Ultrasound in Med. & Biol., Vol. 40, No. 2, pp. 422–433, 2014 Copyright Ó 2014 World Federation for Ultrasound in Medicine & Biology Printed in the USA. All rights reserved 0301-5629/$ - see front matter
http://dx.doi.org/10.1016/j.ultrasmedbio.2013.07.008
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Original Contribution APPLICATION OF ACOUSTIC BESSEL BEAMS FOR HANDLING OF HOLLOW POROUS SPHERES MAHDI AZARPEYVAND* and MOHAMMAD AZARPEYVANDy * Department of Engineering, University of Cambridge, Cambridge, United Kingdom; and y Department of Materials Engineering, Isfahan University of Technology, Isfahan, Iran (Received 22 September 2012; revised 2 July 2013; in final form 15 July 2013)
Abstract—Acoustic manipulation of porous spherical shells, widely used as drug delivery carriers and magnetic resonance imaging contrast agents, is investigated analytically. The technique used for this purpose is based on the application of high-order Bessel beams as a single-beam acoustic manipulation device, by which particles lying on the axis of the beam can be pulled toward the beam source. The exerted acoustic radiation force is calculated using the standard partial-wave series method, and the wave propagation within the porous media is modeled using Biot’s theory of poro-elasticity. Numerical simulations are performed for porous aluminum and silica shells of different thickness and porosity. Results indicate that manipulation of low-porosity shells is possible using Bessel beams with large conical angles, over a number of broadband frequency ranges, whereas manipulation of highly porous shells can occur over both narrowband and broadband frequency domains. (E-mail: m.azarpeyvand@ bristol.ac.uk) Ó 2014 World Federation for Ultrasound in Medicine & Biology. Key Words: Acoustic manipulation, Bessel beam, Biot’s theory, Porous shell, Microporous.
et al. 2004; Slowing et al. 2008; Zhao et al. 2008) and magnetic resonance imaging (MRI) contrast agents (Campbell et al. 2011; Davis 2002; Gao et al. 2008). In the context of drug delivery carriers, the inner cavity of such particles can store a large amount of drug, and the encapsulating porous shell provides a delivery pathway for drug molecule diffusion. The porous-shelled drug carriers have also been found to be mechanically more stable than some other drug carriers, such as those made of polymers, which have exhibited natural burst release behavior (Jing et al. 2011). Another promising application for porous shells is in MRI contrast agents, used as a coating for high-spin toxic or hazardous metals. Numerous core/porous shell combinations have been tested, and for some, such as zeolite- or clay-enclosed gadolinium complexes and magnetite/silica core-shell (Mag@SiO2) or FePt@Fe2O3 yolk-shell nanoparticles, the results are encouraging (Balkus and Shi 1996a, 1996b; Balkus et al. 1991, 1992). Contact-free handling, trapping and precise transport of small suspended objects are essential in many fields of science and technology, such as bioengineering, chemical engineering and pharmaceutical sciences. In biologic applications, in particular, the ability to trap and manipulate micro- and nano-particles is of great importance. For example, the performance of drug delivery systems can be
INTRODUCTION Although a great deal of research has been directed toward acoustic handling of rigid, elastic and porous solid particles and shells by means of sonic beams (Azarpeyvand 2012; Azarpeyvand and Azarpeyvand 2013; Marston 2006, 2007, 2009; Mitri 2008, 2009a, 2009b; Zhang and Marston 2011), almost no pertinent studies can be found for porous shells. Acquiring knowledge of the interaction of acoustic fields with spherical and cylindrical porous shells is of great importance, because of the continuing development of new applications in various engineering and medical fields. For instance, periodically arranged cylindrical porous shells can be used as sonic crystal structures to suppress sound propagation for some frequency bands (Sanchez-Dehesa et al. 2011; Umnova et al. 2005). On the micro and nano scale, porous shells have found numerous applications in modern medicine, pharmacology, biotechnology and chemistry. For example, porous shells are now widely used as drug delivery carriers (Andersson et al. 2004; Cheng et al. 2009; Lai et al. 2003; Mal et al. 2003; Radu
Address correspondence to: Mahdi Azarpeyvand, Department of Mechanical Engineering, University of Bristol, Bristol BS8 1TR, UK. E-mail: [email protected] 422
Acoustic Bessel beams for handling hollow porous spheres d M. AZARPEYVAND and M. AZARPEYVAND
significantly improved by use of a trapping/transporting system (Suwanpayak et al. 2011). For manipulation of a suspended particle, a force must be applied on its body. This force can be produced optically, electrokinetically, hydro-dynamically or acoustically. In the latter case, manipulation can be achieved in two different ways, using either a standing-wave field or one single focused beam (Haake and Dual 2004; Liu and Hu 2009; van West et al. 2007; Wu 1991; Yamakoushi and Noguchi 1998; Yasuda et al. 1995). In the standing-wave method, particles are subject to the mechanical force of a standing acoustic wave, generated by one (and one reflector) or more (van West et al. 2007; Vandaele et al. 2005) ultrasonic transducers. In this article, however, we focus our attention on the second technique. In the single-beam technique, as the name implies, only one highly focused ultrasonic transducer is required, and particle handling is made possible by production of a negative axial force, toward the source. Chen and Apfel (1997) and Marston (2006, 2007, 2009) reported that for some material properties and beam types, the acoustic radiation force for a spherical or cylindrical particle can change from repulsion to attraction. This, however, occurs only at certain frequencies and beam operating conditions. Although much research has been conducted on the viability of using single acoustic beam devices, particularly Bessel beams, for handling particles with different mechanical properties in various media, the research in this area has remained limited to very simple cases and has not yet led to an adequate understanding of the mechanism of particle manipulation when more complex particles are of concern (Azarpeyvand 2012; Marston 2006, 2007, 2009; Mitri 2008, 2009a, 2009b). As stated above, despite the growing attention now being given to different aspects of the application of porous shells, their dynamical behavior when illuminated by an acoustic beam has been the subject of very little research. In this study, we extend the previous investigations by Marston (2006, 2007, 2009), Mitri (2008, 2009a, 2009b) and Azarpeyvand (2012) to the more realistic case of porous shells. The remainder of the paper is organized as follows: The next section is dedicated to the mathematical modeling of the problem. The formulation of a helicoidal Bessel beam is presented, and the radiation force formulations are derived. Biot’s theory of motion in poro-elastic media is presented, and the relevant parameters are defined. The numerical results for hollow aluminum and silica spheres of different shell thickness and porosity are presented and discussed.
