Acoustic radiation force control: Pulsating spherical carriers

Ultrasonics xxx (2017) xxx–xxx

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Acoustic radiation force control: Pulsating spherical carriers Majid Rajabi ⇑, Alireza Mojahed Sustainable Manufacturing Systems Research Laboratory, School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 7 December 2016 Received in revised form 22 May 2017 Accepted 1 June 2017 Available online xxxx Keywords: Acoustic manipulation Handling Negative radiation force Smart acoustic driven mechanism Radiation force cancellation Radiation force control

a b s t r a c t The interaction between harmonic plane progressive acoustic beams and a pulsating spherical radiator is studied. The acoustic radiation force function exerted on the spherical body is derived as a function of the incident wave pressure and the monopole vibration characteristics (i.e., amplitude and phase) of the body. Two distinct strategies are presented in order to alter the radiation force effects (i.e., pushing and pulling states) by changing its magnitude and direction. In the first strategy, an incident wave field with known amplitude and phase is considered. It is analytically shown that the zero- radiation force state (i.e., radiation force function cancellation) is achievable for specific pulsation characteristics belong to a frequency-dependent straight line equation in the plane of real-imaginary components (i.e., Nyquist Plane) of prescribed surface displacement. It is illustrated that these characteristic lines divide the mentioned displacement plane into two regions of positive (i.e., pushing) and negative (i.e., pulling) radiation forces. In the second strategy, the zero, negative and positive states of radiation force are obtained through adjusting the incident wave field characteristics (i.e., amplitude and phase) which insonifies the radiator with prescribed pulsation characteristics. It is proved that zero radiation force state occurs for incident wave pressure characteristics belong to specific frequency-dependent circles in Nyquist plane of incident wave pressure. These characteristic circles divide the Nyquist plane into two distinct regions corresponding to positive (out of circles) and negative (in the circles) values of radiation force function. It is analytically shown that the maximum amplitude of negative radiation force is exactly equal to the amplitude of the (positive) radiation force exerted upon the sphere in the passive state, by the same incident field. The developed concepts are much more deepened by considering the required power supply for distinct cases of zero, negative and positive radiation force states along with the frequency dependent asymmetry index. In addition, considering the effect of phase difference between the incident wave field and the pulsating object, and its possible variation with respect to spatial position of object, some practical points about the spatial average of generated radiation force, the optimal state of operation, the stability of zero radiation force states and the possibly of precise motion control are discussed. This work would extend the novel concept of smart carriers to and may be helpful for robust singlebeam acoustic handling techniques. Furthermore, the shown capability of precise motion control may be considered as a new way toward smart acoustic driven micro-mechanisms and micro-machines. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction The contact-free manipulation, trapping and levitation of small-sized particles has attracted a lot of attention in medical applications such as drug delivery systems, contrast agents, etc. and engineering applications such as particle manipulation in microgravity and normal gravity conditions, hazardous material safe delivery systems, etc., using non-contact power suppliers based on optics [1–6], electro-kinetics [7,8],and acoustics [9–25].

⇑ Corresponding author.

In acoustic manipulation techniques, the acoustic radiation force (RF) exerted on the object plays the role of driver which is generated by acoustic wave fields. The handling is accomplished if both pushing and pulling effects can be generated. Due to always positive (pushing) radiation forces produced by common plane progressive wave field, novel patterns for incident wave fields (e.g., standing waves, Bessel beams, cross-plane beams, Helicoidal beams, Gaussian beams) were introduced and developed in order to generate negative (pulling) radiation force upon a target object [9–13,17,26–42]. The possibility of negative radiation force generation by means of Bessel beams on spherical objects has been shown. Using the analogy between optics and acoustics, Zhang

E-mail address: [email protected] (M. Rajabi). http://dx.doi.org/10.1016/j.ultras.2017.06.002 0041-624X/Ó 2017 Elsevier B.V. All rights reserved.

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and Marston [43] presented a geometrical interpretation of generation negative radiation force on spherical objects due to its interaction with Bessel beams. In their work, they correlated the radiation force to the three distinct and meaningful concepts of powers (See details in Ref. [43]). In their further step [44], they showed that in the case of actively driven targets, generation of negative radiation force required power supplied to the system. Just recently, the new subject of smart carriers has been introduced in which the complete handling is achieved by changing the surrounding acoustic field through stimulating the carriers [45,46], without adjusting the pattern of progressive plane incident wave field. The feasibility of presented suggestions to alter the radiation force has been proved for objects with spherical configuration. In one of the suggested configurations, the spherical carrier is actuated with its inner piezoelectric layer which has been stimulated by a spatially uniform harmonic voltage in order to excite the breathing mode of the object [45]. It has been shown that the negative, zero and positive radiation force states may be achieved by adjusting the magnitude of voltage and its phasedifference with respect to the phase of the progressive plane incident wave. In another configuration, the fundamental case of progressive plane wave interaction with an oscillating spherical rigid body has been investigated [46]. In this work, it has been shown that the oscillation direction of the body can become aligned with the incident wave propagation direction and the handling (i.e., pushing or pulling the object on the desired path) is obtainable for determined values of prescribed amplitude and phase of oscillations. Further investigation about the required power for oscillation of the body in order to cancel or reverse the radiation force on the object proved the feasibility of the proposed configuration. The significance and practical applications of robust single beam acoustic manipulation technique along with the feasibilities of such smart (controllable) carriers which were discussed before, inspired us to extend the basic knowledge of smart carriers, in order to form its theoretical structure. In the current work, the fundamental problem of acoustic manipulation of a monopole radiator as the smart carrier is investigated in two distinct manners. The simple mathematical model of ideal pulsating spherical radiator, insonified by an incident progressive plane wave field, is used with the hope of its capability to reverse the direction of the resultant momentum exchange between object and medium and making it under control. Considering the superposition principle, it seems impossible due to these facts that an isolated spherical pulsating radiator would generate no forces on itself. On the other hand, illumination of a rigid spherical body by a simple progressive plane wave field leads to always pushing forces. Nevertheless, with great thanks to the nonlinear nature of the radiation force, this idea is formed that the desired change in resultant acoustic radiation force may be attained by making change in surrounding acoustic field of body via well-adjusted harmonic pulsation (i.e., specific amplitude and phase) of the radiator. Since this method may seem somewhat intricate to apply due to this fact that a remote control system is required, another method is presented to reduce the complexity of the first method. In this method, a constant amplitude/ phase of displacement with respect to the frequency of system is applied to the sphere and the radiation force on the sphere is altered via change of amplitude and phase of the incident wave.

