Study of axial acoustic radiation force on a sphere in a Gaussian quasi-standing field

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Accepted Manuscript Study of axial acoustic radiation force on a sphere in a Gaussian quasi-standing field Rongrong Wu, Kaixuan Cheng, Xiaozhou Liu, Jiehui Liu, Xiufen Gong, Yifeng Li PII: DOI: Reference:

S0165-2125(15)00169-9 http://dx.doi.org/10.1016/j.wavemoti.2015.12.005 WAMOT 2003

To appear in:

Wave Motion

Received date: 11 March 2015 Revised date: 24 December 2015 Accepted date: 24 December 2015 Please cite this article as: R. Wu, K. Cheng, X. Liu, J. Liu, X. Gong, Y. Li, Study of axial acoustic radiation force on a sphere in a Gaussian quasi-standing field, Wave Motion (2016), http://dx.doi.org/10.1016/j.wavemoti.2015.12.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

*Research Highlights




Highlights 
The Gaussian quasi-standing waves are expressed in terms of spherical wave functions. 
The radiation force on a sphere in Gaussian quasi-standing waves is expressed. 
The radiation force is computed when the sphere is translated axially. 
The feasibility of a Gaussian standing wave trapping is demonstrated theoretically.





*Manuscript (Clear) Click here to view linked References


Study of axial acoustic radiation force on a sphere in a Gaussian quasi-standing field Rongrong Wua, Kaixuan Cheng a, Xiaozhou Liua,b1, Jiehui Liua, Xiufen Gong a , Yifeng Lic a)

Key Laboratory of Modern Acoustics, Institute of Acoustics, school of Physics, Collaborative

Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China b)State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China c) College of Computer science and Technology, Nanjing Tech University, Nanjing 211800, China

A bstract Based on the finite series method, the Gaussian standing or quasi-standing beam is expressed in terms of spherical wave functions and a weighting parameter, which describe the beam shape and location relative to the particle. An expression is derived for the radiation force on a sphere centered on the axis of a Gaussian standing or quasi-standing wave propagating in an ideal fluid. Rigid, fluid, elastic, and viscoelastic spheres immersed in water are treated as examples. In addition, a method is proposed to compute the axial acoustic radiation force when the sphere is translated axially. Results indicate the capability of the proposed method to manipulate and separate spheres based on their mechanical and acoustical properties. The interaction of a Gaussian quasi-standing beam with a sphere can result in periodic axial force under specific operating conditions. The results presented here may provide a theoretical basis for the development of acoustical tweezers in a Gaussian standing beam, which would be useful in micro-fluidic lab-on-chip applications.




























































1 Corresponding author at: Key Laboratory of Modern Acoustics, Institute of acoustics, Nanjing University, Nanjing 210093, China. Tel.: +86 2583593613; fax:+86 2583315557. E-mail address: [email protected] (X. Liu).








Keywords: Acoustic radiation force, Gaussian standing field, Spherical particles, Scattering PACS:43.25.+y, 43.80.Gx, 43.80.Jz

1.

Introduction

Acoustic radiation force [1] is the result of a momentum transfer from a sound wave to a particle as a consequence of second-order (nonlinear) effects. With the development of ultrasonic biological science and technology, the acoustic radiation force has been widely used in particle manipulation applications [2, 3], micro-fluidic lab-on-chip applications [4, 5], biomedical imaging [6], and drug delivery [7]. The acoustic radiation force on a spherical particle can be predicted approximately by the ray acoustics approach model when the dimensionless frequency bandwidth ka

1

(where k is the wave number, a is the radius of a sphere) [8-10], whereas

the static and dynamic acoustic radiation force on the particle can be predicted by the multipole expansion method (MEM) or the partial-wave expansion method, which is applicable to any frequency range both in spherical and cylindrical coordinates [11 32]. This theory has been discussed for various types of waves in spherical coordinates, such as plane waves [11 - 14], amplitude-modulated waves [15, 16], Bessel waves [17 - 20], circular focused beam [21] and Gaussian progressive waves [22, 23]. Furthermore, the theories for an arbitrarily shaped beam on a sphere have been proposed based on this method [24 - 27].










