Acoustic radiation force and torque on spheroidal particles in an ideal cylindrical chamber
Jose P. Leao-Neto, Mauricio Hoyos, Jean-Luc Aider, Glauber T. Silva
AAcoustic radiation force and torque on spheroidalparticles in an ideal cylindrical chamber
Jos´e P. Le˜ao-Neto, Mauricio Hoyos, Jean-Luc Aider, and Glauber T. Silva a Campus Arapiraca/Unidade de Ensino Penedo, Universidade Federal de Alagoas, Penedo, AL57200-000, Brazil Laboratoire de Physique et M´ecanique des Milieux H´et´erog`enes, UMR7636 CNRS, UMPC, ESPCI,10 rue Vauquelin, 75005 Paris, France Physical Acoustics Group, Instituto de F´ısica, Universidade Federal de Alagoas, Macei´o, AL 57072-970, Brazil (Dated: 21 September 2020)
We theoretically investigate how the acoustic radiation force and torque arise on a smallspheroidal particle immersed in a nonviscous fluid inside an ideal cylindrical chamber. Theideal chamber comprises a hard top and bottom (rigid boundary condition), and a soft orhard lateral wall. By assuming the particle is much smaller than the acoustic wavelength, wepresent analytical expressions of the radiation force and torque caused by an acoustic waveof arbitrary shape. Unlike previous results, these expressions are given relative to a fixedlaboratory frame. Our model is showcased for analyzing the behavior of an elongated metallicmicrospheroid (with a 10 : 1 aspect ratio) in a half-wavelength acoustofluidic chamber witha few millimeters diameter. The results show the radiation torque aligns the microspheroidalong the nodal plane, and the radiation force causes a translational motion with a speedof up to one body length per second. At last, we discuss the implications of this study topropelled nanorods by ultrasound.c (cid:13) [https://doi.org(DOI number)][XYZ] Pages: 1–11
I. INTRODUCTION
Techniques for particle manipulation in acoustoflu-idic chambers (acoustic resonators at millimeter-scaleand smaller) have been extensively used in cell separa-tion and sorting, microparticle patterning, and vesicledeformation. At the core of these methods is the radi-ation force of acoustic waves. This phenomenon is a sta-tionary force caused by the linear-momentum flux changeduring the scattering of an incoming acoustic wave bya particle.
Another related effect is the acoustic ra-diation torque caused by the angular-momentum fluxchange due to the presence of an anisotropic or absorp-tive particle.
Computing the radiation force and torque inacoustofluidic settings is essential to developing applica-tions for cell analysis and analytical chemistry. On thatmatter, the forces and torques caused by a standing-wavefield have been investigated considering spherical parti-cles only.
There is an increasing interest in studyingthe behavior of elongated particles in acoustofluidic res-onators such as fibers, microrods, nanorods,
C. elegans , and E. coli . Geometrically speaking, an elongated particle can bemodeled as a prolate spheroid with a high aspect ra-tio. The analytical solution of the radiation force and a gtomaz@fis.ufal.br torque exerted on a prolate spheroid by a standing planewave has been recently derived. In that sense, theeffects of particle compressibility and density have beenaccounted for by using a method based on the Bornapproximation.
Also, the acoustic spin-torque trans-fer to a spheroid has also been studied. Another re-sort to compute acoustic forces and torques on complex-shaped particles rely on numerical methods.
It isworth mentioning that the well-known T -matrix ap-proach has also been applied to compute these fields. In this article, we present a theoretical model to cal-culate the radiation force and torque on spheroidal parti-cles in an ideal acoustic chamber filled with a nonviscousfluid. Our approach is based on the exact expressionsof these fields to the dipole approximation as obtainedin Ref. 42. We transform the radiation force and torqueexpressions to a fixed laborat ory frame in which theparticle dynamics can be analyzed. Thus, we focus ourinvestigation on a chamber that produces a single levita-tion plane (half-wavelength trapping device) with radi-ally symmetric modes. This appears to be more suitablefor studying living matter and developing techniques ofcell culture. We apply the developed model to study artificial mi-croswimmers (micro/nanorods) propelled by ultrasoundwithin a cylindrical chamber. The synthetic microswim-mers have attracted attention due to their potential usefor drug delivery and activation inside living cells. However, the propulsion mechanism of microswimmers
J. Acoust. Soc. Am. / 21 September 2020 Radiation force and torque in a cylindrical chamber 1 a r X i v : . [ phy s i c s . f l u - dyn ] S e p ropelled by ultrasound is still a matter of debate. Nadaland Lauga proposed an acoustic streaming model basedon the asymmetry of a near-spherical particle that is vi-brating at the wave frequency. Collins et al. includeddensity asymmetry to this model. However, a recentarticle questioned the validity of the acoustic stream-ing model for a vibrating near-sphere at low Reynoldsnumber. In our model, we consider an artificial mi-croswimmer as a slender microspheroid. We predict themicroswimmer is trapped in a levitation plane, not nec-essarily a nodal plane, due to the axial radiation force.When the levitation and nodal planes coincide, the ra-diation torque aligns the microspheroid perpendicularlyto the chamber’s principal axis. The radial radiationforce causes an in-plane particle movement with a speedof about one body length per second (BL s − ). Thissuggests the radiation force minimally contributes tothe observed fast speeds of microswimmers, e.g., up to70 BL s − . Although our model does not explain mi-croswimmers’ propulsion mechanism, it presents someuseful insights into the dynamics of these objects in acylindrical chamber.
