Acoustic spin transfer to a subwavelength spheroidal particle
Jose H. Lopes, Everton B. Lima, Jose P. Leao-Neto, Glauber T. Silva
AAcoustic spin transfer to a subwavelength spheroidal particle
Jos´e H. Lopes, Everton B. Lima, Jos´e P. Le˜ao-Neto, and Glauber T. Silva ∗ Grupo de F´ısica da Mat´eria Condensada, N´ucleo de Ciˆencias Exatas,Universidade Federal de Alagoas, Arapiraca, AL 57309-005, Brazil Physical Acoustics Group, Instituto de F´ısica, Universidade Federal de Alagoas, Macei´o, AL 57072-970, Brazil Campus Arapiraca/Unidade de Ensino Penedo, Universidade Federal de Alagoas, Penedo, Alagoas 57200-000, Brazil (Dated: March 31, 2020)We demonstrate that the acoustic spin of a first-order Bessel beam can be transferred to a sub-wavelength (prolate) spheroidal particle at the beam axis in a viscous fluid. The induced radiationtorque is proportional to the acoustic spin, which scales with the beam energy density. The analysisof the particle rotational dynamics in a Stokes’ flow regime reveals that its angular velocity varieslinearly with the acoustic spin. Asymptotic expressions of the radiation torque and angular velocityare obtained for a quasispherical and infinitely thin particle. Excellent agreement is found betweenthe theoretical results of radiation torque and finite element simulations. The induced particle spinis predicted and analyzed using the typical parameter values of the acoustical vortex tweezer andlevitation devices. We discuss how the beam energy density and fluid viscosity can be assessed bymeasuring the induced spin of the particle.
I. INTRODUCTION
The spin angular momentum is a universal featurepresent in different contexts of nature. In classical elec-tromagnetic waves and photons, the spin is caused by thecircular polarization of electric and magnetic fields [1].The electron spin can be regarded as due to a circulatingflow of energy in the Dirac wave field [2]. More recently,the spin of acoustic beams was proposed and measured asa circulation of the fluid velocity field [3]. Subsequently,the spin and orbital angular momenta were theoreticallyanalyzed in monochromatic acoustic wave fields in a ho-mogeneous medium [4]. Before these studies, it was no-ticed that the longitudinal spin, in which the axis of rota-tion is parallel to the propagation direction of an acousticBessel beam, could induce the acoustic radiation torqueon a subwavelength absorbing spherical particle [5].The acoustic radiation torque is the time-averaged rateof change of the angular momentum caused by an acous-tic wave on an object [6]. This subject was extensivelystudied for spherical particles in Refs. [7–13]. In a non-viscous fluid, the radiation torque on a spherical particleonly occurs if the particle absorbs acoustic energy [8].Albeit, nonabsorbing particles without spherical symme-try may develop the radiation torque. Notable examplesare microfibers [14] and nanorods [15]. Some numeri-cal methods have been employed to study the radiationtorque on spheroids [16, 17].Despite the importance of the aforementioned numer-ical studies, they do not reveal the full physical pictureof the acoustic radiation torque. Also, no investigationon the acoustic spin transfer to a spheroidal particle in aviscous fluid was performed to date. We are not the firstto theoretically investigate the acoustic radiation torqueeffects on spheroids. However, the previous work by Fan ∗ gtomaz@fis.ufal.br et al. [18] is mainly devoted to developing a general the-oretical scheme for arbitrarily shaped particles.The goal of this paper is to put the acoustic radiationtorque on a spheroidal particle in a new perspective byestablishing its connection with the acoustic spin. To thisend, we consider a first-order Bessel vortex beam (FOBB)in broadside incidence to a subwavelength spheroidal par-ticle in the beam axis. Our choice relies on the fact thatthe acoustic FOBB possesses spin, which corresponds tothe local expectation value of a spin-1 operator [4]. Thisbeam not only may produce a radiation torque on theparticle but also a time-averaged force, known as theacoustic force [19–21]. Besides, some symmetry consider-ations have motivated