T-matrix evaluation of acoustic radiation forces on nonspherical objects in Bessel beams

Transcription

1 T-matrix evaluation of acoustic radiation forces on nonspherical objects in Bessel beams Zhixiong Gong, 1,2 Philip L. Marston, 2 Wei Li 1,a) 1 School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 4374, China 2 Department of Physics and Astronomy, Washington State University, Pullman, Washington , USA hustgzx@hust.edu.cn; marston@wsu.edu; hustliw@hust.edu.cn Abstract Acoustical radiation force (ARF) induced by a single Bessel beam with arbitrary order and location on a nonspherical shape is studied with the emphasis on the physical mechanism and parameter conditions of negative (pulling) forces. Numerical experiments are conducted to verify the T-matrix method (TMM) for axial ARFs. This study may guide the experimental set-up to find negative axial ARF quickly and effectively based on the predicted parameters with TMM, and could be extended for lateral forces. The present work could help to design acoustic tweezers numerical toolbox, which provides an alternate to the optic tweezers. Keywords: Acoustic pulling forces; Single Bessel beam; T-matrix method; acoustic tweezers numerical toolbox; Numerical experiments a) Author to whom correspondence should be addressed. Electronic mail: hustliw@hust.edu.cn. 1

2 Acoustic tweezers, an appropriate counterpart to optical tweezers 1,2, could be used for levitation 3,4, pulling forces 5-8, and lateral trapping 9,1. Compared with optic tweezers, acoustic tweezers tend to exert a larger force over larger length scales with the same intensity since the radiation force is proportional to the ratio of the intensity to the velocity in the medium 11,12. In general, there are two main schemes to design acoustic tweezers: (quasi)standing wave schemes with dual beams 4,13 and single beam structures 1,14. The single-beam tweezers could be superior to standing-wave tweezers in some respects, for instance, the single-beam tweezers can continuously pull or push a particle over a large position because there are no multiple equilibrium positions 1,15. Negative radiation force single-beam device could pull the target towards the source, which is of interest in both acoustical 5-8 and optical fields 16. The physical mechanism is due to the asymmetric scattering of the incident fields on the target such that the scattering into the forward direction (red arrows in Fig.1 (a)) is relatively stronger than the scattering into the backward direction (blue arrows in Fig.1 (a)) 5-7,16,17. This is understood by the conservation of momentum and the Newton s third law regarding reaction force. 17,18 Beams having the local properties of acoustic Bessel beams are candidates for single-beam tweezers which have been examined in theoretical 5-7,19 and experimental approaches 1,2. The ordinary Bessel beam (OBB) possesses the axial maximum and azimuthal symmetry, while the helicoidal Bessel beams (HBB) have an axial null and 2

3 azimuthal phase gradient. Hefner and Marston applied the experimental demonstration for the acoustical vortices by using four piezoelectric transducers. 21 The physical phenomenon and mechanisms of the negative ARF exerted on the spherical objects in an on-axis incident Bessel beam have been investigated theoretically using the exact series solutions 5-7 with a geometrical explanation 17. In this letter, several numerical experiments are conducted based on the T-matrix method (TMM) with the emphasis on nonspherical objects which are common in engineering practice. This will extend the previous studies of the ARF to cases of an arbitrary-shaped object placed in a Bessel beam with arbitrary location and order using the multipole expansion method 22. This work will focus on the parameter conditions for the pulling force and the related mechanisms. Numerical experiments shown here are an alternate to direct experimental approaches and more versatile than analytical investigations. The radiation stress tensor approach 5,23,24 is widely employed to compute the static radiation force by integrating the time-averaged radiation stress tensor over a spherical surface with an adequately large radius. By using the relations between the velocities (pressures) and complex velocity potentials for both the incident and far-field scattered fields, the expression of force in terms of velocity potentials is i i * * F k Re i s s s ds 2 S n (1) k r where is the fluid density, k is the wave number, * denotes complex conjugation. i, s denote the incident and scattered complex velocity potentials, Re means the real 3

4 part of complex number. From the view of numerical computation, the TMM is an efficient tool to compute acoustic scattering on nonspherical objects. At present, this method will be further introduced for radiation forces which is closely related to the incident and scattered fields. In the TMM formulation, the velocity potentials of the incident and scattered fields could be expanded as 22,25-29 a j kr Y (2) i nm n nm, nm 1 f h kr Y (3) s nm n nm, nm where a nm and f nm are the incident and scattered coefficients, is the beam amplitude, jn kr and 1 n h kr are the spherical Bessel and Hankel functions, respectively., Y denotes the normalized spherical harmonics. The transition nm relationship between a nm and f nm is given by fnm Tnm, n' m' an' m', where T nm, n' m' denotes the transition matrix which only depends on the properties of the object, including the geometrical shape, the material composition and the boundary conditions at the interface, and otherwise is independent of the sources. For the exact series solution, the transition matrix could be considered as T= s 1 2 without dependence on the n azimuthal index m for spheres, which is in fact the partial-wave coefficients a n with s n known for a wide variety of spheres 3 and may be taken as a special case for the TMM. It is noteworthy that both the TMM 26,28 and the series solution for scattering by a sphere can be truncated at appropriate indices in computations, which make the 4

