Numerical Study on the Dynamics of a Charged Bubble in the Acoustic Field

In this paper, the dynamics of an acoustic bubble with a constant charge in compressible liquid are investigated numerically, which is based on the Gilmore-NASG model to estimate the radial oscillations. The cavitation eﬀects are enhanced due to the presence of the charge on the bubble surface. The obtained results from the present model are compared with that calculated by the previous model within a wide range of parameters (e.g., charge, acoustic pressure amplitude, ultrasound frequency, and liquid temperature). The similar inﬂuences of these parameters on bubble collapse intensity can be observed from both models. Since the present model fully considers the compressibility of gas and liquid, it can be applied to a wider parameter range and leads to the larger predicted values. The research in this paper can provide important insights about the eﬀects of charge on bubble dynamics and the acoustic cavitation applications (e.g., sonochemistry, water treatment, and food industry).


Introduction
Cavitation bubbles exist widely in nature, e.g., the snapping shrimp uses cavitation bubbles formed by the rapid closure of its claws to stun its prey; in a fast flowing system, cavitation bubbles are prone to occur if the channel suddenly narrows and then widens.The phenomenon of cavitation in the liquid is concerned because of its damage to hydraulic machinery and ship propellers (Sezen et  ) (e.g., liquid jet, free radicals formation, radiation pressure, and acoustic microstreaming) caused by the bubble oscillations, cavitation has been contributed to various applications such as water treatment (Ferkou et al., 2015), petroleum hydrocarbons degradation (Lei et al., 2020), nanoparticle synthesis (Pokhrel et al., 2016), and so on.
The charges carried on the bubble surface have been reported by many research groups.The zeta potential of the bubble measured experimentally by using the microelectrophoresis technique indicated that the effect of the PH solution on the variation of the bubble zeta potential depended not only on the type of the metal ions but also on the electrolyte concentration (Yang et al., 2001).They also found the charge polarity varied at different solutions (e.g., the bubble was the negative charge in NaCl solutions, while its charge polarity reversed in multivalent metal ions solutions).Takahashi (2005) found that the bubbles were negatively charged under a wide range of the PH condition and positively charged under strongly acidic conditions.In (Lee, Choi, 2020), the stable light emission of a single bubble sonoluminescence (SBSL bubble) with charge in water was investigated for the first time.The results revealed that the SBSL bubble was positively charged and suggested that it was necessary for analyses of the SBSL bubble to take the electrical properties into consideration.The behaviour of lasergenerated bubbles in an electric field was studied by Phukan et al. (2023).They found the maximum bubble radius increased with the increase of the electric field intensity.This effect was more pronounced in the presence of an acetone medium and decreased successively in ethanol and water media owing to their varying magnitudes of electrical conductivity.In addition to experimental researches of the bubbles in an electric field, theoretical studies have also made great progress.The volume mode and shape model dynamics were examined in a weakly viscous dielectric fluid under the uniform and the axisymmetric straining electric field (Oh et al., 2001).A model was built describing the violent collapse of the bubble in the homogeneous, irrotational, solenoidal, and unsteady electric field (Spelt, Matar, 2006).The research group optimized the bubble model in the electric field, and further analyzed the bubble dynamics of violent collapse, translation and shape deformations (Shaw et al., 2009).
Nevertheless, there are few studies on the charged bubbles that undergo acoustic cavitation, which has been applied in many fields (e.g., ultrasonic cleaning, drug delivery, and inactivate viruses).Based on the study of the stability of a charged bubble in the dielectric liquid in (Grigor'ev, Zharov, 2000), the model of a charged bubble excited by ultrasound wave has been deduced by Hongray et al. (2014;2015).The study results proved that the effective surface was reduced due to the presence of charge.Compared with the uncharged bubble oscillations under acoustic excitation, the bubble expanded to a larger radius and compressed to a smaller size, which in turn caused the bubble collapse to be stronger.The bifurcation diagrams have also been studied to some extent, with the presence of charge leading to advance bifurcations.
The theoretical and experimental researches have been proved a lot on the charge bubbles in liquid.The theoretical study of bubble dynamics in the acoustic field is mainly based on uncharged bubbles, which will lead a bias in calculation results.The model of charged bubbles was derived from the Keller-Miksis equation (Hongray et al., 2014;2015).When the effect of charge is taken into account, the bubble collapse will be enhanced resulting in the significant increase of the speed of a bubble wall, and the Mach number can easily reach 1, which is the critical value that the Keller-Miksis equation is safely adopted (Zilonova et al., 2018).Therefore, it is necessary to establish a model that can be used in large parameter intervals, especially the high pressure amplitude and low frequency excitation.In this paper, based on the Gilmore-NASG model (Denner et al., 2021), a more applica-ble dynamic model of charged bubbles is established and compared with the model (Hongray et al., 2014;2015) in detail within a wide range of parameters.Subsequently, the sections of present paper are organized as follows.In Sec. 2, the model of a charged bubble and the numerical method are introduced.In Sec. 3, the calculation results of the present model are quantitatively compared with those of the previous model.In Sec. 4, the main findings of the present paper are summarized.