the choice for a prolate spheroidalparticle. This object has axial symmetry (i.e., it is in-variant to a rotation around the major axis). In particlephysics terms, we may classify the prolate spheroid as aspin-0 particle concerning axial rotations. On the otherhand, rotations around the minor axis (transverse rota-tions) can be described by the interfocal vector, whichhas a 2 π rotational symmetry. Under this circumstance,the prolate spheroid can be regarded as a spin-1 particle.At this point, we contemplate that the FOBB spin canonly induce a transverse spin on the spheroid, which is aspin-1 particle.Our paper is outlined as follows. First, we calculatethe spin of a Bessel beam. Afterward, we obtain the ra-diation torque considering a nonviscous fluid by solvingthe related scattering problem in spheroidal coordinatesand integrating the result in a far-field spherical surface.We then establish the spin-torque relation and obtainsimple asymptotic expressions of the torque as the parti-cle geometry approaches a sphere and an infinitely thinspheroid. Assuming a Stokes’ flow as the particle spinsaround its minor axis [22], we derive the relation betweenthe acoustic spin and angular velocity. We predict the an-gular velocity of microparticles using the typical param-eter values of the acoustic levitation [23] and acousticalvortex tweezer [24] devices. Additionally, the theoretical a r X i v : . [ phy s i c s . c l a ss - ph ] M a r predictions are in excellent agreement with finite-elementresults of the radiation torque. II. ACOUSTIC SPIN
Assume that a Bessel vortex beam of order (cid:96) (alsoknown as vortex charge) and angular frequency ω prop-agates in fluid of density ρ , adiabatic speed of sound c , and compressibility β = 1 /ρ c . The beam interactswith a subwavelength prolate spheroidal particle, e.g. theparticle dimensions are much smaller than the acousticwavelength. A fixed laboratory coordinate system O (cid:48) co-incide to the particle center which lies in the beam axisas depicted in Fig. 1.In the laboratory system, the incident Bessel beam isdescribed in cylindrical coordinates ( (cid:37) (cid:48) , ϕ (cid:48) , z (cid:48) ) by the ve-locity potential φ in = φ J (cid:96) ( k(cid:37) (cid:48) sin β )e i kz (cid:48) cos β e i (cid:96)ϕ (cid:48) , (1)where ‘i’ is the imaginary unit, φ = p /kρ c (with p being the beam peak pressure) is the potential magni-tude, J (cid:96) is the cylindrical Bessel function of (cid:96) th-order, k = ω/c , β is the beam half-cone angle. The beamwavevector is k = k (sin β e (cid:37) (cid:48) + cos β e z (cid:48) ), with e (cid:37) (cid:48) and e z (cid:48) being the radial and axial unit vectors. The time-dependent term e − i ωt is omitted for simplicity. The inci-dent pressure and velocity fields are given, respectively,by v in = ∇ φ in and p in = i kρ c φ in .The acoustic spin density of the incident beam is de-fined by [4] S = ρ ω Im [ v ∗ in × v in ] , (2)where ‘Im’ means the imaginary-part of a quantity. Theacoustic spin is an intrinsic local property of the beam.Inasmuch as the velocity field is irrotational ∇ × v in = ,the spin satisfies the conservation law ∇ · S = 0.By substituting Eq. (1) into Eq. (2), we find the axialspin as S z (cid:48) ( (cid:37) (cid:48) ) = 2 E sin βωkr J (cid:96) ( k(cid:37) (cid:48) sin β ) ˙ J (cid:96) ( k(cid:37) (cid:48) sin β ) , (3)where E = β p / | (cid:96) | = 1). For simplicity we consider (cid:96) = 1. Referring to Eq. (3), the axial spin is given by S z (cid:48) (0) = E sin β ω e z (cid:48) , (4)We note that the beam energy E can be assessed bymeasuring the acoustic spin of the FOBB. FIG. 1. A Bessel vortex beam of (cid:96) th-order with half-cone an-gle β interacting with a spheroidal particle. The beam propa-gates along the x axis toward −∞ and along the z (cid:48) axis toward+ ∞ . The center of both coordinate systems O ( x, y, z ) (blueaxes) and O (cid:48) ( x (cid:48) , y (cid:48) , z (cid:48) ) (red axes) are located in the particlegeometric center. A 90 ◦ -counterclockwise rotation around y and y (cid:48) axis maps O (cid:48) onto O system. III. SCATTERING IN THELONG-WAVELENGTH LIMIT
