Numerical investigation on acoustic cavitation characteristics of an air-vapor bubble: Effect of equation of state for interior gases

Highlights

• •Effects of equation of state on acoustic cavitation characteristics are assessed.
• •Peng–Robinson (PR) equation of state can accurately predict gas thermodynamics.
• •Gilmore-PR model predicts a stronger bubble collapse rather than Gilmore-vdW model.
• •The stronger the bubble collapse is, the differences between two models increase.

Abstract

The cavitation dynamics of an air-vapor mixture bubble with ultrasonic excitation can be greatly affected by the equation of state (EOS) for the interior gases. To simulate the cavitation dynamics, the Gilmore-Akulichev equation was coupled with the Peng–Robinson (PR) EOS or the Van der Waals (vdW) EOS. In this study, the thermodynamic properties of air and water vapor predicted by the PR and vdW EOS were first compared, and the results showed that the PR EOS gives a more accurate estimation of the gases within the bubble due to the less deviation from the experimental values. Moreover, the acoustic cavitation characteristics predicted by the Gilmore-PR model were compared to the Gilmore-vdW model, including the bubble collapse strength, the temperature, pressure and number of water molecules within the bubble. The results indicated that a stronger bubble collapse was predicted by the Gilmore-PR model rather than the Gilmore-vdW model, with higher temperature and pressure, as well as more water molecules within the collapsing bubble. More importantly, it was found that the differences between both models increase at higher ultrasound amplitudes or lower ultrasound frequencies while decreasing as the initial bubble radius and the liquid parameters (e.g., surface tension, viscosity and temperature of the surrounding liquid) increase. This study might offer important insights into the effects of the EOS for interior gases on the cavitation bubble dynamics and the resultant acoustic cavitation-associated effects, contributing to further optimization of its applications in sonochemistry and biomedicine.

Nomenclature

a (b)

Coefficients for the EOS

A_h (B_h, C_h)

Constants in Henry’s law

B (m)

Specific constants

$\overline{c}$

Average velocity of molecules [K·mol·g⁻¹]

C₀

Equilibrium speed of sound [m·s⁻¹]

C_l

Speed of speed at the bubble surface [m·s⁻¹]

c_p

Heat capacity at constant pressure [J·m⁻³·K⁻¹]

c_s

Transient concentration of molecules [mol·m⁻³]

c_∞

Concentration of dissolved gas [mol·m⁻³]

C_v

Heat capacity at constant volume [J·K⁻¹]

D

Diffusion coefficient [m²·s⁻¹]

e_th

Thermal energy of molecules [J]

E

Energy inside a bubble [J·mol⁻¹]

f

Ultrasound frequency [Hz]

f_i

Degree of freedom of species i

h_w

Molecular enthalpy [J]

H

Molar specific enthalpy [J·mol⁻¹]

k₁ (k₂)

Specific constants for the EOS

k_B

Boltzmann constant [J·K⁻¹]

K_H

Henry’s constant

L_diff

Diffusion layer thickness

L_th

Thermal boundary layer thickness

${\dot{m}}_g$

Mass flux of non-condensable gases [mol·s⁻¹]

${\dot{m}}_{\mathit{vap}}$

Evaporation–condensation rate of water [mol·s⁻¹]

M

Molar mass [kg·mol⁻¹]

n

Molar amount [mol]

N

Number of particle [mole]

P₀

Static pressure [Pa]

P_A

Ultrasound amplitude [Pa]

P_ac

Driving acoustic pressure [Pa]

P_in

Gas pressure inside bubble [Pa]

P_w

Liquid pressure at the interface [Pa]

$\dot{Q}$

Heat exchange [W·m⁻³]

R

Instantaneous bubble radius [μm]

R₀

Ambient bubble radius [μm]

R_m

Compression ratio

R_g

Universal gas constant [J·mol⁻¹·K]

t

Time [s]

T

Temperature inside a bubble [K]

T₀

Ambient temperature [K]

T_s

Interface temperature [K]

V

Volume of the bubble [m³]

V_m

Molar volume of the mixture [cm³·mol⁻¹]

