Mean-Acoustic Fields Exerted on a Subwavelength Axisymmetric Particle

Mean-acoustic ﬁelds exerted on a subwavelength axisymmetric particle

Everton B. Lima1 and Glauber T. Silva1, a Physical Acoustics Group, Instituto de F´ısica, Universidade Federal de Alagoas, Maceió,AL 57072- 970, Brazil (Dated: 29 June 2021) The acoustic radiation force produced by ultrasonic waves is the “workhorse” of particle manipulation in acoustofluidics. Nonspherical particles are also subjected to a mean torque known as the acoustic radiation torque. Together they constitute the mean-acoustic fields exerted on the particle. Analytical methods alone cannot calculate these fields on arbitrarily shaped particles in actual fluids and no longer fit for purpose. Here, a semi-analytical approach is introduced for handling subwavelength axisymmetric particles immersed in an isotropic Newtonian fluid. The obtained mean-acoustic fields depend on the scattering coefficients that reflect the monopole and dipole modes. These coefficients are determined by numerically solving the scattering problem. Our method is benchmarked by comparison with the exact result for a subwavelength rigid sphere in water. Besides, a more realistic case of a red blood cell immersed in blood plasma under a standing ultrasonic wave is investigated with our methodology. ©2021 Acoustical Society of America. [https://doi.org(DOI number)] [XYZ] Pages:1–11

I. INTRODUCTION acoustic fields have to be re-calculated at each time step in the interval of interest. As a consequence, particle Acoustofluidics is extensively based on the acoustic 1 dynamics analysis in acoustofluidics might be a very in- radiation force produced by ultrasonic waves to control tense computational task. We emphasize that such anal- and pattern micro/nanoparticles (cells, microorganisms, ysis is of fundamental importance for understanding and and viruses) in a liquid medium. Nonspherical particles 33 2 designing micro/nanorobots propelled by ultrasound. are also accompanied by the acoustic radiation torque. In acoustofluidics, subwavelength nonspherical par- The time-averaged radiation force and torque are referred ticles, which are much smaller than the wavelength, have to as the mean-acoustic fields that drive the particle dy- been conveniently modeled as small spheres mainly be- namics in most acoustofluidic settings. cause they fit the simple analytical expressions of the The analytical solutions for the mean-acoustic fields radiation force34 and torque.4 obtained in the scattering depend, apart from the incoming wave, on the geomet- dipole approximation. Recently, analytical results con- ric shape3–15 and material properties16–22 of the particle. 23–26 27 sidering a subwavelength spheroidal particle were also Moreover, the viscous and thermodynamic prop- derived in the dipole approximation.35 We stress that erties of the surrounding fluid can significantly change a complete theoretical description of the mean-acoustic the mean-acoustic fields on the particle. The solutions fields for subwavelength particles of arbitrary geometry mentioned above are found for particles with a simple in an actual fluids seems to be unattainable. geometric shape, such as spheres and spheroids. Gen- This article proposes a semi-analytical approach to erally, these solutions are derived with the multipole ex- describe the mean-acoustic fields interacting with an ar- pansion and angular spectrum methods. The equivalence bitrary axisymmetric particle in Newtonian thermovis- of these theoretical approaches have been demonstrated cous fluids. The obtained fields are related to the pres- in Ref. 28. sure and fluid velocity of the incident wave and the coeffi- The mean-acoustic fields exerted on particles with cients of the monopole and dipole modes of the scattered arXiv:2104.06090v2 [physics.flu-dyn] 26 Jun 2021 more elaborate geometries are usually obtained through waves. These coefficients are obtained from the projec- numerical techniques such as the finite element29–31 and 32 tion of the scattered pressure onto the angular part of boundary element method. Nevertheless, numerical the corresponding mode (monopole and dipole). In turn, methods compute the mean-acoustic fields for a given the scattered pressure is obtained via a finite element spatial configuration (position and orientation) of the solver. The numerical part of our method is verified particle relative to the incoming wave. To determine the against the exact solution for a rigid spherical particle particle position and orientation versus time, the mean- in a lossless fluid. Besides, we apply our method to compute the mean-acoustic fields exerted on a red blood cell by a standing plane wave in blood plasma. The results, agtomaz@fis.ufal.br such as trap location and spatial orientation of the RBC,

J. Acoust. Soc. Am. / 29 June 2021 Axisymmetric particle dynamics 1 are in good agreement with previously reported experi- ments.

