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 Y_{s} that depends on kR. Here, standard results for Φ are extended as a power series in kR using prior Y_{s} 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.

## 1. Introduction

^{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 Φ

_{0}where the force is normalized as follows:

^{3,4}

_{ac}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,

_{0}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 gives

^{3,4}

_{i}/ρ is the ratio of density of the sphere to that of the surrounding fluid and γ = β

_{i}/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, Φ

_{0}< 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 F

_{z}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.

^{8}The effect of that change of shape is neglected in the discussion which follows.) General relationships between the F

_{z}on spheres in standing waves and the partial wave series (PWS) for the scattering were given in a series of papers by Hasegawa

^{9}as reviewed in Refs. 5 and 10. In the notation of Eq. (2), the radiation force is expressed as

_{s}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 Φ

_{0}in Eq. (3) is to replace Φ

_{0}in Eq. (1) by

_{s}, 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 F

_{z}even when kR is not large. Consequently, the leading terms in expansions of Y

_{s}for different types of spheres have been determined.

^{5,10,11}When combined with Eq. (5), these correspond to expansions of the form

_{0}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, Φ

_{2}follows from comparison of results reviewed here in Secs. 2 and 3. Section 4 gives Φ

_{4}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 Φ

_{2}and Φ

_{4}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 = ρ_{i}/ρ = 1.101 and γ = β_{i}/β = 1/(λσ^{2}), where σ = c_{i}/c = 0.848. The kR independent curve with long dashes is Φ_{0} from Eq. (3); the solid curve is from the exact PWS result for Y_{s} using Eq. (5). The curve with the shorter of the long dashes is based on expressions for Φ_{0} and Φ_{2} that follow from results in Ref. 5 reviewed in Sec. 3. The curve with the shortest dashes is from including the result for Φ_{4} 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 Φ_{0} 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 *μ*m. Readers primarily interested in the results of this method may proceed to Sec. 3.

## 2. The relevant expansion of partial wave phase shifts

^{12}

^{,}

_{n}(cos θ) is a Legendre polynomial. The complex s

_{n}(kR) depends on the material properties. If p

_{inc}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 p

_{sca}= p

_{inc}(R/2r)f exp(ikr). For the situations considered here, there is no energy dissipation so that the complex parameters s

_{n}in Eq. (7) are such that |s

_{n}| = 1 and s

_{n}= −D

_{n}

^{*}/D

_{n}, where the asterisk denotes complex conjugation.

^{13}[Here, D

_{n}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 δ

_{n}such that s

_{n}= exp(i2δ

_{n}). The expressions for s

_{n}are well known for fluid spheres,

^{12}solid spheres,

^{10}and empty shells.

^{11}When kR is below that of all resonances, the δ

_{n}may be expressed from the s

_{n}and δ

_{n}= Ln(s

_{n})/(2i), where the natural logarithm Ln is evaluated in such a way that δ

_{n}= 0 when kR vanishes. Denoting kR by x and using the known results for the s

_{n}it is found that

^{12}

^{,}

_{n}

_{j}= 0 if n >

*j*+ 1. The a

_{n}

_{j}with j < 2 for fluid spheres were previously determined and are reviewed in the supplementary material. The relationship between the Y

_{s}in Eq. (4) and the δ

_{n}is

^{5,12}

_{s}in terms of the complex s

_{n}is

^{9,12,14}

_{n}= Re(s

_{n}– 1)/2 and β

_{n}= Im(s

_{n})/2. [In Eq. (27) of Ref. 9, set A = B and convert the coefficients to the exp(-iωt) convention used here for the s

_{n}.] It is noteworthy in the present context that the utility of expressing radiation forces using the δ

_{n}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.

## 3. The leading order terms in Eq. (6) and applications

^{5}Y

_{s}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),

_{00}, a

_{10}, a

_{01}, a

_{11}, and a

_{21}as functions of λ and σ for fluid spheres. The result for Φ

_{0}is identical to Eq. (3). For the example in Fig. 1, the curve with intermediate dashes shows Φ

_{0}+ (kR)

^{2}Φ

_{2}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 F

_{z}(Refs. 6, 7, 17)] is modified when Φ

_{2}$\u2260$ 0.

_{n}

_{j}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

_{21}= 0 for the density-matched fluid system. That is not the case for solid spheres where a

_{21}= 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 Φ

_{0}+ (kR)

^{2}Φ

_{2}for a solid PMMA sphere in water. The n = 2 (quadrupole mode) has an easily observed resonance near

^{19,20}kR = 1.73. (A corresponding resonance is easily observed for polystyrene spheres in water.

^{21}) From Eq. (13), in that example Y

_{s}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 Φ

_{0}(Ref. 22) and it must also be included in the evaluation of Φ

_{2}.

^{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}

## 4. The new term in Eq. (6) applied to the example in Fig. 1

_{n2}are found in the way previously described. This gives, for example,

_{12}, a

_{22}, and a

_{32}listed in the supplementary material. Inspection of Fig. 1 shows that including Φ

_{4}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 SD_{6} = [(Φ_{0} + x^{2}Φ_{2} + x^{4}Φ_{4}) – Φ_{E}]/x^{6}, where Φ_{E} is the exact result from the numerical series, Eqs. (10) and (5), and x = kR. It is found that SD_{6} approaches a constant as x decreases down to approximately 0.08. (Below x of 0.08, round off error becomes significant when evaluating SD_{6}. Including the contribution a_{32} associated with the n = 3 partial wave is essential.) Similar numerically scaled difference evaluations were previously used to verify formulas for the a_{n}_{j} coefficients in Eq. (8) applied to radiation force and scattering problems.^{11}

## 5. Application to weakly responsive spheres algebraically evaluated

_{i}/ρ and σ = c

_{i}/c are selected to give Φ

_{0}= 0. From Eq. (11), that occurs when a

_{00}= 3a

_{10}, corresponding to a cancellation of the monopole (n = 0) and dipole (n = 1) contributions to Φ

_{0}. For fluid spheres, that corresponds to selecting σ to be

_{0}= 3δ

_{1}[see Eq. (21) of Ref. 12]. A partially analogous suppression of Φ

_{0}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), Φ

_{2}and Φ

_{4}become functions of a single parameter, the density ratio λ. These are plotted in Fig. 2 for a range of λ. Though Φ

_{2}≤ 0, Φ

_{4}is nonnegative. If λ = 1 then Φ

_{0}, Φ

_{2}, and Φ

_{4}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.

## 6. Conclusions and discussion

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 Φ_{0}, Φ_{2}, and Φ_{4} 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 Φ_{0} 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}

## Supplementary Material

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

## Acknowledgments

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.

## Author Declarations

### Conflict of Interest

The author has no conflicts to disclose.

## Data Availability

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