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tioned on the beam axis, and is submerged into and filled with linearly compressible, irrotational and nonviscous ideal fluids. The density and speed of sound in the outer medium are denoted by r and c, and those in the core medium by r* and c*, respectively. The shell is illuminated by a helicoidal Bessel beam, radiating at frequency f (5u/2p), with a conical (or half-cone) angle of b. Figure 1 is a schematic of the problem. In what follows, the Roman numerals I, II, and III designate, the surrounding medium, the porous shell medium and the inner inclusion medium, respectively. The incident Bessel beam, propagating in free space and in the positive z direction, can be expressed in cylindrical coordinates (R, z, 4) as (Hernandez-Figueroa et al. 2008) FðincÞ ðR; z; 4Þ 5 F0 Jw ðzRÞeiw41igz2iut ;
(1)
where F0 is the incident field amplitude, g 5 k cos b and z 5 k sin b are the longitudinal and traverse wavenumber components of the incident field, with k 5 u=c, and Jw ð$Þ is the Bessel function of order w (Abramowitz and Stegun 1972). The plane wave field can be restored by setting w 5 0 and b 5 0, whereas the beam vanishes if w 5 1 and b 5 0. It is interesting to note that an axially symmetric Bessel beam is essentially the result of the superposition of plane waves whose wave vectors lay on the surface of a cone having the propagation axis as its symmetry axis and an angle equal to b (conical angle) (Hernandez-Figueroa et al. 2008). General intrinsic properties of Bessel beams, such as self-healing, diffractionfree or phase singularity and angular momentum for higher-order Bessel beams, have been explained in Azarpeyvand (2012), Azarpeyvand et al. (2012) and Hernandez-Figueroa et al. (2008). To obtain a closed-form solution to the problem using the partial-wave expansion method, it is necessary to re-express the incident field, eqn (1), in the coordinate system of the particle. Using a standard wave transformation technique (Stratton 1941), one can rewrite the incident sound field in the spherical coordinate system ðr; q; 4Þ as
MATHEMATICAL FORMULATION Let us consider a porous spherical shell with outer radius a and inner radius b (h 5 b/a). The particle is posi-
Fig. 1. Acoustic Bessel beam incident on a porous spherical shell.
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FðincÞ ðr; q; 4Þ 5 F0 e2iut
N X
il Lwl jl1w ðkrÞPwl1w ðcos qÞeiw4 ;
l50
(2) with Lwl 5 ð2l12w11Þðl!=ðl12wÞ!ÞPwl1w ðcos bÞ. In eqn (2), jq ð$Þ is the spherical Bessel function of order q, and Ppq ð$Þ is the associated Legendre function (Abramowitz and Stegun 1972). The reflected sound wave, propagating radially outward, may be represented in terms of a series of spherical Hankel functions and Legendre polynomials as FðiÞ ðr; q; 4Þ 5 F0 e2iut
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il Lwl xl hl1w ðkrÞPwl1w ðcos qÞeiw4 ;
l50
(3) where hq ðxÞ 5 jq ðxÞ1iyq ðxÞ is the spherical Hankel function of the first kind of order q, and jq and yq are the spherical Bessel functions of the first and second kind, respectively. The unknown scattering coefficients xl have to be determined by imposing appropriate boundary conditions at the particle’s inner and outer surfaces. This is dealt with later. As the incident wave interacts with the shell, part of the incident sound energy will be transmitted into the particle. According to Biot’s model of sound propagation in porous media, there exist two bulk compressional waves, known as the fast and slow compressional waves, and one shear wave (Bourbie et al. 1987). Thus, the wave field within the porous shell can be described as ðIIÞ
Ffast ðr; q; 4Þ 5 F0 e2iut
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where k 5 u=c : The wave propagation in a fluid-saturated porous medium can be studied using Biot’s theory, which itself is constructed using the equations of linear elasticity, Navier-Stokes equations and Darcy’s law for flow of fluid through the porous matrixes. Consider a homogenous, isotropic, porous solid of density rs, with porosity (pore volume fraction) f0 . The solid frame is saturated with an incompressible Newtonian fluid of density rfl and saturating fluid viscosity h. For such a two-component material, two vectors may be defined to describe the displacement of the skeletal frame (u) and the fluid (U). In simple words, a poro-elastic problem consists of four constitutive relations—stress ðsij Þ, strain ðeij Þ, pore pressure ðpp Þ and increment fluid content ðxÞ—given by (Bourbie et al. 1987) sij 5 lf e2bK Mx dij 12meij (8.1) pp 5 Mðx2beÞ;
(8.2)
e 5 V$u; ε 5 V$U
(8.3)
x 5 V$w 5 2f0 ðε2eÞ
(8.4)
where m is the shear modulus of the skeletal frame in a vacuum. The parameter x gives the quantity of fluid that enters or leaves unit volume attached to the skeletal frame; e 5 div u; ε 5 div U are the dilations of the solid and fluid phases, respectively; and
il Lwl al jl1w kf r 1 bl yl1w kf r Pwl1w ðcos qÞeiw4 ;
(4)
l50
ðIIÞ
Fslow ðr; q; 4Þ 5 F0 e2iut
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il Lwl ½cl jl1w ðks rÞ1 dl yl1w ðks rÞPwl1w ðcos qÞeiw4 ;
(5)
l50
ðIIÞ
Jshear ðr; q; 4Þ 5 F0 e2iut
N X
il Lwl ½el jl1w ðkt rÞ1 fl yl1w ðkt rÞPwl1w ðcos qÞeiw4 ;
(6)
l50
where kf, ks and kt are the fast, slow and shear wavenumbers, respectively, which will be derived later. Equations (4–6) are used later to describe the motion and stresses within the porous medium and apply the boundary conditions. Finally, because the core medium (III) is assumed to be an inviscid compressible fluid, the transmitted sound field in this medium can be characterized by a single scalar potential as FðIIIÞ ðr; q; 4Þ 5 F0 e2iut
N X
ilLwl gl jl1w ðk rÞPwl1w ðcos qÞeiw4;
l50
(7)
21 2m K0 bK 2f0 f0 lf 5 Kf 2 ; bK 5 12 ; M 5 1 ; (9) 3 Ks Ks Kfl where Ks is the bulk modulus of the material constituting the elastic matrix, K0 is the bulk modulus of the dry skeleton (the explicit description is given later), Kfl is the bulk modulus of saturating fluid, and the bulk modulus of the closed system is given by f0 K1s 2K1fl 1K1s 2K10 ; Kf 5 (10) f0 1 1 1 1 1 2 2 1 K0 Ks Kfl Ks Ks K0
Acoustic Bessel beams for handling hollow porous spheres d M. AZARPEYVAND and M. AZARPEYVAND
Combining eqns (8.1) through (8.4) with Darcy’s law for flow through a porous medium, a pair of coupled displacement equations of motion can be obtained that govern the rotational and dilational motions in poro-elastic media (Deresiewicz 1960): r11 utt 1r12 Utt 1bðuÞðr11 ut 2r12 U t Þ 5 VðPe1QεÞ2mV2 u; (11.1) r12 utt 1r22 Utt 2bðuÞðr11 ut 2r12 U t Þ 5 VðQe1RεÞ; (11.2) Here, the subscript t denotes the time derivative, and P 5 l12m; Q 5 f0 MðbK 2f0 Þ; R 5 f20 M;