II. PHYSICAL MODEL

A. General assumptions Consider a subwavelength axisymmetric particle, i.e., with rotational symmetry around its principal axis, in a fluid of density ρ0 and compressibility β0. The particle scatters an incoming wave of pressure amplitude pin, angular frequency ω, wavelength λ, and wavenumber k. In Fig.1, we sketch the the acoustic scattering problem for an axisymmetric particle. The scattering is conveniently described concerning the right-handed reference frame Oxpypzp fixed in the geometric center of the particle (centroid), referred to as the p-frame. It is also useful to obtain the mean-acoustic fields in a fixed laboratory reference frame Oxyz (l-frame) attached to the particle’s centroid. The cartesian unit vectors of both ref- FIG. 1. Problem sketch: an incoming wave (background im- erence systems are denoted by ei and eip , with i = x, y, z and ip = xp, yp, zp. Since the particle is invariant under age with a positive and negative crest in purple and red, re- infinitesimal rotations around the zp axis, we need two spectively) of wavelength λ is scattered by a small axisymmet- angles (α, β), known as the Euler angles, to transform ric particle (gray cylinder). Two Euler angles (α, β) describe a vector from the p-frame to the l-frame. The transfor- the particle orientation regarding a laboratory frame denoted mation structure is formed by two elementary rotations: by blue axes. The particle frame is represented by red axes. a counterclockwise rotation of an angle α around the z The dotted concentric circles and black arrows denote the out- axis, followed by a counterclockwise rotation of an angle going scattered wave. In the inset, we show the elementary 0 β around new y axis–see Fig.1. We thus introduce the rotations that map a vector from the particle frame (red axes) rotation tensor as into the laboratory frame (blue axes).

R = cos α cos β exexp − sin α exeyp + cos α sin β exezp

+ sin α cos β eyexp + cos α eyeyp + sin α sin β eyezp distance to the particle surface are, respectively,

− sin β ezexp + cos β ezezp , (1) a = max Rp(θp), (3a) 0≤θp≤π where the product eieip is a dyadic, which forms the standard basis of second rank tensors in the Euclidean b = min Rp(θp). (3b) 0≤θ ≤π space. The Euler angles are given in the intervals 0 ≤ p α < 2π and 0 ≤ β ≤ π. We define the particle orientation In case of a spherical particle, a = b. The character- in both the p-frame and l-frame as istic size parameter of the subwavelength axisymmetric particle can be defined as ep = ezp , (2a) 2πa el = R · ezp = cos α sin β ex + sin α sin β ey + cos β ez, ka = 1. (4) (2b) λ where the center dot represents the scalar product be- B. Scattering normal modes tween a tensor and vector or two vectors. Note The scattered pressure by the particle is obtained also in deriving Eq. (2b), we have used the expressions by solving the Helmholtz equation with boundary con- e e · e = e e · e = 0, and e e · e = e . The i xp zp i yp zp i zp zp i ditions on the particle surface and the Sommerfeld ra- inverse transformation from the l-frame to the p-frame is diation condition at the farfield (r → ∞). The series given in AppendixA. p solution represents the scattering pressure as an infinity We assume that the largest dimension of the parti- sum of partial waves. It is convenient to express the par- cle, denoted by a, is much smaller than the wavelength, tial waves in spherical coordinates (r , θ , ϕ ). Hence, a λ, the so-called long-wavelength approximation. To p p p the nth-order partial wave is h (kr )Y m(θ , ϕ ), where determine a for an axisymmetric particle, we first notice n p n p p h is the spherical Hankel function of first type, Y m is that the particle occupies region, given in spherical co- n n the spherical harmonic, m is the number representing the ordinates, by Ω = {(r , θ , ϕ ) | 0 ≤ r ≤ R (θ ), 0 ≤ p p p p p p p wave orbital angular momentum. In the case of subwave- θ ≤ π, 0 ≤ ϕ < 2π}, with R being the radial distance p p p length particles much smaller than the wavelength, the to the particle surface Sp. The largest and smallest radial

