Silva, G. , & Drinkwater, B. (2018). Acoustic radiation force exerted on a small spheroidal rigid particle by a beam of arbitrary wavefront: Examples of traveling and standing plane waves. Journal of the Acoustical Society of America, 144(5), [EL453]. https://doi.org/10.1121/1.5080529

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This document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/red/research-policy/pure/user-guides/ebr-terms/ Acoustic radiation force exerted on a small spheroidal rigid particle by a beam of arbitrary wavefront: Examples of traveling and standing plane waves

Glauber T. Silva1, a) and Bruce W. Drinkwater2

1)Physical Group, Instituto de F´ısica, Universidade Federal

de Alagoas, Macei´o,AL 57072-970, Brazil

2)Department of Mechanical Engineering, University of Bristol, Bristol,

BS8 1TR, United Kingdom

[email protected], [email protected]

1 Acoustic radiation force on spheroidal rigid particles

1 Abstract: We present the analytical solution of the acoustic radiation

2 force exerted by a beam of arbitrary shape on a small spheroidal rigid

3 particle suspended in an ideal fluid. We consider the long-wavelength

4 approximation in which the particle is much smaller than the wave-

5 length. Based on this theoretical development, we derive closed-form

6 expressions for the radiation force of a traveling and standing plane

7 wave on a prolate spheroidal particle in the dipole approximation. As

8 validation, we recover the previous analytical result considering a stand-

9 ing wave interacting with a spheroid in axisymmetric configuration, as

10 well as numerical results obtained with the boundary-element method.

c 2018 Acoustical Society of America.

a)Author to whom correspondence should be addressed.

2 Acoustic radiation force on spheroidal rigid particles

11 1. Introduction

12 Nonspherical cells and microorganisms are routinely investigated in biological assays per-

13 formed in acoustofluidic and acoustical tweezer devices. Some examples which significantly

14 deviate from spherical shape include rod-like bacteria and biconcave red blood cells. In these

15 acoustical devices, translational manipulation of small particles, which are much smaller than

16 the wavelength, is performed employing the acoustic radiation force caused by ultrasonic

1 17 waves.

18 Most theoretical analyses of the acoustic radiation force in fluids assume that the par-

2–11 19 ticles have spherical shape. More structured objects such as core-shell spherical particles

12 20 have also been considered. Besides, the secondary radiation force developed between two or

13 21 more spherical particles has been analyzed. All these studies are based on the partial-wave

22 expansion of the incident and scattered waves in spherical coordinates. Analytical expres-

23 sions of the radiation force are derived concerning the incident and scattering expansion

24 coefficients. For a small particle, it need mean only the monopole and dipole scattering

25 coefficients, i.e., the so-called dipole approximation, which are calculated from the boundary

26 conditions on the particle surface. Moreover, the orthogonality of spherical wavefunctions

27 used in the expansion are employed to decouple each mode and obtain a system of linear

28 equations involving the expansion coefficients. However, when considering a nonspherical

29 object, the boundary conditions formulated in spherical coordinates are no longer trivial.

30 This happens because the radial distance to a particle surface point depends on the coordi-

3 Acoustic radiation force on spheroidal rigid particles

31 nate angles. Consequently, the spherical wavefunctions are not necessarily orthogonal, which

32 renders closed-form exact solutions of the radiation force be difficult to be attained.

33 The acoustic radiation force exerted on a rigid spheroidal particle by a standing wave

14 34 was derived by Marston et al. This result relies on the solution of the acoustic scatter-

15 35 ing by a spheroid in the long-wavelength limit. However, it is limited to axisymmetric

36 particles with respect to the incident wave. Aside from Marston’s analytical result, seminu-

37 merical techniques that combine partial-wave spherical expansion and quadrature methods

16 38 have been used to compute an approximate solution. Additionally, numerical techniques

39 such as boundary- and finite-element methods were also employed to calculate the radiation

17–19 40 force.

41 In this paper, we derive an analytical solution of the acoustic radiation force on a

42 spheroidal particle with an arbitrary spatial orientation in the dipole approximation. The

43 method is valid for an incident beam of an arbitrary wavefront. In so doing, we first recognize

44 that the incident and scattered partial-wave expansions in spheroidal coordinates asymptot-

45 ically match expansions in spherical coordinates at the farfield. Hence, the radiation force is

46 calculated through the farfield method in which the radiation stress is integrated on a farfield

20 47 control surface of spherical shape. A closed-form expression is obtained for the force on a

48 rigid prolate spheroid generated by a traveling and standing plane wave. In comparison with

49 the case of a spherical particle of the same volume as the spheroid, the maximum radiation

50 force deviation are 1/3 and 1/5 for, respectively, the traveling and standing plane wave.

