of small droplets, Anal. Bioanal.Chem. 379, 982 (2004).
  * Shi  _et al._ [2009] J. Shi, D. Ahmed, X. Mao, S.-C. S. Lin, A. Lawit, and T. J. Huang, Acoustic tweezers: patterning cells and microparticles using standing surface acoustic waves (SSAW), Lab Chip 9, 2890 (2009).
  * Guo  _et al._ [2016] F. Guo, Z. Mao, Y. Chen, Z. Xie, J. P. Lata, P. Li, L. Ren, J. Liu, J. Yang, M. Dao, S. Suresh, and T. J. Huang, Three-dimensional manipulation of single cells using surface acoustic waves, Proc. Natl. Acad. Sci. U.S.A. 113, 1522 (2016).
  * Wu  _et al._ [2017] M. Wu, Y. Ouyang, Z. Wang, R. Zhang, P.-H. Huang, C. Chen, H. Li, P. Li, D. Quinn, M. Dao, S. Suresh, Y. Sadovsky, and T. J. Huang, Isolation of exosomes from whole blood by integrating acoustics and microfluidics, Proc. Natl. Acad. Sci. U.S.A. 114, 10584 (2017).
  * Ahmed  _et al._ [2016] D. Ahmed, A. Ozcelik, N. Bojanala, N. Nama, A. Upadhyay, Y. Chen, W. Hanna-Rose, and T. J. Huang, Rotational manipulation of single cells and organisms using acoustic waves, Nat. Commun. 7, 1 (2016).
  * Li  _et al._ [2016] S. Li, F. Ma, H. Bachman, C. E. Cameron, X. Zeng, and T. J. Huang, Acoustofluidic bacteria separation, J. Micromech. Microeng. 27, 015031 (2016).
  * Dow  _et al._ [2018] P. Dow, K. Kotz, S. Gruszka, J. Holder, and J. Fiering, Acoustic separation in plastic microfluidics for rapid detection of bacteria in blood using engineered bacteriophage, Lab Chip 18, 923 (2018).
  * Urbansky  _et al._ [2017] A. Urbansky, P. Ohlsson, A. Lenshof, F. Garofalo, S. Scheding, and T. Laurell, Rapid and effective enrichment of mononuclear cells from blood using acoustophoresis, Sci. Rep. 7, 1 (2017).
  * Aghakhani  _et al._ [2022] A. Aghakhani, A. Pena-Francesch, U. Bozuyuk, H. Cetin, P. Wrede, and M. Sitti, High shear rate propulsion of acoustic microrobots in complex biological fluids, Sci. Adv. 8, 10.1126/sciadv.abm5126 (2022).
  * Tian  _et al._ [2020] Z. Tian, Z. Wang, P. Zhang, T. D. Naquin, J. Mai, Y. Wu, S. Yang, Y. Gu, H. Bachman, Y. Liang, Z. Yu, and T. J. Huang, Generating multifunctional acoustic tweezers in Petri dishes for contactless, precise manipulation of bioparticles, Sci. Adv. 6, 10.1126/sciadv.abb0494 (2020).
  * Jooss  _et al._ [2022] V. M. Jooss, J. S. Bolten, J. Huwyler, and D. Ahmed, In vivo acoustic manipulation of microparticles in zebrafish embryos, Sci. Adv. 8, 10.1126/sciadv.abm2785 (2022).
  * Lima and Silva [2021] E. B. Lima and G. T. Silva, Mean acoustic fields exerted on a subwavelength axisymmetric particlea), J. Acoust. Soc. Am. 150, 376 (2021).
  * Sepehrirahnama  _et al._ [2021] S. Sepehrirahnama, S. Oberst, Y. K. Chiang, and D. Powell, Acoustic radiation force and radiation torque beyond particles: Effects of nonspherical shape and Willis coupling, Phys. Rev. E 104, 065003 (2021).
  * Sepehrirahnama  _et al._ [2022] S. Sepehrirahnama, S. Oberst, Y. K. Chiang, and D. A. Powell, Willis Coupling-Induced Acoustic Radiation Force and Torque Reversal, Phys. Rev. Lett. 129, 174501 (2022).
  * Sepehrirahnama and Oberst [2022] S. Sepehrirahnama and S. Oberst, Acoustic Radiation Force and Torque Acting on Asymmetric Objects in Acoustic Bessel Beam of Zeroth Order Within Rayleigh Scattering Limit, Front. Phys. 10 (2022).