2 J. Acoust. Soc. Am. / 29 June 2021 Axisymmetric particle dynamics partial wave expansion can be truncated at the dipole- coefficients and incident fields as35 moment term (n = 1). If we take a spherical particle rad as an example, we see that the truncation error is of the Fp = Re (exp exp + eyp eyp )Dxpyp + ezp ezp Dzpzp order of O[(ka)2].21 ∗ · vin,p , (9a) In the dipole approximation, the partial wave expan- rp=0 2πi3ρ sion of the scattering pressure is D = − 0 s v · ∇ + s v ∂ xpyp k2 k 11 in,p xpyp 10 in,zp zp 0 psc = p0 a00s00Y0 (θp, ϕp)h0(krp) ∗ pin −1 0 − is00 (1 + 2s11) , (9b) + [a1,−1s1,−1Y1 (θp, ϕp) + a10s10Y1 (θp, ϕp) c0 + a s Y 1(θ , ϕ )] h (kr ), (5) 2πi3ρ 11 11 1 p p 1 p D = − 0 s v · ∇ + s v ∂ zpzp k2 k 11 in,p xpyp 10 in,zp zp where p0 and anm are the pressure amplitude and beam- shape coefficient of the incoming wave, respectively. ∗ pin − is00 (1 + 2s10) . (9c) We have four scattering coefficients representing the c0 monopole (s00), transverse dipole (s1,−1) and (s11), and where vin,p = (vin,x , vin,y , vin,z ) is the fluid velocity axial dipole (s10) modes. In Eq. (5), the time-dependent p p p term e−iωt is omitted for simplicity. regarding the p-frame, the asterisk denotes complex conjugation, and ∇ = e ∂ + e ∂ . The acoustic It is useful to introduce a function that represents xpyp xp xp yp yp the angular distribution of the (n, m) mode on a spheri- fields should be evaluated at rp = 0. We should bear cal surface of radius R > a and centered at the particle in mind that the acoustic fields in (9) correspond to the centroid, incoming wave for which thermoviscous effects can be ne- glected. Therefore, the lossless pressure-velocity relation

psc(kR, θp, ϕp) pnm(θp, ϕp) = . (6) i∇p p0anmhn(kR) v = − (10) ρ0c0k Using the orthogonality of the spherical harmonics, the is used into Eq. (9a) to obtain a simpler radiation force scattering coeﬃcients are given as the projection of the expression normalized scattered pressure onto the corresponding an- rad gular mode, Fp = 1 (m)∗ ∗ (d)∗ ∗ snm = hn, m|pnmi Re β α · p ∇ p + ρ α · v · ∇ v , 2 0 p in p in 0 p in,p p in,p Z 2π Z π rp=0 m∗ = dϕp dθp sin θp pnm(θp, ϕp)Yn (θp, ϕp), (11) 0 0 (7) where ∇p = exp ∂xp + eyp ∂yp + ezp ∂zp , and with asterisk denoting complex conjugation. The Dirac’s 4πi α(m) = − s [I + 2s∗ (e e + e e ) bra-ket notation h|i is used for simplicity. p k3 00 11 xp xp yp yp In AppendixB, we show that the transverse dipole + 2s∗ e e ], (12a) modes of an axisymmetric particle are degenerated, 10 zp zp 12πi s1,−1 = s11. Thus, the scattering coeﬃcients of the prob- (d) αp = − [s11(exp exp + eyp eyp ) + s10ezp ezp ] lem are k3 (12b)

s00 = h0, 0|p00i , s10 = h1, 0|p10i , s11 = h1, 1|p11i . (8) are the monopole and dipole acoustic polarizability tensors (in units of volume) of the particle, and I = exp exp +

eyp eyp + ezp ezp is the unit tensor. Note that the radia- C. Acoustic radiation force tion force in Eq. (11) can also be understood in terms of the scattered and absorbed power by the particle in the The radiation force imparted on a subwavelength ax- dipole approximation (see Refs. 36–39). isymmetric particle is given in terms of the scattering After deriving the radiation force in the p-frame, we can obtain the equivalent expression in the l-frame. Using the rotational tensor described in Eq. (1), we ﬁnd

rad rad F = R · Fp (13)

