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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Glauber T. Silva1, a) and Bruce W. Drinkwater2
1)Physical Acoustics 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 acoustic wave of angular frequency ω and wavelength λ propagating
55 in a nonviscous fluid of density ρ0 and adiabatic speed of sound 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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