Researches have suggested the possibility of contact-free trapping or manipulating tiny sized objects (such as particles or biological cells) in a progressive acoustic wave.^[1–3] Particles suspended in acoustic fields are subjected to time-averaged forces from scattering of the acoustic waves, and these forces are named acoustic radiation force (ARF) theoretically.^[4] The problem of ARF on rigid and elastic particles has attracted much attention.^[5–9] Several theoretical models have been developed to calculate the ARF on objects in different types of acoustical fields.^[10–13] King proposed his formula of ARF on a rigid sphere,^[14] which could help us to estimate the absolute sound intensity by the radiation force method. However, Hasegawa and Yosioka^[15] found that King’s theory deviated from the experimental values of the radiation force for spheres of brass, steel and stainless steel in the range of ka > 2 (a is the radius of the sphere, and k is the wavenumber in the liquid). Further, an improved ARF theory is developed by taking the elasticity of the solid sphere into account, and the results of the evaluation are in good agreement with the measurements.^[15]

The multiple features of the objects were considered in the investigations of acoustic radiation force, such as prolate spheroids, cylindrical particles, soft fluid spheres, and air bubbles.^[16–23] It is important to understand the radiation force related to the nonlinearity of sound propagation, and some potential applications could be extended. Objects could be pushed towards nodal or anti-nodal planes by radiation force in acoustic standing wave,^[16] and negative forces on spheres are found to be correlated with reduced acoustic backscattering.^[13,17] The manipulation of pulsating spherical carrier^[10] has potential applications in drug delivery and the handling of the radiation force has proved feasible for spheres.

The scattering of floating objects submerged in liquid can give much information about the internal structure or compositions of materials, which is related to the multiple interaction between object and the surrounding medium.^[24,25] Usually, the object can be considered as solid particle or shell. For example, bacteria cells or spherical carriers were simplified as “shell model” to investigate the acoustic response.^[12,26] The acoustic radiation pressure on thin shell is different from that on solid sphere. Junger^[27] developed the theory of sound scattering of an acoustic plane wave by thin elastic shell submerged in liquid, and pointed out that the scattering pattern is the result of synthesis of rigid body scattering and elastic radiation scattering. Accordingly, the scattering effect could be enhanced by the sound radiation of the shell undergoing forced vibrations. The scattering of two elastic spherical shells in a boundless acoustic medium has been previously studied,^[28,29] and two physical reasons were proposed to explain the phenomenon that the function of total scattering is not in the form of the sum of the two individuals: multiple interactions between the two shells and the observed position of scattering wave. The first antisymmetric Lamb wave could be activated by the multiple interactions, and the enhancement band of the form function curve was observed.^[28] Therefore, it has been shown that the surface vibration modes play an important role in characterizing the sound backscattering.

While there have existed many publications on the interaction of acoustic waves with objects including researches focusing on arbitrary scattering and radiation force, the acoustic dynamic behavior of thin elastic shell differs considerably from that of solid objects. According to the present sound scattering theories, we will explore the acoustic radiation force on thin elastic shells in an unbounded nonviscous liquid in this investigation. A simple mathematical model of two coupled thin shells is introduced, considering the effects of the surface deformation and mutual interactions of shells. This work expands the theories of acoustic radiation force, and conduces to a better understanding of the acoustic multi-interactions between different particles.

We consider a pair of spherical shells with separation distance D_O1O2 = R₁ + R₂ + d, where R₁ and R₂ are the radii of the two shells, respectively. The two shells are immersed in an unbounded liquid as shown in Fig. 1. The origins O₁ and O₂ of the two spherical coordinate systems ((r_j,θ_j, (j = 1,2)) are located at the centers of the left shell and right shell, respectively. The propagating direction of the incident plane wave is parallel to the z axis, and the wave is reflected and diffracted into and out of the shells. The mass density and sound speed of the fluid within and outside the shell are, respectively, (ρ,c) and (ρ_in,c_in). The mass density of shell is denoted by ρ_s. We treat the acoustic radiation force on spherical shell by taking the thin shell approximation^[27] into account. This problem is based on the following assumptions: (i) the fluid medium satisfies the simplified wave equation; (ii) the material is isotropic, devoid of damping, and the deformation of the shell is subject to Hooke’s law; (iii) the shell thickness is much less than other dimensions.