Mathematical model and simulation method
For simplicity, the following assumptions are used in physical models: (1) the bubble is spherically symmetric; (2) the fluid is Newtonian and compressible; (3) the buoyancy force and gravity are neglected; (4) the bubble-bubble interaction is neglected; (5) the thermal conductivity, phase change and mass transport across the bubble-liquid interface are neglected.The radial dynamics of the bubble is governed by the Gilmore equation (Denner, 2021): where R is the instantaneous radius of the bubble, the overdot denotes the time derivative, c l is the speed of sound in the liquid at the bubble wall, H is the difference between the enthalpy of the liquid at the bubble wall and at infinity.The state of gas and vapor inside the bubble, and the liquid outside the bubble are described by the NASG equation of state, and the expression is (Denner, 2021): where p is the pressure, v is the specific volume, T is the temperature, Γ is the polytropic exponent, C v is the heat capacity at a constant volume, b is the co-volume that represents the volume occupied by the individual molecules, B is a pressure constant that models molecular attraction.
H and c l are defined as Eqs.( 3) and ( 4), respectively: where Γ l is the liquid polytropic exponent, p l is the pressure in the liquid at the bubble wall, B l is the liquid pressure constant, p ∞ is the liquid pressure at infinity, ρ l and ρ ∞ are the densities of liquid at the bubble wall and at infinity, respectively.The expressions of p l , p ∞ , ρ l , and ρ ∞ are given as Eqs.( 5)-( 8): where σ is the surface tension coefficient of the liquid, µ is the viscosity of the liquid, Q is the charge at a bubble surface, ε (i.e., ε = 85ε 0 , ε 0 is the vacuum permittivity) is the liquid permittivity, P l,0 is an ambient pressure in the liquid, P a is the acoustic pressure, f is the ultrasonic frequency, K l and p g are the constants representing the liquid reference state and gas pressure inside the bubble, respectively, and expressions are defined as Eqs.( 9) and ( 10): where ρ l,0 is the predefined reference liquid density, P g,0 is the predefined reference gas pressure, ρ g,0 is the predefined reference gas density, R 0 is the initial bubble radius, Γ g is the gas polytropic exponent, B g is the gas pressure constant.The temperature of the gas and the liquid at the bubble wall can be calculated by Eq. ( 11): where T 0 and P 0 are the reference temperature and pressure, respectively.Equations ( 1)-( 11) constitute the model of present study, which is called G-M-N-C model.The bubble radial dynamics, sound velocity, liquid density at the bubble wall, gas pressure, gas temperature and liquid temperature are obtained from Eqs. (1), ( 4), ( 7), (10), (11), respectively.The fourth term on the right side of Eq. ( 5) is introduced by considering the charge on the bubble surface.If this term is ignored, the G-M-N-C model is reduced to the model in (Denner, 2021), which is called the G-M-N model in this paper.
To check the validation of the simulation results, the bubble dynamics obtained by the G-M-N-C model is compared with previous models.A single gas bubble with an initial radius of 3.5 µm oscillating in the water is considered.If not specified, the parameters in present studies are used in Table 1.The models are solved using the method of the Runge-Kutta 4-5 order formula with a variable step length.To obtain the results satisfying the precision requirement, both of the absolute error and relative error are 1 × 10 −12 .