An acoustic standing field can be useful in micro-fluidic lab-on-chip applications, and researchers have calculated the radiation force function for plane and Bessel standing waves [12, 14, 18 and 19]. Similarly to plane and Bessel standing waves, a Gaussian standing or a quasi-standing acoustic field is the result of propagating two equal or unequal amplitude Gaussian beams respectively along the same axis, but in opposite directions. A Gaussian standing beam has been used to develop acoustical tweezers that can immobilize latex particles and frog eggs in a fixed position [2], and can also be very useful in particle manipulation applications and acoustical levitation. Based on the finite series method [22, 30], a Gaussian standing or quasi-standing acoustic wave can be expressed as a partial wave series involving the scattering angle relative to the beam axis. Analytical and numerical analyses are undertaken here to calculate the acoustic radiation force experienced by a spherical object immersed in an ideal fluid placed in an axisymmetric ideal Gaussian standing or quasi-standing waves. Examples for the radiation force on rigid, fluid, elastic, and viscoelastic spheres are discussed. In addition, the axial acoustic radiation force on different spheres when the sphere is translated axially is calculated, which demonstrates the manipulation and separation capability of Gaussian standing waves. It is anticipated that these results can be useful in understanding the mechanism of radiation forces of Gaussian standing waves on spheres for potential applications in particle manipulation and entrapment.

2.




T heory







2.1. Acoustic scattering and radiation force calculation for a Gaussian standing wave In the case of a tightly focused Gaussian beam, the phase front of the fundamental mode propagating in the Gaussian beam is not a plane wave. Using the complex source point method, exact solutions of the scalar Helmholtz equation that describe the tightly focused Gaussian beam can be obtained without any approximations in spherical coordinates [33]. For the sake of simplicity, it is assumed here that the Gaussian beam is weakly focused, with a large width W0 , which will let the phase front propagate like a plane wave. Moreover, the phase front of the fundamental mode propagating in the Gaussian beam is almost equal to the plane wave near the beam waist. This means that the fundamental mode can be regarded approximately as an acoustic plane wave with Gaussian amplitude distribution in that region [22, 30]. Figure 1 shows, a sphere of radius a centered in an acoustic axisymmetric Gaussian standing wave propagating along the z-axis and immersed in a fluid. Assume that the 

s center is at distance h from each beam waist of a Gaussian stationary wave field. Let the incident wave fronts be parallel to the z-axis, and let A be the amplitude of the incident velocity potential ( A is a real number). Assume also a homogeneous and isotropic fluid of infinite extent with adiabatic speed of sound c and density  . According to the representation of the incident velocity potential for a Gaussian wave, the incident velocity potential for a Gaussian standing waves should be expressed as: i  A e


x2  y 2  

W 2  0  

e ik
z  h   e  ik
z  h   e  it



(1) 





where k is the wave number and  is the angular frequency of the wave. Notice that Eq. (1) is not an exact solution of the Helmholtz equation, and hence the scattering problem cannot be solved in general. However, the finite series method can be used to remodel the waves, making the result an exact solution of the Helmholtz equation both in spherical and cylindrical coordinates [22, 30]. In a spherical coordinate system, the incident velocity potential may be expanded into a generalized Rayleigh wave series as in [12, 22]:

 i  
2n  1i n Gn n jn
kr  Pn
cos  e  it

(2)

n 0

where: n  n  A  e ikh 
1 e  ikh 



G2 p 

(3)




p  1 p 
p  j  1 2  2  4
1 kW0   
p  1 2  j 0 
p  j !j!

G2 p 1 






j


p  1 p 
p  j  3 2  2  4
1 kW0   
p  3 2  j 0 
p  j !j!




(4)



j

(5)

where jn () is the spherical Bessel function of order n and Pn () is the Legendre function of order n , and () is the Gamma function. From Eqs. (4) and (5), the beam coefficient can be determined by the dimensionless parameter 1 kW0 . The convergent items are different for different ratios of W0 to . Table I gives the beam coefficients for different beam waists. The beam coefficients are convergent when the values of the beam coefficients Gn 0.0001 [30]. From the simulation results listed in Table I, it is clear that the larger the ratio between W0 and , the greater is the number of the convergent items. Moreover, when W0

,

G n  1 , which from Eq. (2) should be expected in the case of a plane standing waves.










