Transition mechanisms of translational motions of bubbles in an ultrasonic field

Abstract

The translation behaviors of oscillating bubbles are closely related to the polymerizations and dispersions between them, which are crucial for the ultrasonic cavitation effect. In this study, six types of translational motion of bubbles with a wide range of sizes (2–100 μm) in the R₀₁-R₀₂ plane are investigated. Our results demonstrate that in addition (to the 2nd order harmonic), the 1/2 order subharmonic can change the bubble pairs from the three states of the attraction, stable after attraction, and repulsion to that of the repulsion, coalescence, and attraction, respectively. Furthermore, within the range of the main resonance radius and the 1/2 order subharmonic resonance radius, the chaotic bubble pairs with alternating attractive and repulsive forces appear in the region between the coalescence pairs and stable pairs after attraction. Finally, the corresponding physical mechanisms of the chaotic translational motions are also revealed.

Nomenclature

C

sound velocity in liquid (m/s)

D

inter-bubble distance (μm)

D₀

initial inter-bubble distance (μm)

F_B

secondary Bjerknes force between two bubbles (N)

F_ex1,F_ex2

Levich viscous drag acting on bubble 1 and bubble 2 (N)

F_h1

resultant external force acting on bubble 1 (N)

F

frequency of the sound field (kHz)

f_c-res1,f_c-res2

linear resonance frequencies of the coupled bubbles (kHz)

f_nat1, f_nat2

natural frequencies of the two individual bubbles (kHz)

f_res1, f_res2

linear resonance frequencies of the two individual bubbles (kHz)

f_res

linear resonance frequency of the individual bubble (kHz)

f₀

ultrasonic driving frequency (kHz)

F₁₂

radiation pressure acting on bubble 2 originated from bubble 1 (N)

P_A(t)

sinusoidal ultrasonic pressure (bar)

P_a

ultrasonic pressure amplitude (bar)

P_s1, P_s2

pressure emitted by the coupled pulsating bubbles (bar)

P_v

saturated vapor pressure (bar)

P₀

ambient atmospheric pressure (bar)

P₁, P₂

pressure at the walls of the coupled bubbles (bar)

R_max1, R_max2

maximum radii of the coupled bubbles (μm)

R_res

linear resonance radius corresponding to the driving frequency f₀ (μm)

R₁,R₂

instantaneous radii of the coupled bubbles (μm)

${\dot{R}}_1$,${\dot{R}}_2$

radial velocity of the coupled bubbles (μm/s)

${\ddot{R}}_1$,${\ddot{R}}_2$

radial acceleration of the coupled bubbles (μm/s²)

R₁₀, R₂₀

equilibrium radius of the bubble 1 and bubble 2 (μm)

T

time (s)

T₀

driving period (s)

Ρ

density of liquid medium (kg/m³)

V₁, V₂

instantaneous volume of the coupled bubbles

v₁, v₂

liquid velocity generated by the coupled bubbles (μm/s)

x₁, x₂

translational displacements of the coupled bubbles (μm)

${\dot{x}}_1$,${\dot{x}}_2$

translational velocities of the coupled bubbles (μm/s)

${\ddot{x}}_1$,${\ddot{x}}_2$

translational accelerations of the coupled bubbles (μm/s²)

Μ

viscosity of liquid medium (Pa·s)

κ

gas polytrophic exponent

σ

surface tension coefficient (N/m)

1. Introduction

When an ultrasound propagates through an aqueous solution, the gas bubbles oscillate, expand and collapse, which is termed as acoustic cavitation [1], [2]. The fluid flow effects (microjetting and shear force) and extreme environment (high temperature and pressure) caused by oscillation and collapse of bubbles [3], [4] can be widely utilized in applications such as food processing [5], (heavy oil) viscosity reduction [6], cleaning [7], chemical reactions [8], ultrasonic welding [9], medical ultrasound (ultrasonic imaging and guide treatments) [10], [11], [12], [13], [14], [15], [16], among others. In addition, the translational motions of oscillating bubbles, triggered by primary and secondary Bjerknes forces [17], [18], also have profound research and application significances. For instance, the polymerization and dispersion between bubbles caused by translational motion have a significant influence on the bubble cloud dynamics and cavitation effect [19], [20], [21], [22]. Experimentations and numeric evaluations demonstrated that the packing of bubbles changes the attenuation of the bubbly medium [23]. Thus, a prediction of the translational motion of the bubbles and possible stable regions will also help to better understand the wave propagation in bubbly media. In this study, the dynamic behaviors of bubbles caused by secondary Bjerknes forces are emphasized and addressed.

According to the classical linear theory, if a driving frequency lies among the linear resonance frequencies, they will mutually attract; otherwise, they mutually repulse [17], [24], [25]. In the context of this linear theory, Zabolotskaya et al. [26] revealed that the secondary Bjerknes force transits from a state of mutual attraction to mutual repulsion as the inter-bubble distance decreases because the driving frequency falls between the increased linear resonance frequencies. This idea is reaffirmed while considering the multiple scattering of sound [27]. Furthermore, a similar sign inversion mechanism was discovered by Barbat [28] for near-resonant pairs of bubbles. Afterwards, Harkin [25] and Ida et al. [29], [30] also confirmed that a change in the inter-bubble distance can lead to a sign change of the secondary Bjerknes force.

The sign reversal of the secondary Bjerknes force (or translational motion) owing to the nonlinear effect of bubble oscillations has attracted scientists [31], [32], [33], [34], [35], [36], [37]. Oguz et al. [31] indicated that even slight nonlinear oscillations of bubbles can change the sign of the secondary Bjerknes force predicted by the linear theory. Considering the non-spherical oscillations of coupled bubbles, Pelekasis et al. [32] revealed that the decreased distance in the inter-bubbles may reverse the direction of the secondary Bjerknes force. Although the non-spherical oscillations of only a small individual bubble was considered, Zhang et al. [33] concluded that the translational motions (inter-bubble distance) can significantly change the magnitude and sign of a bubble–bubble interaction.