In the j-th local spherical coordinates (j = 1, 2), the velocity potential of the incident wave and the outgoing waves scattered from the shells can be represented as follows:

$$ \begin{eqnarray}\begin{array}{lll} & & {\varphi }_{I}^{(j)}={\varphi }_{0}\displaystyle \sum _{n=0}^{\infty }(2n+1){{\rm{i}}}^{n}{{\rm{j}}}_{n}(k{r}_{j}){{\rm{P}}}_{n}\cos ({\theta }_{j}),\end{array}\end{eqnarray}$$ (1)

$$ \begin{eqnarray}\begin{array}{lll} & & {\varphi }_{{\rm{s}}}^{(j)}={\varphi }_{0}\displaystyle \sum _{n=0}^{\infty }(2n+1){{\rm{i}}}^{n}{A}_{n}^{(j)}{{\rm{h}}}_{n}^{(2)}(k{r}_{j}){{\rm{P}}}_{n}\cos ({\theta }_{j}),\end{array}\end{eqnarray}$$ (2)

The total field is a result of the superposition of waves scattered on the surfaces of the system. Therefore, velocity potential around the shells in the surrounding liquid can be written as

$$ \begin{eqnarray}{\varphi }_{{\rm{o}}}^{(j)}={\varphi }_{0}\displaystyle \sum _{n=0}^{\infty }{G}_{n}^{(j)}(\omega )(2n+1){{\rm{i}}}^{n}{{\rm{P}}}_{n}(\cos {\theta }_{j}),\end{eqnarray}$$ (3)

$$ \begin{eqnarray}\begin{array}{lll}{G}_{n}^{(j)} & = & \left(1+\displaystyle \sum _{m=0}^{\infty }{Q}_{nm}^{(3-j)}\displaystyle \frac{2m+1}{2n+1}{{\rm{i}}}^{m-n}{A}_{m}^{(3-j)}\right){{\rm{j}}}_{n}(k{r}_{j})\\ & & +{A}_{n}^{(j)}{{\rm{h}}}_{n}^{(2)}(k{r}_{j}),\end{array}\end{eqnarray}$$ (4)

The velocity potentials in the internal fluids of the shells are

$$ \begin{eqnarray}{\varphi }_{{\rm{in}}}^{(j)}={\varphi }_{0}\displaystyle \sum _{n=0}^{\infty }(2n+1){{\rm{i}}}^{n}{B}_{n}^{(j)}{{\rm{j}}}_{n}({k}_{{\rm{in}}}{r}_{j}){{\rm{P}}}_{n}\cos ({\theta }_{j}),\end{eqnarray}$$ (5)

For the linearity problems, one has a..jn=−ω2ajn and b..jn=−ω2bjn, and the solutions of a_jn and b_jn can be expressed as follows:

$$ \begin{eqnarray}\begin{array}{lll} & & {a}_{jn}=-{\varphi }_{0}{\bar{\varphi }}_{jn}/{Z}_{jn},\\ & & \end{array}\end{eqnarray}$$ (11)

$$ \begin{eqnarray}\begin{array}{lll} & & {b}_{jn}=\displaystyle \frac{E}{1-\nu }{\left[\displaystyle \frac{E}{1-\nu }\displaystyle \frac{n(n+1)-(1-\nu )}{1+\nu }-{\rho }_{{\rm{s}}}{c}^{2}{x}_{j}^{2}\right]}^{-1}{a}_{jn},\,\end{array}\end{eqnarray}$$ (12)

$$ \begin{eqnarray}{Z}_{jn}=2{\rm{i}}\displaystyle \frac{{h}_{j}}{{R}_{j}}\left[{\rho }_{{\rm{s}}}c{x}_{j}-\displaystyle \frac{E}{c{x}_{j}}\displaystyle \frac{(n-1)(n+2)E-2{\rho }_{{\rm{s}}}{c}^{2}{x}_{j}^{2}(1+\nu )}{[n(n+1)-(1-\nu )]E-{\rho }_{{\rm{s}}}{c}^{2}{x}_{j}^{2}(1-{\nu }^{2})}\right].\end{eqnarray}$$ (13)

The vibration of the shell is determined by the inner and outer sound fields, and the relation among An(j), Bn(j), and a_jn can be obtained from the boundary conditions. Accordingly, the continuity of normal velocity at the surface requires that

$$ \begin{eqnarray}{-\partial {\varphi }_{{\rm{o}}}/\partial {r}_{j}|}_{{r}_{j}={R}_{j}}={-\partial {\varphi }_{{\rm{in}}}/\partial {r}_{j}|}_{{r}_{j}={R}_{j}}={\rm{i}}\omega {a}_{jn}.\end{eqnarray}$$ (14)