Greek symbols

θ_η

Characteristic vibrational temperature [K]

λ

Thermal conductivity [W·m⁻¹·K]

ε

Mass transport coefficient

κ

Thermal diffusivity [m²·s⁻¹]

μ

Liquid viscosity [Pa·s]

ρ_l

Density of the medium [kg·m⁻³]

ρ_vap

Time-varying density [kg·m⁻³]

${\rho }_{\mathit{vap}}^{\mathit{sat}}$

Saturated density of vapor [kg·m⁻³]

σ

Surface tension of liquid water [N·m⁻¹]

Subscript

air

Air

Ar

Argon

c

Critical state

i

Species

mix

air-vapor mixture

N₂

Nitrogen

O₂

Oxygen

tot

Total

vap

Water vapor

Superscript

·

Time derivative

1. Introduction

Acoustic cavitation refers to the emergence, expansion, and violent collapse of small gas–vapor bubbles in liquid medium under sufficiently strong ultrasonic excitation [1]. At the end of violent collapse, extremely high temperature and pressure (up to thousands of Kelvins and hundreds of Megapascals) are reached inside the bubble, which plays an essential role in practical applications [2], [3], [4], [5]. These extreme conditions can further lead to the decomposition of the entrapped water molecules within the bubbles to form various free radicals, which have been extensively utilized in the chemical industry and environmental applications, such as extraction [6], distillation [7], ultrasonic cleaning [8], sonolysis [9], [10], etc. These cavitation-associated physical and chemical effects are closely related to the strength of acoustic cavitation in liquids. Therefore, gaining a deep understanding of the acoustic cavitation characteristics is of great significance, especially the extremely high temperature and pressure generated instantaneously within the violently-collapsing cavitation bubble.

Up to now, numerous experiments have been carried out to measure the temperature and pressure during the collapse of cavitation bubbles with ultrasonic excitation. Through the sonoluminescence experiment, Gaitan et al. [11] measured a maximum temperature of 6000 K and a maximum pressure even up to 2000 MPa during the bubble collapse at a frequency of 21 kHz. Similarly, it was also reported that the internal temperature is estimated to be between 5000 K and 20,000 K by fitting the experimental sonoluminescence spectra of a single bubble [12]. Moreover, Kanthale and Ashokkumar [13] examined multi-bubble sonoluminescence and sonochemistry, obtaining maximum temperatures of 10,300 K at 213 kHz and 700 K at 1056 kHz respectively. By using comparative rate thermometry, Suslick et al. [14] obtained that the temperature in the collapsing cavity of the alkane solutions is 5200 ± 650 K, and the pressure is 50 MPa. Overall, these studies provide experimental estimations of the temperature within a wide range of 700–20,000 K and the measured pressures are within the range of 50–2000 MPa. The obvious discrepancies in the experimentally measured temperature and pressure might be due to the different experimental methods and conditions. Additionally, acoustic cavitation of bubbles occurs at extremely short temporal scales (μs ∼ ms) and small spatial scales (μm ∼ mm). It can considerably increase the difficulty of experiments with high equipment requirements for more accurate measurements of the peak temperature and pressure at the end of violent bubble collapse.