In a strong ultrasonic field (>1.0 bar), sign reversal would also occur as the pressure amplitudes increase, as a result of the changes in the dynamical Blake threshold [18], [34]. Using the Lagrangian formalism, Doinikov et al. [35] derived a model coupling the radial and translational motions to better understand the bubble dynamics, and reported that bubbles would form a steady pair instead of coalescing. Pelekasis and Doinikov et al. [34], [36] revealed the relevant mechanism by which the 2nd order harmonic can prevent the bubbles from coalescing. Furthermore, Pelekasis summarized four types of translational motions. It was pointed out that the 1/2 subharmonic and combination resonances (unique resonance type for dual-frequency case) can affect the values and signs of the secondary Bjerknes force [37].

In addition, Zhang et al. [38] demonstrated that the translational motions not only lead to chaotic pulsations and secondary Bjerknes forces, but also improve the chaotic degree of secondary Bjerknes force. They also indicated that the repulsive secondary Bjerknes force is mainly due to the large radius catching the rebound point of another bubble [39]. Wang et al. [40] concluded that decreased frequency or increased burst ultrasound amplitude can increase the translational velocities.

However, despite the growing interest of the influences of nonlinear effect on the translational motions (or secondary Bjerknes force), the relative mechanisms caused by subharmonic waves remain obscure, especially those by chaotic dynamics that have not been reported yet. Therefore, in this study, a wide R₀₁-R₀₂ space is scanned to investigate the distribution of the types of translational motions. Then, the transition mechanisms of the translational motions triggered by the 1/2 order subharmonic, and the translational and radial motion characteristics of chaotic pairs are analyzed. Finally, the influence of the initial inter-bubble distance and driving frequency on the translational motions are analyzed.

2. Model and solutions

2.1. Dynamic model

Two coupled bubbles were assumed to be located sufficiently distant compared to their radii. The influence of higher-order spherical harmonics of surface modes was neglected and the bubble oscillations were considered completely spherically symmetric [41], [42]. In particular, the subharmonic emissions were experimentally demonstrated to be affected by non-spherical oscillations [42]. Thus, the influences of the surface modes were neglected for simplification of the study. The time delay effect of sound wave propagation owing to the inter-bubble distance was ignored in this study. The study conducted by the Hamilton et al. [43], [44], [45], [46] regarding the time delay effects can be referred for further details. The possible influence of time delays on the dynamic behaviors of the interacting bubbles are reviewed in Ref. [47]. The bubble oscillations were considered to be stable and the destruction threshold was neglected. In many scenarios of high amplitude oscillations where R/R₀ > 2, a bubble may be unable to stably oscillate for several cycles [48]. This appears to be the minimum threshold for inertial bubble collapse and possible destruction [49]. In addition, this present study did not account for the mass transfer effects, which are important at a higher number of cycles, such as those studied in reference [50]. Similar to the previous studies regarding the dynamic behavior of interacting bubbles, the effect of mass transfer was neglected in this study.

Fig. 1 provides the schematic diagram of the dynamic interactions between the two coupled bubbles. R₀₁ and R₀₂ are the equilibrium radii of bubble 1 and bubble 2, respectively.

Fig. 1.

Schematic diagram of the two coupled bubbles system.

The pressure emitted by the adjacent pulsating bubbles can be calculated as follows [18]:

$$p_{s1}=\frac{\rho }{D}\frac{\text{d}}{\text{d}t}\left(R_2^2{\dot{R}}_2\right)$$ (1)

$$p_{s2}=\frac{\rho }{D}\frac{\text{d}}{\text{d}t}\left(R_1^2{\dot{R}}_1\right)$$ (2)

where P_s1 is the pressure emitted by pulsating bubble 2 acting on bubble 1, and P_s2 is the pressure emitted by pulsating bubble 1 acting on bubble 2. R₁ and R₂ are the instantaneous radii of the coupled bubbles, and ${\dot{R}}_1$ and ${\dot{R}}_2$ are the radial velocities of the coupled bubbles. ρ is the density of the liquid medium, D denotes the inter-bubble distance.

The radiation pressure F₁₂ acting on bubble 2, which originates from the oscillating bubble 1, can be expressed as follows:

$$F12=-V_2\mathrm{\nabla }{p}_{s1}$$ (3)

With the volume of bubble 2:

$$V_2\text{= 4}\pi R_2^3/3$$ (4)

Substituting Eqs. (1–4) into Eq. (3) obtains followings:

$$F_{12}=-V_2\frac{\partial p_{s1}}{\partial D}=V_2\frac{\rho }{D^2}\frac{d}{\text{d}t}\left(R_1^2{\dot{R}}_1\right)=\frac{\rho }{4\pi D^2}{V}_2\frac{d^2{V}_1}{d{t}^2}$$ (5)

By the integration of Eq. (5) within one period of bubble oscillation, the secondary Bjerknes force F_B can be obtained as follows:

$$F_{\text{B}}=〈F_{12}〉=-\frac{\rho }{4\pi D^2}〈{\dot{V}}_1{\dot{V}}_2〉$$ (6)

Here,${\dot{V}}_1=4\pi R_1^2{\dot{R}}_1$,${\dot{V}}_2=4\pi R_2^2{\dot{R}}_2$. The positive F_B indicates that the bubbles are mutually repulsive, while the negative F_B indicates that the two bubbles are mutually attractive.

In this study, a model derived by Doinikov [35] based on the Keller-Miksis equations [51] are employed to describes the coupled pulsations and translational motions of bubbles

$$\begin{array}{c}(1-\frac{{\dot{R}}_1}{c})R_1{\ddot{R}}_1+{\dot{R}}_1^2(\frac{3}{2}-\frac{{\dot{R}}_1}{2c})-\frac{P_1}{\rho }(1+\frac{{\dot{R}}_1}{c})-\frac{R_1}{\rho c}\frac{d{P}_1}{\mathit{dt}}\\ =\frac{{\dot{x}}_1^2}{4}-\frac{R_2^2{\ddot{R}}_2+2R_2{\dot{R}}_2^2}{D}+\frac{R_2^2({\dot{x}}_1{\dot{R}}_2+R_2{\ddot{x}}_2+5{\dot{R}}_2{\dot{x}}_2)}{2D^2}-\frac{R_2^3{\dot{x}}_2({\dot{x}}_1+2{\dot{x}}_2)}{2D^3}\end{array}$$ (7)