Substituting Eqs. (3), (5), and (11) into Eq. (14), one obtains

$$ \begin{eqnarray}\begin{array}{lll}{A}_{n}^{(j)} & = & {\alpha }_{n}^{(j)}+{\rm{i}}{\beta }_{n}^{(j)},\end{array}\end{eqnarray}$$ (15)

$$ \begin{eqnarray}\begin{array}{lll}{B}_{n}^{(j)} & = & \displaystyle \frac{{c}_{{\rm{in}}}}{c{{j}^{^{\prime} }}_{n}({x}_{ij})}\left[(1+\displaystyle \sum _{m=0}^{\infty }\displaystyle \frac{2m+1}{2n+1}{Q}_{nm}^{(3-j)}{{\rm{i}}}^{m-n}{A}_{m}^{(3-j)})\right.\\ & & \left.\times {{{\rm{j}}}^{^{\prime} }}_{n}({x}_{j})+{A}_{n}^{(j)}{{{\rm{h}}}^{^{\prime} }}_{n}^{(2)}({x}_{j})\right],\end{array}\end{eqnarray}$$ (16)

$$ \begin{eqnarray}\begin{array}{lll} & & {\alpha }_{n}^{(j)}=-\displaystyle \frac{1+{\delta }_{1n}^{(j)}-{\delta }_{2n}^{(j)}\tan {\gamma }_{n}^{(j)}}{1+{(\tan {\gamma }_{n}^{(j)})}^{2}},\\ & & {\beta }_{n}^{(j)}=-\displaystyle \frac{{\delta }_{2n}^{(j)}+\tan {\gamma }_{n}^{(j)}\left(1+{\delta }_{1n}^{(j)}\right)}{1+{(\tan {\gamma }_{n}^{(j)})}^{2}},\\ & & \tan {\gamma }_{n}^{(j)}=\displaystyle \frac{{n}_{n}({x}_{j})+{\bar{Z}}_{jn}{{n}^{^{\prime} }}_{n}({x}_{j})}{{{\rm{j}}}_{n}({x}_{j})+{\bar{Z}}_{jn}{{j}^{^{\prime} }}_{n}({x}_{j})},\\ & & {\bar{Z}}_{jn}={(\rho c)}^{-1}\left[{\rho }_{{\rm{in}}}{c}_{{\rm{in}}}\displaystyle \frac{{{\rm{j}}}_{n}({x}_{ij})}{{{\rm{j}}}^{^{\prime} }{}_{n}({x}_{ij})}+{\rm{i}}{Z}_{jn}\right],\\ & & {\delta }_{1n}^{(j)}={\rm{Re}}\left[\displaystyle \sum _{m=0}^{\infty }{Q}_{nm}^{(3-j)}\displaystyle \frac{2m+1}{2n+1}{{\rm{i}}}^{m-n}{A}_{m}^{(3-j)}\right],\\ & & {\delta }_{2n}^{(j)}={\rm{Im}}\left[\displaystyle \sum _{m=0}^{\infty }{Q}_{nm}^{(3-j)}\displaystyle \frac{2m+1}{2n+1}{{\rm{i}}}^{m-n}{A}_{m}^{(3-j)}\right].\end{array}\end{eqnarray}$$ ()

The time averaged force acting on a spherical particle immersed in an infinite and ideal fluid is given by^[15]

$$ \begin{eqnarray}\begin{array}{lll}{\boldsymbol{F}} & = & -\left\langle \displaystyle {\iint }_{{S}_{0}}\rho ({\upsilon }_{r}{{\boldsymbol{e}}}_{r}+{\upsilon }_{\theta }{{\boldsymbol{e}}}_{\theta }){\upsilon }_{r}{\rm{d}}S\right\rangle \\ & & +\left\langle \displaystyle {\iint }_{{S}_{0}}\displaystyle \frac{\rho }{2}{|{\boldsymbol{\upsilon }}|}^{2}{{\boldsymbol{e}}}_{r}{\rm{d}}S\right\rangle \\ & & -\left\langle \displaystyle {\iint }_{{S}_{0}}\displaystyle \frac{\rho }{2{c}^{2}}{\left(\displaystyle \frac{\partial \mathop{\psi }\limits^{.}}{\partial t}\right)}^{2}{{\boldsymbol{e}}}_{r}{\rm{d}}S\right\rangle,\end{array}\end{eqnarray}$$ (19)

