Resonance scattering characteristics of double-layer spherical particles

Resonance scattering characteristics of double-layer spherical particles

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Particuology 10 (2012) 117–126

Contents lists available at SciVerse ScienceDirect

Particuology journal homepage: www.elsevier.com/locate/partic

Resonance scattering characteristics of double-layer spherical particles Xuejin Dong, Mingxu Su ∗ , Xiaoshu Cai Institute of Particle and Two-Phase Flow Measurement, University of Shanghai for Science and Technology, Shanghai 200093, China

a r t i c l e

i n f o

Article history: Received 29 September 2009 Received in revised form 9 July 2011 Accepted 25 August 2011 Keywords: Ultrasonic Resonance scattering Double-layer Polydisperse Visco-elastic

a b s t r a c t Based on the principle of ultrasonic resonance scattering, sound-scattering characteristics of double-layer spherical particles in water were numerically studied in this paper. By solving the equations of the scattering matrix, the scattering coefficient determined by the boundary conditions can be obtained, thus the expression for the sound-scattering function of a single double-layer spherical particle can be derived. To describe the resonance scattering characteristics of a single particle, the reduced scattering cross section and reduced extinction cross section curves were found through numerical calculation. Similarly, the numerically calculated sound attenuation coefficient curves were used to depict the resonance scattering characteristics of monodisperse and polydisperse particles. The results of numerical calculation showed that, for monodisperse particles, the strength of the resonance was mainly related to the particle size and the total number of particles; while for polydisperse particles, it was primarily affected by the particle size, the coverage of the particle size distribution and the particle concentration. © 2011 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

1. Introduction Double-layer spherical particles are encountered in various applications (Wang, 2000), such as in pharmaceutical industry, chemical industry, biological and other scientific research. Measurement of the concentration and the particle size distribution of double-layer particles is, therefore, becoming more and more important. Ultrasound is of advantages on the measurement of particles with its strong penetration capability, non-contact measurement and the ability to measure particles of a wide range of sizes. For instance, ultrasound contrast agents (UCA) have been used to aid medical diagnoses by enhancing the ultrasonic backscattering signal due to size-related acoustic resonance scattering of the double-layer bubbles. Therefore, the study on the acoustic resonance scattering properties of double-layer particles is of particular importance. According to the literature, resonance scattering theory (Ayres & Gaunaurd, 1987; Choi, 1997; Gaunaurd & Uberall, 1978) has mostly focused on the resonance spectrum of a target, with some study concerning the dynamics of acoustic resonance scattering (Uberall et al., 1979). Resonance scattering theory (RST) was applied to the problem of sound scattering from an elastic transversely isotropic solid sphere suspended in an ideal acoustic fluid medium; in this case, the target’s spectrum of resonances was obtained by

∗ Corresponding author. E-mail address: [email protected] (M. Su).

subtracting the spectrum of a rigid background from the relevant modal backscattering form functions (Gaunaurd & Uberall, 1978). The relationship between the singularity expansion method (SEM) and RST was theoretically explained by Tang (1991). Taking acoustic scattering from a solid elastic cylinder and from a sphere immersed in water as examples, he proved that RST can be directly derived from the SEM. The analogue of this aforementioned result is not available for measurements of the particle size distribution because the relationship between the particle distribution and the spectrum of resonance scattering cannot be derived directly. In another study, based on the variation of the acoustic target strength of yellowfin tuna as their swimbladder volumes vary at depth for low-frequency acoustic detection systems, Schaefer and Oliver (2000) established a predictive model that can be used to estimate yellowfin tuna swimbladder resonance frequencies for fish lengths and fish depths. Recently, other researchers also have focused on the resonance characterization of swimbladders from the perspectives of theoretical predictor and experimental measurement (Foote, Francis, & Atkins, 2007; Francis & Foote, 2003) and, in doing so, have developed a methodology which can detect the size and position of fish in the ocean. These methods, however, mainly studied on large double-layer targets, and the resonance frequencies were very low, so they are not suited for measurement of small particles. Fortunately, another method of characterizing resonance scattering has been developed, which is directly connected to particle radii and particle concentrations. The effective sphere theory was first advanced by Gaunaurd and Uberall (1982, 1983). In this theory, they revealed the relationship between the