$$\begin{array}{c}(1-\frac{{\dot{R}}_2}{c})R_2{\ddot{R}}_2+{\dot{R}}_2^2(\frac{3}{2}-\frac{{\dot{R}}_2}{2c})-\frac{P_2}{\rho }(1+\frac{{\dot{R}}_2}{c})-\frac{R_2}{\rho c}\frac{d{P}_2}{\mathit{dt}}\\ =\frac{{\dot{x}}_2^2}{4}-\frac{R_1^2{\ddot{R}}_1+2R_1{\dot{R}}_1^2}{D}-\frac{R_1^2({\dot{x}}_2{\dot{R}}_1+R_1{\ddot{x}}_1+5{\dot{R}}_1{\dot{x}}_1)}{2D^2}-\frac{R_1^3{\dot{x}}_1({\dot{x}}_2+2{\dot{x}}_1)}{2D^3}\end{array}$$ (8)

$$\frac{R_1{\ddot{x}}_1}{3}+{\dot{R}}_1{\dot{x}}_1+\frac{1}{D^2}\frac{d}{\mathit{dt}}(R_1{R}_2^2{\dot{R}}_2)-\frac{R_2^2(R_1{R}_2{\ddot{x}}_2+R_2{\dot{R}}_1{\dot{x}}_2+5R_1{\dot{R}}_2{\dot{x}}_2)}{D^3}=\frac{F_{ex1}}{2\pi \rho R_1^2}$$ (9)

$$\frac{R_2{\ddot{x}}_2}{3}+{\dot{R}}_2{\dot{x}}_2-\frac{1}{D^2}\frac{d}{\mathit{dt}}(R_2{R}_1^2{\dot{R}}_1)-\frac{R_1^2(R_2{R}_1{\ddot{x}}_1+R_1{\dot{R}}_2{\dot{x}}_1+5R_2{\dot{R}}_1{\dot{x}}_1)}{D^3}=\frac{F_{ex2}}{2\pi \rho R_2^2}$$ (10)

Here, ${\ddot{R}}_1$ and ${\ddot{R}}_2$ are the radial accelerations of the coupled bubbles. x₁ and x₂ are the translational displacements of the coupled bubbles 1 and 2, ${\dot{x}}_1$ and ${\dot{x}}_2$ are the translational velocities of the coupled bubbles, and ${\ddot{x}}_1$ and ${\ddot{x}}_2$ are translational accelerations of bubbles 1 and 2. c is the sound velocity in liquid, P_v denotes the vapor pressure, P₀ denotes the atmospheric static pressure, κ is the polytrophic exponent, σ is the surface tension coefficient, and μ is the liquid viscosity. P₁ and P₂ denote the pressures at the bubble walls, which can be expressed as follows:

$$P_1=(P_0-P_v+\frac{2\sigma }{R_{10}}){(\frac{R_{10}}{R_1})}^{3\kappa }+P_v-\frac{2\sigma }{R_1}-\frac{4\mu {\dot{R}}_1}{R_1}-P_0-P_A(t)$$ (11)

$$P_2=(P_0-P_v+\frac{2\sigma }{R_{20}}){(\frac{R_{20}}{R_2})}^{3\kappa }+P_v-\frac{2\sigma }{R_2}-\frac{4\mu {\dot{R}}_2}{R_2}-P_0-P_A(t)$$ (12)

Where P_A(t) denotes the sinusoidal ultrasonic pressure

$$P_A(t)=P_a\sin (2\pi f_{0}t)$$ (13)

The external forces F_axi acting on bubble i are equal to the Levich viscous drag [52] and given by the followings:

$$F_{ex1}=-12\pi \mu R_1({\dot{x}}_1-{\nu }_2)$$ (14)

$$F_{ex2}=-12\pi \mu R_2({\dot{x}}_2-{\nu }_1)$$ (15)

Here, v_i is the liquid velocity generated by bubble i and can be expressed as follows:

$${\nu }_1=\frac{R_1^2{\dot{R}}_1}{D^2}+\frac{R_1^3{\dot{x}}_1}{D^3}$$ (16)

$${\nu }_2=-\frac{R_2^2{\dot{R}}_2}{D^2}+\frac{R_2^3{\dot{x}}_2}{D^3}$$ (17)

The 4–5 order Runge-Kutta integration method is used to solve this coupled dynamic model (Eqs. (7–10)). The parametric values of the liquid medium are as follows: μ = 0.001 Pa·s, σ = 0.0728 N/m, c = 1500 m/s, κ = 4/3, ρ = 998 kg/m³, P_v = 0.0233 bar, and P₀ = 1 bar.

For a spherical bubble in infinite medium, without considering the interaction between the bubbles varying with the inter-bubble distance [10], [23], [47], [53], the natural frequency for its volume oscillations f_nat for Keller-Miksis model reads as follows [54]:

$$f_{\mathit{nat}}=\frac{1}{2\pi R_0}\sqrt{\frac{3\kappa (P_0+\frac{2\sigma }{R_0}-P_v)-\frac{2\sigma }{R_0}-\frac{4{\mu }^2}{\rho R_0^2}}{\rho }}$$ (18)

When the value of P_a is low, the value of the linear resonance frequency f_res of the individual bubble is approximately equal to that of its natural frequency f_nat. Fig. 2 provides the frequency response curves of an individual bubble when the pressure amplitudes are as follows: P_a = 0.005, 0.1, 0.3, 0.6 and 0.9 bar. According to Lauterborn [55], when the frequency of the sound field f is close to a rational number “m/n” (m, n = 1, 2, 3,…) times the linear resonance frequency f_res, all the resonances in Fig. 2 can be classified into four types according to the “order” (the inverse of “m/n”):

• i)Main resonance (1/1): around f / f_res = 1;
• ii)Harmonics resonance (2/1, 3/1, 4/1,…): around f / f_res = 1/2, 1/3, 1/4,….
• iii)Subharmonic resonance (1/2, 1/3, 1/4,…): around f / f_res = 2, 3, 4,….…
• iv)Ultraharmonic resonance (3/2, 5/2, 7/2,…): around f / f_res = 2/3, 2/5, 2/7,… (m ＜ n).

Fig. 2.

Frequency response curves of the individual bubble when P_a = 0.005, 0.1, 0.3, 0.5 and 0.7 bar. R₀ = 10 μm. The numbers labelled above the peaks of the curves are the orders of resonances.