The spheroidal particle has a major and minor axis de-noted by 2 a and 2 b , respectively, with interfocal distancebeing d = 2 √ a − b . The acoustic scattering is now de-scribed in a coordinate system O fixed in the geometriccenter of the particle at rest. The major axis lies in the z direction–see Fig. 1. For symmetry reasons, we describethe particle in prolate spheroidal coordinates to which ξ ≥ − ≤ η ≤
1, and 0 ≤ ϕ ≤ π isazimuth angle. In this case, the particle corresponds tothe surface defined by ξ = ξ = 2 a/d = const. The par-ticle aspect ratio is defined as the major-to-minor axisratio, which relates to the particle geometric parameter ξ as ab = 1 (cid:113) − ξ − . (5)The particle volume is V = 4 πab / πd ξ ( ξ − / . A spherical particle of radius r is recovered as ξ → ∞ ,with ξ d → r . Whereas a slender particle correspondsto the minor semiaxis being much smaller than the majorsemiaxis, a/b (cid:29) ξ ≈ (cid:15) = kd kaξ (cid:28) . (6)We emphasize that the other size parameter related tothe minor semiaxis b , say kb/ξ , is also much smaller thanone, as b < a . In this case, only the monopole and dipolemodes of the incident and scattered waves are needed todescribe the particle-wave interaction. Accordingly, thepartial wave expansions of the incident and scattering po-tential velocities in the particle frame are given in prolatespheroidal coordinates by [25] φ in = φ (cid:88) n =0 n (cid:88) m = − n a nm S nm ( (cid:15), η ) R (1) nm ( (cid:15), ξ )e i mϕ , (7a) φ sc = φ (cid:88) n =0 n (cid:88) m = − n a nm s nm S nm ( (cid:15), η ) R (3) nm ( (cid:15), ξ )e i mϕ , (7b)where S nm is the angular function of the first kind, and R (1) nm and R (3) nm are the radial functions of the first andthird kind, respectively. The quantities a mn and s nm arethe beam-shape and scaled scattering coefficients.We assume that the particle behaves as a rigid andimmovable spheroid. Hence, the velocity normal compo-nent is zero on the particle surface, ∂ ξ ( φ in + φ sc ) ξ = ξ = 0.Using (7) in this condition, one obtains the scattering co-efficient as s nm = − ∂ ξ R (1) nm ∂ ξ R (3) nm (cid:12)(cid:12)(cid:12)(cid:12) ξ = ξ . (8)We shall see in Sec. IV that in the long-wavelengthlimit, only the dipole scattering coefficients contributeto the acoustic radiation torque. Hence, after Taylor-expanding the radial functions given in (A1) around (cid:15) =0, we obtain the dipole scattering coefficients as [26] s = i (cid:15) f − (cid:15) f , (9a) s , − = s = i (cid:15) f − (cid:15) f , (9b)where f = 23 (cid:34) ξ ξ − − ln (cid:32) ξ + 1 (cid:112) ξ − (cid:33)(cid:35) − , (10a) f = 83 (cid:34) − ξ ξ ( ξ −
1) + ln (cid:32) ξ + 1 (cid:112) ξ − (cid:33)(cid:35) − (10b)are the scattering factors.In the far-field kξ (cid:29)
1, the spheroidal expansion in(7) asymptotically approaches the expansion in sphericalcoordinates ( r, θ, ϕ ) as follows [26] φ in = φ kr (cid:88) n =0 n (cid:88) m = − n a nm sin (cid:16) kr − nπ (cid:17) Y mn ( θ, ϕ ) , (11a) φ sc = φ e i kr kr (cid:88) n =0 n (cid:88) m = − n i − n − a nm s nm Y mn ( θ, ϕ ) , (11b)where Y mn ( θ, ϕ ) is the spherical harmonic of n th-orderand m th-degree. Here the beam-shape coefficient a nm describes an incident wave in spherical coordinates.Hereafter, we shall consider the beam-shape coefficientsin spherical coordinates. IV. ACOUSTIC RADIATION TORQUE
The density of linear momentum flux carried by anacoustic wave is well-known from the fluid mechanicstheory [27] P = −L I + ρ vv , with the over bar denotingtime-average over a wave period and I being the unit ten-sor. The acoustic fields L and ρ vv are the Lagrangiandensity and Reynolds’ stress tensor. The density of an-gular momentum flux is then L = r × P . The radiationforce exerted by the incident wave on an surface elementd S of the particle is d F rad = P · n d S , with n being theoutwardly unit vector at the particle surface S , whereasthe moment of the infinitesimal radiation force is givenby d τ rad = r × d F rad = L · n d S . Therefore, the acousticradiation torque on the particle is expressed by τ rad = (cid:90) S L · n d S. (12)As the angular momentum flux satisfies the conservationlaw [6] ∇ · L = , the integral can be