1674-2001/$ – see front matter © 2011 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

doi:10.1016/j.partic.2011.08.004

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Nomenclature a outer radius of the shell (␮m) an , bn , cn , dn , en , fn unknown scattering coefficients A undetermined coefficient inner radius of the shell (␮m) b c sound speed (m/s) f scattering function (2) spherical Hankel function of the second kind h2 jn spherical Bessel function of the first kind k wave number shape constraint parameter  n number density of the particles N0 total number of particles n0 total number of particles within a certain range Pn Legendre polynomial r particle size (␮m) r0 mean radius in Eq. (20) (␮m) r0 characteristic size in Eq. (22) (␮m) Greek letters  density (kg/m3 ) e real parts of the elastic Lamé constant (Pa) real parts of the visco-elastic Lamé constant (Pa s) v e imaginary parts of the elastic Lamé constant (Pa) v imaginary parts of the visco-elastic Lamé constant (Pa s)  standard deviation e extinction cross section s scattering cross section absorption cross section a ˛ attenuation coefficient (dB/m) ω angular frequency of the incident wave ϕ concentration of particles ˚ velocity potential amplitude of the incident wave ˚0 azimuthal angle

effective velocity and the particle concentration. Roy, Carey, Nicholas, Schindall, and Crum (1992) experimentally showed that the scattering characteristics of a bubble cloud are not determined by the bubble size and number density but by the free-gas volume fraction and the length scale of the cloud. For the model of a single double-layer particle, Church (1995) derived a Rayleigh–Plessetlike equation incorporating the effects of an elastic solid layer that separates the gas from the bulk Newtonian liquid. The analytical expressions for the resonance frequencies and the first and second harmonic scattering cross sections were presented in a closed form by Church. For the model of Ye (1996), the scattering coefficient determined by boundary conditions was obtained by solving the equation of the scattering matrix, and thus the expression for the sound-scattering function of a single double-spherical particle was derived. Through numerical study of a shelled bubble, the relationship between the radius of the shelled bubble and the resonance frequencies was presented by Ye. Recently, according to the acoustic scattering of UCA, Allen, Kruse, and Ferrara (2001) showed that contrast agents also support shell resonance responses in addition to the monopole response, and that the inclusion of damping affects the lower-frequency dipole peaks but is less important for responses occurring above approximately 30 MHz. The resonance of a bubble with a visco-elastic shell is affected by the visco-elastic Lamé constant; the dispersion phenomenon of an effective sound speed and the resonance phenomenon of an effective attenuation

Fig. 1. Physics model of acoustic scattering for a double-layer spherical particle.

coefficient for shelled bubbles of the same radius were presented by Chen and Zhu (2005). A finite-element model of wave propagation using COMSOL Multiphysics (Falou, Kumaradas, & Kolios, 2006) has been developed to solve the problem of ultrasound scattering by spherical structures. This model can be used to predict ultrasound backscattering from cells for ultrasound tissue characterization as well as scattering from UCAs. Through experimental studies of ultrasound contrast agents, Pauzin, Mensah, and Lefebvre (2007) created a numerical model of UCAs that can be used to predict the resonance of a simple gas core micro-bubble. Resonance scattering properties for a single double-layer particle and monodisperse double-layer particles have been the main focus of research, while research on the resonance scattering properties of polydisperse double-layer particles is rarely found. In actual measurements (Su, Cai, Xu, Zhang, & Zhao, 2004), however, most of the particles studied were polydisperse particles (Wang, 2000). Therefore, the study of polydisperse particles is of great significance for comparison to actual measurement. The main focus of the present paper is a study of the resonance scattering characteristics of polydisperse particles, based on the previous physical models for the acoustic resonance scattering properties of a single double-layer particle. The relationships among the resonance properties of the sound attenuation coefficient, the particle size, the coverage of the particle size distribution and the total particle number are discussed and analyzed in detail for polydisperse particles. Additionally, in view of the fact that most particle size distributions for polydisperse particles follow the Rosin–Rammler distribution function and the Gaussian distribution function (Barth, 1984), these two functions are used as particle size distributions in numerical calculations of the acoustic resonance scattering properties of particles.

2. Principles The physics model of acoustic scattering for a double-layer spherical particle is presented in Fig. 1, where a and b are the outer and inner radii of the shell, respectively. The inner medium and surrounding medium (water) are separated by the shell. Now, considering a plane wave which spreads along the direction of the X axis, when the wave encounters the particle, the phenomenon of sound scattering around the particle will occur.