J. Acoust. Soc. Am. / 29 June 2021 Axisymmetric particle dynamics 3 Using Eq. (A2b) into this equation, we arrive at section area. The acoustophoretic vector is expressed by h i rad 1 (m)∗ ∗ (d)∗ ∗ k (m) (d) F = Re β α · p ∇p + ρ α · v · ∇v , Φ = Re α − α · ez 0 in in 0 in in A 2 r=0 (14a) 2π ∗ ∗ = 2 cos α sin 2β Re[3i(s10 − s11) − 2is00(s10 − s11)]ex (m) (m) −1 k A α = R · αp · R , (14b) 2π ∗ ∗ (d) (d) −1 + 2 sin α sin 2β Re[3i(s10 − s11) − 2i(s10 − s11)]ey α = R · αp · R , (14c) k A 4π 2 2 ∗ 2 + Re[3i(s10 cos β + s11 sin β) − is00(1 + 2s cos β where vin is the fluid velocity in the l-frame. This k2A 10 exquisite expression describes the acoustic radiation force + 2s∗ sin2 β)]e . (18) on a particle immersed in a Newtonian fluid concern- 11 z ing the l-frame. Thermoviscous effects in the particle We see the radiation force depends on the particle orien- and fluid can be accounted for by solving the scattering tation angles α and β. In this regard, we shall see later problem considering temperature fields and appropriate that the possible equilibrium configurations are the axial 27 boundary conditions. Importantly, Eq. (14) is a gen- (β = 0) and transverse orientation (β = π/2). In either eralization of the canonical form of the radiation force orientation, the radiation force becomes an axial force, derived for spherical particles.24,40 rad We now proceed to the radiation force analysis of F = F0Φ sin 2kz0 ez, (19) stationary and traveling waves. As the pressure amplitude of a stationary wave is a real-valued function, we where Φ jointly represents the axial Φ|| and transverse ∗ 2 have, pin∇pin = (1/2)∇|pin| . Also, from Eq. (10) we see Φ⊥ acoustophoretic factors. They are expressed by ∗ 2 that vin · ∇vin = (1/2)∇|vin| . Thereby, for a stationary wave, Eq. (14) reduces to a gradient force 4π ∗ Φ|| = 2 Re[3is10 − is00(1 + 2s10)], (β = 0), k A|| rad h (m)i h (d)i F = Re α · ∇Epot + Re α · ∇Ekin (20a) r=0 (15) 4π ∗ Φ⊥ = Re[3is11 − is00(1 + 2s )], (β = π/2). 2 2 11 This equation shows that the kinetic Ekin = ρ0|vin| /4 k A⊥ 2 (20b) and potential Epot = β0|pin| /4 energy densities pumped into the fluid are transformed into the acoustic force of radiation through their gradient projection onto the real The position where the particle will be trapped depends part of the particle polarizabilities. on the sign of Φ. The trap is located in a pressure A nearly ubiquitous stationary wave employed in node if Φ > 0, and in a pressure antinode, otherwise. acoustofluidic devices is a standing plane wave whose For a spherical particle, the axial and transverse dipole acoustic fields are modes are degenerate s10 = s11–see Eq. (37). Hence, the axial and transverse acoustophoretic factors equal- ize, Φ = Φ . This result is also in agreement with pre- pin = p0 cos[k(z + z0)], (16a) || ⊥ viously obtained solutions.4,34,41 The equations in (20) ip0 vin = sin[k(z + z0)] ez, (16b) also agree with the analytical expressions derived for a ρ0c0 prolate spheroidal particle.6 Now we focus our analysis on a traveling plane wave where z0 is the particle position relative to the pressure antinode at z = 0. Replacing (16) into Eq. (15), we with pressure and velocity fields given by obtain the radiation force as ikz pin = p0e , (21a) rad p F = F0Φ sin 2kz0, (17a) 0 ikz vin = e ez. (21b) 1 ρ0c0 F = AE ,E = β p2, (17b) 0 0 0 4 0 0 where E0 is the acoustic energy density, and F0 is the characteristic force. The characteristic area A can be either the axial (A||) and transverse (A⊥) particle cross-

4 J. Acoust. Soc. Am. / 29 June 2021 Axisymmetric particle dynamics In this particular case, the radiation force in Eq. (14a) The factor Φ jointly represents becomes ∗ Φ|| = 8π Re[3s10 + s00(1 + 2s10)], (β = 0), (25a) rad F = F0Φ, ∗ Φ⊥ = 8π Re[3s11 + s00(1 + 2s11)], (β = π/2). (25b) 2k h i Φ = − Im α(m) + α(d) · e (22) A z It is worth noticing that (25) covers previous results for 4π a spherical particle.4,34,41 = cos α sin 2β Re[3(s − s ) + 2s (s∗ − s∗ )]e k2A 10 11 00 10 11 x 4π ∗ ∗ D. Acoustic radiation torque + 2 sin α sin 2β Re[3(s10 − s11) + 2s00(s10 − s11)]ey k A rad 2 2 ∗ 2 The acoustic radiation torque τp generated by an + 8π Re[3(s10 cos β + s11 sin β) + s00(1 + 2s cos β 10 acoustic wave relative to the p-frame is expressed by11,35 ∗ 2 + 2s11 sin β)]ez. (23) Here the acoustophoretic vector is a result of the scat- rad 6πρ0 ∗ 38,39 τ = Im γ⊥ vin,y ex − vin,x ey v tered and absorbed momentum ﬂuxes. The radiation p k3 p p p p in,zp force dependence with particle orientation is also noted. Again, the possible axial and transverse equilibrium ori- + γ v v∗ e , (26a) || in,xp in,yp zp entations of the particle are β = 0, π/2, which leads to rp=0 ∗ ∗ an axial radiation force, γ⊥ = s10 + s11 + 2s11s10, (26b) ∗ 2 rad γ|| = s11 + s + 2|s11| . (26c) F = F0Φez. (24) 11