evaluated over avirtual surface S of a sphere in the far-field kr (cid:29) τ rad = − (cid:82) S ( r × ρ vv ) · e r d S , with e r being the unit-vector in radial direction. The fluid ve-locity is the sum of the incident and scattered velocities, v = v in + v sc . Substituting the total fluid velocity intothe far-field expression of the radiation torque and notingthat vv = (1 / vv ∗ ], we find [28] τ rad = − ρ r (cid:90) Ω s r × ( v in v ∗ sc + v sc v ∗ in + v sc v ∗ sc ) · e r dΩ s , (13)where ‘Re’ means the real part of, the asterisk denotescomplex conjugation, Ω s represents the unit-sphere, anddΩ s is solid angle. No torque is formed in the absenceof the particle; hence, Re (cid:82) Ω s r × v in v ∗ in · e r dΩ s = .Using the partial wave expansion in the far-field as givenin (11) into Eq. (13), one can show that the Cartesiancoordinates of the radiation torque is expressed by [8] τ rad ,x = − E k √ (cid:20) ( a , − + a )(1 + s ) a ∗ s ∗ + a (1 + s )( a ∗ , − + a ∗ ) s ∗ (cid:21) , (14a) τ rad ,y = − E k √ (cid:20) i ( a , − − a )(1 + s ) a ∗ s ∗ − i a (1 + s )( a ∗ , − − a ∗ ) s ∗ (cid:21) , (14b) τ rad ,z = E k Re (cid:2) ( | a , − | − | a | )(1 + s ) s ∗ (cid:3) . (14c)Clearly, the radiation torque is caused by the nonlin-ear interaction between the incident and scattered dipolemodes.To compute the radiation torque from (14), the beam-shape coefficients of the incident wave should be known a priori . Notable examples are plane waves, and Besselvortex [29] and Gaussian beams [30]. Numerical schemesand the addition theorem of spherical functions have beenemployed to compute the coefficients for different typesof beam [19, 28, 31–33].We now proceed to calculate the FOBB radiationtorque in broadside incidence to the particle. In this case,the Bessel beam propagates along the x axis toward −∞ in the particle system O . We see in Fig. 1 that the labo-ratory system O (cid:48) can be mapped onto the particle system O through a 90 ◦ -counterclockwise rotation around the y (cid:48) axis.In the laboratory system O (cid:48) , the beam-shape coeffi-cient of the FOBB is given by [32] a (cid:48) n(cid:96) = 4 π i n − m Y mn ( β, H ( n − m ) δ m(cid:96) , (15)where H ( n − m ) is the unit-step function, which is equalto 0 for n − m < n − m ≥
0. Accordingto (14), we have to compute the dipole beam-shape co-efficient a ,m in the particle system O . The relation be-tween the beam-shape coefficient in the laboratory andparticle system is given through the Wigner D -function D nmµ ( α, ψ, ζ ) as [34] a nm = n (cid:88) µ = − n a (cid:48) nµ D nmµ ( α, ψ, ζ ) , (16)where α , ψ, and ζ are the Euler angles. Mapping system O onto O (cid:48) corresponds to the Euler angles α = 0, ψ = − π/
2, and ζ = 0. To obtain a ,m we need only the dipolebeam-shape coefficient in system O (cid:48) , a (cid:48) ,µ . According toEq. (16), this coefficient is a (cid:48) ,µ = − δ µ, √ π sin β. Byreplacing it into Eq. (16), we find the dipole beam-shapecoefficients in the particle system as a , − = a = − (cid:114) π β, a = √ π sin β. (17)Using this result into (14), we find the radiation torquealong the x -axis as τ rad ,x = 3 πk E sin β Re [ s + s + 2 s s ∗ ] . (18)Using the scattering coefficients of (9) into this expres-sion, we find τ rad = − ( ka ) χπa E sin β e x , ka (cid:28) , (19a) χ = ( f − f ) ξ , (19b)where χ is related to the difference of the dipole factors.The asymptotic gyroacoustic expressions as the parti-cle geometric parameter describes a spherical ( ξ (cid:29) ξ ≈
1) are given, respectively, by χ = 3400 (cid:18) ξ − ξ (cid:19) , (20a) χ = 427 ( ξ − + 49 ( ξ − (cid:18) (cid:20) ( ξ − (cid:21)(cid:19) . (20b) The radiation torque vanishes as the particle geometryapproaches a sphere, lim ξ →∞ τ rad = 0. This is sup-ported by the fact that no torque is produced on a non-absorbing sphere [8].Importantly, both asymptotic expansions of the gy-roacoustic factor in (20) approach to zero. This sug-gests that the geometric torque factor χ should havean extreme value in the interval 1 ≤ ξ < ∞ . Usingthe Nelder-Mead numerical method through NMaximize function of Mathematica Software [35], we find the max-imum value χ max = 0 .