  * Tang  _et al._ [2023] T. Tang, C. Shen, and L. Huang, Acoustic rotation of non-spherical micro-objects: Characterization of acoustophoresis and quantification of rotational stability, J. Sound Vib. 554, 117694 (2023).
  * Ren  _et al._ [2019] L. Ren, N. Nama, J. M. McNeill, F. Soto, Z. Yan, W. Liu, W. Wang, J. Wang, and T. E. Mallouk, 3D steerable, acoustically powered microswimmers for single-particle manipulation, Sci. Adv. 5, eaax3084 (2019).
  * Liu  _et al._ [2020] T. Liu, R. Xu, P. Yu, Z. Wang, and J. Takahara, Multipole and multimode engineering in Mie resonance-based metastructures, Nanophotonics 9, 1115 (2020).
  * Kislov  _et al._ [2021] D. A. Kislov, E. A. Gurvitz, V. Bobrovs, A. A. Pavlov, D. N. Redka, M. I. Marqués, P. Ginzburg, and A. S. Shalin, Multipole Engineering of Attractive-Repulsive and Bending Optical Forces, Adv. Photonics Res. 2, 2100082 (2021).
  * Gladyshev  _et al._ [2020] S. Gladyshev, K. Frizyuk, and A. Bogdanov, Symmetry analysis and multipole classification of eigenmodes in electromagnetic resonators for engineering their optical properties, Phys. Rev. B 102, 075103 (2020).
  * Wu  _et al._ [2021] H.-W. Wu, W. Li, S.-L. Cheng, Y.-Q. Yin, and Z.-Q. Sheng, Acoustic generalized Kerker effect, Appl. Phys. Express 14, 095501 (2021).
  * Baresch  _et al._ [2016] D. Baresch, J.-L. Thomas, and R. Marchiano, Observation of a Single-Beam Gradient Force Acoustical Trap for Elastic Particles: Acoustical Tweezers, Phys. Rev. Lett. 116, 024301 (2016).
  * Tsimokha  _et al._ [2022] M. Tsimokha, V. Igoshin, A. Nikitina, I. Toftul, K. Frizyuk, and M. Petrov, Acoustic resonators: Symmetry classification and multipolar content of the eigenmodes, Phys. Rev. B 105, 165311 (2022).
  * Chaumet and Rahmani [2009] P. C. Chaumet and A. Rahmani, Electromagnetic force and torque on magnetic and negative-index scatterers, Opt. Express 17, 2224 (2009).
  * Nieto-Vesperinas  _et al._ [2010] M. Nieto-Vesperinas, J. J. Sáenz, R. Gómez-Medina, and L. Chantada, Optical forces on small magnetodielectric particles, Opt. Express 18, 11428 (2010).
  * Chen  _et al._ [2011a] J. Chen, J. Ng, Z. Lin, and C. T. Chan, Optical pulling force, Nat. Photonics 5, 531 (2011a).
  * Bliokh  _et al._ [2014] K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, Extraordinary momentum and spin in evanescent waves, Nat. Commun. 5, 1 (2014).
  * Bekshaev  _et al._ [2015] A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, Transverse Spin and Momentum in Two-Wave Interference, Phys. Rev. X 5, 011039 (2015).
  * Antognozzi  _et al._ [2016] M. Antognozzi, C. R. Bermingham, R. L. Harniman, S. Simpson, J. Senior, R. Hayward, H. Hoerber, M. R. Dennis, A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, Direct measurements of the extraordinary optical momentum and transverse spin-dependent force using a nano-cantilever, Nat. Phys. 12, 731 (2016).
  * Yevick  _et al._ [2016] A. Yevick, D. B. Ruffner, and D. G. Grier, Tractor beams in the Rayleigh limit, Phys. Rev. A 93, 043807 (2016).
  * Hayat  _et al._ [2015] A. Hayat, J. P. B. Mueller, and F. Capasso, Lateral chirality-sorting optical forces, Proc. Natl. Acad. Sci. U.S.A. 112, 13190 (2015).
  * Swartzlander  _et al._ [2011] G. A. Swartzlander, T. J. Peterson, A. B. Artusio-Glimpse, and A. D. Raisanen, Stable optical lift, Nat. Photonics 5, 48 (2011).