II. PHYSICAL MODELA. Acoustic equations
The interaction between an acoustic wave and a par-ticle takes place inside a cylindrical chamber filled with aliquid of density ρ , adiabatic speed of sound c , and com-pressibility β = 1 /ρ c . The chamber has radius R andheight H . The acoustic excitation has angular frequency ω , with corresponding wavenumber k = ω/c = 2 π/λ ,where λ is the acoustic wavelength. We use the complex-phase representation to express the acoustic pressure andfluid velocity, p ( r , t ) = p ( r )e − i ωt and v ( r , t ) = v ( r )e − i ωt ,respectively. Here i is the imaginary unit, r is positionvector, and t is time.The wave dynamics in a nonviscous fluid is describedby the well-known acoustic equations (cid:0) ∇ + k (cid:1) p = 0 , (1a) v = ∇ p i ρ c k . (1b)The term e − i ωt is omitted for readability. The acousticequations are complemented by boundary condition atthe top, bottom, and walls of the cavity. B. Prolate spheroidal particle
We assume the interacting particle with the acousticwave is a prolate spheroid, which is generated by rotatingan ellipse around its major axis. Let us define the particleframe of reference as a right-handed system O p ( x p , y p , z p )placed in the geometric center of the spheroid. The cor-responding unit vectors of the system are e x p , e y p , and e z p . The spheroid foci are at (0 , , ± d/ r and r being the distance from the foci to a field point–seeFig. 1. The prolate spheroidal coordinates ( ξ p , η p , ϕ p ) FIG. 1. (a) The cylindrical acoustic chamber with a (yel-low) spheroid located at r regarding the laboratory frame O in the center of the chamber’s bottom. (b) The prolatespheroid with major and minor semiaxis denoted by a and b , respectively. The interfocal distance is d . The quantities r and r are the distance from the foci to a field point. (c)The rotational transformations through the Euler angles α and β , which take the laboratory ( x, y, z ) to particle frame( x p , y p , z p ). are defined by ξ p = r + r d , ξ p ≥ , (2a) η p = r − r d , − ≤ η p ≤ , (2b) ϕ p = tan − (cid:18) y p x p (cid:19) , ≤ ϕ p < π, (2c)with the isosurface ξ p = ξ = 1 (cid:113) − ( ba ) (3)corresponding to the particle surface. Also, the parti-cle major and minor axis are denoted by 2 a and 2 b ,respectively. While the interfocal distance and particlevolume are given, respectively, by d = 2 √ a − b and V p = 4 πab /
3. The spheroid orientation in the particleframe coincides to the z p axis, d p = d e z p .Note that a sphere of radius a is recovered by setting d → ξ → ∞ , and ξ d/ → a . Whereas, a slenderspheroid corresponds to the limit ξ → d . In contrast, a slender spheroid results from ξ ∼ C. Particle versus laboratory frame of reference
It is convenient to describe the wave-particle inter-action in an inertial frame O ( x, y, z ) referred to as thelaboratory system. In Fig. 1(a), we see the origin of thelaboratory frame is positioned at the center of the cham-ber’s bottom. And the particle position is denoted by ector r . Since the spheroidal particle is invariant un-der rotations around its major axis, we need only twoEuler angles ( α, β ) to transform one frame to the other–see Fig. 1(c). The transformation from the laboratoryto particle frame is constructed as follows. A positiverotation of an azimuthal angle α around the z p axis isfollowed by a rotation of a polar angle β about the new y p axis. By a positive rotation