14 at ξ = 1 . O (cid:48) ) as τ rad = γπa ( ka ) E in (0) e z (cid:48) . The torque ispositive given that e z (cid:48) = − e x , i.e., the z (cid:48) and x axishave opposite orientation. Using Eq. (4), we find at thespin-induced radiation torque on the particle as τ rad = χπa ω S z (cid:48) (0) . (21) V. PARTICLE ANGULAR VELOCITY
In broadside incidence, a FOBB may set the spheroidalparticle to spin around its minor axis. Here we considerthat the particle is immersed in a viscous incompressiblefluid with dynamic viscosity µ . To simplify our analysis,we assume that the yielded flow due to the particle spinhas a small Reynolds number Re (cid:28)
1, i.e., the so-calledStokes’ flow. It is worth noticing that by solving theacoustic scattering problem we have considered a com-pressible fluid.In the laboratory frame (system O (cid:48) ), the rotation dy-namics is described by the Newton’s second law, I p ˙Ω = τ rad − τ drag (Ω) , (22)where I p is the particle moment of inertia relative to theminor axis, Ω is the particle angular velocity, and τ drag is the drag torque that counteracts the radiation torque.Assuming the no-slip boundary condition at the par-ticle surface ξ = ξ , one can find the drag torque as [22] τ drag = πµ d ˜ τ drag Ω , (23a)˜ τ drag = 43 1 − ξ ξ − (1 + ξ ) ln (cid:16) ξ +1 ξ − (cid:17) . (23b)The dimensionless drag torque ˜ τ drag depends only of thegeometry of the particle. As the particle asymptoticallyapproaches ( ξ (cid:29)
1) a sphere of radius r , we recover theclassical result of the drag torque for a spherical geome-try, τ spheredrag = 8 πµ r Ω , with r ≈ ξ d/ FIG. 2. The radiation torque exerted on a particle in (a) water and (b) air as a function of the particle geometric parameter ξ .The torque is evaluated with Eq. (19a) with β = π/
4. The parameters for water are a = 120 µ m, f = 1 MHz, and p = 500 kPa;while for air, we have a = 680 µ m, f = 40 kHz, and p = 3 . ξ = 1 .
31 ( a/b = 1 . ξ = 1 . , . ,
4) are depicted in panel (a).TABLE I. The acoustic parameters of acoustofluidics and lev-itation systems at room temperature [36].Medium Density ρ [kg / m ] Speed of sound c [m / s] Dynamic viscosity µ [Pa · s]Air 1 .