Fig. 2 demonstrated that with P_a increasing, all resonances lean towards the lower frequencies [10], [23], [47], [56], [57]. Furthermore, the detailed work on the comprehensive oscillation characteristics of the 2nd and 3rd order harmonically resonant bubbles could be referred to Ref. [58]. By a novel method [59], Sojahrood et al. comprehensively characterized the evolution of 2nd, 3rd harmonic and subharmonics as a function of pressure [60]. The onset of the first subharmonic (1/2 order) resonance was proposed as a method to determine the cavitation threshold [55]. One of the main applications of the 1/2 order subharmonic emissions is in the medical ultrasound. In imaging applications, they are used to enhance the detection of blood vessels with superior contrast in real-time [10], [11], [12], [13], [14], [15], [16]. Noted, ultraharmonics can also be used for ultrasonic imaging [61]. In therapy, 1/2 order subharmonics are used to monitor and guide treatments [10], [11], [12], [13], [14], [15], [16]. The pulses in therapy are long; and the conditions for the generation, amplification or disappearance of the 1/2 order subharmonic will help optimize and understand the treatment process.

The generation of 1/2 order subharmonics from spherical bubble oscillations was recently comprehensively studied [10], [47], [49], [62], [63], [64], [65], [66]. The 1/2 order subharmonic emissions can be generated when the bubble is sonicated at its resonance [49], [66], and the 1/2 order subharmonic emissions can be generated at twice the resonance frequency at the lowest acoustic pressure [10], [47], [49], [64], [65]. However, when full thermal dissipation was considered, the lowest pressure threshold occurred when the bubble was sonicated at a frequency below its resonance (sonication with the 3/2 or 5/2 order subharmonic resonance) [66]. Sojahrood et al. [49] indicated that when the bubble was sonicated at its resonance frequency, the 1/2 order subharmonic resonance is mainly generated when R/R₀ > 2 and during violent collapses.

2.2. Solution procedure

The dynamic behaviors of two-bubble system are mainly affected by the pressure amplitude and driving frequency of the ultrasonic wave, initial inter-bubble distance, and equilibrium radii of bubbles. In this study, if not indicated otherwise, the liquid physical parameters remain unchanged. The secondary Bjerknes forces within a pressure amplitude range of 0.005–0.9 bar are examined. In this manner, full considerations of the influences of the resonance effect (main, harmonic, and subharmonic resonances) are presented. To ensure that the dynamic behaviors of the two bubbles have reached a stable state, 15,000 periods of driving frequencies were employed for each calculation; 50 × 50 (the step size is 2 μm) calculations were performed in each R₁₀-R₂₀ plane. The initial inter-bubble distance was set as 1500–7500 μm, which is significantly larger than the equilibrium radii of bubbles. The parameter values and ranges employed in this study are summarized in Table 1.

Table 1.

The parameter settings for a two-bubble system.

Parameters	Range
Pressure amplitude Pa/bar	0.005/0.1/0.3/0.6/0.9
Driven frequency f0/kHz	45/75/105/135
Initial inter-bubble distance D0/μm	1500/3500/5500/7500
Equilibrium radius of bubble 1 R01/μm	2–100
Equilibrium radius of bubble 2 R02/μm	2–100

3. Results and discussion

The translational motions are mainly governed by the interaction force between the bubbles. Therefore, we first examine the case when R₀₁ is invariant (R₀₁ = 10 and 44 μm), the variations of F_B with R₀₂, as shown in Fig. 3. The linear resonance radius R_res (≈ 45 μm) corresponding to the driving frequency (f₀ = 75 kHz) is marked with an arrow on the coordinate. The positive F_B indicates that the bubbles are mutually repulsive, while the negative F_B indicates that the two bubbles are mutually attractive.

Fig. 3.

The variation of the secondary Bjerknes force F_B with R₀₂. (a)R₀₁ = 10 μm; (b)R₀₁ = 44 μm. P_a is 0.01, 0.1, 0.3, and 0.6 bar. f₀ = 75 kHz and D₀ = 3500 μm. R_res: linear resonance radius corresponding to the driving frequency f₀.

Fig. 3 shows that the F_B reaches a maximum at R₀₂ ≈ R_res, which indicates that the strongest attraction and repulsion forces are generated when the bubble performs its main resonance. If the driving frequency f₀ lies between f_res1 and f_res2, (R₀₁ < R_res < R₀₂; R₀₂ < R_res < R₀₁), the bubbles repulse each other; otherwise (R_res < R₀₁ and R_res < R₀₂; R₀₁ < R_res and R₀₂ < R_res), they attract each other [17], [24], [25]. Therefore, in Fig. 3(a), the direction of F_B transits from negative to positive at R₀₂ ＞ R_res, while transits from positive to negative values at the same range in Fig. 3(b). With P_a increasing, the presence of the 2nd and 3rd order harmonics (resonance radii corresponding to 2f₀ and 3f₀) triggers the sign reversal of F_B, whereas the 1/2 order subharmonic (resonance radius corresponding to 1/2f₀) only significantly enhances F_B instead of reversing its sign. In addition, the position of the resonance radii corresponding to the main, 2nd (3rd), and 1/2 order resonance frequencies of the coupled bubbles gradually moves towards the left with P_a increasing, which is also reflected in the R₀₁-R₀₂ planes in the following sections. This owing to the fact that all the resonance frequencies lean towards lower frequencies due to the “bending phenomenon” in nonlinear dynamics as the P_a increases [37], [55].

Fig. 4 provides the schematic diagram of six types of translational motions of bubbles under the action of F_B: (a) pair with invariant inter-bubble distance, (b) stable pair after attraction, (c) coalescence pair after mutual attraction, (d) stable pair after mutual repulsion, (e) constantly repulsive pair, and (f) chaotic pair. It should be explained that the invariant inter-bubble distance refers to the variation of inter-bubble distance within 15,000 periods of driving frequency is within the initial inter-bubble distance of 10^-4, so it can be ignored.

Fig. 4.

Schematic diagram of the six types of translational motions.

A large range of equilibrium radii of the bubbles was employed to study the translational motions when P_a was 0.1, 0.3, 0.6, and 0.9 bar, as shown in Fig. 5. f₀ and D₀ are 75 kHz and 3500 μm. As shown in Fig. 5(a), for P_a = 0.1 bar, there are four regions divided by R_res in the R₀₁-R₀₂ plane [37]:

• IAttractive regions (F_B < 0): R_res > R₀₁ and R_res > R₀₂; R_res < R₀₁ and R_res < R₀₂;
• IIRepulsive regions (F_B > 0): R₀₁ < R_res < R₀₂; R₀₁ < R_res < R₀₂.