To validate the finite series method, the incident Gaussian beam has been calculated by exact solution and the finite series solution for different beam widths [22]. The results show that when W0  2 , the wave beam computed by the finite series is in agreement with the exact solution. The scattered velocity potential should be expressed as:

 s  
2n  1i n An Gn n hn
kr  Pn
cos   e  i t

(6)

n 0

where An  n  i n represents the unknown scattering coefficients determined by the appropriate boundary conditions at the surface of the sphere; these coefficients are known for a wide variety of spheres [34]. n , n are respectively the real part and the imaginary part of the scattering coefficient An , and hn () is the spherical Hankel function of the first kind of order n . The total field is the sum of the incident and scattered fields, and can be expressed as:

 t  
2n  1i n Gn n
U n  iVn  Pn
cos   e  i t

(7)

n 0

where U n and Vn are given by the following equations: U n 
1  n  jn
kr   n nn
kr 

(8)

Vn  n jn
kr   n nn
kr 

(9)

where nn () is the spherical Neumann function of order n . For a continuous sinusoidal Gaussian standing beam (described by e  i t ), the radiation force is defined as a time-averaged quantity over at period T , and can be calculated by integrating the excess of pressure over the surface S of the sphere at rest. The force is obtained by integrating the excess pressure P  P0 generated by the sound field over the instantaneous surface S t of the sphere. The averaged radiation










force can be expressed as [11]: F   

S( t )

P  P0 ndS  

 (vn n  vt t )vn dS



1

 2 v ndS 2



1  t 2 ( ) ndS 2

t

 2 c

(10)

where the parameters vn r a   t / r, vt r a   t /
r r  are respectively the normal and tangential components of the particle velocity at the boundary;
n is the outward-pointing unit normal vector of dS ; and t is the outward-pointing unit tangential vector of dS , where dS  2 r 2 sin d . In the direction of wave propagation (the axial z-direction), the total radiation force on the sphere can be expressed as: Fz  Fr  F  Fr ,  Ft

(11)

2

    Fr   a 2   t  sin cos d 0  r  r  a

(12)

2

    F     t  sin cos d 0  r  r  a

       Fr ,  2 a   t   t  sin 2 d 0 r

  r  a   r  a

(13)

(14)

2

Ft  

 a 2    t  sin 2 cos d c 2 0  t  r  a

(15)

Note that  t in Eq. (10) represents the real part of  t in Eqs. (7), which can be represented as follows:

 t  
2n  1Rn Pn
cos 

(16)

n 0

where Rn  Re  G n n i n
U n  iVn 
cos t  i sin t 

(17)

The function Rn satisfies the following relations: Rn  Rn 1 
1




n 1

A2 
gni gni 1  gnr gnr 1 
U nU n 1  VnVn 1  
g ni g nr 1  g nr g ni 1 
VnU n 1  U nVn 1  sin
2kh 







(18) Rn  Rn1 
1

n 1

A2 
gni gni 1  gnr gnr 1 
U nU n1  VnVn1  
g ni g nr 1  g nr g ni 1 
VnU n1  U nVn1  sin
2kh 

(19) Rn  Rn1 
1

n 1

A2 
gni gni 1  gnr gnr 1 
U nU n1  VnVn1  
g ni g nr 1  g nr g ni 1 
VnU n1  U nVn1  sin
2kh 

(20) Rn  Rn 1 
1

n 1

A2 
gni gni 1  gnr gnr 1 
U nU n 1  VnVn 1  
g ni g nr 1  g nr g ni 1 
VnU n 1  U nVn 1  sin
2kh 

(21) Rn Rn 1   2 Rn  Rn 1 t t

(22)

where gnr  Re  Gn , gni  Im  Gn 

(23)

Substituting Eq. (16) into Eq. (11), manipulating the results using the properties of the 1 2

Legendre functions, and denoting by E  k 2 A2 the characteristic energy density, the final expression for the radiation force in the direction of wave propagation (the axial z-direction) can be simplified and expressed as: Fst   a 2 EYst  sin
2kh 


(24)



where the function Yst represents the maximum force per unit incident energy density and per unit cross section of the sphere which is called the radiation force function in a Gaussian standing waves and, can be expressed as: Yst 



8


ka 

2


n  1
1  n 0

n 1


gnr gnr 1  gni gni 1   n 1
1  2 n   n
1  2 n       i r r i  
gn gn 1  gn gn 1 
n  n 1  2 n n 1  2 n n 1   

(25)