(12)
in which l and m denote Lame’s moduli of the material, Q is a measure of the coupling between the volume changes of the solid and of the liquid and R is a measure of the pressure that must be exerted on the fluid to force a given volume of it into the aggregate with the total volume remaining constant (Allard and Atalla 2009; Deresiewicz 1960). The dynamical mass coefficients are defined as r11 5 r0 1f0 rfl ðaN 22Þ;
(13.1)
r12 5 f0 rfl ð12aN Þ;
(13.2)
r22 5 aN f0 rfl ;
(13.3)
where aN is the tortuosity of the porous medium, and r0 is the density of the fluid-saturated material, that is, r0 5 ð12f0 Þrs 1f0 rfl . In the equations of motion, (11), the parameter b is the viscosity coupling coefficient between both phases, defined as (Allard and Atalla 2009; Hasheminejad and Badsar 2004) bðuÞ 5
f20 h k
FðuÞ:
(14)
The quantity f20 h=k corresponds to the ratio of the total frictional force between fluid and solid, per unit volume of bulk material and per unit average relative velocity in the steady-state flow (Poiseuille flow, that is, at zero frequency), and k characterizes the absolute permeability of the porous medium. The frequency-dependent correction FðuÞ is a measure of the deviation from Poiseuille flow (Allard and Atalla 2009; Azarpeyvand 2012) 1=2 4a2 k2 r u ; FðuÞ 5 12i N 2 2fl hL f0
(15)
withpthe viscous characteristic length defined as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Lz 8aN k=f0 . It is worth mentioning here that two frequency regimes can be defined using Biot’s critical
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frequency, defined as ucr 5 f0 h=rfl kaN . At frequencies smaller than the critical frequency, the viscous effects dominate the inertial effects and the fluid inside the pores is of Poiseuille type (i.e., boundary layer is thick compared with pore size). As a result, the fluid viscosity causes the fluid motion to lock on to the solid motion. Thus, the relative solid-fluid motion required for slow wave propagation cannot be realized and the slow wave becomes a filtration wave, which is characterized by a diffusion-type behavior (non-propagating) (Bourbie et al. 1987; Hasheminejad and Badsar 2005). On the other hand, at frequencies greater than the critical frequency, inertial effects begin to dominate the shear forces, resulting in an ideal flow profile, except in the viscous boundary layer. Above the critical frequency, both fast and slow waves may be expected to propagate (Bourbie et al. 1987). A more detailed description of these frequency regimes and the behavior of different wave components in each regime can be found in Cowin and Cardoso (2011). Also, readers interested in the modeling of wave propagation within porous spheres submerged in a fluid medium or buried in a porous medium are directed to Kargl and Lim (1993). It is more convenient to solve eqns (11) by representing the velocity fields, u and U, in terms of scalar and vector potentials. Four potentials are used, the scalar potentials f and c for the compressional waves in the solid and fluid media, respectively, and two vector potentials, c and Q, representing the transverse wave contributions in solid and fluid media. These potentials relate to u and U through (Zimmerman and Stern 1993) u 5 Vf1V 3 j;
(16.1)
U 5 Vc1V3 Q:
(16.2)
By insertion of eqns (16.1) and (16.2) into equations of motion (11), a pair of Helmholtz equations can be obtained for the compressional and shear wave components (Allard and Atalla 2009; Zimmerman and Stern 1993): (17) V2 1kj2 fj 5 0; j 5 kf ; s ;
V2 1kt2 j 5 0;
(18)
where the ff ;s and j are, respectively, the fast, slow and shear waves, as defined in eqns (4–6). The fast and slow wavenumbers ðkf ;s Þ and shear wavenumber ðkt Þ are defined as h pffiffiffiffii u2 kf2;s 5 P~r22 1R~r11 22Q~r12 H D ; (19) 2 2ðPR2Q Þ kt2
u2 r~11 ~r22 2~r212 5 ; ~r22 m
(20)
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Ultrasound in Medicine and Biology
in which
r11 22Q~ r12 Þ2 2 4 PR2Q2 ~ r22 2~r212 ; D 5 ðP~ r22 1R~ r11 ~ (21) where the modified mass density functions are defined by ~ rij 5 rij 2ibðuÞ=u. Using the displacement representation, eqns (16.1) and (16.2), and the coupled displacement equations of motion, eqns (11.1) and (11.2), one can easily see that the total sound field due to compressional components in the fluid and solid parts is given by f 5 ff 1fs ;
(22)
c 5 mf ff 1ms fs ;
(23)
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4. wr;t 5 2ks ðp2pp Þ, indicating the consistency of the pressure drop and the normal component of filtration velocity, where ks is the interface hydraulic permeability and can vary from zero (sealed interface or no flow) to infinity (open interface or zero pressure drop).The value of the interface permeability ðks Þ, for both the inner and outer surfaces, is assumed to be close to zero, that is, an almost sealed interface. The strain-displacement relations in spherical coordinates are (Kausel 2006)
ur 5 vf 1 1 v ðj sin qÞ; vr r sin q vq 21 v ðrjÞ; uq 5 1r vf vq r vr Ur 5 vc 1 1 v ðQ sin qÞ; vr r sin q vq
(25)
Uq 5 1r vc 21 v ðrQÞ: vq r vr
where mf ;s 5
ð~r11 R2~ r12 QÞ2kf2;s ½PR2Q2 : ð~ r22 Q2~ r12 RÞ
(24)
It can be readily seen that the shear wave component in the fluid is related to that of the solid frame through r22 Þc. Q 5 ð~ r12 =~ To determine the eight unknown scattering coefficients involved in the modeling, eqns (3–7), eight boundary conditions must exist (Bourbie et al. 1987). The following boundary conditions must be satisfied on both the outer ðr 5 aÞ and the inner ðr 5 bÞ surfaces of the shell: 1. srr 5 2p; to show the compatibility of the normal stress with the acoustic pressure in the surrounding medium 2. srq 5 0; vanishing of the tangential stress component 3. wr;t 5 f0 ðUr;t 2ur;t Þ 5 sr;t 2ur;t ; which implies the continuity of the normal component of filtration
2
U ;1
U1;hl 6 UUl ;1 6 2;h 6 UUl ;1 6 3;h
U ;1 6 U4;h R56 6 0 6 6 0 6 4 0 l
0
U ;1
U ;1
U ;1
The pressure and stress components in a poro-elastic medium are also given by (Bourbie et al. 1987) pb 5 Mbf kf2 ff 1Mbs ks2 fs ; srr 5 af kf2 ff 1as ks2 fs 12m vuvrr ; vuq r 1r 2u srq 5 mr vu ; q vq vr
where af ;s 5 2lf 1f0 bK Mð12mf ;s Þ and bf ;s 5 bK 1f0 ðmf ;s 21Þ. Finally, substituting eqns (2–7) into the above four boundary conditions for the inner and outer surfaces, and using the field equations (25), and (26), we obtain a set of eight linear equations for each vibrational mode ðl; wÞ, at a given frequency, which can then be cast into a matrix form as ½R½X 5 ½I, where ½R838 is the matrix of coefficients, given by
U ;1
U ;1
U ;1
Y1;jf U ;1 Y2;jf U ;1 Y3;jf
