Recently researchers often normalize the radiation force on spheres in standing waves in inviscid fluids using an acoustic contrast factor (typically denoted by Φ) that is independent of kR where k is the wave number and R is the sphere radius. An alternative normalization uses a function Ys that depends on kR. Here, standard results for Φ are extended as a power series in kR using prior Ys results. Also, new terms are found for fluid spheres and applied to the kR dependence of Φ for strongly responsive and weakly responsive examples. Partial-wave phase shifts are used in the derivation.

The standard expression for the acoustic radiation force on small idealized-fluid spheres in standing waves in inviscid fluids was published by Yosioka and Kawasima in 1955.1 A result consistent with theirs was subsequently given by Gor'kov.2 During the past two decades, those results are often expressed using a dimensionless contrast factor Φ0 where the force is normalized as follows:3,4
(1)
where R is the radius of the sphere, k = ω/c, where c is the speed of sound in the surrounding fluid, ω is the radius acoustic frequency, Eac is the time-averaged acoustic energy density of the standing wave, and h is the distance of the center of the sphere from a pressure antinode (PAN) of the standing wave. For comparison purposes it is convenient to specify the standing wave as in Eq. (1) of Ref. 5,
(2)
where the peak magnitude is designated as 2p0 and the axial coordinate is z, with z = 0 being the center of the sphere. (A PAN of the incident standing wave is located at z = −h. For trapping in normal gravity, the z axis is taken to be vertical.) The result of Ref. 1 gives3,4
(3)
where λ = ρi/ρ is the ratio of density of the sphere to that of the surrounding fluid and γ = βi/β is the corresponding compressibility ratio. The usefulness of this approach was confirmed by Crum who measured the location of trapped drops in a vertical standing wave in water and the pressure amplitudes required to trap those drops.6,7 In Crum's experiments, Φ0 < 0 so that drops were attracted toward a PAN, the drop being trapped above or below the PAN, depending on the sign of (1-λ). In equilibrium the buoyancy of the drop is balanced by Fz so that the trapping location is predicted to be independent of R when Eq. (3) is applicable. For larger drops, however, the assumption in Ref. 1 that kR ≪ 1 no longer applies, and modified expressions discussed in the present article may be needed.
The approaches to extending the approximation of the contrast factor to larger objects discussed here is limited to the case of spherical or nearly spherical objects. (For example, for drops or bubbles the spatial distribution of the radiation pressure can be balanced by surface tension producing a change in the equilibrium shape.8 The effect of that change of shape is neglected in the discussion which follows.) General relationships between the Fz on spheres in standing waves and the partial wave series (PWS) for the scattering were given in a series of papers by Hasegawa9 as reviewed in Refs. 5 and 10. In the notation of Eq. (2), the radiation force is expressed as
(4)
where Ys is a dimensionless radiation force function that depends on kR and on properties of the sphere and of the surrounding fluid. One approach to generalizing the contrast factor Φ0 in Eq. (3) is to replace Φ0 in Eq. (1) by
(5)
from a comparison of the pre-factors in Eqs. (1), (2), and (4). The PWS expressions for Ys, while exact for the idealized situations considered, make use of spherical Bessel and Hankel functions and can be inconvenient for anticipating how material properties influence Fz even when kR is not large. Consequently, the leading terms in expansions of Ys for different types of spheres have been determined.5,10,11 When combined with Eq. (5), these correspond to expansions of the form
(6)
where Φ0 corresponds to the result in Eq. (3), provided the density ratio λ and compressibility ratio γ are generalized appropriately for the types of spheres considered, and from prior results, Φ2 follows from comparison of results reviewed here in Secs. 2 and 3. Section 4 gives Φ4 for the case of a fluid sphere. The form of the expansion used in Eq. (6) assumes that kR is sufficiently small as to be below all the resonances of the fluid loaded sphere. The evaluation of Φ2 and Φ4 requires going beyond the limitation to Rayleigh scattering implicit in Eq. (3).