  * Simpson  _et al._ [2012] S. H. Simpson, S. Hanna, T. J. Peterson, and G. A. Swartzlander, Optical lift from dielectric semicylinders, Opt. Lett. 37, 4038 (2012).
  * Westervelt [1951] P. J. Westervelt, The Theory of Steady Forces Caused by Sound Waves, J. Acoust. Soc. Am. 23, 312 (1951).
  * Bruus [2012a] H. Bruus, Acoustofluidics 7: The acoustic radiation force on small particles, Lab on a Chip 12, 1014 (2012a).
  * Toftul  _et al._ [2019] I. D. Toftul, K. Y. Bliokh, M. I. Petrov, and F. Nori, Acoustic Radiation Force and Torque on Small Particles as Measures of the Canonical Momentum and Spin Densities, Phys. Rev. Lett. 123, 183901 (2019).
  * Bruus [2012b] H. Bruus, Acoustofluidics 2: Perturbation theory and ultrasound resonance modes, Lab on a Chip 12, 20 (2012b).
  * Williams [1999] E. G. Williams,  _Fourier Acoustics_ (1999).
  * Blackstock [2000] D. T. Blackstock,  _Fundamentals of Physical Acoustics_ (Hoboken, NJ, USA, 2000).
  * Silva [2014] G. T. Silva, Acoustic radiation force and torque on an absorbing compressible particle in an inviscid fluid, J. Acoust. Soc. Am. 136, 2405 (2014).
  * Kerker  _et al._ [1983] M. Kerker, D.-S. Wang, and C. L. Giles, Electromagnetic scattering by magnetic spheres, JOSA 73, 765 (1983).
  * Wei and Rodríguez-Fortuño [2020] L. Wei and F. J. Rodríguez-Fortuño, Far-field and near-field directionality in acoustic scattering, New J. Phys. 22, 083016 (2020).
  * Long  _et al._ [2020] Y. Long, H. Ge, D. Zhang, X. Xu, J. Ren, M.-H. Lu, M. Bao, H. Chen, and Y.-F. Chen, Symmetry selective directionality in near-field acoustics, Natl. Sci. Rev. 7, 1024 (2020).
  * Xu  _et al._ [2012] S. Xu, C. Qiu, and Z. Liu, Transversally stable acoustic pulling force produced by two crossed plane waves, Europhys. Lett. 99, 44003 (2012).
  * Zhang and Marston [2016] L. Zhang and P. L. Marston, Acoustic radiation force expressed using complex phase shifts and momentum-transfer cross sections, J. Acoust. Soc. Am. 140, EL178 (2016).
  * Meng  _et al._ [2020] Y. Meng, X. Li, Z. Liang, J. Ng, and J. Li, Acoustic Pulling with a Single Incident Plane Wave, Phys. Rev. Appl. 14, 014089 (2020).
  * Fan and Zhang [2021] X.-D. Fan and L. Zhang, Phase shift approach for engineering desired radiation force: Acoustic pulling force example, J. Acoust. Soc. Am. 150, 102 (2021).
  * Chen  _et al._ [2011b] J. Chen, J. Ng, Z. Lin, and C. T. Chan, Optical pulling force, Nat. Photonics 5, 531 (2011b).
  * Jiang  _et al._ [2016] Y. Jiang, J. Chen, J. Ng, and Z. Lin, Decomposition of optical force into conservative and nonconservative components, arXiv 10.48550/arXiv.1604.05138 (2016), 1604.05138 .
  * Evlyukhin  _et al._ [2012] A. B. Evlyukhin, C. Reinhardt, U. Zywietz, and B. N. Chichkov, Collective resonances in metal nanoparticle arrays with dipole-quadrupole interactions, Phys. Rev. B 85, 245411 (2012).
  * Le Ru  _et al._ [2013] E. C. Le Ru, W. R. C. Somerville, and B. Auguié, Radiative correction in approximate treatments of electromagnetic scattering by point and body scatterers, Phys. Rev. A 87, 012504 (2013).
  * Nieto-Vesperinas [2015a] M. Nieto-Vesperinas, Optical torque on small bi-isotropic particles, Opt. Lett. 40, 3021 (2015a).
  * Nieto-Vesperinas [2015b] M. Nieto-Vesperinas, Optical torque: Electromagnetic spin and orbital-angular-momentum conservation laws and their significance, Phys. Rev. A 92, 043843 (2015b).