X. Dong et al. / Particuology 10 (2012) 117–126

2.1. Sound-scattering model of a single particle

the interfaces. Thus, the equations of boundary conditions can be formulated as

For a single double-layer spherical particle, the incident plane wave in the direction of can be expressed in spherical coordinates as ˚i = ˚0 e−ik1 z+iωt = ˚0

∞ 

n

(2n + 1)(−i) jn (k1 r)Pn (cos )ejωt ,

(1)

n=0

where ˚ represents the velocity potential. The pressure is related to the velocity potential by p = ∂˚/∂t. Here we use the following notation: jn is a spherical Bessel function of the first kind; ˚0 and ω refer to the amplitude and the angular frequency of the incident wave, respectively; Pn is a Legendre polynomial; k1 is the wave number in the surrounding medium and equals ω/c1 , with c1 being the sound speed in the surrounding medium; and 1 , 2 , 3 indicate the densities of the surrounding medium, the shell material, and the medium inside the shell, respectively. The velocity potential of the scattered waves can be formulated as ˚s = ˚0

∞ 

n

(2)

(2n + 1)(−i) an h2 (k1 r)Pn (cos )eiωt ,

(2)

n=0

where an are the unknown scattering coefficients to be determined (2) by boundary conditions, and h2 is a spherical Hankel function of the second kind. Similarly, the scalar and vector velocity potentials of the shell can be expressed as ˚2 = ˚0

∞ 

n

(2n + 1)(−i) [bn jn (kc r) + cn nn (kc r)]eiωt Pn (cos ),

2 3 2 u2r = u3r , rr = rr , r = 0 (r = b),

(8a)

1 4 2 2 u1r + u4r = u2r , rr + rr = rr , r = 0 (r = a),

(8b)

where the superscripts indicate that the values correspond to the wave in different regions as follows: 1, incident wave; 2, the wave in the shell; 3, the wave in the inner medium; and 4, scattered wave. Using the above boundary conditions, the scattering coefficient an is determined to be an = −

n

(2n + 1)(−i) [dn jn (ks r) + en nn (ks r)]

n=0

dPn (cos ) iωt e , (5) d

a62

,

(9)



a23 a33 a43 a53 a63

a24 a34 a44 a54 a64

a25 a35 a45 a55 a65

0  0  a46  a56   0

a13 a33 a43 a53 a63

a14 a34 a44 a54 a64

a15 a35 a45 a55 a65

0  0  a46  a56   0

,

(10)

where the coefficients aij are given in the literature (Ye, 1996). Substituting Eq. (10) into Eq. (2), we can solve for the scattered wave. At large distances (i.e., k1 r  1), we can derive the scattering function to be f ( ) =

1 n (2n + 1)(−i) an ei(n+1) /2 Pn (cos ), k1

(11)

ω2 cc2 (1 − iωM)

,

ks2 =

ω2 cs2 (1 − iωN)

,

(6)

where M = v + 2v /e + 2e , N = v /e , cc2 = e + 2e / and cs2 = e /. e and e are the real and imaginary parts of the elastic Lamé constant, respectively. v and v are the real and imaginary parts of the visco-elastic Lamé constant, respectively. The velocity potential in the inner medium can be described as



where the scattering function is defined, as usual, from ˚s e−ik1 r−iωt , ≡ f ( ) r r→∞ ˚ lim

(12)

and the following asymptotic expansion is used:

where kc and ks are the wavenumbers of the compression and shear waves in the shell, respectively; nn is a spherical Bessel function of the second kind; and the unknown coefficients bn , cn , dn , and en are to be determined by the boundary conditions. The wave numbers of the compression and shear waves in the shell can be defined as



˚3 = ˚0

(2)



(3) (4)



kc2 =

(2)

Fn hn (k1 a) − k1 ah n (k1 a)

n=0

Ar = A = 0,



Fn jn (k1 a) − k1 ajn (k1 a)

  a22 a  32  a42   a52  a62 Fn = −1   a12 a  32  a42   a52 

n=0

A˚ = ˚0

119

n

(2n + 1)(−i) fn jn (k3 r)Pn (cos )eiωt ,

(2)

hn (x) →

1 −ix+i(n+1) /2 e x

(x  1).

(13)

Considering the symmetry of spherical particles, the scattering function should be independent of , so the differential scattering cross section is defined as



2

( ) = f ( ) .