The quantities γ⊥ and γ|| are the transverse and axial gyroacoustic functions. To obtain the radiation torque in the l-frame, we ﬁrst need to consider the velocity ﬁeld relation −1 vin,p = R · vin. (27) Using Mathematica software (Wolfram, Inc., USA), we insert Eq. (27) into Eq. (26a) and apply the rotational tensor to obtain

rad rad τ = R · τp 6πρ = 0 Im γ (v cos β − v sin α sin β) v∗ cos α sin β + v∗ sin α sin β + v∗ cos β k3 ⊥ in,y in,z in,x in,y in,z ∗ ∗ + γ|| cos α sin β vin,y cos α − vin,x sin α (vin,x cos α cos β + vin,y sin α cos β − vin,z sin β) ex ∗ ∗ + γ|| sin α sin β vin,y cos α − vin,x sin α (vin,x cos α cos β + vin,y sin α cos β − vin,z sin β) ∗ ∗ ∗ − γ⊥(vin,x cos β − vin,z cos α sin β) vin,x cos α sin β + vin,y sin α sin β + vin,z cos β ey ∗ ∗ ∗ + γ⊥ sin β(vin,x sin α − vin,y cos α) vin,x cos α sin β + vin,y sin α sin β + vin,z cos β ∗ ∗ + γ|| cos β vin,y cos α − vin,x sin α (vin,x cos α cos β + vin,y sin α cos β − vin,z sin β) ez . (28) r=0

We move on to analyze the acoustic radiation torque The unit vector eα is orthogonal to the particle axis produced by a standing plane wave along the z axis as of symmetry, eα · el = 0. Substituting Eq. (16b) into described in (16). With no transverse ﬂuid velocity com- Eq. (29a) yields ponents, vin,x = vin,y = 0, the radiation torque in (28) rad i 2 becomes τ = τ0γ⊥ sin kz0 sin 2β eα, (30a) 12π 3πρ i rad 0 2 τ0 = VE0, γ⊥ = Im[γ⊥], (30b) τ = Im[γ⊥]|vin,z| sin 2β eα, (29a) k3V k3 eα = cos α ey − sin α ex. (29b)

J. Acoust. Soc. Am. / 29 June 2021 Axisymmetric particle dynamics 5 i where γ⊥ is the transverse gyroacoustic parameter. In Accordingly, we have acoustoﬂuidics, the typical characteristic torque scale is

τ0 ∼ 1 nN µm. (pin + psc − n · σp)|rp∈S0 = 0, (34a) As the particle will be trapped in either a pressure i n · (v + v ) − u = 0, (34b) node or antinode, the pre-sine factor is reduced to one, ω in,p sc,p p 2 rp∈S0 sin kz0 = 1. So the particle will be transversely aligned

(β = π/2) to the wave axis as the gyroacoustic parameter t · σp|rp∈S0 = 0. (34c) i is positive, γ⊥ > 0. The axial orientation parallel to the i where t is the tangential unit vector, σp and up are the wave axis (β = 0) occurs as γ⊥ < 0. Let us switch gears toward the radiation torque due stress tensor and displacement vector inside the particle, to a traveling plane wave. By replacing Eq. (21b) into respectively. the radiation torque expression in (28), we ﬁnd TABLE I: Geometrical and physical parameters used in the τ rad = τ γi sin 2β e . (31) 0 ⊥ α ﬁnite element simulations of the scattering problem. 2 Apart from the pre-sine factor sin kz0, this result is iden- Parameter Description tical to the radiation torque by a standing wave given in Computational domain Cylindrical Eq. (30a). Therefore, the particle orientation follows the same conclusion as the standing wave case. Diameter and height 100a Radius of the integration surface (R) 5a/4