  * Zhang and Marston [2014] L. Zhang and P. L. Marston, Acoustic radiation torque on small objects in viscous fluids and connection with viscous dissipation, J. Acoust. Soc. Am. 136, 2917 (2014).
  * Zhang and Marston [2011a] L. Zhang and P. L. Marston, Acoustic radiation torque and the conservation of angular momentum (L), J. Acoust. Soc. Am. 129, 1679 (2011a).
  * Maidanik [1958] G. Maidanik, Torques Due to Acoustical Radiation Pressure, J. Acoust. Soc. Am. 30, 620 (1958).
  * Quan  _et al._ [2018a] L. Quan, Y. Ra’di, D. L. Sounas, and A. Alù, Maximum Willis Coupling in Acoustic Scatterers, Phys. Rev. Lett. 120, 254301 (2018a).
  * Bohren and Huffman [1998] C. F. Bohren and D. R. Huffman,  _Absorption and Scattering of Light by Small Particles_ (1998).
  * Draine [1988] B. T. Draine, The Discrete-Dipole Approximation and Its Application to Interstellar Graphite Grains, Astrophys. J. 333, 848 (1988).
  * Simpson and Hanna [2010] S. H. Simpson and S. Hanna, Orbital motion of optically trapped particles in Laguerre–Gaussian beams, J. Opt. Soc. Am. A, JOSAA 27, 2061 (2010).
  * Belov  _et al._ [2003] P. A. Belov, S. I. Maslovski, K. R. Simovski, and S. A. Tretyakov, A condition imposed on the electromagnetic polarizability of a bianisotropic lossless scatterer, Tech. Phys. Lett. 29, 718 (2003).
  * Kelly  _et al._ [2003] K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, The Optical Properties of Metal Nanoparticles: The Influence of Size, Shape, and Dielectric Environment, J. Phys. Chem. B 107, 668 (2003).
  * Wokaun  _et al._ [1982] A. Wokaun, J. P. Gordon, and P. F. Liao, Radiation Damping in Surface-Enhanced Raman Scattering, Phys. Rev. Lett. 48, 957 (1982).
  * Westervelt [1957] P. J. Westervelt, Acoustic Radiation Pressure, J. Acoust. Soc. Am. 29, 26 (1957).
  * Gong and Baudoin [2021] Z. Gong and M. Baudoin, Equivalence between angular spectrum-based and multipole expansion-based formulas of the acoustic radiation force and torque, J. Acoust. Soc. Am. 149, 3469 (2021).
  * Bobylev  _et al._ [2020] D. A. Bobylev, D. A. Smirnova, and M. A. Gorlach, Nonlocal response of Mie-resonant dielectric particles, Phys. Rev. B 102, 115110 (2020).
  * Sieck  _et al._ [2017a] C. F. Sieck, A. Alù, and M. R. Haberman, Origins of Willis coupling and acoustic bianisotropy in acoustic metamaterials through source-driven homogenization, Phys. Rev. B 96, 104303 (2017a).
  * Melnikov  _et al._ [2019] A. Melnikov, Y. K. Chiang, L. Quan, S. Oberst, A. Alù, S. Marburg, and D. Powell, Acoustic meta-atom with experimentally verified maximum Willis coupling, Nat. Commun. 10, 1 (2019).
  * Quan  _et al._ [2018b] L. Quan, Y. Ra’di, D. L. Sounas, and A. Alù, Maximum Willis Coupling in Acoustic Scatterers, Phys. Rev. Lett. 120, 254301 (2018b).
  * Sieck  _et al._ [2017b] C. F. Sieck, A. Alù, and M. R. Haberman, Origins of Willis coupling and acoustic bianisotropy in acoustic metamaterials through source-driven homogenization, Phys. Rev. B 96, 104303 (2017b).
  * Willis [1985] J. R. Willis, The nonlocal influence of density variations in a composite, Int. J. Solids Struct. 21, 805 (1985).
  * Poleva  _et al._ [2023] M. Poleva, K. Frizyuk, K. Baryshnikova, A. Evlyukhin, M. Petrov, and A. Bogdanov, Multipolar theory of bianisotropic response of meta-atoms, Phys. Rev. B 107, L041304 (2023).