(14)

When the inner radius of a particle is r = b, the dimensionless reduced scattering cross section is defined as r ( ) =

( ) . b2

(15)

2.2. Scattering function of a group of double-layer spherical particles

(7)

n=0

where fn is the coefficient that can be determined by the boundary conditions and k3 is the wave number of the inner medium, equal to ω/c3 , with c3 being the sound speed in the inner medium. At the interfaces r = b and r = a, the above unknown coefficients an , bn , cn , dn , en and fn can be determined by the following boundary conditions: (1) the normal components of the sound speed must be continuous at the interfaces; (2) the normal components of the shear stress must be continuous at the interfaces; and (3) the tangential components of the shear stress must be continuous at

Once the sound-scattering function of a single double-layer spherical particle has been obtained, the sound-scattering function of a group of double-layer spherical particles can be considered. When sound waves propagate through a medium consisting of a number of particles, the acoustic energy will be weakened by the particles in the medium because of acoustic scattering and absorption. This phenomenon is described by the extinction cross section ( e ), which is the sum of the scattering cross section ( s ) and the absorption cross section ( a ): e = s + a .

(16)

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X. Dong et al. / Particuology 10 (2012) 117–126

For a volume containing many particles, the extinction can be characterized effectively by the attenuation coefficient, which describes the sound attenuation per unit distance. The attenuation coefficient (˛) is related to the extinction cross section as follows (Clay & Medwin, 1977): ˛ = 4.34ne ,

e = s + a = −

4 Im f ( = 0), k1

(18)

where there is a negative sign because of the complex conjugate operation on the propagating waves. Clearly,  e and f are functions of several quantities, e.g., the frequency and the size of particles. For polydisperse particles, we can formulate the expression for the extinction cross section per unit volume se as follows: r2

se = r1

4. Numerical results and discussions

(17)

where n is the number density of the particles. The forward scattering theorem states that the extinction cross section is proportional to the imaginary part of the scattering function in the forward direction (Ishimaru, 1978):



where r0 is the characteristic size,  is the shape constraint parameter, r1 is the lower limit of the particle size distribution and r2 is the upper limit of the particle size distribution.



4 − Im k1



1 n (2n + 1)(−i) an ei(n+1) /2 n(r)dr, k1 ∞

(19)

n=0

where n(r)dr = (number of particles of radius between r and r + d)/unit volume. 3. Particle size distribution functions of polydisperse particles Some studies have shown that, for a group of particles formed by the break-up of a larger object, the particle size distribution essentially follows the Rosin–Rammler distribution function, while for particles formed in other ways, the particle size distribution at least partially follows the Gaussian distribution function. For this reason, the Rosin–Rammler function and the Gaussian function were assumed to represent the distribution of polydisperse particles in the numerical calculation of the acoustic resonance scattering properties of these particles. 3.1. Gaussian distribution function

The numerical computation was performed using the MATLAB software package. As a validation of the programs and equations, the results for the reduced scattering cross section of a single particle (shown in Fig. 2) were obtained from Eq. (15) and compared with the results in the paper of Chen and Zhu (2005). 4.1. Physical properties of particles for numerical calculation Now we consider a double-layer spherical particle with an inner medium of air shelled by a visco-elastic layer in 15 nm thickness. Unless otherwise specified, the physical properties given in Table 1 are used for all calculations. 4.2. Resonance scattering characteristics of a double-layer spherical particle The reduced backscattering cross section as a function of driving frequency of a double-layer spherical particle with a visco-elastic Lamé constant is plotted for four different inner radii in Fig. 2, and the reduced backscattering cross section as a function of driving frequency of a double-layer spherical particle with no visco-elastic Lamé constant is plotted for same inner radii in Fig. 3. Four inner radii are 3.3, 2.75, 2.05 and 1.7 ␮m, respectively. The following features of these response curves can be seen from Figs. 2 and 3: (1) All the reduced backscattering cross sections have resonance features, and the resonance peak for a given radius corresponds with a particular driving frequency, known as the resonance frequency. (2) The resonance frequency increases as the particle radius decreases; this phenomenon can also be clearly seen in Table 2. For example, a particle radius of 3.3 ␮m corresponds to a resonance frequency of 3.5 MHz, and a particle radius of 1.7 ␮m corresponds to a resonance frequency of 9.43 MHz. (3) The amplitude of the resonance peak is greatly affected by the particle radius and the visco-elastic Lamé constant. Due to the damping function of the visco-elastic