III. FINITE ELEMENT MODEL Minimum element size (emin) a/200 Maximum element size (e ) a/10 to a/3 To compute the scattering coefficients given in (8), max we numerically solve the scattering of a traveling plane Mesh growth rate 10 % wave using the finite element (FE) method in the com- PML type Polynomial mercial software Comsol Multiphysics (Comsol, Inc., PML scaling factor 1 USA). The choice for plane wave scattering is justified PML curvature parameter 2 due to its simplicity and for having analytical expressions of the corresponding beam-shape coefficients. Once the PML thickness 50a scattering coefficients are determined, they can be used PML layers 30–50 in conjunction with any incident wave. The p-frame Frequency 2–12 MHz is adopted in the finite element model. Besides, we use the Acoustics Module and Solid Mechanics Module, where Fluid medium Water −3 the equations of linear acoustics and elastodynamics are Mass density (ρ0) 998 kg m implemented. −1 Compressibility (β0) 0.4560 GPa The pressure amplitude p = p eik·rp , with wavevec- in 0 Particle Rigid sphere tor k = k(sin β exp + cos β ezp ), is set as the background pressure in the computational domain. To obtain Radius (a) 3.91 µm the scattering coefficients in (8), we need the monopole Fluid medium Blood plasma43 and dipole beam-shape coefficients of a traveling plane Mass density (ρ ) 1026 kg m−3 wave,42 0 Compressibility (β ) 0.4077 GPa−1 √ √ √ 0 a00 = 2 π, a10 = 2i 3π cos β, a11 = −i 6π sin β. Particle Red blood cell (32) 43 −3 Mass density (ρp) 1100 kg m For a rigid particle, the normal component of the Young’s modulus44 (E) 26 kPa fluid velocity should vanish at the particle surface S0, 43 −1 Compressibility (βp) 0.34 GPa 45 (vin,p + vsc,p)|rp∈S0 · n = 0, (33) Geometric parameters:

Major semiaxis (a) R0 = 3.91 µm where vsc,p is the scattered ﬂuid velocity and n is the outward normal vector of the particle. In the case of Minor semiaxis (b) C0/2 = 0.405 µm an elastic solid particle, the boundary conditions across Constant C2 7.83 µm the particle surface are the continuity of normal stresses Constant C4 −4.39 µm and displacements, and zero tangential stress condition.

The fluid domain corresponds to a mesh within a cylindrical region. We use a perfectly matched layer (PML) in a concentric cylindrical shell of a quarter- wavelength thickness to absorb the outgoing scattered wave, and thus avoid reflection back into the fluid do-

6 J. Acoust. Soc. Am. / 29 June 2021 Axisymmetric particle dynamics main. At the fluid-PML interface, the acoustic fields are continuous. The outer surface of the PML is a rigid wall. To proceed with the FE modeling, we need to choose the appropriate mesh symmetry: 2D axisymmetric or 3D model. The 2D axisymmetric model is less computation- ally intense and corresponds to a plane wave propagat- ing along the particle symmetry axis with β = 0. It can be is used to compute the monopole (s00) and axial (s10) dipole coefficients only. The 2D mesh model is not suitable for computing the transverse dipole coefficient (s11), because β = 0 implies a11 = 0, which leads to a not well-defined outcome of Eq. (6). We are, in principle, left with a full 3D scattering model to find FIG. 2. The 2D axisymmetric mesh for the plane wave s , which usually do not fit in low-memory computers 11 scattering by (a) a rigid spherical particle in water and (b) (< 16 GB). Nevertheless, the Comsol FE solver transforms a 3D acoustic scattering by an axisymmetric ob- a RBC (modeled as an elastic solid) in blood plasma. The ject into a collection of 2D scattering problems using the coarse mesh is for illustration purposes only. The particles, built-in Plane Wave Expansion method.46 We, therefore, shown in gray, have the same diameter 2a = 7.82 µm. The use this method with five terms in the plane wave expan- white dashed circle illustrates the integration surface; upon it, the scattering coefficients are computed. The plane wave sion to compute s11 with β = π/2. Here, the acoustic scattering by a rigid spherical particle and a red blood travels in the vertical direction along the z axis. The back- cell (RBC) is solved via the FE model. The simulation ground image corresponds to the real part of the computed parameters used in the numerical solutions are presented scattered wave, where blue and red regions mean negative and in TableI. positive pressures, respectively. In the upcoming analysis, the RBC is modeled as a solid elastic material with a biconcave disk-shaped geometry that is described in cylindrical coordinates (%, ϕ, z) by the implicit equation45

" 2#" 2 4#2 2 1 % % % z = 1 − C0 + C2 + C4 , 4 R0 R0 R0 (35) where 2R0 and C0 are, respectively, the cell transverse diameter and central thickness, and C2, and C4 are geometric parameters given in TableI. Now we deﬁne the mesh convergence parameter as

≡ ( ref ref ) Re snm − Re [snm] Im snm − Im [snm] max ref , ref . |Re [snm]| |Im [snm]| (36) FIG. 3. The scattering coeﬃcients of a rigid spherical particle much smaller than the wavelength. The solid and dashed ref The reference solution snm for the spherical particle is lines are the corresponding exact solution from (37). related to the exact coeﬃcients given by