Y1;yf U ;1 Y2;yf U ;1 Y3;yf
Y1;js Us ;1 Y2;j Us ;1 Y3;j
Y1;ys Us ;1 Y2;y Us ;1 Y3;y
Y1;jt YU2;jt ;1 YU3;jt ;1
Y1;yt YU2;yt ;1 YU3;yt ;1
Y4;jf
Y4;yf
Y4;js
Y4;ys
Y4;jt
Y4;yt
U ;1
hU ;2 Y1;j f hU ;2 Y2;j f hU ;2 Y3;j f hU ;2 Y4;j f
U ;1
hU ;2 Y1;yf hU ;2 Y2;yf hU ;2 Y3;yf hU ;2 Y4;yf
U ;1
hU ;2 Y1;j s s ;2 YhU 2;j hU ;2 Y3;j s s ;2 YhU 4;j
velocity (subscript t designates the time derivative), where sr is the ambient fluid particle displacement
(26)
U ;1
hU ;2 Y1;ys s ;2 YhU 2;y hU ;2 Y3;ys s ;2 YhU 4;y
U ;1
hU ;2 Y1;j t t ;2 YhU 2;j hU ;2 Y3;j t t ;2 YhU 4;j
U ;1
hU ;2 Y1;yt t ;2 YhU 2;y hU ;2 Y3;yt t ;2 YhU 4;y
3 0 0 0 0
7 7 7 7 7 7 ;2 7; 7 ;2 7 7 ;2 5
hU U1;j II hUII U2;j hU U3;j II hUII ;2 U4;j
(27)
and the matrices of the unknown parameters ½X831 and the incident field contribution ½I831 are also written as
Acoustic Bessel beams for handling hollow porous spheres d M. AZARPEYVAND and M. AZARPEYVAND
2 2UUI ;1 3 3 xl 1;j 6 2UU2;jI ;1 7 6 al 7 6 6 7 7 6 2UUI ;1 7 6 bl 7 6 6 7 3;j 7 6 6 cl 7 7 6 2UU4;jI ;1 7; 7 X56 6 dl 7; I 5 6 7 6 0 7 6 7 6 6 el 7 7 6 0 7 6 7 4 4 fl 5 5 0 gl 0 2
NUMERICAL RESULTS AND DISCUSSION
(28)
where Ul 5 kl a, with l 5 {I, II, f, s, t}; the auxiliary functions used in the above matrices ðY and UÞ are defined in the Appendix. The unknown scattering coefficients can 21 then be readily calculated using ½X 5 ½R ½I. Once the known coefficients have been determined, relevant acoustic quantities, such as pressure, particle velocity, intensity and acoustic radiation force, can be computed. The acoustic radiation force acting on a particle can be calculated by performing an integration of the excess of pressure over the surface of the object. The average force vector is expressed as (Hasegawa 1979; Hasegawa et al. 1981) ð 1 2 1 2 F 5 2r hyyi$dS þ r jyj 2 2 hFt i dS; (29) 2 2s ð s
s
where S is the boundary at its equilibrium position, y is the first-order fluid particle velocity at the surface (i:e:2VF), F is the total velocity potential at the boundary, and dS is directed radially outward. The pointed brackets are used for the temporal average and the subscript t denotes the time derivative. A detailed derivation of the force vector and the radiation force function can be found in Hasegawa (1979) and Hasegawa et al. (1981). Subsequently, the axial radiation force Fz is found to be given by (Mitri 2009b) Fz ðu; bÞ 5 Sc Ec Yw ðu; bÞ;
(30)
where Sc is the cross-sectional area ðpa2 Þ; Ec 5 12 rk2 jF0 j is the characteristic energy density and the radiation force function on the surface of the target sphere is given by (Mitri 2009b) 2
Yw ðu; bÞ 5 2
To gain a better understanding of the performance of the proposed acoustic manipulation device and the dynamical behavior of a porous shell when exposed to a helicoidal Bessel beam, some numerical examples are provided. Because of the large number of parameters involved in the present model, we restrict our attention to the emergence of negative radiation force (NRF) caused by the interaction with Bessel beams of w 5 0 (zeroth-order) and w 5 1 (first spinning mode), acting on a spherical aluminum shell with the outer radius a 5 0.02 m. Simulations are performed for four shell thickness ratios ðh 5 b=aÞ, h 5 0 (no void), 0.3, 0.6 and 0.95, and three porosities, f0 5 0 (solid), 0.3 and 0.6. In addition to aluminum, discussions are also provided for silica because of its wide range of applications and extensive use in drug delivery, gene transfection and biosensing. The critical frequency for the aluminium foams with h 5 0.3 and 0.6 are ðkaÞcr 5 1:37 and 2.75, respectively. The mechanical properties of aluminum and silica, required for our model, are summarized in Table 1. The surrounding ambient medium and the void medium are assumed to be water at atmospheric pressure and 300 K ðr 5 r 5 997 kg=m3 ; c 5 c 5 1497 m=sÞ. As mentioned earlier, the thermal and viscous losses in the surrounding and inner liquid media are assumed to be negligible and disregarded in this analysis. A MATLAB (The Mathworks, Natick, MA, USA) code is developed for treating the boundary conditions and to calculate the unknown scattering coefficients and the radiation force at selected beam half-cone angles ðbÞ and incident wave frequencies ðka 5 ua=cÞ. The computations are performed on a dual-core personal computer with a truncation constant of Nmax 5 30 to ensure the convergence of the simulation at high frequencies. The mechanical properties of porous materials can change with porosity. The porosity dependence of some of the parameters used in Biot’s model, such as bulk and shear moduli and tortuosity, are provided here. It is assumed that the dependence of tortuosity on porosity is given by (Berryman 1980)
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Gðl12Þ w w al 1al11 12 awl awl11 1b wl b wl11 Pwl1w ðcos bÞPwl1w11 ðcos bÞ; ðkaÞ l 5 0 Gðl12w11Þ 4
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2
where Gð$Þ denotes the Gamma function (Abramowitz and Stegun 1972); the scattering coefficient components ðawl ; b wl Þ are defined as awl 5
http://dx.doi.org/10.1016/j.ultrasmedbio.2013.07.008
d
Original Contribution APPLICATION OF ACOUSTIC BESSEL BEAMS FOR HANDLING OF HOLLOW POROUS SPHERES MAHDI AZARPEYVAND* and MOHAMMAD AZARPEYVANDy * Department of Engineering, University of Cambridge, Cambridge, United Kingdom; and y Department of Materials Engineering, Isfahan University of Technology, Isfahan, Iran (Received 22 September 2012; revised 2 July 2013; in final form 15 July 2013)
Abstract—Acoustic manipulation of porous spherical shells, widely used as drug delivery carriers and magnetic resonance imaging contrast agents, is investigated analytically. The technique used for this purpose is based on the application of high-order Bessel beams as a single-beam acoustic manipulation device, by which particles lying on the axis of the beam can be pulled toward the beam source. The exerted acoustic radiation force is calculated using the standard partial-wave series method, and the wave propagation within the porous media is modeled using Biot’s theory of poro-elasticity. Numerical simulations are performed for porous aluminum and silica shells of different thickness and porosity. Results indicate that manipulation of low-porosity shells is possible using Bessel beams with large conical angles, over a number of broadband frequency ranges, whereas manipulation of highly porous shells can occur over both narrowband and broadband frequency domains. (E-mail: m.azarpeyvand@ bristol.ac.uk) Ó 2014 World Federation for Ultrasound in Medicine & Biology. Key Words: Acoustic manipulation, Bessel beam, Biot’s theory, Porous shell, Microporous.