4 Acoustic radiation force on spheroidal rigid particles

51 Additionally, our theoretical prediction are in excellent agreement the numerical solution

52 obtained with the boundary-element method given in Ref. 19.

53 2. Scattering theory

54 Consider an arbitrary of angular frequency ω and wavelength λ propagating

55 in a nonviscous fluid of density ρ0 and adiabatic speed of c0. The wave is scattered by

56 a prolate spheroidal particle with major and minor axis denoted by 2a and 2b, respectively,

p 2 57 as depicted in Fig.1.a. The particle eccentricity is defined by ε0 = 1 − (b/a) . The

58 coordinate system is set to the geometric center of the particle with the z-axis coinciding

59 to the particle rotation axis. The particle defines a prolate spheroidal coordinate system

60 through the transformations

d d dξη x = p(ξ2 − 1)(1 − η2) cos ϕ, y = p(ξ2 − 1)(1 − η2) sin ϕ, z = , (1) 2 2 2

61 where d is the interfocal distance, ξ ≥ 1 is the spheroidal radial coordinate, −1 ≤ η ≤ 1,

−1 62 and 0 ≤ ϕ ≤ 2π (see Fig.1.b). The particle surface is determined by ξ = ξ0 = ε0 = 2a/d,

3 2 63 while its volume is V = 4π(d/2) ξ0(ξ0 − 1)/3. A spherical particle is recovered by setting

64 ε0 = 0. The connection between the spheroidal coordinates with radial distance r and polar

65 angle θ are

dp ηξ r = ξ2 + η2 − 1, cos θ = . (2) 2 pξ2 + η2 − 1

5 Acoustic radiation force on spheroidal rigid particles

The velocity potential function of the incident beam and scattered waves can be

expressed by21

X (1) imϕ φin = φ0 anmSnm(, η)Rnm(, ξ)e , (3a) n,m

X (3) imϕ φsc = φ0 anmsnmSnm(, η)Rnm(, ξ)e , (3b) n,m

P P∞ Pn 66 where φ0 is a constant, n,m = n=0 m=−n, Snm is the angular function of the first kind,

(1) (3) 21 67 Rnm and Rnm are the radial functions of the first and third kind. The time-dependence

−iωt 68 of the potentials e is omitted for simplicity. The particle size parameter is defined by

69  = kd/2, with k = 2π/λ being the wavenumber of the incident beam.

70 To solve the scattering problem, the beam-shape coefficient is known a priori. Where

71 as the scattering coefficient is to be determined from the boundary conditions in the particle

72 surface ξ = ξ0. For for a rigid particle, one requires that the normal component of fluid

73 velocity be zero on the particle surface, ∂ (φ + φ ) = 0. Therefore, using the potentials ξ in sc ξ=ξ0

74 in (3) we find

(1) 0 Rnm (, ξ0) snm = − , (4) (3) 0 Rnm (, ξ0)

7576 with the prime symbol denoting differentiation with respect to ξ0.

77 We limit our analysis to small particles compared to the wavelength, i.e. the so-called

78 long-wavelength limit, which implies   1. In this approximation, only the monopole n = 0

22 79 and dipole n = 1 are relevant in the scattered wave description . After substituting the

80 spheroidal radial functions given in Eqs. (10) and (18) of Ref. 23 into Eq. (4), we Taylor-

6 Acoustic radiation force on spheroidal rigid particles

Fig. 1. (a) An arbitrary incident wave is scattered by a prolate spheroidal particle with major

and minor axis denoted, respectively, by 2a and 2b, and interfocal distance d. The wave incidence

angle is denoted by α. (b) A system of prolate spheroidal coordinates with d = 1 and rotational

symmetry around the z-axis.

81 expand the result around  = 0 to obtain the scattering coefficients,

i3 6 i3 6 i3 6 s = − f − f 2 , s = f − f 2 , s = s = f − f 2 . (5) 00 3 00 9 00 10 6 10 36 10 1,−1 11 12 11 144 11

The scattering factors are expressed by

−1 3 " !# 2  3k V 2 ξ0 ξ0 + 1 f00 = ξ0 ξ − 1 = , f10 = − ln , 0 4π3 3 ξ2 − 1 p 2 0 ξ0 − 1 −1 " 2 !# 8 2 − ξ0 ξ0 + 1 f11 = + ln , (6) 3 ξ (ξ2 − 1) p 2 0 0 ξ0 − 1

7 Acoustic radiation force on spheroidal rigid particles

82 where V is the particle volume.