The Gaussian distribution function is



A (r − r0 )2 √ exp − 2 2  2

n(r) =



,

(20)

and the expression for the total number of particles within a certain range can be written as



r2

n0 =

n(r)dr,

(21)

r1

where A is the undetermined coefficient, r0 is the mean radius,  is the standard deviation that presents the width of the size distribution, r1 is the lower limit of the particle size distribution and r2 is the upper limit of the particle size distribution. 3.2. Rosin–Rammler (R–R) distribution function The Rosin–Rammler (R–R) distribution function is n(r) =

3 4 r 3

−1  r r0

r0

exp



r −

r0

,

(22)

and the concentration of particles can be written as 4 ϕ= 3



r2

n(r)r 2 dr, r1

(23)

Fig. 2. Reduced backscattering cross section as a function of driving frequency for different inner radii of a particle with a visco-elastic Lamé constant.

X. Dong et al. / Particuology 10 (2012) 117–126

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Table 1 Physical properties of the particles and surrounding medium used for calculation.

Water Visco-elastic layer Air

 (kg/m3 )

c (m/s)

1000.0 1100.0 1.23

1500.0

e (Pa)

e (Pa)

v (Pa s)

v (Pa s)

88.84 × 106

6.1 × 1010

0.5

50

340.0

Table 2 Particle radii and corresponding resonance frequencies (data in parentheses are the results of Chen). Particle radius (␮m) Resonance frequency (MHz) Particle radius (␮m) Resonance frequency (MHz)

1.2 16.4 3.3 3.5(3.5)

1.5 11.4 3.6 3.05

1.7 9.43(9.43) 4.0 2.65

Lamé constant, the amplitude of the resonance peak is reduced significantly, and it also diminishes with the decrease of the radius of particles. (4) As the particle radius increases, the resonance width of the reduced backscattering cross section apparently decreases. We observed that the general resonance features in the sound scattering for a double-layer spherical particle and the resonance positions (resonance frequencies) are almost the same as those of Chen and Zhu (2005). When the visco-elastic Lamé constant are not considered, the general resonance features in Fig. 3 are almost the same as the results presented in the paper of Ye (1996). In Fig. 4, the reduced backscattering cross section as a function of driving frequency for a shell layer with an inner radius of 3.3 ␮m is plotted for different visco-elastic Lamé constants 0.5, 1.2, 2.2 and 4.0 Pa s, respectively. We can observe the following traits from response curves in Fig. 4: (1) The reduced cross sections all exhibit resonance behavior at the same resonance frequency. (2) The amplitude of the resonance peak sharply decreases as the viscoelastic Lamé constant increases. This figure clearly shows that the visco-elastic Lamé constant of the layer affects the strength of the resonance. In Fig. 5, the reduced extinction cross section as a function of driving frequency is plotted for inner radii as 3.3, 2.75, 2.05 and 1.7 ␮m, respectively. Comparing this figure with Fig. 2, we can see that, for the most part, the response curves in both figures share the same traits. The main distinction between the two graphs is

Fig. 3. Reduced backscattering cross section as a function of driving frequency for different inner radii of a particle with no visco-elastic Lamé constant.

2.0 7.38 4.5 2.22

2.05 7.13(7.18) 5.0 1.92

2.5 5.22 6.0 1.48

2.75 4.63(4.65) 7.0 1.18

Fig. 4. Reduced backscattering cross section as a function of driving frequency for different visco-elastic Lamé constants in the shell layer with an inner radius of 3.3 ␮m.

Fig. 5. Reduced extinction cross section as a function of driving frequency for different inner radii.

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X. Dong et al. / Particuology 10 (2012) 117–126

Fig. 6. Sound attenuation coefficient as a function of driving frequency for different inner radii of monodisperse particles. Fig. 7. Sound attenuation coefficient as a function of driving frequencies for different total particle numbers of monodisperse particles with inner radii of 3.3 ␮m.