Fig. 5.

The R01-R02 plane distribution diagram for the six types of translational motions. (a) Pa = 0.1 bar, (b) Pa = 0.3 bar, (c) Pa = 0.6 bar, and (d)P_a = 0.9 bar respectively. f₀ = 75 kHz and D₀ = 3500 μm. Different regions are: constantly repulsion pair; stable pair after repulsion; pair with invariant inter-bubble distance, stable pair after mutual attraction; coalescence pair after mutual attraction, chaotic pair.

As shown in Fig. 5, all distributed regions are characterized by a diagonal symmetry. Most of the green region is located at the left bottom corner of the R₁₀-R₂₀ plane. This is because when R₀₁ and/or R₀₂ are small, their radial oscillations are significantly weak, resulting in the nearly negligible translational displacements under low F_B values. As P_a increases, the green region is slightly constricted, which can be explained by the apparent translational velocities caused by the stronger radial oscillations because the translational motions follow the radial oscillations [34].

Additionally, the coalescence bubble pairs are mainly concentrated in two regions: 1) where R₀₁ and R₀₂ are similar (near the diagonal), and 2) where the main, 2nd (3rd, 4th) order harmonics (dotted black circles) and 1/2 order subharmonic (dotted red circles) appear, and the sharply increased F_B causes the bubbles to coalesce. As P_a increases, the 2nd (3rd, 4th) order harmonics and 1/2 order subharmonic led to a sign reversal of the translational motions, which will be further explained. Note, between the region signifying coalescence pair and stable pair after attraction, the chaotic pair (yellow region) with an alternate attraction and repulsion force can be observed.

For large amplitude of acoustic pressure, the strong-nonlinear oscillations make the phases of the bubble oscillations highly uncertain. The collapse and rebound stages of the bubble oscillations lead to a complex and changeable phase relationship between the two bubbles. In particular, when the pressure amplitude is greater than the dynamic Blake threshold (P_a ≈ 1.3 bar), the nonlinear effects of the bubble oscillations are significantly apparent, which causes the difference between the linear resonance frequency and natural frequency of the bubble to be very large. Therefore, it may not be appropriate to divide the different regions of the secondary Bjerknes force (or transitional motions) by the linear resonance radius.

3.1. Transitions of the translation motions of the bubbles

Six pairs of bubbles, indicated by filled circles in Fig. 5(b), are selected to explore the transformation mechanism of the translational motions triggered by the 2nd order harmonic, 1/2 order subharmonic, and chaos. For point a (R₀₁ = 14 & R₀₂ = 22 μm), the effect of the 2nd order harmonic increases as P_a increases, which leads to the aggregated bubbles becoming repulsive [34]. Fig. 6 presents the time-dependent inter-bubble distance of four bubble pairs (point b, c, d and e) when P_a = 0.1, 0.3, 0.6, and 0.9 bar. The translational motions shift from attraction to repulsion or vice versa as P_a increases, indicating that the behavior of bubbles is closely related to P_a. Note, when P_a = 0.1 bar, the translational motions have not reached a stable state within the limited periods; however, they are the candidates for the formation of stable pairs after attraction because significantly longer time is required for the viscous drag force to decelerate the translational velocity and reach a the steady state [34].

Fig. 6.

Variation in the inter-bubble distance of bubble pairs with R₀₁ and R₀₂(a) 50 & 22 μm (point b), (b) 90 & 82 μm (point c), (c) 82 & 48 μm (point d), and (d) 86 & 24 μm (point e) with normalized periods. P_a = 0.1, 0.3, 0.6, and 0.9 bar, f₀ = 75 kHz and D₀ = 3500 μm.

Fig. 7 presents the noise spectra of the radial variations of the bubbles (R₀₁ = 50 μm, R₀₂ = 22 μm) over time and the frequency response curves of the two bubbles when P_a = 0.005 bar (linear condition). Note, when P_a is low and the interaction force between the bubbles is ignored, the values of the linear resonance frequencies of the individual bubble (f_res1 and f_res2) are approximately equal to that of their natural frequencies (f_nat1 = 65 kHz and f_nat2 = 149 kHz). In this case (the inter-bubble distance is significantly larger that their radii), the values of linear resonance frequencies of the coupled bubbles (f_c-res1 and f_c-res2) are also approximately equal to that of f_nat1 and f_nat2. When P_a = 0.1 bar, f_c-res1 (=65 kHz) and f_c-res2 (=149 kHz) are located at the two sides of the frequency of the sound field f (=f₀ = 75 kHz) (Fig. 7(c)), leading to the mutually repulsive translational motions because the F_B caused by the out-of-phase radial oscillations is repulsive. When P_a = 0.3, 0.6 and 0.9 bar, the 2nd order harmonic (2f₀ = 150 kHz) is increasingly pronounced (Fig. 6(b)). f_c-res1 (=65 kHz) and f_c-res2 (=149 kHz) are located at the left side of f (=150 kHz) (Fig. 7(c)), which causes the bubbles to oscillate in phase and finally inverts the translational motions from repulsion to attraction.

Fig. 7.

Noise spectra of the radial variations over time: (a) bubble 1, R₀₁ = 50 μm; (b) bubble 2, R₀₂ = 22 μm when P_a = 0.1, 0.3, 0.6, and 0.9 bar; and (c) frequency response curves of the two bubbles when P_a = 0.005 bar. D₀ = 3500 μm. f₀: (=75 kHz), ultrasonic driving frequency (fundamental frequency). 2f₀: (=150 kHz), 2nd order harmonic frequency. f: frequency. R_max1 and R_max2: maximum radii of the coupled bubbles. f_c-res1 and f_c-res2: linear resonance frequencies of the coupled bubbles. f_nat1 and f_nat2: natural frequencies of the two individual bubbles.