83 In the farfield ξ → ∞, we find from (2) that the spheroidal system reduces to the

−1 −2 84 spherical coordinate ξ = kr + O (ξ ), η = cos θ + O (ξ ) , where r and θ are spherical

21 85 coordinates. Moreover, the spheroidal functions become

ikr (1) 1  nπ  (3) −n−1 e m Rnm(, ξ) ∼ sin kr − ,Rnm(, ξ) ∼ i ,Snm(, η) ∼ Pn (cos θ), ξ→∞ kr 2 ξ→∞ kr →0 (7)

m 86 with Pn being the associated Legendre polynomial.

87 It is useful to introduce the spherical harmonics as

s 2n + 1 (n − m)! Y m(θ, ϕ) = P m(cos θ)eimϕ. (8) n 4π (n + m)! n

After substituting (7) into (3) and using Eq. (8), we obtain the incident and scattered velocity

potentials in the farfield kr  1 as

1 X  nπ  ψ = a˜ sin kr − Y m(θ, ϕ), (9a) in kr nm 2 n n,m 1 n eikr X X ψ = i−n−1a˜ s Y m(θ, ϕ). (9b) sc kr nm nm n n=0 m=−n

88 Here, the potential functions are normalized to φ0. Besides, the beam-shape coefficient in

89 the spherical function basis is

s 4π (n + m)! a˜ = a . (10) nm 2n + 1 (n − m)! nm

90 Importantly, the coefficienta ˜nm has been derived for different types of acoustic beams such

24 91 as a traveling plane wave, Bessel vortex, Gaussian, and Bessel-Gaussian beam. Numer-

8 Acoustic radiation force on spheroidal rigid particles

92 ical schemes can also be used to compute the beam-shape coefficients of off-axial Bessel

25–27 93 beams.

94 The spheroidal wave functions asymptotically match the spherical wave functions in

95 the farfield: φin → ψin and φsc → ψsc, as ξ → kr → ∞. We shall discuss next the use

96 the farfield wavefunctions given in (9) to derive the acoustic radiation force acting on the

97 spheroidal particle.

98 3. Acoustic radiation force

99 The acoustic radiation force exerted on an object is related to the net linear momentum

28 100 removed from the incident beam by absorption and scattering processes. Due to the con-

101 servation of linear momentum, we can obtain the radiation force by integrating the radiation

102 stress tensor over a control spherical surface in the farfield centered at the particle. Taking

20 103 this approach, the acoustic radiation force exerted on a particle is given by

E F = 0 Q , (11) rad k2 rad

2 2 104 where E0 = ρ0k φ0/2 is the characteristic energy density of the wave. The radiation force

105 efficiency is given in terms of the farfield wave functions by

Z 2π Z π     2 ∗ i 2 Qrad = −(kr) Re ψsc 1 − ∂r ψin + |ψsc| er sin θ dθ dϕ. (12) 0 0 k

where er = sin θ cos ϕ ex +sin θ sin ϕ ey +cos θ ez is the radial unit-vector, with ex, ey, and ez

being Cartesian unit-vectors. The Cartesian components of the efficiency Qrad are derived

by replacing (9) into Eq. (12) and performing the angular integrations. Accordingly, we find

9 Acoustic radiation force on spheroidal rigid particles

to the dipole approximation

i r2 Q + iQ = (s + s∗ + 2s s∗ )˜a a˜∗ + (s∗ + s + 2s∗s )˜a∗ a˜  x y 2 3 00 11 00 11 00 11 00 1,−1 0 1,−1 00 1,−1 1 r X (2 + m)(3 + m)  + s a˜ a˜∗ + s∗ a˜∗ a˜  , (13a) 15 1,m 1,m 2,m+1 1,−m 1,−m 2,−m−1 m=−1 " 1 r # 1 X (2 − m)(2 + m) Q = Im √ (s + s∗ + 2s s∗ )a ˜ a˜∗ + s a˜ a˜∗ . z 00 10 00 10 00 10 15 1,m 1,m 2,m 3 m=−1

(13b)

106 These equations are valid for an acoustic beam of arbitrary shape granted that its beam-shape

107 coefficienta ˜nm is known. Also, there is no restriction in the spheroid spatial orientation re-

108 garding the beam propagation direction. Furthermore, the method is suitable for an spheroid

109 of any material composition (rigid, void, fluid, or viscoelastic solid) to which the scattering

110 coefficients snm are to be determined by appropriate boundary conditions across the particle

111 surface ξ = ξ0.