et al. 2004; Slowing et al. 2008; Zhao et al. 2008) and magnetic resonance imaging (MRI) contrast agents (Campbell et al. 2011; Davis 2002; Gao et al. 2008). In the context of drug delivery carriers, the inner cavity of such particles can store a large amount of drug, and the encapsulating porous shell provides a delivery pathway for drug molecule diffusion. The porous-shelled drug carriers have also been found to be mechanically more stable than some other drug carriers, such as those made of polymers, which have exhibited natural burst release behavior (Jing et al. 2011). Another promising application for porous shells is in MRI contrast agents, used as a coating for high-spin toxic or hazardous metals. Numerous core/porous shell combinations have been tested, and for some, such as zeolite- or clay-enclosed gadolinium complexes and magnetite/silica core-shell (Mag@SiO2) or FePt@Fe2O3 yolk-shell nanoparticles, the results are encouraging (Balkus and Shi 1996a, 1996b; Balkus et al. 1991, 1992). Contact-free handling, trapping and precise transport of small suspended objects are essential in many fields of science and technology, such as bioengineering, chemical engineering and pharmaceutical sciences. In biologic applications, in particular, the ability to trap and manipulate micro- and nano-particles is of great importance. For example, the performance of drug delivery systems can be
INTRODUCTION Although a great deal of research has been directed toward acoustic handling of rigid, elastic and porous solid particles and shells by means of sonic beams (Azarpeyvand 2012; Azarpeyvand and Azarpeyvand 2013; Marston 2006, 2007, 2009; Mitri 2008, 2009a, 2009b; Zhang and Marston 2011), almost no pertinent studies can be found for porous shells. Acquiring knowledge of the interaction of acoustic fields with spherical and cylindrical porous shells is of great importance, because of the continuing development of new applications in various engineering and medical fields. For instance, periodically arranged cylindrical porous shells can be used as sonic crystal structures to suppress sound propagation for some frequency bands (Sanchez-Dehesa et al. 2011; Umnova et al. 2005). On the micro and nano scale, porous shells have found numerous applications in modern medicine, pharmacology, biotechnology and chemistry. For example, porous shells are now widely used as drug delivery carriers (Andersson et al. 2004; Cheng et al. 2009; Lai et al. 2003; Mal et al. 2003; Radu
Address correspondence to: Mahdi Azarpeyvand, Department of Mechanical Engineering, University of Bristol, Bristol BS8 1TR, UK. E-mail: [email protected] 422
Acoustic Bessel beams for handling hollow porous spheres d M. AZARPEYVAND and M. AZARPEYVAND
significantly improved by use of a trapping/transporting system (Suwanpayak et al. 2011). For manipulation of a suspended particle, a force must be applied on its body. This force can be produced optically, electrokinetically, hydro-dynamically or acoustically. In the latter case, manipulation can be achieved in two different ways, using either a standing-wave field or one single focused beam (Haake and Dual 2004; Liu and Hu 2009; van West et al. 2007; Wu 1991; Yamakoushi and Noguchi 1998; Yasuda et al. 1995). In the standing-wave method, particles are subject to the mechanical force of a standing acoustic wave, generated by one (and one reflector) or more (van West et al. 2007; Vandaele et al. 2005) ultrasonic transducers. In this article, however, we focus our attention on the second technique. In the single-beam technique, as the name implies, only one highly focused ultrasonic transducer is required, and particle handling is made possible by production of a negative axial force, toward the source. Chen and Apfel (1997) and Marston (2006, 2007, 2009) reported that for some material properties and beam types, the acoustic radiation force for a spherical or cylindrical particle can change from repulsion to attraction. This, however, occurs only at certain frequencies and beam operating conditions. Although much research has been conducted on the viability of using single acoustic beam devices, particularly Bessel beams, for handling particles with different mechanical properties in various media, the research in this area has remained limited to very simple cases and has not yet led to an adequate understanding of the mechanism of particle manipulation when more complex particles are of concern (Azarpeyvand 2012; Marston 2006, 2007, 2009; Mitri 2008, 2009a, 2009b). As stated above, despite the growing attention now being given to different aspects of the application of porous shells, their dynamical behavior when illuminated by an acoustic beam has been the subject of very little research. In this study, we extend the previous investigations by Marston (2006, 2007, 2009), Mitri (2008, 2009a, 2009b) and Azarpeyvand (2012) to the more realistic case of porous shells. The remainder of the paper is organized as follows: The next section is dedicated to the mathematical modeling of the problem. The formulation of a helicoidal Bessel beam is presented, and the radiation force formulations are derived. Biot’s theory of motion in poro-elastic media is presented, and the relevant parameters are defined. The numerical results for hollow aluminum and silica spheres of different shell thickness and porosity are presented and discussed.