Similarly, the noise spectra of radial variations of the bubbles (R₀₁ = 90 μm, R₀₂ = 82 μm) over time (Fig. 6(b)) and the frequency response curves are also examined in Fig. 8. When P_a = 0.1 or 0.3 bar, the fundamental frequency (f₀ = 75 kHz) is the most predominant among the frequency domain (Fig. 7(a) and (b)). The frequency of the sound field f (=f₀) is greater than f_c-res1 (=36 kHz) and f_c-res2 (=40 kHz) (Fig. 8(c)), which leads to the mutually attractive translational motions caused by the in-phase radial oscillations. When P_a = 0.6 or 0.9 bar, the 1/2 order subharmonic (1/2f₀ = 37.5 kHz) dominates the noise spectra (Fig. 8(a) and (b)). The frequency of the sound field f (=1/2f₀) falls between f_c-res1 and f_c-res2 (Fig. 8(c)), which indicates that the bubbles would repel each other.

Fig. 8.

Noise spectra of the radial variations over time: (a) bubble 1, R₀₁ = 90 μm; (b) bubble 2, R₀₂ = 82 μm, when P_a = 0.1, 0.3, 0.6, and 0.9 bar and (c) the frequency response curves of the two bubbles when P_a = 0.005 bar. D₀ = 3500 μm. f₀ = 75 kHz and D₀ = 3500 μm. f₀: (=75 kHz), ultrasonic excitation frequency (fundamental frequency). 1/2f₀: 1/2 order subharmonic frequency. f: frequency. R_max1 and R_max2: maximum radii of the coupled bubbles. f_c-res1 and f_c-res2: linear resonance frequencies of the coupled bubbles. f_nat1 and f_nat2: natural frequencies of the two individual bubbles.

Fig. 9 further illustrates the variations in the radial oscillations of the bubbles (R₀₁ = 90 & R₀₂ = 82 μm) and the secondary Bjerknes force F_B over the normalization driving period. When P_a = 0.1 or 0.3 bar, the radial oscillation period is equal to the driving period T₀, and F_B is always negative (attraction force) because the phases of the radial oscillations of the two bubbles are synchronized in real-time (Fig. 9(a) and (b)). When P_a = 0.6 or 0.9 bar, the radial oscillations of the bubbles with a period of 2 T₀ indicates that the 1/2 order subharmonic resonance has occurred. The time-averaged F_B in one driving period is positive (repulsion force) because the out-of-phase radial oscillations of the two bubbles are dominant (Fig. 9(c) and (d)).

Fig. 9.

Time evolution of the radial oscillations of the bubbles (R₀₁ = 90 μm & R₀₂ = 82 μm) and the secondary Bjerknes force F_B. (a)P_a = 0.1 bar, (b)P_a = 0.3 bar, (c)P_a = 0.6 bar, and (d)P_a = 0.9 bar. f₀ = 75 kHz and D₀ = 3500 μm.

Meanwhile, the time-dependent radial oscillations of the bubbles (R₀₁ = 82 & R₀₂ = 48 μm) and F_B are also illustrated in Fig. 10. When P_a = 0.1 or 0.3 bar, F_B is always negative (attraction force) owing to the real-time in-phase radial oscillations of the two coupled bubbles (Fig. 10(a) and (b)). When P_a = 0.6 or 0.9 bar, the period of radial oscillation for bubble 2 (blue curves) is equal to T₀, while that of bubble 1 (red curves) is twice as the value of T₀. Despite the aforementioned, the time-averaged F_B in one driving period remains negative (attraction force) because the in-phase radial oscillations are dominant in every driving period (Fig. 10(c) and (d)). Furthermore, the significantly increased F_B caused by the large oscillation amplitudes would result in the polymerization of bubbles (Fig. 6(c)). Note, when P_a = 0.9 bar, the radial oscillation amplitudes are greater than those of P_a = 0.6 bar, which makes the bubbles coalesce within a shorter time than that of P_a = 0.6 bar (Fig. 6(c)). Because the transition mechanisms of the translational motions of bubbles with R₀₁ = 86 & R₀₂ = 24 μm (Fig. 6 (d)) are similar to those shown in Fig. 6(a-c), they are not repeatedly explained.

Fig. 10.

Time evolution of the radial oscillations of the bubbles (R₀₁ = 82 μm & R₀₂ = 48 μm) and the secondary Bjerknes force F_B. P_a = (a) 0.1 bar, (b) 0.3 bar, (c) 0.6 bar, and (d) 0.9 bar. f₀ = 75 kHz and D₀ = 3500 μm.

Fig. 5 shows that the chaotic bubble pair primarily appears between the regions signifying the coalescence pair and stable distance after attraction. For a better understanding, Fig. 11 examines the time evolution of the translational displacements of the bubbles (R₀₁ = 46 μm, R₀₂ = 60 μm) when P_a is 0.1, 0.3, 0.6, and 0.9. When P_a = 0.1 or 0.3 bar, the bubbles constantly attract each one another until an invariable D is reached (Fig. 11 (a) and (b)). However, when P_a = 0.3 bar, the time required to reach a steady state is reduced compared to that of P_a = 0.1 bar. For P_a = 0.6 bar, the translational motions are characterized by the alternate motions of attraction and repulsion (Fig. 11 (c)). For P_a = 0.9 bar, the chaotic translational motions are more violent, and the bubbles eventually coalesce at a certain moment (Fig. 11 (d)).

Fig. 11.

Time evolution of the translational displacements of the bubbles (R₀₁ = 46 μm, R₀₂ = 60 μm) when (a)P_a = 0.1 bar, (b)P_a = 0.3 bar, (c)P_a = 0.6 bar, and (d)P_a = 0.9 bar. f₀ = 75 kHz, D₀ = 3500 μm.

The inter-bubble distances over time are processed based on the Fast-Fourier-transform method, and the corresponding amplitude-frequency curves are presented in Fig. 12. As P_a increases, the noise amplitudes of the translational motions increase in entirety. When P_a = 0.1 bar, only the fundamental (f = f₀) and harmonic spectral lines (f = 2f₀, 3f₀, 4f₀, 5f₀) appear in the spectrum. When P_a = 0.3 bar, in addition to the fundamental and harmonic spectral lines, the broadband noises first appear. When P_a = 0.6 or 0.9 bar, the fundamental and harmonic frequency spectral lines are submerged by the raised broadband noise signals, indicating that the bubbles have possessed chaotic translational behaviors [67], [68]. These analyses are consistent with the results shown in Fig. 11.