we mean a counterclock-wise rotation as seen from the top of the rotation axis.The particle orientation in the laboratory frame is thengiven by d = R ( α, β ) d p = d (cos α sin β e x + sin α sin β e y + cos β e z ) , (4a) R ( α, β ) = cos α cos β − sin α cos α sin β sin α cos β cos α sin α sin β − sin β β , (4b)with 0 ≤ α < π and 0 ≤ β ≤ π . It is worth noticing thegradient operator is transformed as, ∇ = R ( α, β ) ∇ p | r p = r , ∇ p = R − ( α, β ) ∇| r = r p , (5)where R − represents the transformation from the labo-ratory to the particle frame. D. Acoustic modes in a cylindrical cavity
The acoustic modes allowed inside the cavity are thesolutions of Eq. (1a) in cylindrical coordinates r ( (cid:37), ϕ, z ).Accordingly, the pressure inside the chamber is p ( r ) = p J n ( k (cid:37) (cid:37) ) cos( nϕ + ϕ ) cos k z z, (6)where p is the pressure magnitude, J n is the n th-orderBessel function, k (cid:37) and k z are the radial and axial wavenumbers, and ϕ is an arbitrary constant.The radial, angular, and axial modes are determinedfrom boundary conditions. We consider hard boundariesat the bottom ( z = 0) and top ( z = H ) of the cham-ber. While for the lateral wall ( (cid:37) = R ), a hard or softboundary is assumed. Accordingly, the fluid velocity andpressure satisfy v z ( (cid:37), ϕ,
0) = 0 , v z ( (cid:37), ϕ, H ) = 0 , (7a) v (cid:37) ( R, ϕ, z ) = 0 (hard) , p ( R, ϕ, z ) = 0 , (soft) . (7b)Since we do not have a tangential boundary condition,the phase ϕ can be arbitrarily set to zero. The condi-tions in (7) implysin ( k z H ) = 0 , (8a) J (cid:48) n ( k (cid:37) R ) = 0 (hard) , J n ( k (cid:37) R ) = 0 (soft) . (8b)Here the primed symbol denotes ordinary differentiation.The solutions of these equations yield the axial and radialdispersion relations, k z = k l = lπH , (9a) k (cid:37) = k nm = j nm R (soft) , j (cid:48) nm R (hard) , (9b) TABLE I. The first five zeros of the zeroth- and first-orderBessel functions. m j ,m j ,m with n = 0 , , , . . . ; l, m = 1 , , , . . . . The m th positivezero of the n th Bessel function and its derivative are j nm and j (cid:48) nm , respectively. The total wave number is givenby k = (cid:113) k l + k nm . (10)We see the angular frequency ω = kc is quantized.In what follows, we analyze radially-symmetricacoustic modes that forms a half-wavelength acoustoflu-idic chamber, ( nml ) = (0 m k = πH , k m = j ,m R (soft) , k m = j ,m R (hard) . (11)We have used the relation between the zeros of the Besselfunctions j (cid:48) ,m = j ,m . In Table I, we list the first fivezeros of the zeroth- and first-order Bessel functions forreference.We now express the pressure of the radially-symmetric modes, p m = p J ( k m (cid:37) ) cos( k z ) . (12)Substituting this equation into Eq. (1b) yields the radialand axial components of the fluid velocity v (cid:37) = i v k m k J ( k m (cid:37) ) cos( k z ) , (13a) v z = i v k k J ( k m (cid:37) ) sin( k z ) , (13b)where v = p /ρ c is the peak velocity. Note we haveused J (cid:48) ( x ) = − J ( x ). E. Scale analysis
We assume that the particle is a subwavelengthspheroid much smaller than the wavelength, which cor-responds to the so-called Rayleigh scattering limit. Theparticle smallness is quantified through the size factor ka = 2 πaλ (cid:28) . (14)Clearly, the minor semiaxis b also satisfies this condition.We also restrict our analysis to particles much smaller tothe chamber, a, b (cid:28) H, R.