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tioned on the beam axis, and is submerged into and filled with linearly compressible, irrotational and nonviscous ideal fluids. The density and speed of sound in the outer medium are denoted by r and c, and those in the core medium by r* and c*, respectively. The shell is illuminated by a helicoidal Bessel beam, radiating at frequency f (5u/2p), with a conical (or half-cone) angle of b. Figure 1 is a schematic of the problem. In what follows, the Roman numerals I, II, and III designate, the surrounding medium, the porous shell medium and the inner inclusion medium, respectively. The incident Bessel beam, propagating in free space and in the positive z direction, can be expressed in cylindrical coordinates (R, z, 4) as (Hernandez-Figueroa et al. 2008) FðincÞ ðR; z; 4Þ 5 F0 Jw ðzRÞeiw41igz2iut ;
(1)
where F0 is the incident field amplitude, g 5 k cos b and z 5 k sin b are the longitudinal and traverse wavenumber components of the incident field, with k 5 u=c, and Jw ð$Þ is the Bessel function of order w (Abramowitz and Stegun 1972). The plane wave field can be restored by setting w 5 0 and b 5 0, whereas the beam vanishes if w 5 1 and b 5 0. It is interesting to note that an axially symmetric Bessel beam is essentially the result of the superposition of plane waves whose wave vectors lay on the surface of a cone having the propagation axis as its symmetry axis and an angle equal to b (conical angle) (Hernandez-Figueroa et al. 2008). General intrinsic properties of Bessel beams, such as self-healing, diffractionfree or phase singularity and angular momentum for higher-order Bessel beams, have been explained in Azarpeyvand (2012), Azarpeyvand et al. (2012) and Hernandez-Figueroa et al. (2008). To obtain a closed-form solution to the problem using the partial-wave expansion method, it is necessary to re-express the incident field, eqn (1), in the coordinate system of the particle. Using a standard wave transformation technique (Stratton 1941), one can rewrite the incident sound field in the spherical coordinate system ðr; q; 4Þ as
MATHEMATICAL FORMULATION Let us consider a porous spherical shell with outer radius a and inner radius b (h 5 b/a). The particle is posi-
Fig. 1. Acoustic Bessel beam incident on a porous spherical shell.
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Ultrasound in Medicine and Biology
FðincÞ ðr; q; 4Þ 5 F0 e2iut
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il Lwl jl1w ðkrÞPwl1w ðcos qÞeiw4 ;
l50
(2) with Lwl 5 ð2l12w11Þðl!=ðl12wÞ!ÞPwl1w ðcos bÞ. In eqn (2), jq ð$Þ is the spherical Bessel function of order q, and Ppq ð$Þ is the associated Legendre function (Abramowitz and Stegun 1972). The reflected sound wave, propagating radially outward, may be represented in terms of a series of spherical Hankel functions and Legendre polynomials as FðiÞ ðr; q; 4Þ 5 F0 e2iut
N X
il Lwl xl hl1w ðkrÞPwl1w ðcos qÞeiw4 ;
l50
(3) where hq ðxÞ 5 jq ðxÞ1iyq ðxÞ is the spherical Hankel function of the first kind of order q, and jq and yq are the spherical Bessel functions of the first and second kind, respectively. The unknown scattering coefficients xl have to be determined by imposing appropriate boundary conditions at the particle’s inner and outer surfaces. This is dealt with later. As the incident wave interacts with the shell, part of the incident sound energy will be transmitted into the particle. According to Biot’s model of sound propagation in porous media, there exist two bulk compressional waves, known as the fast and slow compressional waves, and one shear wave (Bourbie et al. 1987). Thus, the wave field within the porous shell can be described as ðIIÞ
Ffast ðr; q; 4Þ 5 F0 e2iut
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Volume 40, Number 2, 2014
where k 5 u=c : The wave propagation in a fluid-saturated porous medium can be studied using Biot’s theory, which itself is constructed using the equations of linear elasticity, Navier-Stokes equations and Darcy’s law for flow of fluid through the porous matrixes. Consider a homogenous, isotropic, porous solid of density rs, with porosity (pore volume fraction) f0 . The solid frame is saturated with an incompressible Newtonian fluid of density rfl and saturating fluid viscosity h. For such a two-component material, two vectors may be defined to describe the displacement of the skeletal frame (u) and the fluid (U). In simple words, a poro-elastic problem consists of four constitutive relations—stress ðsij Þ, strain ðeij Þ, pore pressure ðpp Þ and increment fluid content ðxÞ—given by (Bourbie et al. 1987) sij 5 lf e2bK Mx dij 12meij (8.1) pp 5 Mðx2beÞ;
(8.2)
e 5 V$u; ε 5 V$U
(8.3)
x 5 V$w 5 2f0 ðε2eÞ
(8.4)
where m is the shear modulus of the skeletal frame in a vacuum. The parameter x gives the quantity of fluid that enters or leaves unit volume attached to the skeletal frame; e 5 div u; ε 5 div U are the dilations of the solid and fluid phases, respectively; and
il Lwl al jl1w kf r 1 bl yl1w kf r Pwl1w ðcos qÞeiw4 ;
(4)
l50
ðIIÞ
Fslow ðr; q; 4Þ 5 F0 e2iut
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il Lwl ½cl jl1w ðks rÞ1 dl yl1w ðks rÞPwl1w ðcos qÞeiw4 ;
(5)
l50
ðIIÞ
Jshear ðr; q; 4Þ 5 F0 e2iut
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il Lwl ½el jl1w ðkt rÞ1 fl yl1w ðkt rÞPwl1w ðcos qÞeiw4 ;
(6)
l50
where kf, ks and kt are the fast, slow and shear wavenumbers, respectively, which will be derived later. Equations (4–6) are used later to describe the motion and stresses within the porous medium and apply the boundary conditions. Finally, because the core medium (III) is assumed to be an inviscid compressible fluid, the transmitted sound field in this medium can be characterized by a single scalar potential as FðIIIÞ ðr; q; 4Þ 5 F0 e2iut
N X
ilLwl gl jl1w ðk rÞPwl1w ðcos qÞeiw4;
l50
(7)
21 2m K0 bK 2f0 f0 lf 5 Kf 2 ; bK 5 12 ; M 5 1 ; (9) 3 Ks Ks Kfl where Ks is the bulk modulus of the material constituting the elastic matrix, K0 is the bulk modulus of the dry skeleton (the explicit description is given later), Kfl is the bulk modulus of saturating fluid, and the bulk modulus of the closed system is given by f0 K1s 2K1fl 1K1s 2K10 ; Kf 5 (10) f0 1 1 1 1 1 2 2 1 K0 Ks Kfl Ks Ks K0