Fig. 1. Configuration of problem.

2.1. Acoustic field equations Following the standard methods of theoretical acoustics, the linearized continuity equation for an inviscid and ideal compressible medium that cannot support shear stresses may be expressed as [47]

@q þ q$  v ¼ 0: @t

ð1Þ

where q is the fluid density and v is the fluid particles’ velocity. The Euler equation in linear acoustic regime can be written as [47]

q

@v þ $p ¼ 0: @t

ð2Þ

where p denotes the fluid pressure. Combining Eqs. (1) and (2) leads to

@2v ¼ c2 $ð$v_ Þ: @t 2

ð3Þ

Assuming the irrotational nature of the wave propagation phenomenon, the velocity vector can be expressed as a gradient of a scalar potential function, v ¼ rw. Substituting this relation into Eq. (3) yields

@2w ¼ c2 $2 w: @t 2

ð4Þ

Assuming wðr; tÞ ¼ ReðuðrÞeixt Þ, leads to the so-called Helmholtz equation as [47] 2

ð$2 þ k Þu ¼ 0;

ð5Þ

where k ¼ x=c is the wave number for the dilatational wave and c is the speed of sound in fluid. The solution of the above equation according to the boundary conditions of the system leads to evaluation of the velocity vector field by v ¼ $ReðuðrÞ eixt Þ and the acoustic pressure via pðr; h; xÞ ¼ q@ReðuðrÞ eixt Þ=@t. It can be easily shown that the expansions of the scalar velocity potential function of incident plane wave and its corresponding acoustic pressure in spherical coordinate, satisfying Eq. (5), have the forms [47]

uinc: ðr; h; xÞ ¼ u0

1 X n ð2n þ 1Þi jn ðkrÞP n ðcos hÞ; n¼0

2. Formulation Fig. 1, depicts the configuration of problem including a pulsating spherical radiator in the path of plane progressive ultrasonic beams. The propagation medium is an ideal acoustic fluid medium. The radius of radiator is denoted by a. ðx; zÞ is the in-plane Cartesian coordinate system and ðr; hÞ is the corresponding spherical coordinate system, considering the axisymmetry of problem.

1 X @ uinc: nþ1 pinc: ðr; h; xÞ ¼ q ¼ u0 qxð2n þ 1Þi jn ðkrÞP n ðcos hÞ; @t n¼0

ð6Þ where jn is the spherical Bessel function of the first kind of order n, Pn are Legendre polynomials[48]. Notice that the harmonic time variations throughout the manuscript with eixt dependence, suppressed for simplicity. jpinc: j ¼ u0 qx is the amplitude of the inci-

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dent field acoustic pressure and jU inc: j ¼ u0 =c is the amplitude of the incident field displacement. Likewise, keeping in mind the Sommerfeld radiation condition for scattered field [49], the solutions of the Helmholtz equation for the scattered velocity potential functions in the surrounding fluid medium can be expressed as a linear combination of spherical waves as follows [47]

uscatt: ðr;h; xÞ ¼ u0

2.3. Acoustic radiation force In the nonlinear acoustic regime, the time-averaged acoustic radiation force on a body with fixed outer surface at equilibrium, !0 , is expressed as

Z Z

1 X n ð2n þ 1Þi An ðxÞhn ðkrÞP n ðcoshÞ;

hFi ¼

T  d!;

1 X @ uscatt: nþ1 ¼ u0 qxð2n þ 1Þi An ðxÞhn ðkrÞP n ðcoshÞ @t n¼0

ð7Þ where hn ðxÞ ¼ jn ðxÞ þ iyn ðxÞ is the spherical Hankel function of the first kind of order n [48], An which is the unknown modal scattering coefficient determined by employing the appropriate boundary conditions, later. Considering the superposition principle in linear acoustic regimes, the total velocity potential, the velocity field and the acoustic pressure in surrounding medium can be evaluated by

where <  > means time averaged over a cycle of oscillations, 2p=x, the Brillouin radiation stress tensor is defined as T ij ¼  < p > dij  qp < v i v j >, in which dij is Kronecker delta, and v i are the particle velocity components in surrounding fluid medium. It is shown that the acoustic radiation force on the object emerges for higher order terms (at least up to second order) of the velocity potential as