5 asymptotic expressions of scattered fields convergent in the far-field. After using the far-field asymptotic expressions for the scattered velocity potentials and implementing several algebraic manipulations, the ARF could be given briefly in terms of the incident and scattered coefficients, such that ' 1 n n 2 2 i * * F k Re a 2 nm fnm fn' m' Ynm, Yn ' m', ds 2 n (4) S nm n' m' kr which could be applied for radiation force with arbitrary orientation and agrees with Eqs. (7) and (9) in Silva s work 31. Only the axial ARF is considered here. The outward unit normal vector is n= sin cose sin sine cose in Cartesian ordinates. Hence, the integration x y z could be simplified easily by using the Eq. (15.152) in Ref. 32 for the integration involving the spherical harmonics and circular functions. Finally, the axial component of the force could be calculated by the axial projection Fz Fe z with 1 F 2 * * z Im anm fnm fn 1, m cn 1, m fn 1, m cn, m (5) 2 nm where c n mn m n n 12, The dimensionless radiation force nm functions Y p is introduced to coincide with the exact solutions for spheres with the 2 relationship F r I cy, where I c 2k 2 z p with c being the speed of sound in the surrounding fluid and r is the characteristic dimension of the target. The TMM has been demonstrated to calculate the scattering for spherical 33 and nonspherical objects. In addition, the TMM is quite efficient because the transition 5

6 matrix only need to be calculated once for the scattered coefficients that could be computed repeatedly for the radiation force 34. Lateral forces could also be calculated through projections of Eq. (4) in corresponding directions, which is outside the scope of the present work. Several numerical experiments were conducted with the emphasis on the parameter conditions for negative ARF on nonspherical objects in Bessel beams and the related physical mechanisms. The Neumann boundary condition was applied throughout for objects including spheroids, second-order superspheroids and finite cylinders. An arbitrary-shaped object is illuminated by a HBB with arbitrary order and location (Fig.1 (a)). To verify the correctness of the TMM, two examples are implemented for a rigid sphere in the OBB (blue line) and first-order HBB (FHBB, red line) with on-axis incidence, respectively (Fig. 1 (b)). The axial ARF Y p extracted from Fig. 2 of Ref. 5 for the OBB with the half-cone angle 6 and Fig. 1 of Ref. 7 for the HBB with are the exact series solutions 5,7. All the TMM results agree well with the series solutions. The TMM has been demonstrated for the spheroid 25-27, finite cylinder 28,29 cases in plane wave and Bessel beams, and hence it could be applied for the radiation forces for these shapes convincingly. The Y p of the oblate and prolate spheroids versus the dimensionless frequency kr in the FHBB are depicted in panels (c) for ab 12 and (d) for ab 2 of Fig. 1, with 3, 66.42, and 8. a is the polar radius and b is the equatorial radius. 27 r is the larger value between a and b. The ranges 6

7 Fig. 1 (Color online) (a) Schematic of an arbitrary object illuminated by a HBB of an arbitrary order with arbitrary location. (b) Validations of the results calculated using the TMM compared with those of the exact solutions. The blue solid line is for the OBB case and the red solid line is for the HBB. The references for the OBB (blue circles) are extracted in Fig. 2 of Ref. 5 with =6 and the references for the FHBB (red stars) are extracted in Fig. 1 of Ref. 7 with =66.42 for the rigid sphere. (c) The axial ARF of the oblate spheroid with aspect ratio ab 12. r b ( a b ). 3, 66.42, and. The enlarged view of the negative ARF region is given in the top right corners. (d) Like panel (c) except that the object is the prolate spheroid with ab 2. r a ( a b). (e) The angular dependence of the scattered form functions versus the scattered polar angle for the oblate spheroid in the FHBB with =8 for kr 1.8 (red solid line) 8 s and kr 2.1 (blue dash line). The black dotted line denotes the direction of the incident wave vector with s. including the negative ARF in panels (c) and (d) are zoomed in and placed in their top right corners. It implies that a large (sufficiently nonparaxial) may facilitate the pulling force since the negative ARFs appear for both cases with 8 in the considered region, while it fails for 3. Specially, negative ARF is impossible for plane waves ( ) with passive spheres 16,17,35. The term in Eq. (21) of Ref. 17 including cos represents the momentum removed from the incident Bessel beam 7

