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Reversals of Acoustic Radiation Torque in Bessel BeamsUsing Theoretical and Numerical Implementations in

Three DimensionsZhixiong Gong, Philip Marston, Wei Li

To cite this version:Zhixiong Gong, Philip Marston, Wei Li. Reversals of Acoustic Radiation Torque in Bessel Beams UsingTheoretical and Numerical Implementations in Three Dimensions. Physical Review Applied, AmericanPhysical Society, 2019, 11 (6), pp.064022. �10.1103/PhysRevApplied.11.064022�. �hal-03330002�

PHYSICAL REVIEW APPLIED 11, 064022 (2019)

Reversals of Acoustic Radiation Torque in Bessel Beams Using Theoretical andNumerical Implementations in Three Dimensions

Zhixiong Gong,1,2,† Philip L. Marston,2 and Wei Li1,*

1School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology,

Wuhan 430074, China2Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814, USA

(Received 10 January 2019; revised manuscript received 1 May 2019; published 11 June 2019)

Acoustic radiation torque (ART) plays an important role in the subject of acoustophoresis, which couldinduce the spinning rotation of particles on their own axes. The partial wave series method (PWSM)and T matrix method (TMM) are employed to investigate the three-dimensional radiation torques of bothspherical and nonspherical objects in a single Bessel beam over broader frequencies (from Rayleigh scat-tering to geometric-optics regimes), with emphasis on the parametric conditions for torque reversal and thecorresponding physical mechanisms. For elastic objects, the dipole, quadrupole, and even several higher-scattering modes are dominant to induce the axial ARTs beyond the Rayleigh limit with a relatively largeoffset from the particle centroid to the beam axis in Bessel beams. The reversals appear with parametricconditions (including wave number, cone angle, and offset) for the extrema or null values of the corre-sponding cylindrical Bessel functions. The reversal of transverse ART from a rigid nonspherical particleis also investigated in an ordinary Bessel beam and the geometrical surface roughness is briefly studiedfor the effect on the radiation torques. This suggests the possibility of acoustic tweezers controlling thespinning motion of particles within and beyond the Rayleigh limit.

DOI: 10.1103/PhysRevApplied.11.064022

I. INTRODUCTION

Acoustic manipulations have attracted more and moreattention for the peculiar advantages of biocompatibilityand remote contact, which mainly consider the translo-cation and rotation dynamics of the particles immersedin fluids. As compared to its optical counterpart, acous-tic tweezers could provide larger forces and torques withthe same incident wave field intensity and get rid of heat-ing issues [1]. Meanwhile, the transducers in designing theacoustic tweezers could provide a broader band of frequen-cies from kHz to GHz, which could be applied for a largerange of particle sizes [2]. It is noteworthy that the acousticradiation force (ARF) is induced by the transfer of lin-ear momentum, while the acoustic radiation torque (ART)comes from the exchange of angular momentum from theacoustic field to immersed particles. In general, acousticscattering [3–11] and streaming [12–16] are the two typicalkinds of physical mechanisms to move or rotate particlesin the fluids illuminated by vortex beams [17–22] or stand-ing waves [23–27]. For the ARFs in a viscous fluid, theacoustic streaming will dominate for small-sized particles,

*hustliw@hust.edu.cn†Present address: Univ. Lille, CNRS, Centrale Lille, ISEN,

Univ. Polytechniques Hauts-de-France, UMR 8520-IEMN,International laboratory LIA/LICS, F-59000 Lille, France.

whereas the acoustic scattering plays a dominant role forparticles with sizes larger than the critical value [28–30].To date, the ARF has been studied extensively in the fieldof particle manipulation including particle trapping, selec-tivity, sorting, patterning, orbital rotation, and so on, whilethe applications of spinning (i.e., rotation around the par-ticle centroid) induced by ART still need more attention,especially on the reversal conditions and physical mech-anisms for particles with different geometrical shapes andmaterial composition. In addition, most of the theoreticalwork related to acoustic torque are focused on the Rayleighlimitation [31]. It is worthwhile noting that particle dynam-ics are interesting and promising beyond long-wavelengthapproximation, such as in the Mie and geometric-opticsregimes. Hence, it is necessary to develop physical modelsand theoretical or numerical methods to take the broadbandfrequency analysis in acoustofluidics into consideration.

Standing waves and vortex beams are very promisingcandidates in designing acoustic tweezers for biomedicalapplications and material chemistry in the future, whichcould both induce the three-dimensional ARFs to move ortrap particles and the ARTs to spin particles in the acous-tic field under certain circumstances. The vortex beamscould be employed to rotate the particles by the transferprocedure of the orbital angular momentum between theacoustic field and particles. Rotations give extra degreesof particle dynamics other than the translocations in three

2331-7019/19/11(6)/064022(13) 064022-1 © 2019 American Physical Society

GONG, MARSTON, and LI PHYS. REV. APPLIED 11, 064022 (2019)

dimensions. For particles in a single vortex beam, the rel-ative location of the object and beam axis will affect theARFs and ARTs exerted on the object. Zhang and Marstonderived a theoretical relationship between the axial torqueTz on an axisymmetric object located on the axis of a sin-gle vortex beam and the absorption energy Pabs, weightingby the ratio of topological charge M and wave angularfrequency ω, written as Tz = (M/ω)Pabs [32]. However,this theory cannot be applied when the particle moves offthe beam axis. In the following work by Zhang, a sim-ilar expression was given for the off-axis situation [31],which is, in fact, based on the Graf’s addition theoremof Bessel functions [33] to use the summation of on-axistorques to describe the off-axial condition. Under this cir-cumstance, the sign of effective topological charge couldnot be decided intuitively since it depends on the weightingamplitude and cylindrical Bessel functions. As observedin the aforementioned formulas for the on-axis situation,the direction of axial ART can be determined once thesigns of the topological charge are known. The sign of theeffective topological charge is very important for findingthe positive or negative axial torques as well as the rever-sal parameters. Zhang gave the theoretical derivation ofthe effective topological charge for a small particle (i.e.,long-wavelength approximation) and found the conditionfor axial torque reversal in the off-axis incidence [31]. Itis noteworthy that the ARTs from small particles of fluid[34] and viscoelastic [35] materials have been investigatedin the traveling/standing plane wave and Bessel beam withthe on-axis configuration. The secondary radiation torquesfrom multiple particles in arbitrary position have beenstudied in the Mie scattering regime under the illumina-tion of traveling and standing plane waves [36]. However,it still needs to be discussed for a more general case inBessel beams, which will include a broadband frequency(in and beyond the Rayleigh regime). This is discussed andinvestigated in the present work.