Combining Eqs. (3), (12), (17), and (18) with Eq. (20) and making the integration over the outer surface of the j-th shell, the dimensionless radiation force function is obtained as

$$ \begin{eqnarray}\begin{array}{lll}Y{p}_{j} & = & \langle {F}_{jx}\rangle /(\bar{E}{S}_{jc})\\ & = & \displaystyle \frac{4}{{x}_{j}^{2}}\displaystyle \sum _{n=0}^{\infty }\left\{(n+1)\left({{V}^{^{\prime} }}_{n}^{(j)}{{U}^{^{\prime} }}_{n+1}-{{U}^{^{\prime} }}_{n}^{(j)}{{V}^{^{\prime} }}_{n+1}\right){x}_{j}^{2}\right.\\ & & -n(n+1)(n+2)\left({V}_{n}^{(j)}{U}_{n+1}^{(j)}-{U}_{n}^{(j)}{V}_{n+1}^{(j)}\right)\\ & & +{x}_{j}[n(n+1)\left({{V}^{^{\prime} }}_{n+1}^{(j)}{U}_{n}^{(j)}-{{U}^{^{\prime} }}_{n+1}{V}_{n}^{(j)}\right)\\ & & -(n+1)(n+2)\left({V}_{n+1}^{(j)}{{U}^{^{\prime} }}_{n}^{(j)}-{U}_{n+1}{{V}^{^{\prime} }}_{n}^{(j)}\right)]\\ & & \left.+{x}_{j}^{2}(n+1)\left({V}_{n}^{(j)}{U}_{n+1}^{(j)}-{U}_{n}^{(j)}{V}_{n+1}^{(j)}\right)\right\},\end{array}\end{eqnarray}$$ (21)

The acoustic radiation forces on coupled elastic shells in a plane wave field will be numerically studied in this section. The main objective here is to examine the effects related to the mutual interactions of the coupled scattering on the radiation force exerting on a thin elastic shell. Combining Eqs. (17) and (18) with Eqs. (20) and (21), the ARF function Yp_j (j = 1, 2) of the thin shells can be estimated. The results of the evaluation are illustrated in the case of stainless-steel shell, whose physical constants are cited from Ref. [15]. The surrounding liquid and inside medium are assumed to be water and air with properties of ρ = 1000 kg/m³ and c = 1497 m/s for water, and ρ_in = 1.29 kg/m³ and c_in = 344 m/s for air.

To check the validity of this model, the evaluations of the ARF function for a single thin shell, rigid sphere and elastic sphere are shown by the solid, broken and dashed curves in Fig. 2, respectively. King’s formula^[14] and Hasegawa’s formula^[15] are chosen for calculating the ARF function of rigid sphere and elastic sphere made of stainless steel, respectively. In our model, the thin shell has a thickness-to-radius ratio h/R = 0.01. The results show that Yp₁ begins at small value of kR₁ (kR₁ < 0.5), as though the thin shell is rigid. Deviation from the other curves appears with the increase of kR₁ (kR₁ > 0.5) and the Yp₁ arrives at a high resonant peak, which corresponds to the forced vibration of the thin shell activated by ultrasound sound wave. Because of the acoustic response of the thin elastic shell, the nonlinearity is enhanced, and Yp₁ of the thin elastic shell is larger than that of the sphere with the same size at the band of dynamics response of the thin shell when the objects are activated by the same sound wave. Comparing the curves of three models, one finds that each curve ascends in the range of 0 < kR₁ < ≈ 1.6. Therefore, the ARF function for the spherical bodies with the same outside radii has similar physical properties related to the sound scattering. However, the sound scattering can be enhanced significantly by elastic radiation scattering^[27] of the thin elastic shell vibrated forcedly in the range 0.5 < kR₁ < ≈ 5. There are a number of sharp maxima and minima in the curve of solid elastic sphere, which results from the normal modes of the free vibration in the range of kR₁ > ≈ 1.6.^[15] However, a similar phenomenon does not exist in the curve of the thin shell in that kR₁ range, which means the effects of the high-order normal modes of a thin shell on the ARF might be suppressed by other factors. Therefore, there is distinct deviation between the elastic shell and solid sphere.

In Fig. 3, all numerical results of the ARF function Yp_j (j = 1) are presented for the case of two identical stainless-steel shells (R₁ = R₂) with a thickness-to-radius ratio h/R = 0.01 and ratios of separation distance to radius d/R₁ = 2, 2.5, 4, 10, 20, 50, and 100, which are contrasted with those of a single shell for a range of 0 ≤ kR₁ ≤ 8. At the very low kR₁ (kR₁ < 1), there is no distinctive difference among all the cases, which means that the nonlinear effects of interaction on ARF of shells can be omitted on condition that the wave length is much larger than the size of the shells. However, if the scattering system is approaching to acoustical resonance, the peaks of the ARF function are observed. As for a small separation of d/R₁ = 2 or 2.5, there is a sharp resonant peak at kR₁ ≈ 1.3, and ARF function fluctuates within the rough range of 2.8 < kR₁ < 6.0, as shown in Fig. 3(a). The two shells are elastic scatters, and the radiation scattering influences the returned echoes extremely.^[27] The system is capable of pronounced resonant response in most modes. Therefore, when the two shells are very close to each other, the scattering are amplified by the interactions, and a sharp peak of Yp₁ is observed.