22 343 1 . × − Water 998 1483 10 − reaches a stationary angular velocity that can be ob-tained by combining Eqs. (19a), (22) and (23), Ω st = τ rad πµ d ˜ τ drag = ( ka ) ˜Ω st E µ sin β e z (cid:48) , (24a)˜Ω st = χξ τ drag , (24b)with ˜Ω st being the dimensionless angular velocity. Bymeasuring the angular velocity Ω st and knowing the par-ticle and beam parameters, one can determine the fluidviscosity µ through Eq. (24a).For a quasispherical and slender particle, the dimen-sionless angular velocity is, respectively,˜Ω st = 33200 (cid:18) ξ − ξ (cid:19) , ξ (cid:29) , (25a)˜Ω st = −
136 ( ξ − (cid:20) (cid:18) ξ − (cid:19)(cid:21) . ξ ≈ . (25b)The relation between the axial acoustic spin and par-ticle angular velocity follows by replacing Eq. (21) into(24a), S z (cid:48) (0) = γ Ω st (26a) γ = 16 µ ˜ τ drag ( ka ) χξ ω , (26b) where γ is the gyroacoustic ratio of the spin and angularvelocity in the SI units of kg m − . This result describeshow the spin is transferred to a subwavelength spheroidalparticle. It also enables the experimental assessment ofthe acoustic spin by measuring the angular velocity of asubwavelength spheroidal particle. VI. MODEL PREDICTIONS
We provide some predictions for typical experimentalsetups of acoustical vortex tweezers [24] and acoustic lev-itation [23] to which the particle is immersed in a water-like medium and air, respectively. The acoustic parame-ters of these fluids are summarized in Table I. The par-ticle has a fixed major semiaxis of a = 680 µ m in air and a = 120 µ m in water.The theoretical predictions will be compared with 3Dfinite-element simulation results performed in ComsolMultiphysics (Comsol Inc., USA). The radiation torquewas computed by numerical integration of the angularmomentum flux L over the particle surface as describedin Eq. (12). The mean discretization length on the sur-face is b/
50; while in the surrounding fluid, we considerat least λ/
12. The domain has a cylindrical geometrywith 36 b diameter and height. We have also adopted thefirst-order scattering boundary condition at the domainedges.In Fig. 2, we show the radiation torque exerted on aspheroidal particle as a function of the geometric param-eter ξ in water and air. The torque is evaluated withEq. (19a). The pressure peaks are p = 3 . p = 0 . f = 40 kHz (air) and f = 1 MHz (water). The half-coneangle of the beam is β = π/
4. According to Eq. (6) thesize parameter (cid:15) is always smaller than 0 .
51. The radia-
FIG. 3. The stationary angular velocity as a function of theparticle geometric parameter ξ for different peak pressuresin water and air. The velocity is calculated with Eq. (24a)and β = π/
4. The parameters for water are a = 120 µ mand f = 1 MHz; and for air, a = 680 µ m and f = 40 kHz.The maximum value of the angular velocity is at ξ = 1 . a/b = 1 . tion torques peak at ξ = 1 .
31, which corresponds to theaspect ratio a/b = 1 .
54. finite-element results are alsodepicted for comparison. The root mean square error(rms) is about 10 − in both cases.In Fig. 3, we plot the angular velocity versus the parti-cle geometric parameter ξ with different peak pressures p = 1 , p = 100 ,
500 kPa (water). Wenote that the peak velocity is reached at ξ = 1 .
21, whichcorresponds to the aspect ratio a/b = 1 .
77. When com-pared to the radiation torque, this maximum value ap-pears for a different geometric parameter. This happensbecause the viscous drag torque acts on the particle, asshown in Eq. (24b), changing the optimal aspect ratiofor the angular velocity. In water, the angular velocitycan be as large as 100 rpm, whereas in air, it can be tentimes this value.