On the other hand, Baresch et al. recently found that thequadrupolar particle vibration mode could not be neglectedeven in the Rayleigh regime for the viscous dissipation inthe fluid and absorption in the particle [37]. Hence, it isimportant to compute the total radiation torque at a broad-band frequency, which will consider not only the dipolarvibration mode, but also higher-order vibration modes.The partial wave series method (PWSM) is one kind oftheoretical method that considers the necessary individualvibration modes and can correctly compute the scatteringfield of particles immersed in fluid. Based on the angu-lar momentum stress tensor theory, the ARTs could beexpressed in terms of the incident and scattered beamshape coefficients with the explicit derivation given below.It is anticipated that the PWSM could provide a theo-retical method for spherical shapes to calculate the exacttorques induced by the acoustic field. In biomedical appli-cations and physical chemistry, the spherical shapes cannot

always model the particle properly. Hence, the model usingmore accurate geometric shapes will predict more pre-cise torques on the particles of interest. To this end, aversatile numerical method, the T matrix method (TMM)[38–41], is introduced to calculate the three-dimensionalARTs for the nonspherical particle. Note that the TMMcould apply for a spherical shape as a special case ofgeneral three-dimensional shapes. In addition, it is neces-sary to pay attention to the surface shape of the particlesince the geometrical shape will influence the scatteringfield and accordingly the transfer process of both the lin-ear and angular momenta between the particle and theacoustic field. The influence of the surface roughness willbe briefly studied in this work. In combination with thework of ARFs in three dimensions [11], it is promisingto develop the acoustic tweezers numerical toolbox withsix degrees of freedom [42,43], which could provide pre-cise and stable manipulations and will finally facilitate theapplications of acoustofluidics technology in ultrasounddiagnosis and clinical medicine. The orbital rotation of theparticle around the beam axis (due to the lateral ARFs) isoutside the scope of this work, while the emphasis will beon the spin rotation around the particle center induced bythe ARTs.

II. PHYSICAL MODELS AND MATHEMATICALFORMULAS

A. Three-dimensional ARTs in arbitrary acousticwaves

The theoretical methods (partial wave series method)and the numerical method (T matrix method) have beendemonstrated for acoustic scattering in our previous work[3–5,33,39–44], which will not be shown here for simplic-ity. Both the PWSM and TMM provide a linear relation-ship between the incident (anm) and scattered (snm) beamshape coefficients. In the following, the acoustic radia-tion torques in three dimensions are independently derivedbased on the radiation angular momentum stress tensorapproach [32,45,46], which gives a general relationshipbetween the three-dimensional torques and (incident andscattered) beam shape coefficients.

Consider an object surrounded by an ideal fluid (theangular momentum stress tensor meets ∇ · S′

T = 0) illu-minated by an arbitrary acoustic field and based on thedivergence theorem, the integral of the angular momentumstress tensor over the object surface (S) could be extendedto an arbitrary spherical surface in the far-field (S0 withr → ∞) with the center located at the geometric center ofthe particle O (see Fig. 1). This procedure facilitates thetheoretical and numerical methods to compute the ARTson arbitrary shape objects and gives a uniform formulain terms of incident and scattered fields. According to theformula T = ∫ ∫S0

S′TdS and the angular momentum stress

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REVERSALS OF ACOUSTIC RADIATION TORQUE. . . PHYS. REV. APPLIED 11, 064022 (2019)

fluid

FIG. 1. Schematic of an arbitrarily shaped object in a nonvis-cous fluid illuminated by a Bessel beam with arbitrary topologi-cal charge and location.

tensor S′T = r × ST where ST = 〈L〉 − ρ0〈uu〉 is the lin-

ear momentum stress tensor, the torque vector could beexpressed as

T = −ρ0

∫∫S0

〈L〉r × dS − ρ0

∫∫S0

〈(r × u)u〉dS, (1)

where ρ0 is the fluid density, 〈L〉 is the time average ofthe Lagrangian density, and u denotes the total veloc-ity vector. The differential surface area in the far field isdS = n · r2 sin θdθdϕ with n the outward unit normal vec-tor. Introducing the relationship r × dS = 0 and the timeaverage of complex numbers, Eq. (1) could be simplifiedas

T = −ρ0

2Re

∫∫S0

{(r × ∇�) · n∇�∗}dS, (2)

where � is the total velocity potential having u = ∇�

and * denotes complex conjugation. For convenience, theangular momentum operator L from quantum mechanics isintroduced with L = −i(r × ∇) [47], where i is the imag-inary unit. By substituting r × ∇ = iL and � = �i + �sinto Eq. (2), it is rearranged as

T = ρ0

2Im

∫∫S0

{∂�i

∗

∂rL�i + ∂�i

∗

∂rL�s + ∂�s

∗

∂rL�i

+∂�s∗

∂rL�s

}dS, (3)

where Im designates the imaginary part. The far-fieldincident �i and scattered �s potentials are, respectively

�i = �0

∞∑n=0

n∑m=−n

anmjn(kr)Ynm(θ , ϕ), (4)

�s = �0

∞∑n=0

n∑m=−n

snmh(1)n (kr)Ynm(θ , ϕ). (5)