Another effect that may appear in an acoustoflu-idic chamber is the acoustic streaming, which appearnear boundaries. Acoustic streaming close to the cham-ber walls produces causes a drag force on the particle,while near the particle surface, it can alter the radiation
J. Acoust. Soc. Am. / 21 September 2020 Radiation force and torque in a cylindrical chamber 3 orce and produce a viscous torque. As a diffusiveprocess, streaming has a characteristic length known asthe viscous boundary layer, δ = (2 µ /ρ ω ) / , with µ being the dynamic viscosity of the fluid. To avoid stream-ing effects, we should consider particles much larger thanthis parameter, δ (cid:28) a, b. For example, an acoustic waveof a frequency greater than 1 MHz (a typical lower limitfor acoustofluidic devices) in water generates a viscousboundary layer δ < . µ m. III. WAVE-PARTICLE NONLINEAR INTERACTIONA. Acoustic radiation force
The radiation force imparted on a subwavelengthspheroidal particle by a stationary wave is expressed by F radp ( ) = −∇ p U p ( ) , (15a) U p = πa (cid:20) β f | p | − ρ (cid:18) f | v x p | + | v y p | ) + f | v z p | (cid:19)(cid:21) , (15b)where E = β p / v p = ( v x p , v y p , v z p ) is the fluid velocity in the par-ticle frame. Considering a rigid particle, the scatteringamplitudes of the monopole f , axial f and transverse f dipole modes are given by f = 1 − ξ − , (16a) f = 23 ξ (cid:34) ξ ξ − − ln (cid:32) ξ + 1 (cid:112) ξ − (cid:33)(cid:35) − , (16b) f = 83 ξ (cid:34) − ξ ξ ( ξ −
1) + ln (cid:32) ξ + 1 (cid:112) ξ − (cid:33)(cid:35) − . (16c)These factors depend on the particle aspect ratio a/b through the parameter ξ introduced in Eq. (3). After inspecting (16), we find the following inequalities0 < f < f < , < f < f < . (17)As the particle geometry becomes spherical, the dipolefactors turn into f → f . Whereas, slender particlesscatter much less acoustic waves, f , f , f → ξ → . (18)It is more convenient to analyze the radiation forceon the particle in the laboratory frame. To this end, wehave to express the acoustic fields of Eq. (15b) in thelaboratory frame. By inserting the velocity componentsof (A2) into (15b), we obtain the radiation force potentialin this frame as U = πa (cid:20) β f | p | − ρ (cid:18) f | ( v x cos α + v y sin α ) sin β + v z cos β | + 12 f (cid:2) | v x sin α − v y cos α | + | v x cos α cos β + v y sin α cos β − v z sin β | (cid:3)(cid:19)(cid:21) . (19)To find the potential in cylindrical coordinates, we use v x = v (cid:37) cos ϕ , v y = v (cid:37) sin ϕ . Thus, we have U = πa (cid:20) β f | p | − ρ (cid:18) f | v (cid:37) sin β cos( α − ϕ )+ v z cos β | + f (cid:2) | v (cid:37) cos β cos( α − ϕ ) − v z sin β | + | v (cid:37) | sin ( α − ϕ ) (cid:3)(cid:19)(cid:21) . (20)Now, substituting the pressure and fluid velocity compo-nents given in Eqs. (12) and (13) into Eq. (20), we obtainthe potential of the radially-symmetric acoustic modes, U m = U (cid:26) f ( k z ) J ( k m (cid:37) ) − f (cid:20) k k sin( k z ) J ( k m (cid:37) ) cos β + k m k cos( k z ) J ( k m (cid:37) ) cos( α − ϕ ) sin β (cid:21) − f (cid:20)(cid:18) k k sin( k z ) J ( k m (cid:37) ) sin β − k m k cos( k z ) J ( k m (cid:37) ) cos β cos( α − ϕ ) (cid:19) + (cid:18) k m k (cid:19) cos ( k z ) J ( k m (cid:37) )sin ( α − ϕ ) (cid:21)(cid:27) , (21)where U = πa E is the peak potential. For simplic-ity, we drop the sub-index 0 of the particle position incylindrical coordinates, r = ( (cid:37), ϕ, z ).By fixing the height and diameter of the chamber,the normalized potential ˜ U m = U m /U depends only on the particle aspect ratio a/b through the scatteringfactors f , f , and f . The potential also depends onthe orientation angles α and β , and to the azimuthal an-gle ϕ , albeit the (0 m