Results and discussion
Before showing the dynamics obtained with the G-M-N-C model under various conditions, the other models (i.e., G-M-N in (Denner, 2021); K-M-C in (Hongray et al. 2014;2015)) are adopted to calculate the bubble dynamics under a high amplitude driving acoustic wave in order to conduct the comparative analysis.
Figure 1 shows the predictions obtained by three models, i.e., G-M-N-C, K-M-C, and G-M-N.Most of the variables associated with the bubble are periodic oscillations, except that the sound velocity, the density and temperature of liquid at the bubble wall remain constant calculated by the model of K-M-C.From Fig. 1a, the bubble grows slowly in the initial stage and then rapidly expands to its maximum radius (R max ).Correspondingly, the gas temperature and pressure inside the bubble reach the minimum values (Figs.1c  and 1d).At this time, the bubble begins to collapse due to the difference between the inside and outside of the bubble having a maximum value.When the bubble collapse to the minimum radius (R min ), according to Figs. 1b-1g, the variable values (i.e., bubble wall velocity, gas temperature and pressure, the temperature, density and sound velocity of liquid on the bubble wall) reach the maximum.After the first collapse, the bubble oscillates slightly several times.
d) e) f) Comparing to the predictions obtained by the model of G-M-N, the maximum variable values (i.e., the gas temperature and pressure, bubble wall velocity, liquid temperature, density and sound speed) calculated by the model of G-M-N-C are larger.This owes to the larger absolute value of the maximum pressure difference at the bubble wall in the model of G-M-N-C with the consideration of the charge on the bubble surface.The bubble can absorb more energy during its expansion, reaching a larger size, and a smaller size will be acquired as the bubble collapses.
The dynamics of the charged bubble are commonly predicted by the K-M-C model, which is compared in detail with the G-M-N-C model.As reported in Fig. 1a, the lower R min is achieved from the G-M-N-C model (0.42 µm, compared to 0.44 µm for the K-M-C model).Since NASG equation of state is used in G-M-N-C model to describe dynamic features of the gas in the bubble and the liquid at the bubble wall, and the compressibility of the gas in the bubble is fully considered.It is believed that the covolumes of gas molecules should be different with temperature and pressure.The high temperature and pressure environment caused by the bubble collapse results in a smaller covolume of gas molecule, meaning that there is more space for the gas to compress.Therefore, the bubble collapse depth is deeper, and the minimum bubble radius is smaller.Consequently, the bubble can be compressed more, yielding a much higher gas temperature and pressure inside the bubble, as indicated in Figs.1c and 1d.This result matches the finding in (Nazari-Mahroo et al., 2018; 2020).
The G-M-N-C model gives the maximum liquid temperature and density at the bubble wall reaching 414.67 K and 1.37 × 10 3 kg/m 3 , respectively, whereas these two variables keep constant in the K-M-C model from Figs. 1e and 1f.The calculations of the Mach number (Ma) for two models are shown in Fig. 1h.It can be seen that the peak Ma calculated by the G-M-N-C model is significantly lower than that predicted by the K-M-C model.The liquid velocity of sound at the bubble wall varies with time in the G-M-N-C model (Fig. 1g), at the instant of the first bubble collapse, the bubble wall velocity and the liquid velocity of sound at the wall are both large (Figs.1b and 1g), reducing Ma (e.g., Ma = 0.55 from Fig. 1h), and the reliability of the model can be guaranteed.While the liquid velocity of sound at the bubble wall remains constant in the K-M-C model leading to a larger Ma during the first bubble collapse (e.g., Ma = 0.73 from Fig. 1h).
The K-M-C model evolves from the Keller-Miksis equation taking the liquid compressibility into account, which is accurate at Ma < 1 (Zilonova et al., 2018).The G-M-N-C model is derived from the Gilmore equation, which is obtained basing on the variation of liquid sound velocity and integrating the liquid enthalpy directly instead of the liquid pressure.Hence, the bubble dynamics with charge estimated by the G-M-N-C model is accurate for Ma ≤ 2.2 (Zilonova et al., 2018).Figure 2 summarizes the maximum absolute value of Ma in P a (1.2-3 atm)f (20-400 kHz) plane for two models.Within the parameters studied, the Ma calculated by the K-M-C model range from 0.02 to 10.60, while the values obtained by the G-M-N-C model are 0.02-4.76.Under a certain parameter, Ma calculated by the latter model is smaller than that from the former model.The example can be seen in Fig. 1h.The positions of the blue curves are Ma = 1 in Fig. 2a and  2.2 in Fig. 2b, respectively, which are the critical values of Ma for the K-M-C model and the G-M-N-C model applied in numerical studies.Under the excitation parameters on the upper left of the curves, the application of the model is safe, while the lower right is not.It can be seen that the G-M-N-C model has a wider application range than the K-M-C model.