In the above, anm and snm are the incident and scatteredcoefficients of expansion (beam shape coefficients), �0is the beam amplitude, and jn(kr) and h(1)

n (kr) are thespherical Bessel and Hankel functions of the first kind,respectively, Ynm(θ , ϕ) represents the normalized spheri-cal harmonics and k is the wave number. In the absenceof scatterers there is no transfer of angular momentum inthe ideal fluid, and therefore, the incident acoustic field willremain as original so that the term in Eq. (3) only involving�i will vanish. In order to further simplify the ART in Eq.(3) and use incident and scattered beam shape coefficientsto describe the ART, the far-field (kr → ∞) asymptoticexpressions of the spherical Bessel function and Hankelfunction of the first kind with a recursion relation of thespherical Bessel function and its derivative are employed

jn(kr) i−(n+1)eikr/2kr + in+1e−ikr/2kr,

h(1)n (kr) i−(n+1)eikr/kr,

j ′n(kr) = (n/kr)jn(kr) − jn+1(kr). (6)

By inserting Eqs. (4)–(6) into Eq. (3) and conducting sev-eral algebraic operations, the ART vector can be describedbriefly in terms of the incident and scattered beam shapecoefficients, such that

T = −ρ0�20

2kr2 Re∫∫

S0

{∑nm

∑n′m′

in−n′(a∗

nm + s∗nm)sn′m′

×(Y∗nmLYn′m′)

}dS. (7)

To obtain the ARTs theoretically or numerically,the outward unit normal vector n = sin θ cos ϕex +sin θ sin ϕey + cos θez in Cartesian coordinates is appliedfor the projections of ARTs in three dimensions. Thedimensionless ART function Γ is introduced to depict theART with the relationship T = πr3

0I0c−10 Γ, where I0 =

(ρ0c0/2)(k�0)2 with c0 being the speed of sound in the

surrounding fluid and r0 is the characteristic dimension ofthe target (i.e., the radius of the maximum circumscribedsphere). Finally, the dimensionless ART function Γ could

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GONG, MARSTON, and LI PHYS. REV. APPLIED 11, 064022 (2019)

be rewritten as

Γ = − 1π(kr0)

3 Re∫∫

S0

{∑nm

∑n′m′

in−n′(a∗

nm + s∗nm)sn′m′

×(Y∗nmLYn′m′)

}sinθdθdϕ. (8)

In Cartesian coordinates, the angular momentum opera-tor is expressed as L = Lxex + Lyey + Lzez, with the threecomponents [47]

Lx = i sin ϕ∂

∂θ+ i cot θ cos ϕ

∂

∂ϕ,

Ly = −i cos ϕ∂

∂θ+ i cot θ sin ϕ

∂

∂ϕ,

Lz = −i∂

∂ϕ. (9)

To implement the integration of the product of the angu-lar momentum operator and spherical harmonics over thesolid angle, the ladder operators L± = Lx ± iLy are con-sidered, which are related to the spherical harmonics asL±Ynm = √

(n ± m + 1)(n ∓ m)Yn,m+1. In addition, the zcomponent Lz of the angular momentum operator hasa relationship with the spherical harmonics as LzYnm =mYnm. Substituting the relationships of the angular momen-tum operator and spherical harmonics into Eq. (8), thedimensionless ARTs in Cartesian coordinates could bederived as follows:

Γx = − 12π(kr0)

3 Re∑nm

(a∗nm + s∗

nm)(bnm−sn,m−1

+ bnm+sn,m+1),

Γy = − 12π(kr0)

3 Im∑nm

(anm + snm)(bnm+s∗n,m+1

− bnm−s∗n,m−1),

Γz = − 1π(kr0)

3 Re∑nm

m(a∗nm + s∗

nm)sn′m′ , (10)

with coefficients defined as bnm− = √(n − m + 1)(n + m)

and bnm+ = √(n + m + 1)(n − m). Re designates the real

part of the indicated expression. It is noteworthy that theabove formulas, which are derived independently, agreewith the ART results in Silva’s work [48], which areobtained by using the trigonometric functions to expressthe incident potential in the far field. The general expres-sions in Eq. (10) demonstrate the possibility of calculationof the three-dimensional ARTs of objects in arbitrary fieldsonce the incident and scattered beam shape coefficientscan be computed properly, and in fact, could be applied

for arbitrary acoustic waves once the incident beam shapecoefficients are available.

B. Beam shape coefficients of a Bessel beam witharbitrary location

Bessel beams are one kind of typical vortex beams,which are a theoretical solution to the scalar Helmholtzequation. The essential point to describe a vortex beam intheoretical and numerical methods is to obtain the incidentbeam shape coefficients anm on a given basis function. Inthis paper, Bessel beams with arbitrary topological charges(or orders) M and locations OVB relative to the particlecenter (as shown in Fig. 1) are considered with the acousticpotential

�B = �0iM eikz(z−z0)JM (krR′)eiMϕ′, (11)

where kr = k sin β and kz = k cos β depict the radial andaxial components of the wave number with β the cone

angle of the Bessel beam, R′ =√

(x − x0)2 + (y − y0)

2

gives the radial distance of a field point (x, y, z) to the beamorigin OVB,(x0, y0, z0), and ϕ′ = tan−1[(y − y0)/(x − x0)]is the relative azimuthal angle. Note that Eq. (11) degradesinto the special cases of on-axis incidence situation[3,4,44] when the particle center is located on the axisof Bessel beams. By using the Graf’s addition theorem ofBessel functions, the incident beam shape coefficients anmwere first derived theoretically in our previous work [33]

anm = 4πξnmin−m+M Pmn (cos β)Jm−M (σ0)e−ikzz0e−i(m−M )ϕ0 ,

(12)

where ξnm = [(2n + 1)(n − m)!]1/2[4π(n + m)!]−1/2 arethe normalized coefficients, Pm

n are the associated Leg-

endre functions, σ0 = krR0, R0 =√

x20 + y2

0 , and ϕ0 =tan−1(y0/x0).