that the amplitude of the resonance peak for each curve in Fig. 2 is significantly lower than that for the corresponding curve in Fig. 5. For example, the resonance peak amplitudes of the particle radius of 3.3 ␮m correspond to 78 in Fig. 2 and 1300 in Fig. 5. This difference is expected because the extinction cross section is the sum of the scattering cross section and the absorption cross section. 4.3. Resonance scattering characteristics of monodisperse particles Fig. 6 presents the sound attenuation coefficient as a function of driving frequency of monodisperse double-layer spherical particles for four inner radii of 3.3, 2.75, 2.05 and 1.7 ␮m, and the total number of monodisperse particles in each case is 1.3 × 109 . The following common features of these response curves can be observed from Fig. 6: (1) The amplitude of the sound attenuation coefficient is almost equal to zero except at the resonance. (2) The resonance frequency increases as the radius of the monodisperse particles decreases. (3) The amplitude of the resonance peak decreases as the radius of the monodisperse particles decreases. Comparing this graph with Fig. 5, the reduced extinction cross section curves and the sound attenuation coefficient curves exhibit nearly the same traits because the attenuation coefficient is derived from the extinction cross section (the relationship between these two is shown by Eq. (17)). In other words, when complex scattering and multiple scattering are not considered, the sound attenuation of the particles is equivalent to the sum of the sound attenuation of each particle. In Fig. 7, the sound attenuation coefficient as a function of driving frequency for various total particle numbers of monodisperse particles with inner radius of 3.3 ␮m is presented. The different total particle numbers (N0 ) are 1.3 × 1010 , 1.3 × 109 , 1.3 × 108 and 1.3 × 107 . The following traits of the response curves can be found from this figure: the resonance peak amplitude sharply decreases as the total particle number decreases, while the resonance frequency does not vary with the total particle number. We can therefore reach the following conclusion: for monodisperse particles with a given radius, the resonance strength of the sound attenuation coefficient is directly related to the total particle number.

4.4. Resonance scattering characteristics of polydisperse particles The normalized Gaussian particle size distribution, which can be used to describe some groups of polydisperse double-layer spherical particles, is shown for several different standard deviation in Fig. 8, where r0 = 3.3 ␮m, r1 = 0.4 ␮m and r2 = 6.0 ␮m, and the standard deviations are  = 0.6 × 10−6 ,  = 1.3 × 10−6 , and  = 1.8 × 10−6 , respectively. As in the case of the monodisperse particles investigated in the previous section, the total number of particles for each curve is 1.3 × 109 . Under the conditions of the same total particle number and the same mean radius (r0 = 3.3 ␮m), it can be easily seen that the coverage of the particle size distribution differs among the three curves; namely, the coverage becomes wider as the distributed parameter increases in this figure. In Fig. 9, the sound attenuation coefficient is plotted as a function of driving frequency for double-layer spherical particles with the Gaussian particle size distribution in different distributed parameters. The following features can be observed from this graph: (1) As the standard deviation increases, the amplitude of the resonance peak decreases. The reason for this result is that percentage

Fig. 8. Normalized Gaussian distribution functions with different distributed parameters.

X. Dong et al. / Particuology 10 (2012) 117–126

123

Fig. 9. Sound attenuation coefficient as a function of driving frequency for polydisperse particles with the Gaussian distribution function in different distributed parameters.

Fig. 11. Sound attenuation coefficient as a function of driving frequency for polydisperse particles following a Gaussian particle size distribution with mean radii of r0 = 1.2 ␮m and r0 = 2.3 ␮m.

of particles in the resonance frequency extends with the decrease of the standard deviation. (2) The resonance frequency slowly shifts toward lower frequencies as the standard deviation increases. (3) Comparing this figure with those for monodisperse particles, the amplitudes of the resonance peaks in the polydisperse case are significantly lower for the same total particle numbers. In summary, it can be concluded that the resonance strength of the sound attenuation coefficient is influenced by the coverage of the particle size distribution for polydisperse particles. In Fig. 10, the normalized Gaussian particle size distribution is presented for various mean radii, where  = 1.3 × 10−6 , r1 = 0.4 ␮m, r2 = 6.0 ␮m, the different mean radii are r0 = 1.2, 2.3, 3.3, and 5.0 ␮m, and the total number of particles in each case is 1.3 × 109 . It can be clearly seen from this graph that the distribution of particle numbers is markedly different among the four curves under the condition of uniform coverage of the particle size distribution and identical total particle number. For instance, in Fig. 10, the highest particle number occurs at a radius of 1.2 ␮m for the leftmost curve and at a radius of 5.0 ␮m for the rightmost curve.