Acoustic Bessel beams for handling hollow porous spheres d M. AZARPEYVAND and M. AZARPEYVAND
Combining eqns (8.1) through (8.4) with Darcy’s law for flow through a porous medium, a pair of coupled displacement equations of motion can be obtained that govern the rotational and dilational motions in poro-elastic media (Deresiewicz 1960): r11 utt 1r12 Utt 1bðuÞðr11 ut 2r12 U t Þ 5 VðPe1QεÞ2mV2 u; (11.1) r12 utt 1r22 Utt 2bðuÞðr11 ut 2r12 U t Þ 5 VðQe1RεÞ; (11.2) Here, the subscript t denotes the time derivative, and P 5 l12m; Q 5 f0 MðbK 2f0 Þ; R 5 f20 M;
(12)
in which l and m denote Lame’s moduli of the material, Q is a measure of the coupling between the volume changes of the solid and of the liquid and R is a measure of the pressure that must be exerted on the fluid to force a given volume of it into the aggregate with the total volume remaining constant (Allard and Atalla 2009; Deresiewicz 1960). The dynamical mass coefficients are defined as r11 5 r0 1f0 rfl ðaN 22Þ;
(13.1)
r12 5 f0 rfl ð12aN Þ;
(13.2)
r22 5 aN f0 rfl ;
(13.3)
where aN is the tortuosity of the porous medium, and r0 is the density of the fluid-saturated material, that is, r0 5 ð12f0 Þrs 1f0 rfl . In the equations of motion, (11), the parameter b is the viscosity coupling coefficient between both phases, defined as (Allard and Atalla 2009; Hasheminejad and Badsar 2004) bðuÞ 5
f20 h k
FðuÞ:
(14)
The quantity f20 h=k corresponds to the ratio of the total frictional force between fluid and solid, per unit volume of bulk material and per unit average relative velocity in the steady-state flow (Poiseuille flow, that is, at zero frequency), and k characterizes the absolute permeability of the porous medium. The frequency-dependent correction FðuÞ is a measure of the deviation from Poiseuille flow (Allard and Atalla 2009; Azarpeyvand 2012) 1=2 4a2 k2 r u ; FðuÞ 5 12i N 2 2fl hL f0
(15)
withpthe viscous characteristic length defined as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Lz 8aN k=f0 . It is worth mentioning here that two frequency regimes can be defined using Biot’s critical
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frequency, defined as ucr 5 f0 h=rfl kaN . At frequencies smaller than the critical frequency, the viscous effects dominate the inertial effects and the fluid inside the pores is of Poiseuille type (i.e., boundary layer is thick compared with pore size). As a result, the fluid viscosity causes the fluid motion to lock on to the solid motion. Thus, the relative solid-fluid motion required for slow wave propagation cannot be realized and the slow wave becomes a filtration wave, which is characterized by a diffusion-type behavior (non-propagating) (Bourbie et al. 1987; Hasheminejad and Badsar 2005). On the other hand, at frequencies greater than the critical frequency, inertial effects begin to dominate the shear forces, resulting in an ideal flow profile, except in the viscous boundary layer. Above the critical frequency, both fast and slow waves may be expected to propagate (Bourbie et al. 1987). A more detailed description of these frequency regimes and the behavior of different wave components in each regime can be found in Cowin and Cardoso (2011). Also, readers interested in the modeling of wave propagation within porous spheres submerged in a fluid medium or buried in a porous medium are directed to Kargl and Lim (1993). It is more convenient to solve eqns (11) by representing the velocity fields, u and U, in terms of scalar and vector potentials. Four potentials are used, the scalar potentials f and c for the compressional waves in the solid and fluid media, respectively, and two vector potentials, c and Q, representing the transverse wave contributions in solid and fluid media. These potentials relate to u and U through (Zimmerman and Stern 1993) u 5 Vf1V 3 j;
(16.1)
U 5 Vc1V3 Q:
(16.2)
By insertion of eqns (16.1) and (16.2) into equations of motion (11), a pair of Helmholtz equations can be obtained for the compressional and shear wave components (Allard and Atalla 2009; Zimmerman and Stern 1993): (17) V2 1kj2 fj 5 0; j 5 kf ; s ;
V2 1kt2 j 5 0;
(18)
where the ff ;s and j are, respectively, the fast, slow and shear waves, as defined in eqns (4–6). The fast and slow wavenumbers ðkf ;s Þ and shear wavenumber ðkt Þ are defined as h pffiffiffiffii u2 kf2;s 5 P~r22 1R~r11 22Q~r12 H D ; (19) 2 2ðPR2Q Þ kt2
u2 r~11 ~r22 2~r212 5 ; ~r22 m
(20)
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Ultrasound in Medicine and Biology
in which
r11 22Q~ r12 Þ2 2 4 PR2Q2 ~ r22 2~r212 ; D 5 ðP~ r22 1R~ r11 ~ (21) where the modified mass density functions are defined by ~ rij 5 rij 2ibðuÞ=u. Using the displacement representation, eqns (16.1) and (16.2), and the coupled displacement equations of motion, eqns (11.1) and (11.2), one can easily see that the total sound field due to compressional components in the fluid and solid parts is given by f 5 ff 1fs ;
(22)
c 5 mf ff 1ms fs ;
(23)
Volume 40, Number 2, 2014
4. wr;t 5 2ks ðp2pp Þ, indicating the consistency of the pressure drop and the normal component of filtration velocity, where ks is the interface hydraulic permeability and can vary from zero (sealed interface or no flow) to infinity (open interface or zero pressure drop).The value of the interface permeability ðks Þ, for both the inner and outer surfaces, is assumed to be close to zero, that is, an almost sealed interface. The strain-displacement relations in spherical coordinates are (Kausel 2006)
ur 5 vf 1 1 v ðj sin qÞ; vr r sin q vq 21 v ðrjÞ; uq 5 1r vf vq r vr Ur 5 vc 1 1 v ðQ sin qÞ; vr r sin q vq
(25)
Uq 5 1r vc 21 v ðrQÞ: vq r vr
where mf ;s 5
ð~r11 R2~ r12 QÞ2kf2;s ½PR2Q2 : ð~ r22 Q2~ r12 RÞ
(24)
It can be readily seen that the shear wave component in the fluid is related to that of the solid frame through r22 Þc. Q 5 ð~ r12 =~ To determine the eight unknown scattering coefficients involved in the modeling, eqns (3–7), eight boundary conditions must exist (Bourbie et al. 1987). The following boundary conditions must be satisfied on both the outer ðr 5 aÞ and the inner ðr 5 bÞ surfaces of the shell: 1. srr 5 2p; to show the compatibility of the normal stress with the acoustic pressure in the surrounding medium 2. srq 5 0; vanishing of the tangential stress component 3. wr;t 5 f0 ðUr;t 2ur;t Þ 5 sr;t 2ur;t ; which implies the continuity of the normal component of filtration