In order to obtain the three-dimensional ARTs of theparticle in an ideal fluid according to Eq. (10), the scat-tered (beam shape) coefficients need to be calculated witha suitable method. In the present work, both the the-oretical method (PWSM) for spherical shapes and thesemi-analytical and semi-numerical method (TMM) fornonspherical shapes are introduced to study the rotationalcharacteristics of the particles of interest. Both the PWSMand TMM give a linear relationship between the inci-dent (anm) and scattered (snm) beam shape coefficients.For the PWSM, the relation could be described as snm =Ananm, where the partial wave coefficients An are relatedto the complex functions (scattering coefficients) sn: An =(sn − 1)/2. The scattering coefficients sn are known for awide variety of spheres [44] and could be easily extendedfor spherical shapes with different material compositions.On the other hand, the transition relationship between anm

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REVERSALS OF ACOUSTIC RADIATION TORQUE. . . PHYS. REV. APPLIED 11, 064022 (2019)

and snm in TMM is described by snm = Tnm,n′m′an′m′ , whereTnm,n′m′ denotes the transition matrix, which only dependson the properties of the object, including the geometri-cal shape, the material composition, and the boundaryconditions at the interface, and otherwise is independentof the sources. For the PWSM, the partial wave coeffi-cients An could be regarded as the transition matrix Tnn′ =(sn − 1)/2 without the dependence on the azimuthal indexm for spheres. The TMM has been verified for the scat-tering of several typical rigid nonspherical objects [39–43]and hence could be extended for ART computations in thepresent work. In the following computations, the trunca-tion number is set as Nmax = 2 + Int

(8 + kr0 + 4.05 3√kr0

)(Int denotes the integer part of the indicated argument),which could ensure the accuracy and convergence of thepresent computations of PWSM and TMM [38].

III. THEORETICAL AND NUMERICAL RESULTS

A. Validation and resonance scattering explanationwith PWSM

The transfer of angular momentum could come fromdissipation mechanisms by the absorption either in theabsorbing particle or the viscous boundary layer in the sur-rounding fluid [37]. In this work, an ideal fluid withoutthe viscosity and only the spinning motions around theparticle centroid are considered to investigate the three-dimensional ARTs of spherical and nonspherical shapesin a single Bessel beam. To verify the theory and home-made programming, a viscoelastic solid sphere located onthe axis of the first-order Bessel beam is studied. Thesphere is made of polyethylene (PE) [49] with density ρe =957 kg/m3, and longitudinal and transverse sound speedscp = 2430 m/s, cs = 950 m/s, respectively. The normal-ized absorption coefficients are γp = 0.0074 and γs =0.022, respectively. The surrounding fluid is an ideal waterwith density ρ0 = 1000 kg/m3 and speed of sound c0 =1500 m/s. As shown in Fig. 2, the characteristic lengthof the spherical shell is the outer radius r0 = a and theinner radius is b. b = 0 gives the special case for a solidsphere and the shell case will be discussed in Sec. B. Thepartial wave coefficients An for the viscoelastic sphericalshell could be obtained by using the complex wave num-bers instead of those in elastic materials [43,50]. When thespherical shell case turns into a solid sphere, Eq. (18) inRef. [50] [Eq. (A2) in Appendix A] should choose thefirst and second rows with the first and third columnsboth for the numerator and denominator according to theboundary condition and continuity [43], as shown in theAppendix A.

The dimensionless axial ARTs of the solid PE sphereare depicted versus the dimensionless frequency ka andcone angle β of the first-order (M = 1) Bessel beam forthe on-axis incidence [transverse offset (x0, y0) = (0, 0)],as shown in Fig. 2. The present theory could be applied for

FIG. 2. The two-dimensional axial acoustic radiation toque pat-tern of a viscoelastic solid sphere (b/a = 0) vs (ka, β) with ka thedimensionless frequency and β the cone angle of the first-kindBessel beam (order M = 1). The modulus of the dimensionlessresonance backscattering form function (see text) from the solidsphere by an ordinary plane wave is scaled by a ratio of 20with the magenta solid line. The frequencies of the quadrupole(n = 2) and octupole (n = 3) scattering are marked out wherethe axial ART peaks occur.

both the on-axis and off-axis incidences of a Bessel beamilluminating a particle. In other words, when the originof the beam coincides with the particle center, our theorycould degrade into the special case of on-axis incidence,as derived previously [32]. The theory and homemadeprogramming could be validated according to the theoreti-cal expression Tz = (M/ω)Pabs and could be extended formore general situations. To explain the physical mecha-nism for the maximum axial ARTs, the modulus of dimen-sionless resonance backscattering form function (given inthe Appendix A) in an ordinary plane wave is also depictedin Fig. 2 with a scale ratio of 20 (the magenta solid line),which is obtained by subtracting a complex rigid back-ground according to the resonance scattering theory [51].As observed, the frequencies of ART peaks appear beyondthe Rayleigh regime and coincide with those of resonancescattering peaks (especially the quadrupole (n = 2) andoctupole (n = 3) scattering, respectively). The physicalinterpretation may be given as: based on the energy con-servation, the vibration modes at the resonance frequencieswill absorb more energy (i.e., larger Pabs) from the Besselbeam with the same topological charge (M ) and angu-lar frequency (ω), resulting in a larger transfer of orbitalangular momentum and accordingly larger axial ARTs, asillustrated by Tz = (M/ω)Pabs. This result has also beenstudied recently by Baresch et al., who concluded that thedipolar and quadrupolar particle vibration modes domi-nate for the ART even in the Rayleigh scattering regime[37]. Furthermore, as shown in Fig. 2, there is no negativeaxial ARTs (i.e., ART reversal), which is easily under-stood from Tz = (M/ω)Pabs > 0 since M > 0, ω > 0, and

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GONG, MARSTON, and LI PHYS. REV. APPLIED 11, 064022 (2019)

Pabs > 0 for the considered situation. However, the theorywill fail when the particle center moves off the beam axis.The axial ART may reverse under off-axis incidence andwill be further investigated in the following.