ð8Þ

The spherical source is supposed to vibrate radially with a prescribed velocity (i.e., specific frequency dependent amplitude (jVðxÞj) and phase (\VðxÞ)). As mentioned before, the unknown modal coefficient, An ðxÞ, must be determined by imposing the suitable boundary conditions at the interface of radiator and surrounding fluid. Accordingly, the continuity of normal velocities at the surface of the radiator requires that

 @p ðr; h; xÞ  t  ¼ VðxÞ; ixq @r r¼a 1

ð9Þ

where q is the fluid density. Incorporating Eq. (9) in Eq. (8), one will obtain

u0 kð2n þ 1Þin An ðxÞh0n ðkaÞ þ u0 kð2n þ 1Þin j0n ðkaÞ ¼



VðxÞ n ¼ 0 0

n>0

:

hFi ¼ Einc: Sc Y;

An ð x Þ ¼

n¼0

ð11Þ

n>0

2.2. Scattered acoustic field The scattered scattering coefficient, An ðxÞ, may be rewritten as a linear function of the radiator prescribed velocity as ð0Þ

ð0Þ

A0 ðxÞ ¼ Z1 þ Z2 VðxÞ; n ¼ 0 An ð x Þ ¼

ðnÞ Z1 :

ð12Þ

n>0 0

ð0Þ

ðnÞ

0

0

where Z2 ¼ 1=½u0 kh0 ðkaÞ and Z1 ¼ jn ðkaÞ=hn ðkaÞ for n P 0, or may be written as a function of the incident wave amplitude and phase as ð0Þ

ð0Þ

A0 ðxÞ ¼ Z1 þ f2 An ð x Þ ¼

ðnÞ Z1 :

.

u0 ðxÞ; n ¼ 0 n>0

ð16Þ 2

where Einc: ¼ qk u20 =2 is an indicator of incident wave energy density, Sc ¼ pa2 is the cross-sectional area of the spherical body, and Y is the dimensionless radiation force function given as a function of the scattering coefficient, An , as

Y¼

1 4 X ðn þ 1Þ½an þ anþ1 þ 2ðan anþ1 þ bn bnþ1 Þ; 2 ðkaÞ n¼0

ð17Þ

where an and bn are the real and imaginary components of An , respectively.

Solving Eq. (10)for An ðxÞ, yields

0  hjn0 ðkaÞ : n ðkaÞ

ð15Þ

where er and eh are the unit vectors in radial and tangential directions, w ¼ Reðut Þ and v r ¼ @w=@r and v h ¼ @w=r@h are the radial and tangential components of the particle velocity of surrounding fluid medium, q is the density of ambient medium, and v i are the particle velocity components in surrounding fluid medium. Substitution of the potential functions, Eqs. (4) and (5), into Eq. (15) and making the integration over the outer surface of spherical shell (i.e., r ¼ a), yields the time-averaged radiation force function in the

ð10Þ u0 k j00 ðkaÞþVðxÞ ; u0 kh00 ðkaÞ

  1 q 1 2 2 hð@w=@tÞ i  q hðj $ wj Þi er 2 c2 2 !0

Z Z hFi ¼ 

þqhðv r er þ v h eh Þ v r id!;

ut ¼ uscatt: þ uinc: ; vf ¼ rðuscatt: þ uinc: Þ; pt ¼ pscatt: þ pinc: ¼ ixqðuscatt: þ uinc: Þ:

A0 ð x Þ ¼ 

ð14Þ

!0

n¼0

pscatt: ðr;h; xÞ ¼ q

0

ð0Þ

where f2 ¼ VðxÞ=½kh0 ðkaÞ.

ð13Þ

2.4. Interaction of incident field, scattered field and the radiated field 2.4.1. Radiation force manipulation via pulsation control of the radiator Considering Eq. (17), the acoustic radiation force is just dependent on the acoustic scattering coefficient, An which is dependent on the body dynamics and its background reflection effects [50,51]. The scattering coefficient consists of the known sphere passive response and the monopole prescribed radiation effect. The radiation force may be expanded as below

Y¼

4 ðkaÞ

2

fa0 ð1 þ 2a1 Þ þ b0 ð2b1 Þ þ a1 þ

1 X ðn þ 1Þ½an þ anþ1 n¼1

þ 2ðan anþ1 þ bn bnþ1 Þg:

ð18Þ

Substituting Eq. (13) into Eq. (17) yields

 þ C;  þ Bb Y ¼ Aa

ð19Þ

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 ¼ Im½VðxÞ are the real and imaginary  ¼ Re½VðxÞ and b where a parts of VðxÞ and A; B and C are frequency dependent functions as below

A¼ B¼ C¼

4

ð0Þ

ðkaÞ

ð0Þ ½ImðZ2 Þð1 2

4

ð0Þ ReðZ2 Þð2b1 Þ;

ð0Þ

ð20Þ

ð0Þ

ReðZ1 Þð1 þ 2a1 Þ þ ImðZ1 Þð2b1 Þþa1

2

þ

þ 2a1 Þ þ

r þ p

(

ðkaÞ

) 1 X ðn þ 1Þ½an þ anþ1 þ 2ðan anþ1 þ bn bnþ1 Þ : n¼1

For any specified frequency of incident wave, A; B and C are con  bstant values. Therefore, Eq. (19) represents a straight line in a plane, for any specific acoustic radiation force function value, Y. In the case of any desired radiation force, Y ¼ Y d , one can reach