Figure 1 illustrates the utility of Eq. (6) and serves to motivate some of the sections which follow. The properties of the liquid drop considered, chlorobenzene acoustically trapped in water, were taken from Ref. 6: λ = ρi/ρ = 1.101 and γ = βi/β = 1/(λσ2), where σ = ci/c = 0.848. The kR independent curve with long dashes is Φ0 from Eq. (3); the solid curve is from the exact PWS result for Ys using Eq. (5). The curve with the shorter of the long dashes is based on expressions for Φ0 and Φ2 that follow from results in Ref. 5 reviewed in Sec. 3. The curve with the shortest dashes is from including the result for Φ4 given in Sec. 4 and in the supplementary material. Section 5 illustrates another application of this approach. To appreciate the sphere sizes for kR = 0.5, where the deviation from Φ0 is appreciable in Fig. 1, notice that with ω/2π = 50 kHz, R = 2.4 mm but for 1 and 10 MHz, R = 0.12 mm and 12 μm. Readers primarily interested in the results of this method may proceed to Sec. 3.

Fig. 1.

The result of four expressions for the radiation force contrast factor for a spherical drop of chlorobenzene in water are plotted as a function of kR, where R is the radius and k the wave number. The horizontal line with long dashes is the commonly used approximation: Φ0 in Eq. (3). The solid curve is the full numerical partial wave series result. The curve with short dashes shows the algebraic approximation derived here: Φ0 + x2Φ2 + x4Φ4 where x = kR. The curve with longer dashes shows the previously available approximation: Φ0 + x2Φ2. When the deviation from Φ0 is significant, the observed trapping location in a vertical standing wave will depend on the size of the drop in normal gravity. For Φ > 0 the attraction is towards a pressure node.

Fig. 1.

The result of four expressions for the radiation force contrast factor for a spherical drop of chlorobenzene in water are plotted as a function of kR, where R is the radius and k the wave number. The horizontal line with long dashes is the commonly used approximation: Φ0 in Eq. (3). The solid curve is the full numerical partial wave series result. The curve with short dashes shows the algebraic approximation derived here: Φ0 + x2Φ2 + x4Φ4 where x = kR. The curve with longer dashes shows the previously available approximation: Φ0 + x2Φ2. When the deviation from Φ0 is significant, the observed trapping location in a vertical standing wave will depend on the size of the drop in normal gravity. For Φ > 0 the attraction is towards a pressure node.