2
U ;1
U1;hl 6 UUl ;1 6 2;h 6 UUl ;1 6 3;h
U ;1 6 U4;h R56 6 0 6 6 0 6 4 0 l
0
U ;1
U ;1
U ;1
The pressure and stress components in a poro-elastic medium are also given by (Bourbie et al. 1987) pb 5 Mbf kf2 ff 1Mbs ks2 fs ; srr 5 af kf2 ff 1as ks2 fs 12m vuvrr ; vuq r 1r 2u srq 5 mr vu ; q vq vr
where af ;s 5 2lf 1f0 bK Mð12mf ;s Þ and bf ;s 5 bK 1f0 ðmf ;s 21Þ. Finally, substituting eqns (2–7) into the above four boundary conditions for the inner and outer surfaces, and using the field equations (25), and (26), we obtain a set of eight linear equations for each vibrational mode ðl; wÞ, at a given frequency, which can then be cast into a matrix form as ½R½X 5 ½I, where ½R838 is the matrix of coefficients, given by
U ;1
U ;1
U ;1
Y1;jf U ;1 Y2;jf U ;1 Y3;jf
Y1;yf U ;1 Y2;yf U ;1 Y3;yf
Y1;js Us ;1 Y2;j Us ;1 Y3;j
Y1;ys Us ;1 Y2;y Us ;1 Y3;y
Y1;jt YU2;jt ;1 YU3;jt ;1
Y1;yt YU2;yt ;1 YU3;yt ;1
Y4;jf
Y4;yf
Y4;js
Y4;ys
Y4;jt
Y4;yt
U ;1
hU ;2 Y1;j f hU ;2 Y2;j f hU ;2 Y3;j f hU ;2 Y4;j f
U ;1
hU ;2 Y1;yf hU ;2 Y2;yf hU ;2 Y3;yf hU ;2 Y4;yf
U ;1
hU ;2 Y1;j s s ;2 YhU 2;j hU ;2 Y3;j s s ;2 YhU 4;j
velocity (subscript t designates the time derivative), where sr is the ambient fluid particle displacement
(26)
U ;1
hU ;2 Y1;ys s ;2 YhU 2;y hU ;2 Y3;ys s ;2 YhU 4;y
U ;1
hU ;2 Y1;j t t ;2 YhU 2;j hU ;2 Y3;j t t ;2 YhU 4;j
U ;1
hU ;2 Y1;yt t ;2 YhU 2;y hU ;2 Y3;yt t ;2 YhU 4;y
3 0 0 0 0
7 7 7 7 7 7 ;2 7; 7 ;2 7 7 ;2 5
hU U1;j II hUII U2;j hU U3;j II hUII ;2 U4;j
(27)
and the matrices of the unknown parameters ½X831 and the incident field contribution ½I831 are also written as
Acoustic Bessel beams for handling hollow porous spheres d M. AZARPEYVAND and M. AZARPEYVAND
2 2UUI ;1 3 3 xl 1;j 6 2UU2;jI ;1 7 6 al 7 6 6 7 7 6 2UUI ;1 7 6 bl 7 6 6 7 3;j 7 6 6 cl 7 7 6 2UU4;jI ;1 7; 7 X56 6 dl 7; I 5 6 7 6 0 7 6 7 6 6 el 7 7 6 0 7 6 7 4 4 fl 5 5 0 gl 0 2
NUMERICAL RESULTS AND DISCUSSION
(28)
where Ul 5 kl a, with l 5 {I, II, f, s, t}; the auxiliary functions used in the above matrices ðY and UÞ are defined in the Appendix. The unknown scattering coefficients can 21 then be readily calculated using ½X 5 ½R ½I. Once the known coefficients have been determined, relevant acoustic quantities, such as pressure, particle velocity, intensity and acoustic radiation force, can be computed. The acoustic radiation force acting on a particle can be calculated by performing an integration of the excess of pressure over the surface of the object. The average force vector is expressed as (Hasegawa 1979; Hasegawa et al. 1981) ð 1 2 1 2 F 5 2r hyyi$dS þ r jyj 2 2 hFt i dS; (29) 2 2s ð s
s
where S is the boundary at its equilibrium position, y is the first-order fluid particle velocity at the surface (i:e:2VF), F is the total velocity potential at the boundary, and dS is directed radially outward. The pointed brackets are used for the temporal average and the subscript t denotes the time derivative. A detailed derivation of the force vector and the radiation force function can be found in Hasegawa (1979) and Hasegawa et al. (1981). Subsequently, the axial radiation force Fz is found to be given by (Mitri 2009b) Fz ðu; bÞ 5 Sc Ec Yw ðu; bÞ;
(30)
where Sc is the cross-sectional area ðpa2 Þ; Ec 5 12 rk2 jF0 j is the characteristic energy density and the radiation force function on the surface of the target sphere is given by (Mitri 2009b) 2
Yw ðu; bÞ 5 2
To gain a better understanding of the performance of the proposed acoustic manipulation device and the dynamical behavior of a porous shell when exposed to a helicoidal Bessel beam, some numerical examples are provided. Because of the large number of parameters involved in the present model, we restrict our attention to the emergence of negative radiation force (NRF) caused by the interaction with Bessel beams of w 5 0 (zeroth-order) and w 5 1 (first spinning mode), acting on a spherical aluminum shell with the outer radius a 5 0.02 m. Simulations are performed for four shell thickness ratios ðh 5 b=aÞ, h 5 0 (no void), 0.3, 0.6 and 0.95, and three porosities, f0 5 0 (solid), 0.3 and 0.6. In addition to aluminum, discussions are also provided for silica because of its wide range of applications and extensive use in drug delivery, gene transfection and biosensing. The critical frequency for the aluminium foams with h 5 0.3 and 0.6 are ðkaÞcr 5 1:37 and 2.75, respectively. The mechanical properties of aluminum and silica, required for our model, are summarized in Table 1. The surrounding ambient medium and the void medium are assumed to be water at atmospheric pressure and 300 K ðr 5 r 5 997 kg=m3 ; c 5 c 5 1497 m=sÞ. As mentioned earlier, the thermal and viscous losses in the surrounding and inner liquid media are assumed to be negligible and disregarded in this analysis. A MATLAB (The Mathworks, Natick, MA, USA) code is developed for treating the boundary conditions and to calculate the unknown scattering coefficients and the radiation force at selected beam half-cone angles ðbÞ and incident wave frequencies ðka 5 ua=cÞ. The computations are performed on a dual-core personal computer with a truncation constant of Nmax 5 30 to ensure the convergence of the simulation at high frequencies. The mechanical properties of porous materials can change with porosity. The porosity dependence of some of the parameters used in Biot’s model, such as bulk and shear moduli and tortuosity, are provided here. It is assumed that the dependence of tortuosity on porosity is given by (Berryman 1980)
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Gðl12Þ w w al 1al11 12 awl awl11 1b wl b wl11 Pwl1w ðcos bÞPwl1w11 ðcos bÞ; ðkaÞ l 5 0 Gðl12w11Þ 4
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2
where Gð$Þ denotes the Gamma function (Abramowitz and Stegun 1972); the scattering coefficient components ðawl ; b wl Þ are defined as awl 5