 þ C 0 ¼ 0:  þ Bb Aa

ð21Þ

0

where C ¼ C  Y d . In the case of radiation force cancellation, i.e., Y d ¼ 0, this char  b-plane acteristic line divides a into two different regions where positive (i.e., pushing force) and negative (i.e., pulling force) radiation force effects emerges. Moreover, the amplitude and phase of pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi minimum required radiation velocity is jC 0 j= A2 þ B2 and tan1 ðB=AÞ, respectively. 2.4.2. Radiation force manipulation via incident wave control In this section, a known amplitude and phase is considered for pulsation of the radiator and radiation force manipulation is going to be achieved by adjusting amplitude and phase of incident wave field. In order to highlight the contribution of incident wave in radiation force, Eq. (18) can be rewritten as

Y¼

4

ðk1 1r þ k2 1i þ k3 Þ;

2

ðkaÞ

ðqjVj2 Þ, as below

ð0Þ

½ReðZ2 Þð1 þ 2a1 Þ þ ImðZ2 Þð2b1 Þ;

2

ðkaÞ 4

1=2

½ðk21 þ k22 Þ=ð4k23 Þ  hFid =ð2pqk3 Þ . Eq. (25) can be written in terms i ¼ pt =ðqjVj2 Þ ¼ ðpr þ ipi Þ= ¼p r þ i p of normalized pressure, p

ð22Þ

xk2

!2 i  þ p

2k3 jVj2 2

x where #2 ¼ jVj 4

h

ðk21 þk22 Þ 4k23

xk 1 2k3 jVj2

!2 ¼ #2 :

ð26Þ

i d  2hFi pqk3 represents the frequency dependent

radius of the circles and j ¼  tan1 ðk1 =k2 Þ is the phase of the cenr p i -plane. Clearly, Eq. (26) divides p i r p ters of the circles within p plane into two different regions where according to the desired radiation force, the region in the circles of Eq. (26) is associated with the normalized pressures which generate lower radiation force function than the desired radiation force function, hFi < hFid . However, normalized pressures belong to the outer region of the circles cause higher radiation force function than the desired radiation force function, hFi > hFid . Also, the circle itself is locus of the points i  plane which create the desired radiation force. Characr p in the p teristic circles are defined as the circles associated with Eq. (26) while hFid ¼ 0. 3. Results and discussions 3.1. Introduction of the case and validation In this section, a numerical example is considered in order to examine the behavior of the proposed technique for acoustic handling, focusing on the required prescribed velocity amplitude and required power. The surrounding fluid is assumed to be water at atmospheric pressure and ambient temperature with the properties of q ¼ 997:05 kg=m3 and c ¼ 1497 m=s. The radius of radiator is taken as a ¼ 1 mm. To check the validity of the solutions, the radiation force function and form-function for a submerged rigid spherical shell is calculated by setting VðxÞ ¼ 0, a ¼ 1 m, q ¼ 1000 kg=m3 , c ¼ 1500 m=s in our code. The numerical result as shown in Fig. 2(a) and (b), exhibit excellent agreement with Fig. 1 of Ref. [42] and Fig. 8 of Ref. [36], respectively.

where 1r ¼ ur0 =ju0 j2 and 1i ¼ ui0 =ju0 j2 in which ur0 and ui0 are the real and imaginary parts of u0 respectively, and k1 , k2 and k3 are frequency dependent functions as below ð0Þ

ð0Þ

ð0Þ

ð0Þ

k1 ¼ Reðf2 Þð1 þ 2a1 Þ þ Imðf2 Þð2b1 Þ; k2 ¼ Imðf2 Þð1 þ 2a1 Þ  Reðf2 Þð2b1 Þ; ð0Þ

ð0Þ

k3 ¼ ReðZ1 Þð1 þ 2a1 Þ þ ImðZ1 Þð2b1 Þ þ a1 þ

1 X ðn þ 1Þ½an

ð23Þ

n¼1

þ anþ1 þ 2ðan anþ1 þ bn bnþ1 Þ: In order to achieve a desired radiation force, hFi ¼ hFid , by substituting Eq. (22) into Eq. (16), the incident wave phase and amplitude must satisfy the equation below

hFid ¼

4 ðkaÞ

2

ðk1 1r þ k2 1i þ k3 ÞEinc: Sc :

ð24Þ

Since for any specified frequency of incident wave, k1 , k2 and k3 are constant values, Eq. (24), after some simple and straightforward manipulations, can be written in the form below 2

ður0 þ k1 =2k3 Þ þ ðui0 þ k2 =2k3 Þ ¼

ðk21

þ

k22 Þ=ð4k23 Þ

2

 hFid =ð2pqk3 Þ:

ð25Þ

As it is obvious, Eq. (25) represents the equation of a circle in the ur0 ui0 -plane, centered at ðk1 =2k3 ; k2 =2k3 Þ and the radius of

Fig. 2. (a) Acoustic radiation force function, Y, of a rigid sphere submerged in water, as a function of nondimensional frequency. (b) Form function amplitude of a rigid sphere submerged in water, as a function of nondimensional frequency. (c) Average radiation resistance of a simple pulsating spherical radiator as a function of nondimensional frequency. (d) Average radiation reactance of a simple pulsating spherical radiator as a function of nondimensional frequency.