VII. SUMMARY AND CONCLUSION
We have demonstrated that the acoustic spin can betransferred to a subwavelength spheroidal particle. Us-ing the partial wave expansion of the incident and scat-tered velocity potentials in spheroidal coordinates andintegrating the total angular momentum density in thefar-field, we derived a general expression of the radia-tion torque in the long-wavelength limit. Consideringa broadside incidence of a FOBB onto the particle cen-tered at the beam axis, we obtained the correspondingradiation torque. In turn, the torque produces an angu-lar velocity on the particle that rotates around its minoraxis.We offer a more fundamental explanation of thespin-induced torque using a description from quantumphysics. The acoustic FOBB is regarded as a spin-1field [4], whereas a prolate spheroid can be classified as a spin-0 and spin-1 particle under axial and trans-verse rotations, respectively. Therefore, we found thatthe spin can only be transferred from the FOBB inbroadside incidence to the particle inducing a trans-verse rotation. Importantly, axial rotations can be gen-erated by viscous torques caused by tangential stresseswithin the particle boundary layer. However, the vis-cous torque can be neglected as the boundary layer thick-ness, δ = (2 µ /ρ ω ) / is much smaller than the particlesize [37]. Here δ/a ∼ − (in water) and δ/a ∼ − (in air). For this reason, this torque was discarded in ouranalysis.The stationary angular velocity is obtained by takingthe radiation and drag torque balance in Eq. (22). Con-sidering the physical parameters of acoustofluidic and ex-perimental levitation setups, our model predicts that thestationary angular velocity can reach 100 rpm in waterand 1000 rpm in air. Therefore, it is feasible to mea-sure the angular velocity and use the result to obtain theacoustic spin. We can also determine the fluid viscos-ity by measuring the angular velocity. Additionally, bymeasuring the acoustic spin, we can obtain the beam en-ergy density as described in Eq. (2). This may provide ameans of assessing the energy of focused ultrasonic vor-tices in acoustic levitation systems [23] and acousticalvortex tweezers [24].In conclusion, we have established a connection be-tween the acoustic spin and the angular velocity of aspheroidal particle in a viscous fluid. The developedmethod can also be applied to unveil the properties ofother spin-carrying acoustic beams. ACKNOWLEDGMENTS
We thank the National Council for Scientific andTechnological Development–CNPq, Brazil (Grant No.401751/2016-3 and No. 307221/2016-4) for financial sup-port.
Appendix A: Monopole and dipole radial functions
In the long-wavelength limit, the radial spheroidal functions are given to the (cid:15) -order by [38] R (1)00 = 1 + (cid:15) (cid:0) − C (cid:1) + (cid:15) (cid:20) − C + 135 C + (cid:15) (cid:0) − C + 5670 C − C (cid:1)(cid:21) , (A1a) R (1)10 = (cid:15)C + (cid:15) C (cid:20) − C + (cid:15) (cid:0) − C + 875 C (cid:1)(cid:21) , (A1b) R (1)11 = (cid:15)S (cid:15) S (cid:20) − C + (cid:15) − C + 875 C ) (cid:21) , (A1c) R (2)00 = − (cid:15) (cid:26) L − (cid:15) C + L (3 C − (cid:16) (cid:15) (cid:17) (cid:20) C + 9 C + L
60 (1109 − C + 135 C ) (cid:21)(cid:27) , (A1d) R (2)10 = 3 C (cid:15) (cid:26) C − C C − L − (cid:16) (cid:15) (cid:17) (cid:20) C − C C + L (22 − C ) (cid:21) + 1882 (cid:16) (cid:15) (cid:17) × (cid:20) C − C C + 7875 C − L (116073 − C + 7875 C ) (cid:21)(cid:27) , (A1e) R (2)11 = − S (cid:15) (cid:26) C S − L − (cid:16) (cid:15) (cid:17) (cid:20) C (cid:18) − S (cid:19) − L (33 + 5 C ) (cid:21) − (cid:16) (cid:15) (cid:17) (cid:20) C − C + 712 C S − L (106324 − C − C ) (cid:21)(cid:27) , (A1f) R (3) nm = R (1) nm + i R (2) nm , (A1g)where R (2) nm is the radial function of the second-kind. Note that R ( i ) nm = R ( i ) n, − m , with i = 1 , ,