Fig. 12.

Noise spectra of the time-dependent inter-bubble distance (R₀₁ = 46 μm, R₀₂ = 64 μm) when (a)P_a = 0.1 bar, (b)P_a = 0.3 bar, (c)P_a = 0.6 bar, and (d)P_a = 0.9 bar. f₀ = 75 kHz, D₀ = 3500 μm.

Fig. 13, Fig. 14 present the radial motion characteristics of the bubble 1 (R₀₁ = 46 μm) when the pressure amplitude P_a = 0.3 and 0.6 bar. When P_a = 0.3 bar, the period of the radial oscillation is equal to T₀ (Fig. 13(a)). The steady-state solution curve periodically repeats with f₀, and the obtained phase trajectory is a closed curve (Fig. 13(b)). There is only one point on the Poincare section (Fig. 13(c)). Only fundamental and harmonic spectral lines are in the power spectrum (Fig. 13(d)). These analyses indicate that radial motion of the bubble 1 is periodic when P_a = 0.3 bar. When P_a = 0.6 bar, bubble 1 oscillates with irregular radii and frequencies (Fig. 14(a)). The motion trajectory is an extremely dense curve that will never close (Fig. 14(b)). A sequence of points with a special fractal structure occurs in the Poincaré section (Fig. 14(c)), suggesting that this bubble is in the chaotic oscillation mode [54], [69]. In addition to the spectral lines, a continuous spectrum composed by many irregular frequency components occurs (Fig. 14(d)). These analyses indicate that the radial motion of bubble 1 is chaotic when P_a = 0.6 bar [67], [68].

Fig. 13.

Radial oscillation characteristics of bubble 1. P_a = 0.3 bar, f₀ = 75 kHz, D₀ = 3500 μm.

Fig. 14.

Radial oscillation characteristics of bubble 1. P_a = 0.6 bar, f₀ = 75 kHz, D₀ = 3500 μm.

Fig. 15 gives the bifurcation diagram of the normalized radii of bubble 1 (R₀₁ = 46 μm) versus the pressure amplitude P_a. When P_a is within [0.4, 0.46 bar], bubble 1 is in a periodic oscillation state, and the chaotic oscillation can be found when P_a is within [0.46, 0.535 bar]. Different from that periodic oscillation to chaos through period-doubling bifurcation, the bubble considering the interaction force suddenly transits from periodic to chaotic state when P_a = 0.46 bar, which reflects the sensitive dependence of the system on variable parameters and is also an important feature of the binary bubble nonlinear system.

Fig. 15.

Bifurcation diagrams of the normalized radii of bubble 1 (R₀₁ = 46 μm) with the pressure amplitude P_a. f₀ = 75 kHz and D₀ = 3500 μm.

The translational motions of the bubbles are determined by the resultant force F_h1 of the secondary Bjerknes force F_B and viscous resistance F_ex1. Fig. 16 gives the time-variations of the translational velocity ${\dot{x}}_1$ and force analysis of bubble 1 (R₁₀ = 46 μm) when P_a = 0.6 bar. The direction from bubble 1 to bubble 2 is negative, while it is positive from bubble 2 to bubble 1. The chaotic pair is characterized by possessing an alternately attractive and repulsive F_B and opposite F_ex1, eventually leading to an alternate F_h1 and${\dot{x}}_1$.

Fig. 16.

(a) Time evolution of translational velocity ${\dot{x}}_1$ and force analysis of bubble 1 (R₀₁ = 46 μm) when P_a = 0.6 bar. (b) The enlarged view in the dash line area of Fig. 16(a). f₀ = 75 kHz and D₀ = 3500 μm.

The inter-bubble distances corresponding to the points a, b, c and d (Fig. 16 (b)) are approximately 400, 220, 240, and 520 μm. Fig. 17 gives the frequency response curves of the bubble 1 (R₀₁ = 46 μm) when D is 400, 220, 240, and 520 μm. The pressure amplitude is 0.005 bar (linear condition). As shown in Fig. 17 that a bubble interacting acoustically with a neighboring bubble has two linear resonance frequencies, both of which are significantly affected by the inter-bubble distance. The left linear resonance peaks of the curves are caused by the coupling effect between the bubbles, whose value represents the linear resonance frequency of another coupled bubble. Differently, the right linear resonance peaks of the curves (plotted in the black dashed frame) are owing to the directly linear response of the bubble to the external ultrasonic driving, the evolution of which is the focus of our next work.

Fig. 17.

(a) The frequency response curves of the bubble 1 (R₀₁ = 46 μm) when D = 400, 220, 240, and 520 μm. (b) The enlarged view in the dash line area of Fig. 17(a). f₀ = 75 kHz and P_a = 0.005 bar. f_c-res1: linear resonance frequency of the coupled bubble 1 (R₀₁ = 46 μm). f_nat1 (=71 kHz): natural frequency of the individual bubble 1.

For convenience of the analysis, the time-dependent bubble oscillations and F_B in one driving period (a: 491st period; b: 494th period; c: 497th period; d: 500th period, marked with filled circles in Fig. 16(b)) are shown in Fig. 18. Apparently, for point a (D = 400 μm), the linear resonance frequency of the coupled bubble 1 f_c-res1 (=72.5 kHz) is slightly greater than f_nat1 (=71 kHz) (Fig. 17(b)). The frequency of the sound field f (=f₀ = 75 kHz) falls on the same side of f_c-res1 (=72.5 kHz) and f_c-res2 (=50 kHz). Thus the time-averaged F_B is negative (attraction force) within one period because the in-phase radial oscillations are dominant in one driving period (Fig. 18(a)), which causes the bubble to accelerate to translate along the attraction direction under the action of F_h1. In this particular case, with increasing time, the f_c-res1 further increases owing to the decrease of D (D = 220 μm, point b), resulting in f (=75 kHz) falling within the range of f_c-res1 (=78.5 kHz) and f_c-res2 (=48.5 kHz) (Fig. 17(a)). Furthermore, there are many continuous spectral components greater or less than f₀ in the power spectrum of bubble 1 (Fig. 14(d)), indicating that the translational motion is determined by more than one time scale [34]. Eventually, these two factors combined to cause the bubble to oscillate out-of-phase (Fig. 18(b)). Correspondingly the F_B is positive (repulsive force), hence the bubble decelerates in the direction of attraction until ${\dot{x}}_1$ decreases to zero. After then, the bubble begins to accelerate in the direction of repulsive. When D reaches 240 μm (point c), f (=75 kHz) is still within the range of f_c-res1 (=76.5 kHz) and f_c-res2 (=49 kHz). Consequently, the bubbles still exhibit out-of-phase oscillations mode, and F_B is positive (Fig. 18(c)). As D increases to a certain extent (e.g. D = 520 μm, point d), the decrease of f_c-res1 causes f (=75 kHz) on the same side as f_c-res2 (=50 kHz) and f_c-res1 (=72 kHz) (Fig. 17(a)). Correspondingly time-averaged F_B becomes negative (Fig. 18(d)). At this time, the bubbles decelerate in the direction of repulsion until the velocity decreases to zero, and then start the subsequent alternate motions of attraction and repulsion. Based on the aforementioned analysis, we can conclude that the interaction between the bubbles can change the resonance frequencies of the coupled bubbles [10], [23], [47], [53], [70]. Thus, as the bubbles translate and proceed closer or further from one another, their net force can change and sign reversal can occur.