B. Reversal of axial radiation torque and physicalmechanism with PWSM

The PWSM is employed to study the axial ARTs for thePE spherical shell and solid sphere. There is air inside theshell with density ρi = 1.23 kg/m3 and sound speed ci =340 m/s. The reversal of axial ARTs for a polyurethanesphere in a first-order Bessel beam has been observed[48], however, the physical mechanism and parametricconditions have not yet revealed, which is one of themotivations of the present work. As shown in Fig. 3, thetwo-dimensional axial ARTs patterns versus (ka, β) for

a spherical shell (the first row, aspect ratio b/a = 0.99)and solid sphere (the second and third rows, b/a = 0) ina Bessel vortex beam are presented with different offsets:(x0, y0) = (π/ka, π/ka), (2π/ka, 2π/ka), (3π/ka, 3π/ka)

for the first to third columns with the distances in theunit of meters. Obviously, the reversal ARTs could beobserved at certain dimensionless frequencies and coneangles. Note that the axial ART of a sphere will vanish forthe zeroth-order Bessel beam even in the off-axis incidencesince there is no orbital angular momentum in the ordi-nary Bessel beam [31], which is not given here. However,the axial ART may still exist for nonspherical shapes withthe off-axis situation. Similar to the results in Fig. 2, themaximum modulus of axial ARTs lies at the frequenciesof resonance scattering due to the large absorption energy.In addition, when the particle deviates further away fromthe beam axis (with larger offset), the high-order scattering

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

FIG. 3. The two-dimensional axial acoustic radiation toque patterns for (a–c) spherical shell with aspect ratio b/a = 0.99 and (d–i)solid sphere with b/a = 0. The first-order for (a–f) and second-order for (g–i) Bessel beams are considered with off-axis incidence(The offset is in the length unit of meter.): (x0, y0) = (π/ka, π/ka) for the first, (x0, y0) = (2π/ka, 2π/ka) for the second, and (x0, y0) =(3π/ka, 3π/ka) for the third column, respectively. The numbers of scattering modes (n = 2, 3, 4) are also marked out in panels (e) and(f) at frequencies where the axial ART reverses.

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(a)

(b)

FIG. 4. (a) The two-dimensional axial ARTs pattern of a PEsolid sphere (b/a = 0) illuminated by the first-order (M = 1)

Bessel beam with a fixed offset ,(x0, y0) = (5, 5). (b) The cylin-drical Bessel function of the first kind versus krR0 = k sin βR0.The ART reversals occur on the β − ka lines with the param-eters of β and ka, which makes J1(k sin βR0) the extremum orzero values.

modes, including the octupole (n = 3) and hexadecapole(n = 4) or even higher orders as marked in Figs. 3(e)and 3(f), also play a dominant role in the axial ARTfor the off-axis incidence situation. However, the maxi-mum torque will decrease with the increase of offset. Thisphenomenon and principle may be very meaningful foracoustic rotation of elastic particles beyond the Rayleighregime. It is interesting to note that the intervals of boththe thin shell and solid sphere are almost equally spacedalong the dimensionless frequency ka, due to the leak ofelastic wave components, which circumnavigate the shellor sphere repeatedly (the zeroth-order symmetric and anti-symmetric Lamb modes are for thin shells [52], while theRayleigh surface wave is for solid spheres). Comparing theaxial ARTs of the second-order case (third row for M = 2)with the first-order (second row for M = 1), higher topo-logical charges tend to give a larger maximum torque sincebeams with larger topological charges carry more orbitalangular momentum [32].

To further study the physical mechanism and the param-eter conditions of the reversals, the axial ARTs of a PEsphere (b/a = 0) illuminated by the first-order (M = 1)

Bessel beam with a fixed offset (x0, y0) = (5, 5) aredepicted [Fig. 4(a)] as well as the cylindrical Bessel func-tion of the first kind versus krR0 = k sin βR0 [Fig. 4(b)].As shown in Eq. (11), the velocity potential of the Besselbeam depends on the cylindrical Bessel function JM (krR′).It is more important to discuss the properties of JM (krR0)

rather than those of JM (krR′), which is only related tothe offset of the particle to the beam axis and indepen-dent of the arbitrary field point (x, y, z). Similar to theresults in Fig. 3, Fig. 4(a) also depicts the reversals ofaxial ARTs, especially at certain fixed ka and differentβ. To find out the parameter conditions when the rever-sal appears, the ka − β curves are also plotted in Fig.4(a), which meets k sin βR0 = 1.85, 3.832, 5.335, 7.016,8.551, 10.17, and 11.71 [in fact the roots of the extremumor zero values of J1(krR0) from left to right as shown inFig. 4(b)]. It is obvious to observe that the reversals (thenegative and positive ARTs exchange) will occur whenthe parameters of dimensionless frequency ka and coneangle β are located on the ka − β curves in Fig. 4(a). Thisphenomenon has been found independently by numeri-cal experiments for the parameter condition of axial ARTreversal at broadband frequencies (beyond the Rayleighlimit) [42,43], which coincides with the recent theoreticalderivation for a small particle in the Rayleigh limit [i.e.,JM (krR0)J ′

M (krR0) = 0] [31].