In application researches, evaluating the cavitation intensity by calculating the gas temperature or the emitted sound pressure, which is a difficult task, because of the need to obtain the bubble wall velocity, or the acceleration.To this end, it is necessary to measure the cavitation intensity from the radial dynamics of the bubble.There exist several approaches in the literatures, e.g., the compression ratio (R max R min ) (Hongray et al., 2014; Kalmár et al., 2020; Nazari-a)  , where t c is the bubble collapse time.In the present study, the compression ratio is used to describe the cavitation intensity and is represented by C s .
The effect of charge Q on the compression ratio C s is shown in Fig. 3.As can be seen, C s increases linearly with increasing Q from 0 to 0.6 pC for both models.At the coordinate value of (0.49, 77.36), the two prediction curves intersect.At this time, the minimum bubble radii calculated by two models are similar, and the  values are about 0.41 µm, which is near the hard-core radius for the bubble with an initial radius of 3.5 µm.Therefore, the variation curves of C s predicted by two models can only intersect at this point.Also, this result proves the accuracy of the present model.When Q value is away from 0.49 pC, the difference of C s calculated by two models is increasing.At Q < 0.49 pC, the C s obtained by the G-M-N-C model is larger than that predicted by the K-M-C model.It is worth noting that in the K-M-C model, when Q is greater than 0.53 pC, bubble collapse intensity increases, resulting in Ma > 1 (e.g., at Q = 0.55 pC, Ma = 1.03), so that the accuracy of the model is lost.Nevertheless, the range of Q is 0-0.6 pC, and the value of Ma is 0.49-0.61lying in the application range of the G-M-N-C model.
In order to seek the correlation between the compression ratio C s and the pressure amplitude P a , numerical studies have been performed for various P a (1.2-3.0 atm).The observed results are presented in Fig. 4. With the increase of P a , C s calculated by two models increases monotonically.The predicted C s of the G-M-N-C model is always higher than that of the K-M-C model.The difference of C s calculated by two models is about 10.02% at P a from 1.2 to 2.2 atm.When P a is between 2.2-3.0 atm, the difference increases significantly, and the difference value reaches the maximum at P a = 2.4 atm, which is 51.51%.The effect of ultrasound frequency f on the compression ratio C s is explored as presented in Fig. 5.It is observed that C s decreases by increasing f .It is a wellknown trend of cavitation bubble dynamics.With the increase of f , there has no enough time to grow for the bubble, resulting in the decrease of C s .The cavitation intensity is weakened, and the bubble temperature and pressure are reduced.For the two considered models, C s in the G-M-N-C model is larger than that in the K-M-C model.The difference of C s calculated by two models reaches the maximum value of 140.51% at the frequency of around 700 kHz.
Other reports in the literatures supported the variation trends of the compression ratio C s with the pres- Figure 6 shows the variation of the compression ratio C s as a function of liquid temperature T ∞ .The cavitation intensity can be enhanced by the increase of T ∞ .As can be seen from the figure, the difference (around 10%) between predicted values by two models is not strongly affected by T ∞ from 283 to 333 K.The similar results obtained by Merouani and co-workers (Chadi et   ble collapse was enhanced with the increase of liquid temperature (Chadi et al., 2018).But the overall production ratio and sonochemical degradation of nonvolatile organic pollutants in aqueous media reached their maximum values at T ∞ = 50 ○ .The cavitation effect is mainly affected by the single bubble collapse intensity and the number of bubbles, and the latter decreased with the increase in liquid temperature.Dehane et al. (2021b) found the production of RCS, ⋅ OH, ⋅ H, HCl and HOCl increased proportionately with the increase of liquid temperature (from 10 ○ to 50 ○ ) on carbon tetrachloride sono-conversion under acoustic excitation with the intensity of 0.7 W/cm 2 and the ultrasonic frequency of 355 kHz.
Based on the Gilmore-NASG model, the acoustic bubble dynamic model is established in this paper considering the effect of the charge on the bubble surface.
The paper analyze the bubble dynamics and cavitation intensity, along with the accuracy of the present model by comparing with the results in the literature.The paper proves the model has a wider range of application parameters.In the future work, present model can be optimized.For example, the effects of bubble-bubble interaction (Zhang et al., 2016) should be taken into account to investigate the secondary Bjerknes force between two gas bubbles, that coupled with a viscous drag force to analyze the translational motions of the bubbles.The detail results are refered to the works in (Wang et al., 2023).In addition, heat exchange, mass transport and chemical reactions on the dynamics of the bubble with a constant charge are also worth considering (Dehane et