1) acoustic mode in Eq. (12) hascircular symmetry. As the particle becomes spherical f → f ), Eq. (21) reduces to the radiation poten-tial of a spherical particle as given in Ref. 14, Eq. 1, with m = 0 in the reference’s notation.Having discussed how the potential function is ob-tained, we are able to derive the radiation force in the lab-oratory frame. From Eqs. (4b) and (5), we find this forceas minus the gradient of the potential given in Eq. (21), F rad = R − ( α, β ) F radp = − R − ( α, β ) ∇ p U p ( )= −∇ U ( r ) . (22)Thus far, we derived the exact solution of the radia-tion force problem for the particle placed anywhere insidethe chamber. We can distill this solution for two particu-lar cases, namely, along the chamber’s axis of symmetryand at the nodal plane. For the first case, the potentialand radiation force are derived using Eqs. (12) and (13)into Eq. (21) and setting (cid:37) = 0. The obtained result isused in Eq. (22). Accordingly, we arrive at U m = U (cid:20) f cos ( k z ) − (cid:18) k k (cid:19) sin ( k z )(2 f cos β + f sin β ) (cid:21) , (23a) F rad z = F ,z Φ a sin(2 k z ) , (23b)Φ a = 2 f (cid:18) k k (cid:19) (cid:18) f cos β + f β (cid:19) , (23c)with F ,z = k U being the axial force magnitude. Thefunction Φ a is the axial acoustophoretic factor which de-pends on the scattering modes and orientation angle β .Referring to the inequalities in (17), we conclude thatΦ a >
0. When effects of gravity can be neglected, therigid spheroidal particle is trapped in the pressure node, z eq = H/
2. Note the maximum axial force correspondsto F z, max = F ,z Φ a at z = H/ , H/ z eq = H/
2, we see from (13) the pressure and theradial component of the fluid velocity vanish, p m = 0and v (cid:37) = 0. From Eq. (20), we find U m = − πa Φ r ( β ) ρ | v z | , (24a)Φ r ( β ) = f cos β + f β. (24b)The radiation force potential is a function of the ax-ial component of the kinetic energy density. Besides,the acoustophoretic factor Φ r does not depend on themonopole scattering mode f . This happens becausethe pressure vanishes at the nodal plane and so does themonopole term in Eq. (21). After substituting Eq. (13b)into Eq. (24a) and replacing the result into Eq. (22), we obtain the potential and radial radiation force as U m = − (cid:18) k k (cid:19) U Φ r ( β ) J ( k ,m (cid:37) ) , (25a) F rad (cid:37) = − F ,(cid:37) Φ r ( β ) J ( k m (cid:37) ) J ( k m (cid:37) ) , (25b) F ,(cid:37) = 2 (cid:18) k k (cid:19) k m U , (25c)with F ,(cid:37) being the force magnitude. The radial acous-tic traps correspond to the the minima of the potentialfunction, while the largest force occurs at k m (cid:37) = 1 . F rad (cid:37), max = 0 . F ,(cid:37) Φ r . For a rigid particle, the radial acoustophoretic factor ispositive and the potential minima are obtained by solv-ing the equation J (cid:48) ( k m (cid:37) ) = 0. This corresponds tofind the zeros of the first-order Bessel function. Hence,the position of the i th radial trapping point is at (cid:37) i,m = j ,i − j ,m R (soft) , j ,i − j ,m R (hard) , m = 1 , , . . . (26)Here we consider j , = 0. The primary trap correspondsto (cid:37) ,m = 0 regardless the lateral boundary condition,e.g., soft or hard wall. To determine the second trapposition, we refer to Table I, (cid:37) , = 0 . R (soft wall) and (cid:37) , = R (hard wall). We see soft walled chambers areable to produce only a middle trap. Whereas, the secondtrap of a hard walled chamber is located at the lateralwall. B. Acoustic radiation torque
The acoustic radiation torque exerted on thespheroidal particle by the acoustic mode described inEq. (12), is given in the particle frame by τ radp = − πa χ (cid:0) e z p × P p · e z p (cid:1) r p = , (27a) P p = ρ v p v ∗ p ] = ρ (cid:2) v i v ∗ j e i e j (cid:3) , i, j = x p , y p , z p , (27b)where χ = f − f > P p is the time-average of the linear momentum flux (asecond-rank tensor) relative to the particle frame. Weexpress the projection of the linear momentum flux ontothe axial direction as P p · e z p = ( ρ /
2) Re[ v ∗ z p v p ]. Car-rying on the calculations, we arrive at τ radp = πa χ ρ Re (cid:104) v y p v ∗ z p e x p − v x p v ∗ z p e y p (cid:105) . (28)To find the radiation torque in the laboratory frame, weapply the rotation matrix R into Eq. (28), τ rad = R ( α, β ) τ radp = πa χρ Re (cid:2) ( v y p v ∗ z p cos α cos β + v x p v ∗ z p sin α ) e x + ( v y p v ∗ z p sin α cos β − v x p v ∗ z p cos α ) e y − v y p v ∗ z p sin β e z (cid:3) . (29) J. Acoust. Soc. Am. / 21 September 2020 Radiation force and torque in a cylindrical chamber 5 ubstituting the fluid velocity components given inEq. (13) into Eq. (A3) and replacing the result intoEq. (29), we obtain τ x = − πa χE (cid:20)(cid:18) k k (cid:19) sin 2 β sin α sin ( k z ) J ( k m (cid:37) ) + k k m k sin(2 k z ) J ( k m (cid:37) ) J ( k m (cid:37) )[sin β sin α cos( α − ϕ ) + cos β sin ϕ ] − (cid:18) k m k (cid:19) sin 2 β cos ( k z ) sin ϕ J ( k m (cid:37) ) cos( α − ϕ ) (cid:21) , (30a) τ y = πa χE (cid:20)(cid:18) k k (cid:19) sin 2 β cos α sin ( k z ) J ( k m (cid:37) ) + k k m k sin(2 k z ) J ( k m (cid:37) ) J ( k m (cid:37) )[sin β cos α cos( α − ϕ ) − cos β cos ϕ ] − (cid:18) k m k (cid:19) sin(2 β ) cos ( k z ) J ( k m (cid:37) ) cos( α − ϕ ) cos ϕ (cid:21) , (30b) τ z = πa χE (cid:20) k k m k sin 2 β sin( α − ϕ ) sin(2 k z ) J ( k m (cid:37) ) J ( k m (cid:37) ) + (cid:18) k m k (cid:19) sin β cos ( k z ) sin[2( α − ϕ )] J ( k m (cid:37) ) (cid:21) . (30c)When the particle is trapped at z eq = H/
2, we seefrom (13) that the radial component of the fluid velocityvanishes v (cid:37) = 0. Hence, referring to Eqs. (A3) and (29),the radiation torque reduces to τ rad = πa χ sin 2 β ρ | v z | e α . (31)The unit vector e α = cos α e y − sin α e x lies along the mi-nor semiaxis pointing to the counterclockwise directionin the xy plane. The radiation torque is proportional tothe axial component of the kinetic energy density aver-aged in time ρ | v z | /
4. It also depends on the orientationfactor sin 2 β . The particle is set to rotate around the mi-nor axis, since e α · e z = 0. Now we replace v z in Eq. (31)by Eq. (13b) to encounter τ rad ( β ) = τ χJ ( k m (cid:37) ) sin 2 β e α , (32)where τ = πa E k / k is the characteristic torque.The maximum torque τ radmax = τ χ , which occurs at β = π/ (cid:37) = 0. The equilibrium angular positioncorresponds to β = π/ C. Effects of gravity
An actual particle of density ρ p is subjected to effectsof gravity, which changes its axial equilibrium position.The new position can be determined from the force equi-librium equation F rad (0 , z eq ) − ( ρ p − ρ ) V p g = 0, with g being the gravity acceleration. Thus from Eq. (23b), theaxial equilibrium position is z eq = H − H π arcsin (cid:34) ρ p − ρ ) gH π Φ a E (cid:18) ba (cid:19) (cid:35) . (33) To bring the particle close to the nodal plane, we needto increase the acoustic energy density. From Eq. (33),we see the energy density needed to keep the particle inequilibrium is E = 4( ρ p − ρ ) gH π Φ a sin(2 πz eq /H ) (cid:18) ba (cid:19) . (34)We see that slender particles with a (cid:29) b require lessenergy to be axially trapped. D. Translational and angular velocity of the particle
Here we obtain the stationary translational and an-gular velocity achieved by the particle at the nodal plane z eq = H/
2. This analysis is restricted to particles atmicroscale in an aqueous solution.To determine the translational velocity, we assumethe particle is at ( (cid:37), ϕ, H/
2) and aligned to the radialdirection, β = π/ α = ϕ . Hence the velocity isdenoted by ˙ (cid:37) , with dot notation meaning time deriva-tive. As the particle moves, a drag force counteracts theradiation force, F drag = − πaµ g f ˙ (cid:37) e (cid:37) , (35a) g f = 1 ξ [( ξ + 1) arccoth ξ − ξ ] . (35b)The geometric factor g f becomes 3 / ξ → ∞ ), which leads to the well-known Stoke’slaw, F dragsphere = − πµ a ˙ (cid:37) .Using Eq. (25b), we find the equation of motion of aparticle moving along its major axis as¨ (cid:37) + 8 πaµ g f M ˙ (cid:37) = − F ,(cid:37) Φ r M J ( k m (cid:37) ) J ( k m (cid:37) ) . (36) ABLE II. The physical and geometric parameters of themicroswimmer in a submillimeter cylindrical chamber at roomtemperature and pressure.
Parameter ValueMicrospheroid (Au)
Major semiaxis ( a ) 10 µ mMinor semiaxis ( b ) 1 µ mAspect ratio ( a/b ) 10:1Radial parameter ( ξ ) 1 . V p ) 41 . µ m Density ( ρ p ) 19 300 kg m − Moment of inertia ( I ) 16 . µ m Monopole mode ( f ) 0 . f ) 0 . f ) 0 . Water
Density ( ρ ) 1000 kg m − Speed of sound ( c ) 1492 m s − Cylindrical chamber Height ( H ) 180 µ mRadius ( R ) 2 . z eq ) 76 . µ mEnergy density ( E ) 15 . − TABLE III. The theoretical predictions of the microspheroidat the nodal plane considering the parameters of Table II.
Acoustic modesFeature Soft Hard (011) (021) (011) (021)Frequency [MHz] 4 .
150 4 .
177 4 .
160 4 . F rad (cid:37), max [pN] 0 .
407 0 .
921 0 .
645 1 . (cid:37) [ µ m s − ] 8 .
185 18 .
55 12 .
98 23 . t (cid:37) [s] 43 .
05 8 .
277 17 .
04 5 . τ radmax [nN µ m] 0 .
299 0 .
296 0 .
298 0 . β [rad s − ] 23 .
62 23 .
02 23 .