3. The auxiliary functionsare expressed by C n = 12 (cid:104) ( (cid:112) ξ − ξ ) n + ( (cid:112) ξ − ξ ) − n (cid:105) ,S n = 12 (cid:104) ( (cid:112) ξ − ξ ) n − ( (cid:112) ξ − ξ ) − n (cid:105) ,L = 12 ln (cid:32) (cid:112) ξ − ξ ) − − ( (cid:112) ξ − ξ ) − (cid:33) . Appendix B: Wigner D function The Wigner D function D nµm (0 , − π/ ,
0) was evaluated with Mathematica Software (Wolfram Inc., USA). To thedipole approximation, we have D = 1 , (B1a) D − , − = D − , = D , − = D = 12 , (B1b) D − , = D = − √ , (B1c) D , − = D = 1 √ , (B1d) D = 0 . (B1e) [1] H. C. Ohanian, Am. J. Phys. , 500 (1986).[2] F. J. Belinfante, Physica , 887 (1939). [3] C. Shi, R. Zhao, Y. Long, S. Yang, Y. Wang, H. Chen,J. Ren, and X. Zhang, Nat. Sci. Rev. , 707 (2019). [4] K. Y. Bliokh and F. Nori, Phys. Rev. B , 174310(2019).[5] G. T. Silva, J. Acoust. Soc. Am. , 2405 (2014).[6] L. Zhang and P. L. Marston, J. Acoust. Soc. Am. ,1679 (2011).[7] L. Zhang and P. L. Marston, Phys. Rev. E , 065601(2011).[8] G. T. Silva, T. P. Lobo, and F. G. Mitri, Europhys. Lett. , 54003 (2012).[9] F. G. Mitri, T. P. Lobo, and G. T. Silva, Phys. Rev. E , 026602 (2012).[10] L. Zhang and P. L. Marston, Biomed. Opt. Expr. , 1610(2013).[11] F. G. Mitri, Ultrasonics , 57 (2016).[12] L. Zhang, Phys. Rev. Applied , 034039 (2018).[13] Z. Gong, P. L. Marston, and W. Li, Phys. Rev. Appl. , 064022 (2019).[14] T. Schwarz, P. Hahn, G. Petit-Pierre, and J. Dual, Mi-crofluid Nanofluid , 65 (2015).[15] W. Wang, L. A. Castro, M. Hoyos, and T. E. Mallouk,ACS Nano , 6122 (2012).[16] F. B. Wijaya and K.-M. Lim, Acta Acust. united Ac. , 531 (2015).[17] T. S. Jerome, Y. A. Ilinskii, E. A. Zabolotskaya, andM. F. Hamilton, J. Acoust. Soc. Am. , 36 (2019).[18] Z. Fan, D. Mei, K. Yang, and Z. Chen, J. Acoust. Soc.Am. , 2727 (2008).[19] G. T. Silva, J. H. Lopes, and F. G. Mitri, IEEE Trans.Ultrason. Ferroelectr. Freq. Control , 1207 (2013).[20] L. Zhang and P. L. Marston, Phys. Rev. E , 035601(2011).[21] J. P. Le˜ao-Neto and G. T. Silva, Ultrasonics , 1 (2016).[22] D. Kong, Z. Cui, Y. Pan, and K. Zhang, Intl. J. PureAppl. Math. , 455 (2012).[23] A. Marzo, S. A. Seah, B. W. Drinkwater, D. R. Sahoo,B. Long, and S. Subramanian, Nat. Commun. , 8661 (2015).[24] M. Baudoin, J.-C. Gerbedoen, A. Riaud, O. B. Matar,N. Smagin, and J.-L. Thomas, Sci. Adv. , eaav1967(2019).[25] C. Flammer, Spheroidal Wave Functions (Dover Publi-cations, London, 2005).[26] G. T. Silva and B. W. Drinkwater, J. Acoustic. Soc. Am. , EL453 (2018).[27] G. T. Silva, J. Acoust. Soc. Am. , 3541 (2011).[28] J. H. Lopes, M. Azarpeyvand, and G. T. Silva,IEEE Trans. Ultrason. Ferroelectr. Freq. Control , 186(2016).[29] Z. Gong and P. L. Marston, J. Acoust. Soc. Am. ,EL574 (2017).[30] F. G. Mitri and G. T. Silva, Phys. Rev. E , 053204(2014).[31] G. T. Silva, IEEE Trans. Ultrason. Ferroelectr. Freq.Control , 298 (2011).[32] F. G. Mitri and G. T. Silva, Wave Motion , 392 (2011).[33] G. T. Silva, A. L. Baggio, J. H. Lopes, and F. G. Mitri,IEEE Trans. Ultrason. Ferroelectr. Freq. Control , 576(2015).[34] M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scat-tering, Absorption, and Emission of Light by Small Par-ticles
CRC Handbook of Chemistry and Physics ,84th ed. (CRC Press, Boca Raton, FL, 2004).[37] C. P. Lee and T. G. Wang, J. Acoust. Soc. Am. , 1081(1989).[38] J. E. Burke, Stud. Appl. Math.45