  * Evans and Fung [1972] E. Evans and Y.-C. Fung, Improved measurements of the erythrocyte geometry, Microvasc. Res. 4, 335 (1972).
  * Gupta  _et al._ [2018] T. Gupta, R. Ghosh, and R. Ganguly, Acoustophoretic separation of infected erythrocytes from blood plasma in a microfluidic platform using biofunctionalized, matched-impedance layers, Int. J. Numer. Methods Biomed. Eng. 34, e2943 (2018).
  * Dulińska  _et al._ [2006] I. Dulińska, M. Targosz, W. Strojny, M. Lekka, P. Czuba, W. Balwierz, and M. Szymoński, Stiffness of normal and pathological erythrocytes studied by means of atomic force microscopy, J. Biochem. Bioph. Methods 66, 1 (2006), 16443279 .
  * Phillips  _et al._ [2012] K. G. Phillips, S. L. Jacques, and O. J. T. McCarty, Measurement of Single Cell Refractive Index, Dry Mass, Volume, and Density Using a Transillumination Microscope, Phys. Rev. Lett. 109, 118105 (2012).
  * Buzdalov and Shalyto [2015] M. Buzdalov and A. Shalyto, Hard test generation for augmenting path maximum flow algorithms using genetic algorithms: Revisited, in  _Proceedings of IEEE Congress on Evolutionary Computation_ (2015) pp. 2121–2128.
  * Zhang  _et al._ [2016] S. Zhang, C. Qiu, M. Wang, M. Ke, and Z. Liu, Acoustically mediated long-range interaction among multiple spherical particles exposed to a plane standing wave, New J. Phys. 18, 113034 (2016).
  * Bliokh  _et al._ [2023] K. Y. Bliokh, E. Karimi, M. J. Padgett, M. A. Alonso, M. R. Dennis, A. Dudley, A. Forbes, S. Zahedpour, S. W. Hancock, H. M. Milchberg, S. Rotter, F. Nori, Ş. K. Özdemir, N. Bender, H. Cao, P. B. Corkum, C. Hernández-García, H. Ren, Y. Kivshar, M. G. Silveirinha, N. Engheta, A. Rauschenbeutel, P. Schneeweiss, J. Volz, D. Leykam, D. A. Smirnova, K. Rong, B. Wang, E. Hasman, M. F. Picardi, A. V. Zayats, F. J. Rodríguez-Fortuño, C. Yang, J. Ren, A. B. Khanikaev, A. Alù, E. Brasselet, M. Shats, J. Verbeeck, P. Schattschneider, D. Sarenac, D. G. Cory, D. A. Pushin, M. Birk, A. Gorlach, I. Kaminer, F. Cardano, L. Marrucci, M. Krenn, and F. Marquardt, Roadmap on structured waves, J. Opt. 25, 103001 (2023).
  * Zhang and Marston [2011b] L. Zhang and P. L. Marston, Angular momentum flux of nonparaxial acoustic vortex beams and torques on axisymmetric objects, Phys. Rev. E 84, 065601 (2011b).
  * Zhang  _et al._ [2020] P. Zhang, C. Chen, X. Su, J. Mai, Y. Gu, Z. Tian, H. Zhu, Z. Zhong, H. Fu, S. Yang, K. Chakrabarty, and T. J. Huang, Acoustic streaming vortices enable contactless, digital control of droplets, Sci. Adv. 6, 10.1126/sciadv.aba0606 (2020).
  * Muelas-Hurtado  _et al._ [2022] R. D. Muelas-Hurtado, K. Volke-Sepúlveda, J. L. Ealo, F. Nori, M. A. Alonso, K. Y. Bliokh, and E. Brasselet, Observation of Polarization Singularities and Topological Textures in Sound Waves, Phys. Rev. Lett. 129, 204301 (2022).
  * Guo  _et al._ [2022] S. Guo, Z. Ya, P. Wu, and M. Wan, A review on acoustic vortices: Generation, characterization, applications and perspectives, J. Appl. Phys. 132, 10.1063/5.0107785 (2022).
  * Kotelnikova  _et al._ [2021] L. M. Kotelnikova, D. A. Nikolaev, S. A. Tsysar, and O. A. Sapozhnikov, Determination of the Elastic Properties of a Solid Sphere Based on the Results of Acoustic Beam Scattering, Acoust. Phys. 67, 360 (2021).