40 22 . t β [ms] 31 .
44 32 .
26 31 .
73 32 . where M is the particle’s mass. Considering amicrometer-sized particle in water, we see the viscouscontribution overcomes inertia by far. So the inertialterm in Eq. (36) can be neglected. The equation of mo-tion then becomes˙ (cid:37) = − (cid:18) k k (cid:19) k m a Φ r g f E µ J ( k m (cid:37) ) J ( k m (cid:37) ) . (37)We conclude the translational speed increases with theparticle length squared. We find the solution of Eq. (37) for a particle in the vicinity of (cid:37) = 0 with the initialposition at (cid:37) (0) = (cid:37) , (cid:37) ( t ) = (cid:37) e − t/t (cid:37) , (38a) t (cid:37) = (cid:18) kk k m a (cid:19) g f µ Φ r E . (38b)Importantly, the characteristic trapping time t (cid:37) is of theorder of seconds.Turning now to the angular velocity induced by theradiation torque of Eq. (32) on a particle at ( (cid:37), ϕ, H/ β , the angular velocity corresponds to the ratechange of the orientation, ˙ β . Moreover, a drag torquearises on the particle, τ drag = − πa µ g t ˙ β e α , (39a) g t = 43 ξ − ξ ξ − (1 + ξ ) ln (cid:16) ξ +1 ξ − (cid:17) . (39b)The well-known result of the drag torque for a sphere, τ drag = − πa µ ˙ β , is obtained by setting ξ → ∞ .The rotational particle dynamics is described by thedifferential equation¨ β + 8 πa µ g t I ˙ β = τ χJ ( k m (cid:37) ) I sin 2 β, (40)with I = M ( a + b ) / β = τ χJ ( k m (cid:37) ) sin 2 β πa µ g t , (41)which can be solved by the method of separation of vari-ables. Let β be the initial particle orientation. Usingthe expression (cid:82) sin − β d β = ln(tan β ) /
2, we find β ( t ) = arccot (cid:20) exp (cid:18) − tt β J ( k m (cid:37) ) (cid:19) cot β (cid:21) , (42a) t β = (cid:18) kk (cid:19) g t µ χE . (42b)The orientation angle asymptotically approaches β = π/ t → ∞ . Slenderparticles χ → t β is of the order of milliseconds.The rotational-to-translational characteristic timeratio is about t β t (cid:37) ∼ ( k m a ) . (43)This ratio is about 10 − for typical acoustofluidic set-tings. J. Acoust. Soc. Am. / 21 September 2020 Radiation force and torque in a cylindrical chamber 7 - - - -- - - FIG. 2. The radiation force fields (red arrows) of the mi-crospheroid aligned to the x axis. The force is generated bythe (011) acoustic mode with (a) soft and (b) hard lateralwalls. The background contours illustrate the potential func-tion U , given by Eq. (21), normalized to U = 48 .
07 fJ.The force fields are evaluated in the laboratory frame at theaxial position z eq = 0 . H/
2. The physical parameters usedhere are listed in Table II. The bluish regions correspond tothe middle trap, while the dotted-purple circle in panel (b) isthe annular trap.
IV. CASE STUDY: AU MICRORODS
Now, we use the theory to analyze the radiation forceand torque fields in a acoustofluidic chamber wherein theparticles are trapped as described in Ref. 26. In this ref-erence, the chamber operates at nearly 4 MHz, and theparticles are metallic (Au) nanorods with length of fewmicrometers and hundreds of nanometers wide. Theseobjects can be geometrically modeled as microspheroidswith a slender shape. As the particle width is of theorder of the viscous boundary layer, we cannot appliedour method directly to these nanorods. Nevertheless, thetheory can be used to explain the behavior of wider par-ticles with the same aspect ratio (10 : 1) of the nanorods.In doing so, the physical parameters of our analysis aresummarized in Table II. Finally, our choice of the levi-tation plane position at z eq = 0 . H/ , 0 . H/ < z eq < H/
2. Hence, accord-ing to Eq. (34), the corresponding energy density for thechosen height of the levitation plane is E = 15 . − .With all model parameters in place, we can computesome features of the microspheroid behavior at the nodalplane for the (011) and (021) acoustic modes. The resultsare summarized in Table III. The characteristic trap timeis of the order of seconds, and the reorientation time isabout 31 ms. Besides, the microspheroid can be as fastas one body length per second. Note also the rigid walledchamber yields the largest radiation forces. In contrast,the radiation torque does not change with the chamberboundary conditions at all.In Fig. 2, we show the radiation force field (red ar-rows) acting on the microspheroid aligned with the x axis as a function of the scaled coordinates x/R and y/R . The background contour plots corresponds to theforce potential U , which appears radially symmetric at z eq = 0 . H/
2. Panels (a) and (b) display the results forsoft and hard lateral boundary conditions, respectively.The bluish region corresponds to the middle trap, whilethe dotted-purple circle at (cid:37)/R = 1 in panel (b) illus-trates the annular trap.In Fig. 3, we show the radiation torque field (redarrows) on the microspheroid as a function of the scaledCartesian coordinates. Both soft and hard wall chambersare considered with the (011) acoustic mode. The back-ground contour plot is the radiation torque amplitudenormalized to the characteristic torque τ = 48 .