Conclusion
In the present work, the model for a charged bubble under acoustic excitation is proposed.Compared with the previous model, the proposed model can be applied to a wider parameter range, especially high acoustic amplitudes and low ultrasound frequencies excitation.The bubble collapse intensity calculated by two models is enhanced with increasing charge on the bubble surface, acoustic amplitude and liquid temperature, and the decrease of ultrasound frequency.Except for the charge on the bubble surface, the bubble collapse intensity by the present model is larger under the studied parameter range, mainly because the effect of liquid and gas compressibility is fully considered.When the charge is 0.49 pC, the minimum bubble radius calculated by two models is near the hard-core radius, and the bubble collapse intensity is equal.The differences between predicted values by two models reaches the maximum at P a = 2.4 atm or f = 700 kHz, and the liquid temperature has little effect on the difference (around 10%).

Fig. 1 .
Fig.1.Time developments of the bubble radius (a), the bubble wall velocity (b), the gas temperature (c) and pressure (d) inside the bubble, the liquid temperature (e), density (f) and sound velocity (g) at the bubble wall, and the Mach number of the bubble wall (h) as retrieved by G-M-N-C model (solid line), K-M-C model (dash line) and G-M-N model (dash dot line).The bubble driven by an ultrasound wave with frequency of 30 kHz and amplitude of 1.35 atm.

Fig. 2 .
Fig. 2. Maximum absolute value of Mach number of the bubble wall calculated by K-M-C model (a) and G-M-N-C model (b).The numerical results are obtained from more than 200 combinations of ultrasound frequency (20-400 kHz) and acoustic amplitude (1.2-3.0 atm).

Fig. 3 .
Fig. 3. Compression ratio Cs vs. charge on the bubble, calculated by G-M-N-C model (solid line) and K-M-C model (dash line) for ultrasound frequency of 30 kHz and amplitude of 1.35 atm.

Fig. 4 .
Fig. 4. Compression ratio Cs vs. acoustic amplitude, calculated by G-M-N-C model (solid line) and K-M-C model (dash line) for ultrasound frequency of 30 kHz and charge of 0.3 pC on the bubble.

Fig. 5 .
Fig. 5. Compression ratio Cs vs. ultrasound frequency, calculated by G-M-N-C model (solid line) and K-M-C model (dash line) for ultrasound amplitude of 1.35 atm and charge of 0.3 pC on the bubble.

Fig. 6 .
Fig. 6.Compression ratio Cs vs. liquid temperature, calculated by G-M-N-C model (solid line) and K-M-C model (dash line) for ultrasound frequency of 30 kHz, acoustic amplitude of 1.35 atm and charge of 0.3 pC on the bubble.