112 4. Some wave examples

113 4.1 Traveling plane wave

114 Consider an incident plane wave

ik·r φin = φ0e , (14)

115 where k = k (sin α cos β ex + sin α sin β ey + cos α ez) is the wavevector, with α and β being

116 its polar and azimuthal angles. Note that α is also the wave incidence angle regarding the

24 117 particle major axis–see Fig.1. The beam-shape coefficient of the plane wave is given by

n m∗ a˜nm = 4πi Yn (α, β). (15)

10 Acoustic radiation force on spheroidal rigid particles

26,29 118 Other expressions of the beam-shape coefficient are noted for a Bessel vortex beam,

30 31 119 Gaussian beam, and off-axial Bessel beam. By replacing the scattering coefficient given

120 in (5) and Eq. (15) into the equations in (13), we obtain the efficiency components for a rigid

121 spheroidal particle as

Qx = Qk sin α cos β, Qy = Qk sin α sin β, Qz = Qk cos α, (16)

122 where

4π6   3f 2  1  3f 2   Q = f 2 + f f + 10 cos2 α + f f + 11 sin2 α . (17) k 9 00 00 10 4 2 00 11 8

123 is the radiation force efficiency along the wave propagation direction Referring to Eq. (11),

4 124 we find that the radiation force varies with frequency to the forth power, Frad ∼ ω . This

125 dependence arises due to the connection of the radiation force with the Rayleigh scattering

126 power, which also varies with frequency to the forth power. Moreover, the radiation force

4 127 on the spheroid has the same frequency dependence as that exerted on a spherical particle.

128 For small eccentricity 0  1 (ξ0  1), this equation becomes

 1 31  627  Q (α) = Q 1 − ε2 (1 + 3 cos 2α) + ε4 1 − cos 2α , (18) k 0 22 0 3080 0 155

6 2 129 where Q0 = 11k V /16π is the radiation force efficiency of a rigid spherical particle. For a

◦ ◦ 2 4 130 wave of lateral (α = 90 ) and frontal (α = 0), we have Qk(90 ) = 1+(1/11)ε0 +(391/7700)ε0

2 4 131 and Qk(0) = 1 − (2/11)ε0 − (59/1925)ε0. Clearly, lateral incidence produces more radiation

◦ 132 force on the particle, Qk(90 ) > Qk(0). We recover the classical radiation force result on a

2 133 rigid sphere by setting ε0 = 0 and α = 0, which yields Qk = Q0. On the other hand, when

11 Acoustic radiation force on spheroidal rigid particles

134 the particle is a very slender prolate spheroid, the efficiency Qk becomes

  34Q0 5 Qk(α) ∼ 1 − cos 2α . (19) ε0→1 33 17

135 Hence, the largest relative deviation from the radiation force on spherical particle of the

136 same volume as the spheroid are |1 − Qk(0)/Q0| = 3/11 (frontal incidence, α = 0) and

◦ ◦ 137 |1 − Qk(90 )/Q0| = 1/3 (lateral incidence, α = 90 ).

138 In Fig.2.a, we plot the radiation force efficiency Qk/Q0 along the wave propagation

139 direction versus the incidence angle α. We consider a particle of different eccentricities ε0 =

◦ 140 0, 0.55, 0.86. The efficiency monotonically increases with the incidence angle. With 40 <

◦ 141 α < 50 , the efficiencies becomes larger than that of a spherical particle. The approximate

142 solution (dashed lines) becomes inaccurate as the eccentricity increases.

143 4.2 Standing plane wave

144 Consider a standing wave formed by the superposition of two counter-propagating plane

145 waves as described by Eq. (14). The incident wave function is expressed by

φ0 φ = φ cos [k · (r + r )] = eik·(r+r0) + e−ik·(r+r0) , (20) in 0 0 2

146 where r0 is the distance from the particle center to the nearest pressure antinode, which lies

147 in the same direction as the wavevector, thus, k · r0 = kr0. To establish the partial-wave

148 expansion of the standing wave, we notice that the expansion for a traveling plane wave is

149 given by

ik·(r+r0) ikr0 X n m∗ m e = 4πe i Yn (α, β) jn(kr)Yn (θ, ϕ). (21) n,m

12 Acoustic radiation force on spheroidal rigid particles

m∗ m −m 150 By replacing Eq. (21) into Eq. (20) and using the relation Yn = (−1) Yn , the beam-shape

151 coefficient of the standing wave reads

 nπ  a˜ = 4π cos kr + Y m∗(α, β). (22) nm 0 2 n

152 Substituting this coefficient into the equations of (13) along with the scattering coefficients

153 of (5) yields

Qx = Qk sin α cos β, Qy = Qk sin α sin β, Qz = Qk cos α, (23)

154 where 4π 2 1  Q = 3 sin 2kr f + f cos2 α + f sin2 α (24) k 9 0 3 00 10 2 11

155 is the efficiency along the wavevector direction. Referring to Eq. (11), we note that the radi-

156 ation force varies with frequency linearly, Frad ∼ ω. Note that this dependence is the same

4 157 as that for a spherical particle. The particle will be trapped in a pressure node granted that

158 the quantity in the parenthesis of Eq. (24) is positive. Besides being trapped, the spheroidal

32–34 159160particle can be set to spin around the y-axis due to the acoustic radiation torque.