0 0 j0(ka) j1(ka) s00 = − 0 , s10 = s11 = − 0 , (37) IV. RESULTS AND DISCUSSION h0(ka) h1(ka) In Fig.2, we depict the 2D axisymmetric mesh do- where jn is the spherical Bessel function of order n, and main around the spherical particle and RBC. The parti- the prime symbol means differentiation. Whereas for the cles have same diameter 2a = 7.82 m. The integration ref µ RBC, snm is the solution that shows negligible changes surface, where the scattering coefficients are computed, after further mesh refinements. The FE model was exe- is shown as a white circle enclosing the particle. The cuted on a Supermicro workstation (Supermicro Com- real part of the scattered pressure is also illustrated as puter, Inc., USA) based in a dual-processor Intel Xeon the background image. The positive and negative pres- E5-2690v2 @ 3 GHz and 224 GB memory size. The com- sure correspond to blue and yellow-to-red regions, respec- putational time is about 1 min in the 2D mesh and 5 min tively. with the plane wave expansion method. In Fig3, we plot scattering coefficients of the small rigid sphere in water for a routinely frequency range in

J. Acoust. Soc. Am. / 29 June 2021 Axisymmetric particle dynamics 7 FIG. 4. The scattering coefficients of a RBC in blood plasma FIG. 5. The gyroacoustic and acoustophoretic factors of a versus frequency. RBC in blood plasma versus frequency. The green and blue lines represent a linear interpolation of data. acoustofluidics, 2–12 MHz. The numerical result is in excellent agreement with the exact solution. At 2 MHz, the idic device.47 Note also that when the second orientation convergence parameter is = 7 %, with the maximum angle is α = 0, the radiation torque direction follows element size emax = a/3. This value drops to = 1 % at rad τ ∼ −ey. As the l-frame is a right-handed coordinate 3 MHz, and keeps going down as the frequency increases. system, the RBC rotates around its transverse diameter In our numerical studies, the solution convergence be- (y axis) in the clockwise direction. On the other hand, comes an issue at frequencies smaller than 1 MHz. We rad when α = π/2 we have τ ∼ ex, the RBC rotates also found that the transverse and axial dipole coeffi- around its transverse diameter, but this time in counter- cients are degenerated, s11 = s10, within the numerical clockwise direction. error. We note that the convergence error as calcu- Having determined the RBC orientation, we know lated from Eq. (36) is more prominent in the real part of the radiation force by the standing wave is described the scattering coefficient, for which the reference solution ref 6 21 by Eq. (19). Using a linear interpolator, we obtain the is of the order snm = O[(ka) ]. So, as the frequency acoustophoretic factor as decreases, the reference solution is drastically reduced, which may cause a convergence instability at lower fre- −1 Φ|| = −(0.001 871 MHz ) f. (39) quencies. Figure4 shows the scattering coefficients of the RBC 2 where the transverse cross-section area A⊥ = πR0 was in blood plasma. In this study, the mesh convergence used, and the frequency should be given in megahertz. parameter is < 9 %, for the maximum element size The interpolation error reads nrms = 0.1 %. The peak −3 emax = a/10. The parameter approaches the upper radiation force for E0 = 10 J m and f = 2 MHz is bound at lower frequencies. With these coefficients, we 1.79 pN. Moreover, as Φ|| < 0, we conclude the RBC can proceed to compute the radiation force and torque is trapped in a pressure node, kz0 = (2n + 1)π/2, with generated by a standing plane wave. n ∈ Z, which agrees with previous experimental observa- In Fig.5, we show the computed gyroacoustic and tions.47 acoustophoretic factors for the RBC. The obtained data Finally, our results can be compared with those ob- is interpolated with polynomial functions of linear fre- tained by the Born approximation48 for the RBC at quency f. Let us first examine the gyroacoustic factor. 2 MHz, under the same conditions as specified in TableI. Referring to (30), we find For the acoustic radiation force, the relative deviation

i −1 between the results is a mere 0.4 %. In the case of the γ⊥ = −0.001 433 − (0.000 174 MHz ) f, (38) radiation torque, we cannot, unfortunately, draw a comparison because the reference only provides the torque in in which the RBC volume45 V = 94 m3 was used, and µ a pressure antinode, e.g., kd = 0 in the article’s notation. the frequency should be specified in megahertz. The At this position, our method predicts a zero radiation normalized root mean square error between the fitting torque. For the sake of curiosity, we calculate the peak polynomial and data is nmrs = 5 % (normalized to the torque from Ref. 48 as ∼ 10−5τ . This value is two orders data range). For a typical value of the energy den- 0 of magnitude smaller than our outcome for the RBC in sity in acoustofluidics, say 10 J m−3, the radiation torque a pressure node, peak is 1.62 pN m. As γi is negative, we see the radi- µ ⊥ which can be considered close to our zero-torque pre- ation torque aligns the RBC (axis of symmetry) with diction at a pressure antinode. the axial direction (β = 0). This prediction was ex- perimentally observed in a half-wavelength acoustoflu-