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In addition, the radiation effect is validated by calculating the frequency-dependent average radiation impedance components: radiation resistance and radiation reactance curves, of a water submerged pulsating radiator by setting u0 ¼ 0; a ¼ 1 m; VðxÞ ¼ 1 m=s;q ¼ 1000 kg=m3 ; c ¼ 1500 m=s in our code. The results in Fig. 2(c) and (d) are in excellent agreement with Fig. 20.4, page 398 of Ref. [52]. 3.2. Radiation force control by prescribed pulsation and required power Fig. 3(a) and (b) display the normalized amplitude of minimum  ¼ ðVÞ=ðxjU inc: jÞ, and its required displacement of the radiator, U corresponding phase, for radiation force cancellation, Y d ¼ 0, as a function of nondimensional frequency, ka. The frequency bandwidth is selected as 0:1 < ka < 50, which is in accordance with the operational frequency of many practical ultrasonic transducers in the range of 20 kHz < f ¼ x=2p < 10 MHz for a ¼ 1 mm. As it is observable, the normalized displacement amplitude depicts increasing trend with approximate relationships of  / ðkaÞ4 for frequency bandwidth of 0:1 < ka < 1 and U  / ðkaÞ2 U for frequency bandwidth of 1 < ka < 50 which are approximated by calculating the high (i.e., limka!1 ) and low frequency limit pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (i.e., limka!0 ) of jCj= A2 þ B2 .

5

As stated in formulations, a characteristic line in the displace  ment plane (i.e., ReðUÞImð UÞ plane) determines the positive or negative radiation force situations. As it is clear from Eq. (23), this characteristics line (i.e., radiation force cancellation) is frequency dependent. Fig. 4(a) and (b) illustrate the characteristic lines and the positive/negative radiation force regions in normalized displacement domain for selected frequencies of (a) ka ¼ 20 and (b) ka ¼ 24. The minimum required displacement is found from the intersection of the tangent circle and the characteristic line. The radius of the circle denotes the amplitude of the orientation of the normal line determine the phase. Returning to Fig. 3 (a) and (b), the radius of the tangent circles increases and the characteristic line is revolving clockwise around the tangent circle as frequency increases. One of the efficient strategies for acoustic handling of pulsating carriers is to insonify the object non-actively (i.e., no pulsation) in the desired forward motion and run it with extremely low power of pulsation with the appropriate phase difference with respect to the incident field. For further investigation, the required power to drive, to cancel or to reverse the radiation force acting on the object, can be derived. Thanks to the theory developed by Zhang and Marston [43,53,54] which relates the radiation force with the asymmetry of scattered field and the absorbed and scattered acoustical power, we are able to determine the power in order to satisfy the required

Setting a ¼ 1 mm and jpinc: j ¼ 102 Pa, the amplitude of prescribed displacement in the practical frequency range of  < 109 m. 20 kHz < f < 10 MHz is in the range of 1011 m < U These incredibly small values for prescribed amplitude mean that the radiation force cancellation is possible with extremely low powers. Looking at the phase of the minimum required prescribed displacement in Fig. 3(b), an approximate linear variation of phase with the frequency is observed. This linearity can be approximated  ¼ limka!1 ½tan1 ðB=AÞ ¼  tan1 ½tanðkaÞ for high with limka!1 \U frequency region (i.e., ka > 20).

Fig. 3. (a) Normalized amplitude of the minimum prescribed displacement required to eliminate the radiation force on the rigid sphere, submerged in water and insonified by a time-harmonic plane progressive wave, as a function of nondimensional frequency. (b) Phase of the minimum prescribed displacement required to eliminate the radiation force on the rigid sphere, submerged in water and insonified by a time-harmonic plane progressive wave, as a function of nondimensional frequency.

Fig. 4. (a) Normalized real and imaginary parts of the prescribed displacement on the spherical radiator and the negative radiation force zone (attractive effects), the positive radiation force zone (repulsive effects) and the characteristic line corresponding to the zero- acoustic radiation force situation, at selected nondimensional frequency of ka ¼ 20. (b) Normalized real and imaginary parts of the prescribed displacement on the spherical radiator and the negative radiation force zone (attractive effects), the positive radiation force zone (repulsive effects) and the characteristic line corresponding to the zero-acoustic radiation force situation, at selected nondimensional frequency of ka ¼ 24.

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pattern to cancel or reverse or alter the radiation force. In the case of progressive plane wave, Eqs. (17) and (19) of [43] become 2

Q sca ¼ ½4=ðkaÞ 

1 X ð2n þ 1ÞjAn j2 ;

ð27aÞ

n¼0

2

Q abs ¼ ½1=ðkaÞ 

1 X ð2n þ 1Þð1  j2An þ 1j2 Þ;

ð27bÞ

n¼0

where Q sca and Q abs denote the normalized absorbed and scattered powers, which are related to the absorbed and scattered powers as Pabs ¼ pa2 r0 Q abs and Psca ¼ pa2 r0 Q sca , respectively, where r0 ¼ p2inc: =2qc is the incident wave field intensity. In this theory, the radiation force may be represented in terms of absorbed, scattered powers and asymmetry index as

Y ¼ Q sca ð1  hwiÞ þ Q abs ;