161 When the particle eccentricity is small ε0  1, we have

 3 3  Q (α) = Q 1 − ε2(1 + 3 cos 2α) + ε4(5 − 69 cos 2α) , (25) k 0 100 0 7000 0

3 162 where Q0 = (k V/5) sin 2kr0 is the radiation force efficiency for a spherical particle. Impor-

163 tantly, when α = 0 this expression turns to

 3 24  Q = Q 1 − ε2 − ε4 . (26) k 0 25 0 875 0

13 Acoustic radiation force on spheroidal rigid particles

Fig. 2. Radiation force efficiency Qk/Q0 of a rigid prolate spheroid (with different eccentricities)

caused by (a) a traveling and (b) standing plane wave versus the wave incidence angle α. Dots

denote the numerical results obtained with the boundary-element method (BEM) as given in Ref. 19.

164 Writing this equation in terms of the aspect ratio ε1 = b/a − 1, we find

 6 9  Q = Q 1 + ε2 + ε4 . (27) k 0 25 1 875 1

14 165 This result was previously obtained by Marston et al. When the eccentricity approaches

166 one, the efficiency Qk becomes

 1  Q = Q 1 − cos 2α . (28) k 0 5

167 Therefore, the largest relative deviation from a spherical particle occurs |1 − Qk(0)/Q0| =

◦ 168 |1 − Qk(90 )/Q0| = 1/5, which correspond to the frontal and lateral incidence, respectively,

169 In Fig.2.b, we present the radiation force efficiencies Qk/Q0 due to a standing plane

170 wave as a function of the incidence angle α. The spheroidal particle has eccentricity ε0 =

171 0, 0.55, 0.86. The efficiency increases with the incidence angle. The deviation of the spherical

14 Acoustic radiation force on spheroidal rigid particles

172 particle case (ε0 = 0) is more acute for high-eccentricity particles. Moreover, the approximate

173 solution (dashed lines) given in Eq. (25) becomes inaccurate when ε0 > 0.8. Excellent

174 agreement is found between the exact solution and the boundary-element method as given

175 in Ref. 19.

176 5. Summary and conclusions

177 We have calculated the acoustic radiation force exerted by an arbitrary wave on a spheroidal

178 rigid particle in the long-wavelength limit. The general formulation gives the radiation force

179 as a function of the scattering coefficients that represent the weight of each mode in the

180 scattered waves. Exact results to the dipole approximation are presented for a rigid prolate

181 spheroid considering a traveling and standing plane wave of arbitrary spatial orientation. Our

182 results show that the largest relative deviation of the radiation force compared to the case

183 of a spherical particle with the same volume as the spheroid are 1/3 (traveling plane wave)

184 and 1/5 (standing plane wave). Approximate solutions of the radiation force are accurate

185 for particles with eccentricity ε0 < 0.8. Our analysis recovers previous theoretical results

14 186 for a standing wave in the axisymmetric configuration. Additionally, excellent agreement

19 187 is found between the exact solution and the boundary-element method.

188 The developed theoretical framework can be readily applied to spheroidal particles

189 composed of fluid, elastic, and viscoelastic material, by solving the appropriate boundary

190 conditions to find the scattering coefficients. Thermoviscous effects can also be accounted in

191 the present theory. The method can also be promptly adapted to oblate spheroidal particles.

15 Acoustic radiation force on spheroidal rigid particles

192 In conclusion, our work brings the theoretical analysis of the acoustic radiation force

193 on small spheroidal particles to the same level as that for spherical particles. The obtained

194 results further extend our knowledge on the underlying mechanisms of the radiation force

195 phenomenon. Our findings are particularly useful for particle manipulation in acoustofluidic

196 and acoustical tweezer techniques.

197 Acknowledgments

198 This work was supported by CNPq (National Research Council, Brazil), Grant No.

199 303783/2013-3 and 307221/2016-4, Royal Society (Newton Advanced Fellowship, UK), Grant

200 No. NA160200.

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