C. Reversal of transverse radiation torque of rigidspheroid with TMM

For the experimental setup, it may be challenging forthe calibration of the vortex beam and the center of par-ticles. Hence, one technology of the vortex-beam-basedacoustic tweezers may be promising, which could reversethe torque without rotating the devices of the experimen-tal transducers and only need to tune the cone angle orincident frequency. In the long-wave approximation, theinteractions between the acoustic field and particle will notbe sensitive to the particle shapes, while the geometricalshape effect may dominate beyond the Rayleigh scatter-ing regime. To this end, the traditional TMM is introducedto study the ARTs combined with the derived formulasin Sec. II. Considering a rigid spheroid centered on theaxis of the zeroth-order Bessel beam (M = 0), as shown inFig. 5(a), the incident beam shape coefficients are derivedusing the addition theorem of spherical harmonics [39]

anm = 4πξnminPmn (cos θi)Pm

n (cos β)

(cos(mϕi), σ = esin(mϕi), σ = o

),

(13)

where θi and ϕi are the polar and azimuthal angles ofthe incident beam and σ = e, o (even, odd) specifies theazimuthal parity, which is introduced to compute the tran-sition matrix for convenience [41]. The transition matrix

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GONG, MARSTON, and LI PHYS. REV. APPLIED 11, 064022 (2019)

(a) (b)

(d) (c)

FIG. 5. (a) The schematicof a rigid spheroid (aspectratio b/a = 0.5) centered onthe axis of the zeroth-orderBessel beam (M = 0) with theincident polar angle θi. (b, c)the transverse ARTs �y curvesversus the incident angle θi inthe Rayleigh (ka = 0.0628)

and geometric-optics regimes(ka = 10). Different cone anglesare selected with β = 0◦ (planewave case) and β = 30◦, 45◦,60◦, and 80◦. (d) The geometricalray diagram of the zeroth-orderBessel beam with oblique inci-dence to qualitatively explainthe reversal of the transverseART �y .

could be calculated by Eqs. (14) and (15) from Ref. [39],which is given in Appendix B. When the axis of the beamcoincides with that of the rotational axis of the spheroid,the axial ART will vanish for the symmetry and out ofabsorption. However, the transverse ARTs will be inducedbecause of the asymmetrical scattering, which leads to thespinning motions.

A rigid spheroid with the aspect ratio b/a = 0.5 (here,r0 = a) is illuminated by the ordinate Bessel beam withnormal or oblique incidences (i.e., different θi), wherea and b are the polar and equatorial radii as shown inFig. 5(d). The geometrical shape function of the spheroid isr(θ) = (cos2θ/a2 + sin2θ/b2)−1/2 with θ the polar angle,which is independent of the azimuthal angle for the rota-tional symmetry. The transverse ART �y with differentcone angles β are investigated at frequencies lying in boththe Rayleigh limit [Fig. 5(b): ka = 0.0628] and geometric-optics regime [Fig. 5(c): ka = 10]. β = 0◦ gives the specialcase when the Bessel beam turns into an ordinary planewave. The black dashed line describes θi = 45◦. As shownin Fig. 5(b) in the Rayleigh limit, the ART �y vanisheswith normal θi = 0◦ or broadside θi = 90◦ incidence forsymmetry and the maximum modulus occurs at θi = 45◦.The direction of the ART is defined based on the right-handrule. This phenomenon agrees with the theoretical resultsof a rigid spheroid in a plane standing-wave field by Fanet al. with �y proportional to sin(2θi), suggesting the maxi-mum modulus of ART occurs at θi = 45◦ [53]. Since in theRayleigh regime the wavelength is much larger than the

characteristic length r0 = a of the spheroid, the transversetorque �y is not sensitive to the geometrical crosssectionof the spheroid, leading to the symmetry about θi = 45◦of the �y ∼ θi curves depicted in Fig. 5(b). However, thecrosssection will play an important role when the parti-cle size is comparable to the wavelength, for example, inthe geometric-optics regime. The results are shown in Fig.5(c) and the symmetry about θi = 45◦ of the �y ∼ θi curvebreaks as expected for the variety of the crosssection of thespheroid as the incident polar angle changes. In addition,the transverse ART reverses at certain cone angles whenthe incident polar angle θi is larger than a critical value.This could be explained with the acoustic rays diagram asshown in Fig. 5(d): the beam vectors deviate the axis witha cone angle β for Bessel beams and the red-marked raysdominate to produce a larger positive �y . It is easier toobtain the ART reversal for a larger cone angle (e.g., beamwith β = 80◦ reverses at θi = 25◦ while with β = 60◦ itreverses at θi = 48◦). The present work could be extendedfor other kinds of geometrical shapes if the geometricalfunction is available, for example, the finite cylinder withendcaps [40,41].

D. Roughness effect on the transverse ARTs withTMM

The surface roughness of particles will affect the acous-tic scattering in three-dimensional space and thus alterthe transfer of linear and angular momenta between the

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(a)

(b)

FIG. 6. Similar to Figs. 5(b) and 5(c) except that the aspectratio of the rigid spheroid is b/a = 0.8 with rough surfacesat dimensionless frequencies (a) ka = 0.0628 and (b) ka = 10,respectively. The rough spheroid is still rotationally axisymmet-ric. The two-dimensional rough spheroids are depicted at thebottom with the relative roughness δ/a = 0 (smooth spheroid),0.01, 0.03, and 0.05.

acoustic field and particles. This section will discussthe effect of surface roughness of a rigid nonsphericalshape on the dimensional ARTs with the same expres-sion of incident beam shape coefficients (zeroth-orderBessel beam with a fixed cone angle β = 60◦). The geo-metrical shape function of the rough spheroid is r(θ) =(cos2θ/a2 + sin2θ/b2)−1/2 + δ cos(lθ) with δ the ampli-tude of the roughness and l the periodic number ofthe roughness around the spheroid (a multiple of four),as shown in Fig. 6 with aspect ratio b/a = 0.8 (here,r0 = a), l = 32, δ/a = 0 (smooth spheroid), and δ/a =0.01, 0.03, and 0.05 (rough spheroids). Note that the rough-ness amplitude δ could be random over the geometricalshape of the particle, which will be given as a statis-tical distribution. For simplicity, only the fixed surface

roughness is taken into consideration in the present work.The convergence test is conducted for the rough spheroidwith ka = 10, δ/a = 0.05 and demonstrates the correct-ness of our homemade programming. The convergence isachieved at Nmax = 19, which is smaller than the trunca-tion number applied in the present work having Nmax =2 + Int