ð28Þ

where the asymmetry index, hwi, means the angular distribution of scattered acoustic power, calculated from Eq. (20) of Ref. [43]. Following the same procedure as in [43] in case of progressive plane wave, radiation force function can be rewritten as summation of two separate terms as Y ¼ Y 1 þ Y 23 , where 2 P Y 1 ¼ ½8=ðkaÞ  1 n¼0 ðn þ 1Þðan anþ1 þ bn bnþ1 Þ ¼ hwiQ sca is the term of radiation force which is associated with only the scattering from 2 P the sphere and Y 23 ¼ ½4=ðkaÞ  1 n¼0 ðn þ 1Þðan þ anþ1 Þ is the term which is associated with the interaction between incident and scattered wave field. Fig. 5 illustrates the normalized absorbed powers, associated with the spherical object in its passive state (i.e., no pulsation) and active states of radiation force cancellation and reversion. Obviously, since no power is supplied to the system in the passive case, the absorbed power is zero which is in coincidence with the theory of Ref. [44]; however, in other two cases, absorbed powers, take non-zero and negative values due to the fact that the system needs to be supplied with power in order for radiation force function to be manipulated) [43]. It can also be seen that the required power to reverse the radiation force on the sphere, is almost four times greater than the power required to cancel the radiation force. Fig. 6 shows the asymmetry index for three cases of passive sphere, radiation force cancellation and reversion. It can be seen that the asymmetry index tends to zero as the frequency increases for both cases of radiation force cancellation and reversion, while for the passive case, it tends to a nonzero value. It means that with the increase of frequency, pressure field tends to become more symmetric around the sphere. The discussed symmetric pattern can be justified by taking a look at Fig. 3(a). As discussed before, the required displacement of the surface of the sphere (similar to its velocity) increases for radiation force cancellation, as the frequency increases. This pattern is valid for the case of radiation force reversion, since the required displacement to achieve

Fig. 5. Normalized required power for radiation force cancellation (dashed curve) and reversion (dotted curve) versus nondimensional frequency.

Fig. 6. Asymmetry index of passive sphere (solid curve), radiation force cancellation (dashed curve) and reversion (dotted curve) versus nondimensional frequency.

reversed radiation force is merely twice that for radiation force cancellation. The mentioned increase in the radial velocity in both cases, result in increasing the dominancy of the scattered field over the incident field as the frequency increases. Note that due to the existence of the incident wave field, the graphs of asymmetry index tend to zero but never cross hwi ¼ 0. 3.3. Radiation force control by prescribed incident wave field In order to display the main features and results for the second method of acoustic radiation force alteration, we keep the physical properties the same as those for the first method. In this example, it is assumed that the radiator is pulsating with a known steadystate velocity associated with 1 lm normal displacement of the radiator surface, equivalent to normal velocity of the radiator surface, jVðxÞj ¼ 1 06 x and the desired radiation force is set to be zero, hFid ¼ 0. According to Eq. (26), imaginary and real components of incident wave pressure for radiation force cancellation belong to ciri r p cles passing through the origin of real-imaginary plane (i.e. p plane). Fig. 7(a) and (b), display the radius of the discussed circles, #, in terms of normalized pressure and the phase of the centers of r p i -plane, respectively. The selected frethe circles, j, in the p quency bandwidth is selected in MHz region, 5 < ka < 50. As it can be seen in Fig. 7(a), the radius of the circle is decreasing as the frequency increases. It means that as the frequency of operation increases, the region in which the radiation force exerted upon the pulsating sphere can become negative, becomes narrower thus requires more delicate adjustments to achieve negative radiation force on the sphere. Also, the frequency dependency of the radius 2

of the circles can be expressed as # / ðkaÞ . Fig. 7(b) shows the r p i -plane phase of the position of the center of the circles in the p with respect to frequency. Fig. 8(a) and (b) shows three dimensional (3D) and two dimensional (2D) locus of the centers of these r p i -plane circles which form a counterclockwise helix curve in the p around frequency axis, respectively. In addition, Fig. 9(a) and (b) demonstrate 3D continuous and 2D selected locus of circles’ circumferences as a function frequency. Fig. 10(a) and (b) depict the characteristic circles (i.e., circles associated with zero-radiation force on the sphere) at selected frequencies ka ¼ 20 and ka ¼ 24, respectively, and the positive and negative states of radiation force function exerted upon the pulsating sphere. As expected, the inner regions of the circles are associated with negative radiation force on the sphere (pulling force), outer region, on the other hand is associated with positive radiation force (pushing force). Points on the circumference of the circles are the points if chosen as real and imaginary parts of the incident wave pressure, will result in radiation force cancellation (stationary state). Obviously, the characteristic circles pass through r p i -plane. Unlike the case of radiation force the origin of the p

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Fig. 7. (a) Normalized radius of the characteristic circles, as a function of nondimensional frequency. (b) Phase of the position (phase) of the characteristic circles’ center, as a function of nondimensional frequency.

Fig. 9. (a) 3D representation of the characteristic circles. (b) 2D representation of the characteristic circles for various frequencies.

control by pulsation adjustment, in this case, in each selected frequency, there is a finite region in which the generation of negative radiation force are possible. According to Eq. (26), non-zero values of desired radiation forces at any selected frequency occur on the circles which are centered at the same center of the characteristic circle, with larger and smaller radius, respectively. Fig. 11 (a) and (b) depict the circles corresponding to zero, negative and positive desired radiation force states, at selected frequencies of ka ¼ 20 and ka ¼ 24, respectively. A more interesting feature associated with the maximum and minimum radiation forces that may be produced via the presented methodology may be discovered form Eq. (26). The possible values for radiation force according to Eq. (26), belongs to an semi-infinite interval of radiation force amplitude, ½hFimin ðx; pinc ; VÞ; 1Þ, where hFimin means the minimum possible value for radiation force at any specific frequency, incident wave pressure amplitude and velocity of pulsation, which will be discussed later. Obviously, the possibility of existence of limitless upper bound for the desired radiation force is doubted because of this fact that the derived theory is valid only up to second order expansion of the state equation in the nonlinear regime. To determine the maximum value of negative radiation force on the sphere, the radiation force should be represented as a function of incident wave pressure field by substitution, thus Eq. (24) can be rewritten as

hFi ¼ 2qpðk1 ur0 þ k2 ui0 þ k3 ju0 j2 Þ: Fig. 8. (a) 3D locus of the centers of the characteristic circles. (b) 2D locus of the centers of the characteristic circles.