With a slightly larger separation d/R₁ = 4, two peaks are observed in the curve. Comparing the curve in the case d/R₁ = 4 with that of a single shell, the radiation force behind shell differs from that of the single only by a small deviation in the range of kR₁ > ≈ 6.0 as shown in Fig. 3(b). In the electromagnetic end-fire scattering case, this kind of small deviation has been explained as that the frontal shell could hide the back shell when acoustic wave frequency is higher than the resonant frequencies of the pair shells.^[28] With the increase in the distance separation d, the interaction between the identical shells wakens. The ARF function Yp₁ approaches to that of the single when the separation- to-radius ratio d/R₁ > 10, and the multiple scattering effects of the shells can be neglected approximately.

To fully understand the multiple interaction effects on the ARF of thin elastic shells, the dependence of the ARF function Yp₁ on the radii ratio (R₂/R₁ = 0.5, 1, 2, and 5) is shown in Fig. 4 for the cases of d/R₁ = 4 and h/R = 0.01. Figure 4(a) presents that the frontal shell can completely shield the back shell at the lower frequency band kR₁ < 2.4 for the case R₂/R₁ = 0.5, and the ARF of the frontal shell differs from the that of the single only by a small perturbation in the range kR₁ > 2.6. In each case for d/R₁ = 4, two peaks are observed, and the main peak is located at kR₁ ≈ 1.6, while the lower (side) peaks for R₂/R₁ = 0.5, 1, 2, and 5 are situated at 2.6, 1.3, 0.7 and 0.3, respectively as shown in Figs. 4(b) and 4(c). Comparing the kR₁ values of the lower peaks for the case of different radius ratios, one finds that the value of kR₁ is in reverse proportion to the size of the back shell approximately, which reveals that the lower peak might be related to the resonant response of the back shell to the driving acoustic wave. For the cases R₂/R₁ > 1, there exists a narrow kR₁ range in which the ARF is negative, and shells could be attracted to the sound source. According to the calculating results, for the case that the size of the back shell is larger than that of the frontal, its back-scattering is stronger than that of the frontal, which enhances the multiple interaction effects on ARF of the frontal. Therefore, the ARF can be significantly affected by the multiple interactions between shells in the resonant region of the acoustic response of the pair.

To further check the validity, the effects of the shell thickness on the ARF are investigated numerically as shown in Fig. 5. The magnitude of the main peak in ARF function Yp₁ curve decreases with the increase of the shell thickness at the kR₁ band of resonant response to incident progressive wave when R₂/R₁ = 1 and d/R₁ = 4. Elastic scattering of a thin shell^[27] is the important reason for the difference between the scattering action of thin shell and that of a solid elastic body. The acoustic scattering feature of the thick shells is different from that of thin shells, and the elastic waves (transverse wave and longitudinal wave) might be activated in the medium of a thick shell, which relates to the surface vibration (or wave) of the thin elastic shell. Therefore, this model should be limited to the application to predicting the ARF of the thin elastic shell precisely, i.e., shelled medical carrier, micro-capsular ultrasound contrast agent.

This theory is based on the thin elastic shell approximation, and it may be considered as a development of the ARF theory, including the elastic radiation of thin shells and the mutual interactions of multiple particle systems.

Based on the rectified model of sound scattering of a thin elastic shell pair including the effects of elastic scattering of the shell and interaction between shells, the acoustic radiation force of a thin elastic shell suspended in a plane progressive sound wave field is calculated approximately. Due to the nonlinear nature of the acoustic radiation, some new features related to the multiple interactions of two shells in liquid are highlighted. Enhanced resonant peak in the curve of ARF force versuskR₁ reasults from elastic scattering. The interactions between shells is closely related to the identify character of acoustic radiation force, and the negative force is obtained at the band related to the resonant response of the back shell. Acoustic shielding phenomenon of the frontal shell first impinged upon by the incident wave can affect the ARF magnitude of lower peak to some extent for the case R₂/R₁ < 1. The kR₁ value related to the lower peak is in reverse proportion to the size of the back shell approximately. Therefore, the acoustic resonant response of thin shells should be emphasized for the manipulation of multiple particle systems.