The sound attenuation coefficient as a function of driving frequency for polydisperse double-layer spherical particles following a Gaussian particle size distribution is plotted for different mean radii in Figs. 11 and 12. These two graphs clearly show that the resonance width of the sound attenuation coefficient is markedly augmented as the mean radius decreases, while the amplitude of the resonance peak is lessened, for the same coverage of the particle size distribution and total particle number (1.3 × 109 ). Comparing these graphs with Fig. 6, the resonance frequency of the sound attenuation coefficient in the polydisperse case clearly differs from that of the monodisperse case corresponding to the most-abundant particle size. For the third curve in Fig. 10, the largest particle number is at a particle radius of 3.3 ␮m, and the resonance frequency of the corresponding sound attenuation coefficient is at about 3.0 MHz in Fig. 12, which is different from the resonance frequency of 3.5 MHz in Fig. 7. Thus, we can infer that the

Fig. 10. Normalized Gaussian particle size distribution of polydisperse particles for different mean radii.

Fig. 12. Sound attenuation coefficient as a function of driving frequency for polydisperse particles following a Gaussian particle size distribution with mean radii of r0 = 3.3 ␮m and r0 = 5.0 ␮m.

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Fig. 13. Normalized Gaussian particle size distribution of polydisperse particles for various particle concentrations.

resonance peak of the sound attenuation coefficient was not produced by the particles with radius of 3.3 ␮m, but by the particles with radius larger than 3.3 ␮m. This phenomenon can also be observed for the other cases in these two figures. Finally, we can conclude that, under the conditions of the same total particle number and the same coverage of the particle size distribution, the resonance strength of the sound attenuation coefficient is influenced not only by the number of particles at a certain radius, but also by the size of particles. Fig. 13 presents the normalized Gaussian particle size distribution for various particle concentrations of ϕ = 5%, ϕ = 10% and ϕ = 15%. When the range of particle size and the mean radius are held constant ( = 1.3 × 10−6 , r1 = 0.4 ␮m, r2 = 6.0 ␮m and r0 = 3.3 ␮m), it is observed that the particle numbers at all radii markedly increase as the particle concentration increases. In Fig. 14, the sound attenuation coefficient as a function of driving frequency is plotted for these distributions of polydisperse particles. This figure indicates that, as the particle concentration increases, the

Fig. 14. Sound attenuation coefficient as a function of driving frequency for polydisperse particles following a Gaussian particle size distribution with three different particle concentrations (mean radii r0 = 3.3 ␮m).

Fig. 15. Relationship between the particle concentration and the amplitude of the resonance peak of the sound attenuation coefficient for polydisperse particles following a Gaussian particle size distribution with different distributed parameters.

amplitude of the resonance peak of the sound attenuation coefficient increases, while the resonance frequency and the resonance width of the sound attenuation coefficient are not changed. Fig. 15 presents the relationship between the particle concentration and the amplitude of the resonance peak of the sound attenuation coefficient for polydisperse particles following a Gaussian particle size distribution for different standard deviations of  = 0.8 × 10−6 ,  = 1.0 × 10−6 ,  = 1.3 × 10−6 and  = 1.5 × 10−6 . In the figure, there is a linear relationship between the particle concentration and the resonance peak amplitude, and the resonance peak amplitude increases significantly as the distributed parameter decreases. The normalized R–R distribution, which can often be used to describe a non-Gaussian particle size distribution of polydisperse double-layer spherical particles, is presented for various shape constraint parameters () of  = 5,  = 8 and  = 13 in Fig. 16, where the size parameters are r0 = 3.3 ␮m, r1 = 0.4 ␮m and r2 = 6.0 ␮m. This figure shows that, as the shape constraint parameter decreases,

Fig. 16. Normalized R–R particle size distribution of polydisperse particles for various shape constraint parameters.

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Fig. 18. Normalized double-peaked particle size distribution of polydisperse particles. Fig. 17. Sound attenuation coefficient of polydisperse particles following an R–R particle size distribution as a function of driving frequencies for various shape constraint parameters.