Close modal
The analysis makes use of the partial wave series (PWS) in the exp(-iωt) convention for the scattering of a plane traveling wave by the sphere of interest written as12,
(7)
where θ is the scattering angle and Pn(cos θ) is a Legendre polynomial. The complex sn(kR) depends on the material properties. If pinc denotes the complex pressure of the incident plane wave at the location of the center of the sphere (but without the sphere present), the complex far-field scattered pressure at radius r becomes psca = pinc(R/2r)f exp(ikr). For the situations considered here, there is no energy dissipation so that the complex parameters sn in Eq. (7) are such that |sn| = 1 and sn = −Dn*/Dn, where the asterisk denotes complex conjugation.13 [Here, Dn corresponds to a commonly used notation for the denominator of the partial wave series (see supplementary material).] Following the usual approach for PWS expansions of that type in quantum mechanics, it is convenient to define phase shifts δn such that sn = exp(i2δn). The expressions for sn are well known for fluid spheres,12 solid spheres,10 and empty shells.11 When kR is below that of all resonances, the δn may be expressed from the sn and δn = Ln(sn)/(2i), where the natural logarithm Ln is evaluated in such a way that δn = 0 when kR vanishes. Denoting kR by x and using the known results for the sn it is found that12,
(8)
with anj = 0 if n > j + 1. The anj with j < 2 for fluid spheres were previously determined and are reviewed in the supplementary material. The relationship between the Ys in Eq. (4) and the δn is5,12
(9)
The more widely used PWS expansion of Ys in terms of the complex sn is9,12,14
(10)
where αn = Re(sn – 1)/2 and βn = Im(sn)/2. [In Eq. (27) of Ref. 9, set A = B and convert the coefficients to the exp(-iωt) convention used here for the sn.] It is noteworthy in the present context that the utility of expressing radiation forces using the δn goes beyond the direct applicability of the expansion in Eq. (8) as illustrated in Fig. 1 of Ref. 12, developed in Ref. 15, and demonstrated in Ref. 16.
In a prior publication,5 Ys was expanded out to (kR)3 by inserting Eq. (8) into Eq. (9) and grouping terms. When combined with Eq. (5), those results give in Eq. (6),
(11)
(12)
The results reviewed in the supplementary material give a00, a10, a01, a11, and a21 as functions of λ and σ for fluid spheres. The result for Φ0 is identical to Eq. (3). For the example in Fig. 1, the curve with intermediate dashes shows Φ0 + (kR)2Φ2 which is a significant improvement over Eq. (3) when kR is not small. The usual size independence of the trapping location [given by balancing buoyancy with Fz (Refs. 6, 7, 17)] is modified when Φ2 $≠$ 0.
The approximation Φ = Φ0 + (kR)2Φ2 and the corresponding one for Ys predict that Φ and Ys vanish at kR = x0, where5
(13)
For the example in Fig. 1, x0 = 1.769.
The expressions for the anj needed in Eqs. (11) and (12) have been derived for various solid spheres including empty spherical shells.10,11 There is a noteworthy difference between certain of the cases. For the fluid spheres, the contribution from the n = 2 partial wave gives
(14)
indicating a21 = 0 for the density-matched fluid system. That is not the case for solid spheres where a21 = 2/135 independent of the density. The terms in Eqs. (11) and (12) give all contributions through (kR)2, provided kR is below that of any resonance. The breakdown of this approach near a resonance is evident in the red-dotted curve in Fig. 9 of Ref. 18 showing what is analogous to Φ0 + (kR)2Φ2 for a solid PMMA sphere in water. The n = 2 (quadrupole mode) has an easily observed resonance near19,20 kR = 1.73. (A corresponding resonance is easily observed for polystyrene spheres in water.21) From Eq. (13), in that example Ys and Φ vanish near kR = 1.27. (The ordering of the nulls in the approximate and in the numerical cases are consistent in Ref. 18 and Fig. 1 of Ref. 10.) For solid spheres, the dependence on the transverse wave velocity is included in the evaluation of Φ0 (Ref. 22) and it must also be included in the evaluation of Φ2.10 A widely studied example of a resonance at small kR is the monopole mode of a gas bubble in a liquid where the expansions given here are not directly applicable for many kR regions of interest.17,23,24
By combining Eqs. (8) and (9) it is found that
(15)
which should be applicable to a variety of types of spheres. For confirming the results of this expansion, attention is restricted to the case of ideal inviscid spheres surrounded by an inviscid fluid where the an2 are found in the way previously described. This gives, for example,
(16)
with a12, a22, and a32 listed in the supplementary material. Inspection of Fig. 1 shows that including Φ4 in Eq. (6) noticeably improves the agreement with the numerical result (the solid curve) above kR of 0.5.

It was numerically confirmed that the next term of the series in Eq. (6) scales as (kR)6. This was done for the example in Fig. 1 by computing the scaled difference SD6 = [(Φ0 + x2Φ2 + x4Φ4) – ΦE]/x6, where ΦE is the exact result from the numerical series, Eqs. (10) and (5), and x = kR. It is found that SD6 approaches a constant as x decreases down to approximately 0.08. (Below x of 0.08, round off error becomes significant when evaluating SD6. Including the contribution a32 associated with the n = 3 partial wave is essential.) Similar numerically scaled difference evaluations were previously used to verify formulas for the anj coefficients in Eq. (8) applied to radiation force and scattering problems.11