(8 + kr0 + 4.05 3√kr0

) = 29. As is similar to thecase in Sec. III C with the rough spheroid replacing thesmooth one, the dimensionless ARTs around the y axis ofthe rough spheroids with different roughnesses under theillumination of the zeroth-order Bessel beam are describedversus the incident polar angle θi at dimensionless frequen-cies ka = 0.0628 and ka = 10, respectively. Once again, asshown in Fig. 6(a), the symmetries of the �y ∼ θi curvesstill exist in the Rayleigh regime (ka = 0.0628 with awavelength much larger than the particle size and rough-ness δ). This is because the geometrical shapes of therough spheroids are still symmetric. However, a slightincrease of the roughness will lead to a small increase inthe ART, probably due to the interactions between the sub-wavelength structure along the geometrical surface of therough spheroid and the acoustic field. However, the phys-ical mechanism is outside the scope of the present work.For the effect of the roughness on the transverse ARTin the geometric-optics regime (ka = 10), the roughnessplays a dominant role to tune the �y versus the polar angleθi, as shown in Fig. 6(b). This phenomenon suggests thatthe surface shape with geometrical roughness may not beneglected to model several biological cells or bacterial atrelatively high frequencies. At the broadband frequencies,the more accurate model could provide a better forecast ofthe three-dimensional ART as well as acoustic radiationforces to better control particles or cells in acoustics.

IV. DISCUSSION AND CONCLUSION

The spinning rotation of a particle around its own axes inan acoustic wave field describes extra degrees of freedombesides the translocations and orbital rotations for acoustictweezers. The nonviscous fluid is considered in this workand hence the radiation angular momentum stress tensortheory [45,46] could be applied to compute the ARTsby transferring the integrals over the geometrical shapeof the particle to a standard spherical surface in the farfield. In addition, theoretical formulas are derived indepen-dently and proven to express the three-dimensional ARTsin terms of the incident and scattered beam shape coeffi-cients, which could be calculated with the PWSM [3–5]and TMM [33,39–43]. Note that the dissipations both inthe bulk of an absorbing particle and in the viscous bound-ary layer in the surrounding fluid play a dominant role toinduce the ARTs [37]. In addition, the spinning rate � fora Rayleigh particle in a Bessel beam could be evaluatedby keeping the balance between the axial ART and the

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GONG, MARSTON, and LI PHYS. REV. APPLIED 11, 064022 (2019)

viscous-drag torqueTD = 8πa3vρ0�, where a is the par-ticle size, v is the kinematic viscosity, and ρ0 is the fluiddensity outside [31,37]. Viscous effects on the acoustictorque neglected here can be unimportant when the oscil-lating viscous boundary layer is small in comparison withthe object size, the effects of streaming are negligible, anddissipation mechanisms within the particle are dominant.Our approach is similar to that in Ref. [32], which pre-dicted torques on an absorbing disk in an acoustic vortexbeam in agreement with measurements [54]. An approxi-mation for estimating viscous contributions to the acoustictorque for small spheres given in Ref. [55] is supported bynumerical simulation in a relevant situation [56].

Both the axial and transverse ARTs are investigated withemphasis on the parameters’ conditions and correspondingphysical mechanisms of the radiation torque reversals. Thespherical and nonspherical particles are studied by usingthe partial wave series method and the T matrix method,respectively. It is found independently that the axial torquewill reverse at certain parameter conditions verified bythe combination of numerical computations and theoreticalformulas, which indicates that the dipole, quadrupole, andeven higher-order scattering modes have a large contribu-tion to the radiation torque when the absorbing particle isplaced off the beam axis. Note that even in the Rayleighlimit, the quadrupole vibration mode in the particle domi-nates to induce the axial torque, and hence the long-waveapproximation will fail to obtain the total ARTs [37].Meanwhile, higher-order vibration modes (e.g., octupolewith n = 3 and hexadecapole with n = 4) play a dominantrole for the ARTs when the particle moves far away fromthe beam axis (with larger offsets). The resonance scat-tering seems to lead to larger energy absorption from theacoustic field by the energy conservation, resulting in thelarger ARTs at the resonance frequencies.

For the rigid spheroid, the transverse ART shows thesymmetry on the incidence angles in the Rayleigh regimefor both the smooth and rough geometrical surfaces sincethe particle size and roughness is much smaller than thelong wavelength, leading to the fact the induced ARTsare not sensitive to the geometrical shapes of the parti-cles. On the contrary, the transverse ARTs will depend onthe shapes in the geometric-optics regime and the symme-try on the incident angle is hence broken for the differentcrosssections of the spheroid when the incident directionvaries. Furthermore, the transverse ART will reverse witha relatively larger cone angle of the Bessel beam underthe oblique incidence and this can be understood qual-itatively by a geometrical rays diagram. The roughnesson the geometrical shape is also discussed and found tobe important since it affects the ARTs in the geometric-optics regime and can be neglected in the long-wavelengthapproximation.

In the fields of life sciences [25] and microfluidics,acoustic tweezers are playing a more and more important

role for precise control of particles in acoustics, whichmainly consider the translocation and orbital rotationinduced by the acoustic radiation forces or the drag forcefrom acoustic streaming, and the spinning rotation inducedby the acoustic radiation torques. Hence, it is helpful toconsider more degrees of freedom at broadband frequen-cies (including the Rayleigh limit and Mie and geometric-optics regimes) and more precise geometrical models,leading to the applicability to explore new physical mecha-nisms and applications in biomedical science and lab-on-a-chip technology. The present work provides improvementsto some extent on these alternative phenomenon and phys-ical mechanisms for the ART reversal in the spinningmotion dynamics, which may be beneficial to experimentaldesigns and engineering applications.

ACKNOWLEDGMENTS

Work supported by HUST Postgraduate Overseas Short-term Study Program Scholarship and China ScholarshipCouncil (Z.G.), NSFC (W.L. and Z.G.), and ONR (P.L.M).The authors are grateful for helpful discussions with Dr.Likun Zhang at the University of Mississippi.