07 pN µ m.The microspheroid position is at ( (cid:37), ϕ, . H/ α = ϕ and β = π/ e ϕ –see the inset inpanel (b). Also, a larger radiation torque is achieved inthe middle area with nearly the same amplitude in bothchambers. Though the soft chamber develops a more ho-mogeneous torque around the central area of the levita-tion plane. The principal effect of the radiation torque isto reorient the particle to the angular position β = π/ - -- - - -- FIG. 3. The radiation torque fields (red arrows) in thelevitation plane at z eq = 0 . H/ τ = 23 . µ m. The insetof panel (b) shows the microspheroid (in yellow) at the po-sition ( (cid:37), ϕ ), aligned with the radial direction ( α = ϕ ), andwith β = π/
4. The radiation torque is always perpendicularto the particle orientation. The physical parameters used hereare listed in Table II. is about t β = 30 ms. Moreover, it is independent of thelateral boundary conditions.The particle reorientation effect was observed inmillimeter-sized paper fibers caused by a standing planewave at 72 kHz in water. A similar conclusion wasachieved for polystyrene fibers with one-fourth of thewavelength in an acoustic resonator filled with water. Nonetheless, an intriguing experimental observation inmicrogravity shows that a cluster of trapped 3 µ m-longnanorods in water are aligned perpendicularly to thenodal plane inside a cylindrical chamber. On this mat-ter, we offer the following explanation for this effect.Firstly, the fluid viscosity may play a significant role inthe radiation torque changing the orientation equilibriumposition. Secondly, with the inter-particle distances be-ing about the particle dimensions, the secondary radi-ation force becomes dominant.
So one may expectthe rise of secondary radiation torques. In turn, the sec-ondary interaction torques are likely to change the par-ticle orientation equilibrium. Thirdly, both density andgeometric asymmetries seem to have a markedly influ-ence on the nanorods behavior. None of these featuresare taken into account by our approach.
V. CONCLUDING REMARKS
In this study, we present analytical results of theacoustic radiation force and torque developed on a rigid(prolate) spheroidal particle inside an ideal cylindricalchamber. The particle is considered far smaller than theacoustic wavelength and much larger than viscous bound-ary layers. The ideal chamber comprises a rigid bottomand top, with hard or soft lateral walls. The radiationforce and torque expressions are given in the laboratoryframe, paving the way to investigating the particle behav-ior through equations of motion. This approach can alsobe used for an incident wave of arbitrary shape, as longas the beam is expressed (analytically or numerically) inCartesian or cylindrical coordinates.The theory is applied to calculate the radiation forcesand torques acting on a microspheroid. The model pa-rameters are chosen to mimic the experimental setup ofnanorods propelled by ultrasound. As the nonviscousapproximation is assumed, we could not apply theory di-rectly to the nanorods. Notwithstanding, we keep thesame aspect ratio of the nanorods (10 : 1) but considera microspheroid with a diameter of 2 µ m which is largerthan the boundary layer depth. We obtain the character-istic radiation force and torque, and the particle transla-tional and angular velocities of the first acoustic modesof the chamber. Furthermore, the particles in the nodalplane are reoriented to the same direction of this planeby means of the radiation torque. The reorientation timeis of the order of milliseconds. Whereas, the radial trapoccurs after several seconds passed.Our model also predicts translational speeds of upto one body lengths per second (BL s − ). The speedincreases with the particle length squared. Should weapplied the theory to the nanorods of Ref. 26, the speed J. Acoust. Soc. Am. / 21 September 2020 Radiation force and torque in a cylindrical chamber 9 ould be at least ten times smaller. This hints that theradial radiation force does not significantly impact thenanorods’ propulsion mechanism.The present analysis is a solid step toward under-standing the physics behind trapping elongated particlesin acoustofluidic settings. It offers results that can beverified experimentally for systems whose boundary con-ditions can be approximated to ideal conditions (hardor soft walls). Adding thermoviscous properties of thesurrounding fluid to the model is the next level to beattained in future publications.
ACKNOWLEDGMENTS
G. T. Silva thanks the Brazilian National Council forScientific and Technological Development–CNPq (Grantnumber 308357/2019-1), and Chaire Total ESPCI-Paris(2019).
APPENDIX A:
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