The expansion in Eq. (6) enables the display of the kR dependence of Φ for a wide range of fluid parameters in certain situations of interest. The case considered here is where the fluid parameters λ = ρi/ρ and σ = ci/c are selected to give Φ0 = 0. From Eq. (11), that occurs when a00 = 3a10, corresponding to a cancellation of the monopole (n = 0) and dipole (n = 1) contributions to Φ0. For fluid spheres, that corresponds to selecting σ to be
(17)
which corresponds to taking δ0 = 3δ1 [see Eq. (21) of Ref. 12]. A partially analogous suppression of Φ0 has been previously investigated for spherical shells.25 (Unlike the present case, that example requires the specification of multiple parameters.) In the present example, from Eqs. (17), (11), and (15), Φ2 and Φ4 become functions of a single parameter, the density ratio λ. These are plotted in Fig. 2 for a range of λ. Though Φ2 ≤ 0, Φ4 is nonnegative. If λ = 1 then Φ0, Φ2, and Φ4 vanish. Figure 2 shows that the vanishing of the radiation force associated with the condition in Eq. (17) is not maintained at finite kR and that Eq. (6) is applicable in this situation.
Fig. 2.

(a) The coefficients Φ2 (solid curve) and Φ4 (dashed curve) are evaluated as a function of the density ratio λ. The sound velocity ratio σ = ci/c is selected according to Eq. (17) so that Φ0 given by the widely used contrast factor approximation, Eq. (3), vanishes. Finite size corrections are essential in this situation. (b) For the case λ = 1.2 and σ given by σ0 ≈ 0.8416, the solid curve is the partial wave series result. The curves with short and longer dashes give Φ0 + x2Φ2 + x4Φ4 and Φ0 + x2Φ2 where x = kR. The trends for Φ(kR) over a range of λ can be anticipated from the results in (a).

Fig. 2.

(a) The coefficients Φ2 (solid curve) and Φ4 (dashed curve) are evaluated as a function of the density ratio λ. The sound velocity ratio σ = ci/c is selected according to Eq. (17) so that Φ0 given by the widely used contrast factor approximation, Eq. (3), vanishes. Finite size corrections are essential in this situation. (b) For the case λ = 1.2 and σ given by σ0 ≈ 0.8416, the solid curve is the partial wave series result. The curves with short and longer dashes give Φ0 + x2Φ2 + x4Φ4 and Φ0 + x2Φ2 where x = kR. The trends for Φ(kR) over a range of λ can be anticipated from the results in (a).

Close modal

For the case of ideal inviscid fluids considered, the kR dependence of the generalized acoustic contrast factor Φ is given by Eq. (6) when kR is small but finite with Φ0, Φ2, and Φ4 given by Eqs. (3), (12), and (15). For sufficiently large immiscible drops in water, the tapping conditions from Eqs. (3)–(6) will depend on drop size in normal gravity. (Such a dependence has been reported.26 The trapping of immiscible drops in water with kR > 0.5 has been demonstrated but is not widely studied.27,28) It is assumed here that kR is sufficiently small as to be less than the region where resonances first appear in the associated scattering amplitude, Eq. (7). Physical processes which dissipate energy such as viscosity and thermal conductivity are completely neglected in the analysis as are forces on the sphere from acoustic streaming.29,30 Some effects of viscosity, significant for very small spheres, are reviewed in Refs. 17, 29, and 30. Radiation force modifications resulting from reflections of waves scattered by the sphere off of the boundaries of the acoustic chamber have also been neglected.

Though the standing waves here have been taken to be plane waves, an extension based on Eq. (8) allows for the modification of Φ0 in Eq. (3) for the case of cylindrically symmetric standing waves. See Secs. IV–VII of Ref. 12. The incident wave has also been assumed here to be temporarily unmodulated. For approaches to modulated cases see, for example, Refs. 17, 28, and 31. Responses to temporally modulated ultrasound can be complicated and include neuromodulation and other biological applications.30–33

See the supplementary material for formula used in the evaluation of Eqs. (11), (12), and (15).

This work was supported by the U.S. Office of Naval Research Award No. N000142212599. Equation (5) and applications thereof were discussed by the present author at the virtual Acoustofluidics 2020 meeting.

The author has no conflicts to disclose.

The data that support the findings of this study are available within the article.

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