APPENDIX A: PARTIAL WAVE COEFFICIENTSFOR VISCOELASTIC SPHERICAL SHELL

FILLED WITH FLUID

For convenience, several dimensionless frequencies areintroduced as [43]: x = ka, xp = kpa, xs = ksa, zp = kpb,zs = ksb, and zi = kib, where a and b describe the outerand inner radii of the spherical shell and k, kp , ks, and ki arethe wave numbers in the outer fluid, in the elastic shell oflongitudinal and shear components, and in the inner fluid.In the outer fluid, the velocity and density are ρ0 and c0. Inthe elastic shell, the density is ρe and the longitudinal andshear velocities are cp and cs. In the inner fluid, the velocityand density are ρi and ci. The partial wave coefficients An(instead of scattering coefficients sn) are given as [50]

An = − Fnjn(x) − xj ′n(x)

Fnh(2)n (x) − xh(2)

n′(x)

, (A1)

where the coefficients Fn

Fn = −ρ0

∣∣∣∣∣∣∣∣∣

α22 α23 α24 α25 0α32 α33 α34 α35 0α42 α43 α44 α45 α46α52 α53 α54 α55 α56α62 α63 α64 α65 0

∣∣∣∣∣∣∣∣∣

/

∣∣∣∣∣∣∣∣∣

α12 α13 α14 α15 0α32 α33 α34 α35 0α42 α43 α44 α45 α46α52 α53 α54 α55 α56α62 α63 α64 α65 0

∣∣∣∣∣∣∣∣∣, (A2)

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REVERSALS OF ACOUSTIC RADIATION TORQUE. . . PHYS. REV. APPLIED 11, 064022 (2019)

with the matrix elements

α12 = (2ρe/x2s )[(x

2p − 1/2x2

s )jn(xp) + x2p j ′′

n (xp)],

α13 = (2ρe/x2s )[(x

2p − 1/2x2

s )yn(xp) + x2py ′′

n (xp)],

α14 = (2ρe/x2s )n(n + 1)[xsj ′

n(xs) − jn(xs)],

α15 = (2ρe/x2s )n(n + 1)[xsy ′

n(xs) − yn(xs)],

α22 = xp j ′n(xp),

α23 = xpy ′n(xp),

α24 = n(n + 1)jn(xs),

α25 = n(n + 1)yn(xs),

α32 = 2[xp j ′n(xp) − jn(xp)],

α33 = 2[xpy ′n(xp) − yn(xp)],

α34 = (n2 + n − 2)jn(xs) + x2s j ′′

n (xs),

α35 = (n2 + n − 2)yn(xs) + x2s y ′′

n (xs),

α42 = (2ρe/z2s )[(z

2p − 1/2z2

s )jn(zp) + z2p j ′′

n (zp)],

α43 = (2ρe/z2s )[(z

2p − 1/2z2

s )yn(zp) + z2py ′′

n (zp)],

α44 = (2ρe/z2s )n(n + 1)[zsj ′

n(zs) − jn(zs)],

α45 = (2ρe/z2s )n(n + 1)[zsy ′

n(zs) − yn(zs)],

α46 = ρijn(zi),

α52 = zp j ′n(zp),

α53 = zpy ′n(zp),

α54 = n(n + 1)jn(zs),

α55 = n(n + 1)yn(zs),

α56 = −zij ′n(zi),

α62 = 2[zp j ′n(zp) − jn(zp)],

α63 = 2[zpy ′n(zp) − yn(zp)],

α64 = (n2 + n − 2)jn(zs) + z2s j ′′

n (zs),

α65 = (n2 + n − 2)yn(zs) + z2s y ′′

n (zs), (A3)

where jn and yn are the first and second kind of sphericalBessel functions, one (′) and two (′′) primes of functionsrepresent the first- and second-order derivate with respectto the indicated argument. For the viscoelastic materi-als, the complex wave number should be introduced withk∗

p=kp(1 − iγp) and k∗s =ks(1 − iγs) in replacement of the

longitudinal and shear wave numbers in xp ,s and zp ,s whereγp ,s are the normalized absorption coefficients, respec-tively. When the inner fluid is absent, the special case goesto the spherical solid situation. According to the linearequations of the boundary conditions, one has

Fn = −ρ0

∣∣∣∣ α22 α24α32 α34

∣∣∣∣/∣∣∣∣ α12 α14

α32 α34

∣∣∣∣. (A4)

Note that for a vacuum-filled spherical shell, the scatteringcoefficients are given in Refs. [57,58]. The far-field formfunction is defined in the PWSM as

f (ka, θ , β, ϕ) = −2ika

Nmax∑n=0

n∑m=−n

anmAnYnm(θ , ϕ). (A5)

For the nonspherical shape using the TMM, the outerradius of the spherical shape a in Eq. (A5) should bereplaced with the characteristic length r0, and the partialwave coefficients An should be replaced by the transitionmatrix.

APPENDIX B: TRANSITION MATRIX FOR ARIGID ROTATIONALLY SYMMETRIC SHAPE

As discussed in Sec. II B, the scattering coefficientscould be obtained using the transition (T) matrix multi-plying by the incident beam shape coefficients. For a rigidparticle with rotational symmetry, the T matrix could becalculated as T = −ReQQ−1 with the element of the Qmatrix [39]

Qσσ ′nm,n′m′ =

∫ π

0ξn′m′ jn′(kr)Pm′

n′ (cos θ)ξnm

×[∂h(1)

n (kr)∂r

Pmn (cos θ) − rθ

r2 h(1)n (kr)

∂Pmn (cos θ)

∂θ

]

× r2 sin θdθ

∫ 2π

0

(cos m′ϕsin m′ϕ

)(cos mϕsin mϕ

)dϕ,

(B1)

where r(θ) is the geometrical shape function and rθ =dr/dθ is the derivate of r(θ) with respect to the polar angleθ on the particle surface. Further details and simplifiedmethods can be found in Refs. [40,41].

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