Fig. 18.

Time evolution of the radial oscillations and the secondary Bjerknes force F_B. P_a = 0.6 bar, f₀ = 75 kHz and D₀ = 3500 μm.

3.2. Influence of the initial inter-bubble distance

The initial inter-bubble distance is crucial for the secondary Bjerknes forces and dynamic characteristics of bubbles [71], which ultimately results in the changes in translational motions. Fig. 19 examines the distributions of the translational motion types in the R01-R02 plane at D₀ = 1500, 3500, 5500, and 7500 μm. The distribution ranges of the bubble pairs with an invariant D (green region) gradually increase as D₀ increases, which can be explained by the reduced translational velocities caused by the low F_B because F_B is inversely proportional to D [72]. In addition, the chaotic bubble pairs only appear when D₀ = 1500 μm (Fig. 19 (a)) owing to the stronger F_B compared to that of D₀ = 3500, 5500, or 7500 μm (Fig. 19(b, c, or d)). As D₀ increases, the distribution ranges of the coalescence pairs caused by the in-phase radial oscillations of the bubbles, and the main, 2nd order harmonic, and 1/2 order subharmonic, gradually decrease or disappear. These aforementioned phenomena are mainly due to the decreased inter-bubble distances leading to the increased interaction between the two bubbles, which eventually causes the decreased pressure thresholds of the 1/2 order subharmonic resonances and chaos [10], [47], [53].

Fig. 19.

The R01-R02 plane distribution diagram for six types of the translational motions. f₀ = 75 kHz, P_a = 0.3 bar. (a)D₀ = 1500 μm, (b)D₀ = 3500 μm, (c)D₀ = 5500 μm, and (d)D₀ = 7500 μm.

3.3. Influence of the driving frequency

The driving frequency is a key factor affecting the bubble radial dynamic [73]. When P_a = 0.3 bar and D₀ = 3500 μm, Fig. 20 shows the distributions of the translational motion types in the R01-R02 plane when f0 is 45, 75, 105, and 135 kHz. The corresponding R_res values are 75, 45, 32, and 25 μm.

Fig. 20.

The R01-R02 plane diagrams for the translation motion types at (a)f₀ = 45 kHz, (b)f₀ = 75 kHz, (c)f₀ = 105 kHz, and (d)f₀ = 135 kHz. P_a = 0.3 bar and D₀ = 3500 μm.

The boundaries of the divided regions tend to the smaller equilibrium radii because R_res decreases as f₀ increases. The decrease in the distribution range of the coalescence pairs is caused by the main resonance, while the increase in its distribution range is caused by the 1/2 order subharmonic. Nevertheless, the increase caused by the 1/2 order subharmonic is unexpected, since the absorptions and scatterings of the acoustic wave in liquid are mainly caused by large bubbles with a diameter>100 μm formed by bubble aggregations [20], [74]. For f₀ = 45 kHz (Fig. 20(a)), the transitions of the translational motions caused by the 3rd order harmonic resonance (marked by black dashed circle) first appear in the R₀₁-R₀₂ plane. This is because the lower driving frequency leads to more intense nonlinear radial oscillations compared with 75 kHz, 105 kHz, and 135 kHz. Notably, the chaotic pairs (yellow region) also appear in the region between the coalescence pairs and stable pairs after attraction at R₀₁＞R_res and R₀₂＞R_res, as shown in Fig. 20(a).

4. Conclusions

In this study, the transition mechanisms of the translational motions of bubbles caused by the harmonic, subharmonic resonance, and chaos are studied. The influences of the driving frequency f₀ and the initial inter-bubble distance D₀ on the translational motions are also investigated.

The translational motions under the action of F_B are divided into the following: constantly repulsive pair, stable pair after repulsion, pair with invariant inter-bubble distance, stable pair after attraction, coalescence pair after attraction and chaotic pair. As P_a increases, the changes in the oscillation periods of the bubbles lead to the occurrences of harmonic or subharmonic resonances, and eventually cause the phase difference among the nonlinear radial oscillations. Therefore, the translational motions change from the previously repulsed bubbles to attractive bubbles through the action of the 2nd order harmonic wave; meanwhile, the bubbles change from an attractive, stable pair after attraction, repulsive transform to repulsive, coalescence and attractive state through the action of 1/2 order subharmonic wave. Furthermore, the linear resonance frequency of the bubble varying with the inter-bubble distance plus the effect of multi-scale time cause attractive-repulsive alternating transitions of F_B between the two bubbles. These chaotic bubble pairs, whose sizes are within the main and 1/2 order subharmonic resonance radii, usually occur in the region between the coalescence pairs and stable pairs after attraction. Finally, as D₀ increases, the distribution ranges of the bubble pairs with an invariant D gradually increase, while those of coalescence pairs decrease or disappear. With the increase of f₀, the decrease in the distribution range of the coalescence pairs is caused by the main resonance, while the increase in its distribution range is caused by the 1/2 order subharmonic. For f₀ = 45 kHz, the transitions of the translational motions caused by the 3rd order harmonic first appear in the R₀₁-R₀₂ plane, and the chaotic pairs also appear at R₀₁＞R_res and R₀₂＞R_res.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

Data will be made available on request.