2.2. Radiation force calculation for Gaussian quasi-standing waves As shown in Figure 2, let the velocity potential  i of an incident Gaussian quasi-standing wave be:








i

e

( x2

y 2 )/W02

Ae ik z

h

Be

ik z h

e

i t

(26)

where the first and second terms represent Gaussian progressive waves propagating in the  z and  z directions respectively. Here it is assumed that A B ( A and B are both real numbers). Setting  n  Ae ikh 
1n Be  ikh 

(27)

then:  i  
2n  1i n Gn n jn
kr  Pn
cos  e  i t

(28)

Rn  Re  G n n i n
U n  iVn 
cos t  i sin t 

(29)

n 0

where the function Rn satisfies the following relations: Rn  Rn 1 
1 

n 1

AB 
g ni g ni 1  g nr g nr 1 
U n U n 1  VnVn 1  
g ni g nr 1  g nr g ni 1 
Vn U n 1  U nVn 1   sin
2kh 

A2  B 2  i r r i i i r r  
g n g n 1  g n g n 1 
U n U n 1  VnVn 1  
g n g n 1  g n g n 1 
U nVn 1  Vn U n 1   2

(30) Rn  Rn 1 
1 

n 1

AB 
g ni g ni 1  g nr g nr 1 
U n U n 1  Vn Vn 1  
g ni g nr 1  g nr g ni 1 
Vn U n 1  U n Vn 1   sin
2kh 

A2  B 2  i r r i i i r r







 
g n g n 1  g n g n 1 
U n U n 1  VnVn 1  
g n g n 1  g n g n 1 
U nVn 1  Vn U n 1   2

(31) Rn  Rn 1 
1 

n 1

AB 
g ni g ni 1  g nr g nr 1 
U n U n 1  VnVn 1  
g ni g nr 1  g nr g ni 1 
Vn U n 1  U nVn 1   sin
2kh 

A2  B 2  i r r i i i r r





 
g n g n 1  g n g n 1 
U n U n 1  VnVn 1  
g n g n 1  g n g n 1 
U nVn 1  Vn U n 1   2

(32) Rn  Rn 1 
1 

n 1

AB 
g ni g ni 1  g nr g nr 1 
U n U n 1  Vn Vn 1  
g ni g nr 1  g nr g ni 1 
Vn U n 1  U n Vn 1   sin
2kh 

A2  B 2  i r r i i i r r 






g n g n 1  g n g n 1 
U n U n 1  VnVn 1  
g n g n 1  g n g n 1 
U nVn 1  Vn U n 1   2

(33)

Rn Rn 1   2 Rn  Rn 1

t

t

(34)

If the radiation force function F qst for a quasi-standing wave is defined as:










Fqst   a 2 EYqst

(35)

then Yqst reduces to:

Yqst 

8


ka 

2

  
g r g r  g ni g ni 1    n 1
1  2 n    n
1  2 n    
1n 1 B  n n 1  sin
2 kh     A  
g ni g nr 1  g nr g ni 1 
 n   n 1  2 n n 1  2 n  n 1       n 1 
   i r r i n0  A2  B 2 
g n g n 1  g n g n 1    n 1
1  2 n    n
1  2 n         2 r r i i 2 A  
g n g n 1  g n g n 1 
 n   n 1  2 n n 1  2 n  n 1        

(36)

The radiation force function for a Gaussian progressive wave Y p was obtained in a recent study [22]. Y p is the radiation force per unit energy density and unit cross-sectional surface in a Gaussian progressive wave and can be expressed as:

Yp

4

ka

n 1

2

n 0

Re G n G n*

1

Im G n G n*

n 1

n 1

n 1

2

1 2

n n

n 1 n

2

n

n 1

1 2

n 1

(37)

where * denotes the complex conjugate. Like the radiation force function of a plane quasi-standing wave [12], using Eqs. (25) and (37), Yqst can be expressed in terms of Y p and Yst as: B2  B Yqst  1  2  Yp  Yst sin
2kh  A  A 

(38)

Moreover, Eqs. (35) becomes:  B 2  B  Fqst  F p  1  2   R ps sin
2 kh    A  A 

where R ps 

Yst Yp

(39)

(40)

F p   a 2 EYp

Notice that when A  B , the radiation force for the Gaussian quasi-standing wave F qst becomes the radiation force for the Gaussian standing wave F st ,when

B  0,

F qst becomes the radiation force for the Gaussian progressive wave F p .