large particles (greater than 3.3 ␮m) markedly decreases and smaller ones clearly increases, In Fig. 17, the sound attenuation coefficient of polydisperse particles following a R–R particle size distribution as a function of driving frequency is presented for various shape constraint parameters. The total number of polydisperse particles in each case is 1.3 × 109 . From Fig. 17, the following major traits of these response curves can be observed: (1) All of the sound attenuation coefficient curves exhibit resonance peaks, and the driving frequencies of these peaks are almost equal. (2) The amplitude of the resonance peak of the attenuation coefficient decreases as the shape constraint parameter decreases. For example, the amplitude of the resonance peak for  = 13 is about three times as large as the one for  = 5. This phenomenon is related to the resonance scattering properties of a double-layer spherical particle, and it can be inferred that the discrepancy is due to the fact that the  = 13 distribution contains a larger number of particles of the mostabundant size than does the  = 5 distribution. At the same time, the amplitude of the sound attenuation coefficient when  = 5 is larger than that when  = 13 for the frequencies from 5 to 14 MHz, which can be explained in the same way: the  = 5 distribution contains more particles of smaller radius than does the  = 13 distribution. The amplitude of the sound attenuation coefficient is obviously affected by the particle numbers at a certain radius (the particle concentration) for polydisperse particles. In Fig. 18, we present a distribution of polydisperse doublelayer spherical particles that is qualitatively different from those presented previously. The major difference is that this distribution contains two distinct particle-number peaks at radii of 2.0 and 3.6 ␮m, while each of the previous distributions contained only one. The sound attenuation coefficient as a function of driving frequency for this double-peaked distribution of polydisperse double-layer spherical particles is plotted in Fig. 19. There is only one distinct peak in the sound attenuation coefficient. Although the particle distribution of Fig. 18 (ϕ = 15%) is significantly different from it in Fig. 13, the sound attenuation coefficient curve of Fig. 19 is more or less similar to the related curve in Fig. 14; the main difference between these two curves is that the amplitude of the sound attenuation coefficient in Fig. 19 is obviously larger than it in Fig. 14 at frequencies ranging from 6 to 14 MHz. The reason for

Fig. 19. Sound attenuation coefficient as a function of driving frequency for doublepeaked particle size distribution.

this is that there were many smaller particles (less than 3.3 ␮m) in the former case. It should also be noted that, although the particle number at 2.0 ␮m is greater than the particle number at 3.6 ␮m in Fig. 18, the amplitudes of the sound attenuation coefficient corresponding to 2.0 ␮m were obviously smaller than those corresponding to 3.6 ␮m in Fig. 19. 5. Conclusions For a single double-layer spherical particle, the resonance frequency increases as the particle radius decreases, the amplitude of the resonance peak decreases as the particle radius decreases. The amplitude of the resonance peak decreases as the visco-elastic Lamé constant increases, and the resonance width of the reduced scattering cross section is related to the particle size. The resonance strength of the sound attenuation coefficient for monodisperse double-layer spherical particles is directly related to the total particle number and the particle size.

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For polydisperse double-layer spherical particles, the sound attenuation coefficient curve also exhibits the phenomenon of resonance at a certain driving frequency. The resonance width of the sound attenuation coefficient is mainly related to the particle size and the coverage of the particle size distribution. The amplitude of the resonance peak of the sound attenuation coefficient is obviously affected by the particle size, the coverage of the particle size distribution and the particle concentration. Acknowledgments The authors gratefully acknowledge the support from the National Natural Science Foundation of China (50836003, 51076106, 51176128) and the Project of Shanghai Science and Technology Commission (Grant No.: 10540501000). References Allen, J. S., Kruse, D. E., & Ferrara, K. W. (2001). Shell waves and acoustic scattering from ultrasound contrast agents. IEEE Transactions on Ultrasonics, Ferroelectics, and Frequency Control, 48(2), 409–418. Ayres, M., & Gaunaurd, G. C. (1987). Acoustic resonance scattering by viscoelastic objects. The Journal of the Acoustical Society of America, 81(1), 301–311. Barth, H. G. (1984). Modern methods of particle size analysis. Now York: John Wiley & sons. Chen, J. S., & Zhu, Z. M. (2005). Sound scattering characteristics of bubbles with viscoelastic shells. Acta Acustica, 30(5), 385–392 (in Chinese). Choi, M. S. (1997). New formulation of the resonance scattering theory. The Journal of the Acoustical Society of America, 101(5), 2491–2495. Church, C. C. (1995). The effects of an elastic solid surface layer on the radial pulsations of gas bubbles. The Journal of the Acoustical Society of America, 97(3), 1510–1521. Clay, C. S., & Medwin, H. (1977). Acoustical oceanography: Principles and applications. New York: Wiley-Interscience. Falou, O., Kumaradas, J. C., & Kolios, M. C. (2006). Modeling acoustic wave scattering from cells and microbubbles. In Proceedings of the COMSOL users conference 2006 Boston, USA.

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