8 J. Acoust. Soc. Am. / 29 June 2021 Axisymmetric particle dynamics V. SUMMARY AND CONCLUSIONS APPENDIX A: This article introduces a semi-analytical method to compute the mean-acoustic fields (radiation force and The inverse rotational tensor, which transforms a torque) exerted on a subwavelength axisymmetric par- vector from the l-frame to the p-frame, is given by ticle in a Newtonian fluid. The analytical part draws R−1 = cos α cos β e e + sin α cos β e e − sin β e e on the partial wave expansion of the scattering pressure xp x yp x zp x up to the dipole approximation. The mean-acoustic fields − sin α exp ey + cos α eyp ey − cos α sin β exp ez are derived as a function of the scattering coefficients–see + sin α sin β eyp ey + cos β ezp ez. (A1) Eqs. (11) and (28). The monopole and dipole scattering coefficients are computed as the projection of the scat- The gradient operator is transformed as tered pressure onto the angular part of the corresponding multipole mode–see Eq. (8). These coefficients are ∇ = R · ∇p|rp=r, (A2a) then numerically obtained using the finite element (FE) −1 ∇p = R · ∇|r=r . (A2b) method to solve the plane wave scattering by the particle. p We should remember that the scattering coefficients of spherical particles depend intrinsically on the mechanical properties of the particle and surrounding fluid, regard- APPENDIX B: less the incoming ultrasonic wave. This can be seen, for instance, in the case of solid elastic particles.19 In our ap- We want to demonstrate that the transverse dipole proach, the scattering coefficient of axisymmetric parti- mode of axisymmetric particles are degenerated, e.g., cles are also independent of the incident wave–see Eq. (5). s11 = s1,−1. Assume the plane wave travels along

So, after obtaining these coefficients from the plane wave the xp axis, then the wavevector is k = kexp and problem, one can compute the mean-acoustic fields on β = π/2. We note the corresponding√ beam-shape co- 42 the particle caused by any structured wave generated in efficients are a1,−1 = −a11 = i 6π. From Eq. (6), we acoustofluidic devices, using Eqs. (14) and (28). see that the normalized pressure of the dipole mode sat- We performed numerical tests to obtain the scatter- isfies p1,−1 = −p11. Thus, using the spherical harmonic −m m m∗ ing coefficients of a small rigid sphere in water. The symmetry property Yn = (−1) Yn and referring to results are in excellent agreement with the well-known Eq. (7), we have exact scattering solution. The semi-analytical method Z 2π Z π is showcased for a RBC subjected to a standing plane 1 wave in blood plasma. Our predictions for the particle s1,−1 = dϕp dθp sin θp p11(θp, ϕp)Y1 (θp, ϕp). 0 0 orientation and entrapment location agree with previous (B1) 47 experimental results. Inasmuch as the particle rotation symmetry around zp Lastly, our work is a concrete step toward computing axis, the scattered pressure has to be an even func- the mean-acoustic fields on nonspherical particles in more tion of the azimuthal angle ϕp. Hence p11(θp, −ϕp) = realistic acoustofluidics settings. The method is valid for p11(θp, ϕp), otherwise the scattered pressure will not be particles made of fluid, elastic, viscoelastic, and struc- symmetric with respect to the xpzp plane as it should be. tured material. Besides, thermoviscous properties of the Equation (B1) can be re-written as surrounding fluid can be incorporated into the numerical Z 2π Z π model. Not to mention that other numerical techniques 1∗ such as the boundary element method, T-matrix, and fi- s1,−1 = dϕp dθp sin θp p11(θp, ϕp)Y1 (θp, ϕp) 0 0 nite differences can be used here. In conclusion, our the- = h1, 1|p i = s . (B2) ory may serve as the foundation for new investigations 11 11 of RBCs and elongated cell dynamics in acoustofluidic Here Y 1(θ , −ϕ ) = Y 1∗(θ , ϕ ) was used. We conclude devices and also the development of micro/nanorobots 1 p p 1 p p the transverse dipole modes are indeed degenerated for propelled by ultrasound. axisymmetric particles.

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