2.3. Axial radiation force calculation when the sphere deviates from the Gaussian standing wave center Figure 3 shows that, when the position of the sphere deviates by h0 from the Gaussian standing wave center, the final expression for the radiation force on the sphere in the axial z-direction can be simplified and expressed as: Fst   a 2 EYst  sin
2kh 

Yst

8

ka

1

2

n 0

(41)

n

Re G n G n*

cos kh0

1

Im G n G n*

1

cos kh0

i sin kh0 i sin kh0

n 1

1 2

n

n 1

n

2

n

1 2

n

n 1

n

2

(42)

3.

Simulation and discussion

This section presents several numerical examples of the radiation force function for a Gaussian standing wave Yst and a Gaussian quasi-standing wave Yqst . Two liquid materials (oleic acid and animal oil), two elastic materials (stainless steel and brass) and a viscoelastic polymer (polymethylmethacrylate (PMMA)) were selected. Table II lists the mechanical properties of these materials [11, 22]. The results in the following figures were obtained using a wavenumber increment of ka  104 . The series in Eqs. (25), (36), and (42) are truncated for n when G n less than 0.0001.

3.1. Liquid drops in air in a Gaussian standing wave A fundamental example in fluid dynamics applications is manipulating liquid drops in air. This example is particularly useful for simulating the interaction of sound waves







n

n 1




in a reduced-gravity environment. Under this circumstance, a levitated drop can maintain its spherical shape. Water drops in air can be approximated as perfectly rigid objects in water because water is about 813 times denser than air and the acoustic impedances of liquid and air are mismatched [19]. The curves of water drops in air matched perfectly with the curves of rigid spheres in water, as shown in Figure 4, and the radiation force was not sensitive to the density or the speed of sound of the drop. The present results cover a maximum range 0  ka  10 within which the change in the form of the Yst curves is significant. Inspection of Figure 4 reveals that the radiation force increases rapidly with ka when ka  1 . This peak at low ka is a characteristic property of the interaction of a rigid

sphere with a standing wave. The radiation force function reduces rapidly to 0 when ka  1 , and the Yst
ka curve has a series of prominent peaks and valleys, which are caused by the effects of resonant vibration of the rigid sphere. As ka increases, the magnitudes of the peaks and valleys become small. The result here is similar to the radiation force function of a plane standing wave [14]. Figure 4 also shows the radiation force function for different beam widths of Gaussian standing waves. The width of a Gaussian standing wave influences the radiation force function. The magnitudes of the peaks and the valleys increase with increasing W0 . When the beam waist is far greater than the wavelength of the incident wave ( W0  3 ), the radiation force function of the Gaussian beam approaches that of a plane standing wave. Similar results have been also presented in references for the acoustic radiation










force of a Gaussian beam [22, 30].

3.2. Liquid sphere in water in a Gaussian standing wave Another example of particular interest in bioengineering concerns the possibility of manipulating cells according to their acoustic properties. In this research, an oleic acid sphere and an animal oil sphere were chosen to simulate fat beads. Figure 5(a) shows the predictions of the radiation force function Yst with different beam widths of Gaussian standing waves for an oleic acid sphere versus the dimensionless frequency bandwidths 0  ka  10 . The Yst
ka curves have a series of prominent peaks and valleys after a large peak at ka  1 as before, and these peaks and valleys are more prominent than in the rigid case. Moreover, when ka  1 , the peaks in the Yst
ka curves are more distinct than in the rigid case, as are the negative valley. Yst also exhibits increased magnitudes of peaks and valleys in each curve as the width of the Gaussian standing wave increases. Figure 5(b) shows the predictions of the radiation force function Yst in a Gaussian standing wave for an oil sphere. The change from the oleic acid solution is not remarkable because of the difference in the physical properties of the two materials. The value of the radiation force function for an oil sphere is greater than for an oleic acid one, but the value of the radiation force function for a fluid sphere is much less than that of a rigid one (Fig. 4).