8 (which induces positive ARF) and the term including cos gives the axial projection of the momentum transport associated with the scattered field (which may induce positive or negative ARF). The angular dependences of the scattered form functions versus the scattered polar angle s for the oblate spheroid in the FHBB with =8 are plotted in Fig. 1(e) with kr 1.8 and kr 2.1. The black dotted line denotes the direction of the incident wave vector (i.e. = s ). As shown in the enlarged view in panel (c), the ARF is negative at kr 1.8, and otherwise positive at kr 2.1. It can be observed in panel (e) that for kr 1.8, the scattering dominates in the forward directions with, resulting in the negative ARF; for kr 2.1, the scattering in the backward is relatively stronger than that in the forward, leading to the positive ARF. After giving an explicit explanation of the physical mechanism for the negative ARF, the emphasis will be put on the parameter conditions for exerting the pulling force below. Panels (a-d) of Fig.2 study the influence of the topological charges (orders) of the Bessel beams for a second-order superspheroid with ab 2 for an on-axis incidence. The definitions of a and b are analogical with those for spheroid. The 2D plots depict only the negative ARFs in the kr domain and the white domains stand for the positive, ARFs (not shown numerically). The islands of the negative ARF are different between the OBB and HBBs since panel (a) has two subregions, while panels (b-d) have one subregion under consideration. For the HBBs, the frequencies of the negative ARF seem to increase with the increase of the beam order. To discuss the parameter of the aspect 8

9 Fig. 2 (Color online) (a) The 2D plot depicting only the negative ARFs in the kr domain for the rigid second-order superspheroid with ab 2 under the on-axis incidence of the OBB. The white domain stands for the positive ARFs. (b) Like panel (a) except that the FHBB is incident. (c) Like panel (a) except that the second-order HBB is incident. (d) Like panel (a) except that the third-order HBB is incident. (e) Like panel (b) except that ab 3. (f) Like panel (b) except that ab 4. (g) Like panel (c) except that ab 12. r b since a b. This shape may model a red blood cell shape with a dip in the center. (h) Like panel (b) except that a capsule shape ( r l ) is considered with the on-axis incidence. (i) Like panel (h) except with the off-axis incidence. ratio, the 2D plots of a superspheroid with ab 3 and ab 4 in the FHBB are given in panel (e) and (f) (compared with panel (b)), respectively. Obviously, the subregion and the maximum absolute value of the negative ARF decreases when the aspect ratio increases. These results imply that the distributions of the negative ARF depend on the beams and objects. However, the central frequencies do not change heavily with the, 9

10 aspect ratios. The oblate case for the superspheroid is also described in panel (g). This shape is like some biological cells, the red blood cell with a dip in the center for example. Capsule-shaped (cylinder with spherical endcaps 28,29 ) objects are investigated for both the on-axis and off-axis incidences in (h) and (i). The aspect ratio is lb 2, where l is the half length of the total cylinder and b is the radius of the cylindrical portion. 28 The beam axis is shifted off the axis of the object in the transverse plane as.1 kr,.1 kr. There is no need for extra computational cost for the off-axis incidence compared with the on-axis case 22. By comparison, both the area of the island and the absolute value of the negative ARF decrease when the beam is shift off the object s axis. In summary, the numerical experiments demonstrate the effectiveness of the TMM to calculate the ARF for several typical shapes, and the negative axial ARFs are obtained under certain conditions with the corresponding physical mechanisms. The present method is very versatile for both spherical and nonspherical shapes with different material composition 25-29,33 once the geometrical shape functions could be given explicitly, providing an alternate to theoretical and experimental approaches. The aforementioned theoretical formulation could compute forces of arbitrary direction by projecting the force in Eq. (5) in a considered direction and using the integration relationships in Ref.32, which will be studied more explicitly afterwards. Other numerical methods, such as the modified finite element method 36,37, the smoothed particle 1

11 hydrodynamics 38, may combine with the derivation to provide more choices for the computations of ARF in Bessel beams. The TMM also has the potential to calculate the acoustic radiation torques, which has been implemented in optics with the TMM 39 for a Gaussian beam incidence 34 by using the sums of products of the expansion coefficients for the integrals of the angular momentum fluxes 4. The design of the acoustic tweezers numerical toolbox will benefit from the present work. Acknowledgments Work supported by HUST and CSC (Z.G.), NSFC (W.L. and Z.G.) and ONR (P.L.M). 11

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