ð29Þ

Using simple calculus, it can be shown that the maximum amplitude of negative radiation force occurs for

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Fig. 11. (a) Circles associated with zero, negative and positive radiation force, at selected nondimensional frequency of ka ¼ 20. (b) Circles associated with zero, negative and positive radiation force, at selected nondimensional frequency of ka ¼ 24. Fig. 10. (a) Regions of negative radiation force (attractive effects), positive radiation force (repulsive effects) and the characteristic circles, at selected nondimensional frequency of ka ¼ 20. (b) Regions of negative radiation force (attractive effects), positive radiation force (repulsive effects) and the characteristic circles, at selected nondimensional frequency of ka ¼ 24.

ur0 ¼ k1 =2k3 ; ui0 ¼ k2 =2k3 ;

ð30Þ

which leads to hFimin: ¼ pqðk21 þ k22 Þ=2k3 . The point of maximum negative radiation force occurrence in r p i -plane, Eq. (30), corresponds to the center of the characteristics p circle, at any selected frequency of operation. It is interesting to note that this maximum amplitude of negative radiation force state, has the exact value of the positive radiation force exerted upon the spherical object in its passive state (i.e., non pulsating state VðxÞ ¼ 0). This statement can be mathematically represented as

hFimin: =hFi0 ¼ k3 =Y 0 ¼ 1:

that the desired force/motion emerges. Here, we consider the case of a prescribed incident wave field and pulsation of the object. Clearly, due to the object spatial position with respect to the incident field source (i.e., due to the motion of the object), the phase difference between the pulsation and wave field, varies. Fig. 12 depicts the schematic of the issue, in which z is the axis of motion. It is interesting to calculate the spatial average of the radiation force on the body along one wavelength of the incident wave field as below

ð31Þ

where hFi0 is the acoustic radiation force exerted upon a passive sphere and Y 0 is the corresponding radiation force function. 3.4. Radiation force and spatial position of object Our suggested methodology assumes that the characteristics of the object pulsation or the incident wave field can be adjusted so

Fig. 12. Schematic of a radiator placed in an incident wave field.

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M. Rajabi, A. Mojahed / Ultrasonics xxx (2017) xxx–xxx

 ¼ hhFii ¼ ð1=kÞ hFi z

Z

zþk

0

hFi dz ;

ð32Þ

z

where in above expression, hiz means the average on spatial position of the object and k is the wavelength of the incident wave field at the selected frequency of operation. Substitution of Eq. (29) into  ¼ hFi . This finding is not surprising, since the Eq. (32) leads to hFi 0 radiation force of Eq. (29) may be written as hFi ¼ hFi0 þ df ðz; VÞ which means the summation of radiation force in passive state, hFi0 ¼ 2qpk3 ju0 j2 , and a spatial dependent deviation, df ¼ 2qpðk1 ur0 þ k2 ui0 Þ, which is originated from the pulsation characteristics of the object, V. Obviously, hdf iz ¼ 2qpðk1 hur0 iz þ k2 hui0 iz Þ ¼ 0 due to its simple (trigonometric) periodic nature with respect to z. Fig. 13 displays the spatial variation of radiation force exerted on the spherical body in which its spatial harmonic nature can be observed. (This figure is associated with the incident wave field of the amplitude in accordance with OCO which will be introduced in the next section).

3.5. Optimal operation state Turning back to our methodologies and considering practical challenges of implementing a control system embedded in active object, it is suggested to use the second technique of activating the object with a specified pulsation characteristics and handling it with a tunable incident wave field, with specified amplitude of acoustic pressure and adjustable phase. Recalling the importance of the radiation force in both pulling and pushing states for complete axial manipulation, Fig. 14 depicts a suggested circle of operation for incident wave characteristics (i.e., amplitude and phase). r p i  plane, The proposed operation circle, centered at the origin of p qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 with the radius of jpjopt: ¼ x k1 þ k2 =ð2k3 jVj Þ which is equal to the minimum pressure of the incident wave which can generate the maximum negative radiation force state, equivalent to hFi0 and maximum positive radiation force with the value of þ3 hFi0 . Which from here on, the suggested circle is called the ‘‘optimal circle of operation (OCO)”. The tangent point of operating circle with the hFid ¼ 3 hFi0 circle (see short dash circle in Fig. 14) represents the maximum pushing force on the object. In order to achieve pulling state, the point c is the maximum efficiency point of operation which corresponds to maximum negative radiation force on object.