3.3. PMMA sphere in water in a Gaussian standing wave The polymer particles are widely used in drug delivery application. When subjected to










an acoustic field, polymer viscoelastic spheres tend to absorb acoustic energy. Absorption was modeled by introducing complex wave-numbers into the theory [35]. This principle can be directly applied to take into account compressional and shear wave absorption inside the polymers material. The normalized absorption coefficients of compressional  1 and shear waves  2 , respectively, are constant quantities and independent of frequency [15, 30]. For visco-elastic materials, the terms xl  k l a and xs  k s a are replaced by xl , xs , given by: xl  kl a
1  j 1  , xs  ks a
1  j s  , where k l and k s are the wave numbers of compressional and shear waves, respectively.

Figure 6 shows the radiation force function Yst for a PMMA sphere in a Gaussian standing wave for bandwidths 0  ka  10 . The successive peaks and valleys are more prominent and angular than for the rigid sphere, which suggests that a certain tendency is becoming more noticeable as the material becomes more compressible [12]. Moreover, if wave absorption in the spherical material is taken into account, Yst decreases with the magnitudes of the peaks and valleys in each curve except around ka

8.

Therefore, the absorption of

sound was considered frequency dependent describing the behavior of the polymeric material.

3.4. Solid elastic sphere in water in a Gaussian standing wave The ability to exert an acoustic radiation force on a metal would be useful in microstructure assembly. This example is also helpful in predict the response of a heavy solid elastic sphere in a Gaussian standing wave for acoustical levitation.










The calculation results for the acoustic radiation function Yst for an elastic stainless steel sphere and a brass sphere are shown in Figures 7(a) and 7(b) respectively. It was found that the Yst  ka curves have a series of prominent peaks and valleys corresponding to the resonant frequencies of elastic oscillation of the spheres. The waist of the Gaussian standing waves has an influence on the radiation force function in the peaks and valleys which is related to vibrational modes, and the influence decreases with increasing beam waist. When the beam waist is far greater than the wavelength of the incident wave ( W0  3 ), the radiation force function of a Gaussian standing wave is the same as that of a plane standing wave.

3.5. Stainless steel sphere in water in a Gaussian quasi-standing wave Figure 8(a) shows the radiation force function Yqst based on Eq. (36) or Eq. (38) for a stainless steel sphere in a Gaussian quasi-standing wave for bandwidths 0  ka  10 . Here the distance between the center of the sphere and that of the Gaussian quasi-standing wave is equal to the radius of the sphere
h  a  , and the beam waist W0  3 . The magnitude ratio between the waves propagating from directions  z and z

is B A . With decreasing of B A , the magnitude of the peak at ka  1 decreases,

but the following peaks and valleys increase, making Yqst more similar to the radiation force function for a Gaussian progressive wave Y p . Noticed that when

B  A , the radiation force for a Gaussian quasi-standing wave F qst becomes the radiation force for a Gaussian standing wave F st (Fig. 7(a)). When B A  0 , F qst becomes the radiation force for a Gaussian progressive wave F p [22]. Figure 8(b) shows the calculated values of R ps based on Eq. (40) for stainless steel










spheres, which represent the ratio between the radiation force function of the Gaussian standing wave and of the Gaussian progressive wave for bandwidths 1  ka  10 . Note that R ps has no meaning because Yst  Y p  0 at ka  0 . When 0  ka  1 , R ps is especially large because Yst increases rapidly with ka . Therefore,

calculated values of R ps were chosen for bandwidths 1  ka  10 rather than 0  ka  10 . As shown in Figure 8(a), the curve of the Gaussian standing waves has a

greater value than that of the Gaussian progressive wave when ka  1 , and therefore R ps is very large at first. The two curves cross at around ka  1.5 , meaning that

R ps  1 at the same point in figure 8(b). The discussion is similar to that regarding

Figure 7(a); successive resonances appear at the same positions which are related to the vibrational modes of the sphere, the zero points of R ps appear when Yst  0 as shown in Figure 7(a).

3.6. Different spheres in water in a Gaussian standing wave Based on Eq. (42), Fig. 9 shows the radiation force function Yst for two different spheres in water in Gaussian   

 

  

h0 a  . Here, ka  2 and W0  3 .

Obviously, the radiation force function Yst for both spheres resembles sinusoidal waves, which means that the sphere will be pushed or pulled to the node or the antinode by the standing wave. Furthermore, because of the different vibrational modes, Yst for the PMMA sphere is much larger than Yst for the stainless steel sphere when ka  2 . The magnitude and position of the zero points
Yst  0 of the










acoustic radiation force function on the different spheres are different. These observations show the possibility of manipulating cells differentially according to their acoustic properties in Gaussian standing waves.

4.

Conclusion

Based on the finite series method, Gaussian standing waves can be remodeled and
an exact solution of the Helmholtz equation can be obtained. Then the acoustic radiation force function can be expressed for spherical particles in fluid by a Gaussian standing or quasi-standing acoustic field for calculating the beam coefficient. Illustrative numerical results for a Gaussian standing waves incident upon the rigid, fluid and elastic spheres immersed in non-viscous water were obtained and discussed. Results for standing and quasi-standing waves show the possibility of using Gaussian standing acoustic waves to manipulate spheres with known mechanical and acoustical properties; viscoelasticity has also been shown to decrease the radiation force amplitude. This paper has also discussed the influence on the radiation force function of the width of a Gaussian standing wave. Furthermore, the acoustic radiation force function in a Gaussian quasi-standing field shows the connection between the acoustic radiation force function in a Gaussian standing field and the acoustic radiation force function in a Gaussian progressive wave. Moreover, the different acoustical properties of the spheres change the magnitude of the forces, as well as the position of the potential well, which means that particles can be chosen and manipulated in distinct ways using Gaussian standing waves. This condition provides an impetus for further










design work on acoustic tweezers operating with standing or quasi-standing Gaussian acoustic waves. A cknowledgement The authors would like to acknowledge the National Basic Research Program of China (No.2012CB921504, No.2011CB707902),financial support of the National Natural Science Foundation of China (No. 11274166, No.61571222), fundamental Research Funds for the Central Universities (No. 020414380001), State Key Laboratory of Acoustics, Chinese Academy of Sciences(No.SKLOA201401), the priority academic program development of Jiangsu Higher Education Institutions and SRF for ROCS, SEM and project of Interdisciplinary Center of Nanjing University. References [1] L.V. King, On the acoustic radiation pressure on sphere, Proc. R. Soc. London Ser. A 147 (1935) 212-240. [2] J. R. Wu, Acoustical tweezers, J. Acoust. Soc. Am. 89 (1991) 2140
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F igure captions FIG. 1 The diagram of a sphere in Gaussian standing waves FIG. 2 The diagram of a sphere in Gaussian quasi-standing waves FIG. 3 The diagram of a sphere deviates from the center of Gaussian standing waves FIG. 4 The values Yst for a water sphere in air (or a rigid sphere in water) in Gaussian standing waves FIG. 5(a) The values of Yst for oleic acid sphere in water in Gaussian standing waves FIG. 5(b) The values of Yst for animal oil sphere in water in Gaussian standing waves FIG. 6 The values of Yst for PMMA sphere in water in Gaussian standing waves FIG. 7(a) The values of Yst for stainless steel sphere in water in Gaussian standing waves FIG. 7(b) The values of Yst for brass sphere in water in Gaussian standing waves FIG. 8 (a) The values of Yqst for stainless steel sphere in water in Gaussian quasi-standing waves FIG. 8 (b) The values of R ps (Yst Y p ) for a stainless steel sphere in water FIG. 9 Acoustic radiation force function vs. sphere's location










Gn
n


W0



W0
2


W0
3


1










2










3

 





 





 





 





 


 








 


 


 





 
















 


 













 











 











 











 











 











 









































TABLE I. The convergent of the beam coefficients for different beam waists.
































Mass density Material


10 kg 3

Air

m3 

Compressional sound velocity (m/s)

Shear sound velocity

Normalized longitudinal

Normalized shear

(m s )

absorption  1

absorption  2

0.00123

340

-

-

-

Water

1

1500

-

-

-

Oleic acid

0.938

1450

-

-

-

Animal oil

0.8

1400

-

-

-

1.19

2680

1380

0.0119

0.0257

Stainless steel

7.9

5240

2978

-

-

Brass

8.1

3830

2050

-

-

Polymethylmethacrylate

TABLE II. Acoustical parameters of different materials







Figure 1 Click here to download high resolution image

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Figure 4 Click here to download high resolution image

Figure 5a Click here to download high resolution image

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Figure 6 Click here to download high resolution image

Figure 7a Click here to download high resolution image

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Figure 8a Click here to download high resolution image

Figure 8b Click here to download high resolution image

Figure 9 Click here to download high resolution image