3.6. Zero radiation force state, stability analysis and precise motion control Another feature associated with the acoustic handling especially in the interesting cases that precise motion control is

Fig. 14. Optimal circle of Operation (‘‘OCO”) and its associated maximum and minimum radiation force on the radiating sphere.

required (e.g., targeted delivery, micro-machining, microbiological robotics and mechanisms, etc.), is discussed here. Taking a look at Fig. 14, OCO intersects with the circle associated with the zero radiation force state, hFid ¼ 0, at two points. These two points are designated in Fig. 13 as S and U which S/U represents the spatial stable/unstable positions of the object with respect to incident wave field. Both points can be considered as trapping points, however, by analyzing force variation in the vicinity of these two points, it can be found that any small deviation from point S lead to emergence of attractive forces toward S point (i.e., if dz > 0, then dhFi < 0, and if dz < 0, then dhFi > 0), in contrast to U point for which any deviation lead to emergence of repulsive forces from U point(i.e., if dz > 0, then dhFi > 0, and if dz < 0, then dhFi < 0). Existence of the stable point S provides this opportunity to precisely move the object along the desired path, toward the target point, by changing the phase of incident wave field, corresponding to OCO. Emergence of trapping points at the position of object, whether stable or unstable, in our case in which the wave field is progressive (not standing) is fascinating. Clearly, generation of standing or pseudo-standing wave field may be possible when the pulsating acoustic wave field from the object develops and interact with the incident wave field; while at the center of the object (its spatial position), no effect of pulsation field can be defined or in other word, such nodes or antinodes effect of common standing wave fields cannot be applied in this case. Definitely, between the pulsating object and the incident wave transducer, a pseudo-standing wave field may be presumed. The phrase of pseudo- turns back to this fact that the amplitude of radiation field from the object is decreasing in space and also, towards the transducer. Considering the floating condition of the object and the motion created due to the exerted radiation force, such pseudo-standing wave field may be generated if the object stays in its stable trapping point (zero-radiation force state). 3.7. Oscillatory behavior of the object around the stable trapping points The equation of motion of the object for any small spatial deviation from stable point S, can be approximated as

Fig. 13. Radiation force acting on a radiating sphere versus its position on the normalized wave propagation axis (z=k) or incident wave phase (w).

Md€z þ Cdz_ þ Kdz ¼ 0;

ð33Þ

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M. Rajabi, A. Mojahed / Ultrasonics xxx (2017) xxx–xxx

Fig. 15. Normalized frequency of oscillation around stable trapping point, versus nondimensional frequency.

where M  ð4=3Þpa3 q is the mass of the object which can be estimated with the assumption of neutrally buoyant condition  Oðqa3 Þ, C is the equivalent damping coefficient which may be roughly estimated from the drag force, 6pgaðdz_ Þ, due to the viscosity of medium, g, by assuming low Reynolds number regime dominance over the system,  Oð10gaÞ and pffiffiffi K ¼ @hFh=@zjz¼S ¼ ð 3=2Þpqkðk21 þ k22 Þ=k3 is the equivalent spring stiffness coefficient because of the restitution effect of radiation 2

0

force,  OðqV 2 =½kðh0 ðkaÞÞ Þ .The oscillation frequency is calculated pffiffiffi pffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi as xosc: ¼ K=M ¼ ðð3 3=8Þkðk21 þ k22 Þ=ðk3 a3 ÞÞ, with the order of 3

0

2

1=2

 OðqV 2 =½ka ðh0 ðkaÞÞ Þ . Fig. 15 depicts the normalized frequency of oscillation, xosc: =x, at trapping stable point, S, as a function of nondimensional frequency, ka, in logarithmic scale. As it is seen, two distinct regions are observed: (i) low frequency region, ka < 1, where the normalized oscillation frequency is 3=2

, (ii) high fre Oð103 Þ  Oð101 Þ in which ðxosc: =xÞ / ðkaÞ quency region, ka > 1, where the normalized oscillation frequency 1=2

. Taking care is  Oð104 Þ  Oð103 Þ in which ðxosc: =xÞ / ðkaÞ about the time averaged nature of radiation force and the comparable values of oscillation frequency with respect to the frequency of operation, the validity of the results and the presented analysis about the behavior in the vicinity of the trapping points at region (i) in which the wavelength of the incident wave field is in the same order or higher than the size of the object, is questionable; however, at higher frequencies (i.e., the wavelength of the incident wave field is much less than the size of the object), the theory may be valid since the time scale of the oscillatory motion of the object is much higher than the operation frequency where the time-averaging has been performed. 4. Conclusion Due to the nonlinear nature of the radiation force phenomenon, two theoretical methods for acoustic handling of ideal pulsating spherical radiators using plane progressive ultrasonic beams have been investigated. In the first method, an exact expression is obtained for the acoustic radiation force exerted upon the spherical body as a linear function of the radiator’s harmonic velocity characteristics (i.e., amplitude and phase), while it is insonified by incident plane progressive wave field with known amplitude/phase. It has been shown that the pushing effects of radiation force may be reversed to pulling effects by selecting the amplitude and phase of the monopole vibrations of the radiator in specific infinite regions in velocity complex plane which is characterized with frequencydependent straight line equations. It has been shown that in the first methodology, large amplitudes for both positive and negative radiation force are possible.

In the second method, it has been shown that the radiation force on the sphere which is pulsating with prescribed amplitude/phase can be altered by adjusting the amplitude and phase of the incident wave field with respect to characteristic circles in incident wave real-imaginary plane. The characteristic circles determine the zero-radiation force state. In the second methodology, a frequency dependent limit for the maximum amplitude of negative radiation force exists. This work may be considered as a significant step for introduction of novel handling techniques by virtue of stimulated carriers, with great possible applications in medical and engineering. Moreover, considering the spatial dependency of the radiation force due to the phase difference between the incident wave field and the pulsation of the object, the rests (trapping) points are discovered with stable and unstable nature. 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