In addition to the experimental measurements, the researchers have also conducted numerical predictions to further investigate the bubble dynamics as well as the time-varying temperature and pressure during acoustic cavitation, regardless of the experimental limitations. It was reported that Ahmed et al. [15] coupled the Keller–Miksis (KM) model with the ideal gas (IG) law, predicting the maximum temperatures up to 10,000 K and the maximum pressures up to the order of 10¹¹ Pa. Then, Ferkous et al. [16] also coupled the KM model with the IG law and the peak values of the temperature and pressure within the collapsing cavity were estimated to be approximately 5000 K and 100 MPa, respectively. Moreover, Yasui [17] used the Van der Waals (vdW) equation of state (EOS) to study the non-equilibrium effects of evaporation and condensation at the cavitation bubble wall and predicted a maximum temperature of 10⁴ K and a maximum pressure of 10¹⁰ Pa. Furthermore, the most developed Gilmore model was coupled with a van der Waals hard-core law to investigate the bubble dynamics in laser-induced cavitation [18], [19]. It was found that upon the first collapse, the collapsed bubble contained more vapor than gas and had a pressure of 13.5 GPa [19]. In addition, by coupling the Gilmore model with the Noble-Abel-Stiffend-Gas (NASG) EOS to simulate the dynamics of single-bubble cavitation, a maximum temperature inside the bubble of 10⁴ K was estimated at the end of bubble collapse [20]. Most recently, Agrež et al. utilized the Gilmore-NASG model to calculate the cavitation bubble temporal evolution and pressure, which are well consistent with the experimental measurements [21]. In these numerical investigations, different bubble dynamics models (e.g., KM model and Gilmore model, etc.) and different equations of state (e.g., IG law, vdW EOS, and NASG EOS, etc.) were coupled to investigate the time-varying bubble dynamics of acoustic cavitation, consequently predicting the gas temperature and pressure within the bubble. Evidence from these theoretical results has established that there exist obvious deviations in the predicted results using numerical simulations as compared to the experimental data. This difference could be attributed to the inherent limitations of model simplifications and assumptions, the inconsistent parameter settings between experiments and simulations, as well as the potential errors in both experiments and simulations, etc. For an accurate simulation, the proper choice of the EOS to describe the gases within the cavitation bubble is crucial. It not only links the gas temperature, pressure and volume simultaneously, but also can readily obtain the thermodynamic properties of the gases that failed to be measured directly from experiments during the transient acoustic cavitation. More importantly, it has been proved that using the vdW EOS to estimate the state of the gases is more accurate than the IG law [22]. Compared to the vdW EOS, the Peng–Robinson (PR) EOS has more advantages in calculating vapor pressure, saturated liquid volumes, saturated vapor volumes and enthalpies of vaporization [23], [24]. Accordingly, it can be expected that choosing the PR EOS to estimate the thermodynamic properties of the gases within the cavitation bubble can provide more accurate results. However, little attention has been paid to combining the PR EOS with the cavitation model for predicting the acoustic cavitation characteristics with a high accuracy.

This study coupled the more accurate PR EOS with the most developed Gilmore model, i.e., a newly developed Gilmore-PR model, to accurately quantify the cavitation dynamics and the resultant temperature, pressure as well as number of gas molecules within the bubble, with a focus on the impacts of the EOS. Specifically, the Gilmore-Akulichev equation was respectively coupled with the PR and vdW EOS to numerically simulate the cavitation dynamics with considering the effects of mass and heat transfers. The thermodynamics of the interior mixed gases (i.e., water vapor and air) predicted by the two equations of state were first compared. Then, the differences in the acoustic cavitation characteristics (i.e., the bubble collapse strength, maximum gas temperature and pressure, as well as number of gas molecules within the bubble) between the two equations of state were comprehensively examined under various parameters, including the ultrasound amplitude, the ultrasound frequency, the initial bubble radius, as well as the surface tension, viscosity and temperature of the surrounding liquid.

2. Gilmore-PR model

Suppose that the bubble initially consists of water vapor (H₂O) and air (78 % N₂, 21% O₂, 1% Ar) with only radial motions. Moreover, it is assumed that the bubble remains spherical throughout the whole cavitation process and does not break after the collapse [25]. Note that the bubbles may lose their spherical stability and exhibit non-spherical oscillations under certain conditions [26]. The non-spherical oscillations are neglected in the present work for the sake of simplification, similar to previous studies [15], [17], [20], [22]. Due to the geometrical symmetry of the bubble, it can be assumed that the internal temperature and pressure are spatially uniform. The effects of mass and heat transfers at the bubble interface are considered without chemical reactions.

2.1. Radial bubble motion

According to previous studies [27], [28], the Gilmore model and a newly developed bubble wall motion equation with the second-order Mach number are more suitable for investigating bubble oscillations with high Mach number. In this work, a relatively simple generalized Gilmore model is utilized to describe the radial bubble dynamics, which is given by [29]:

$$\left(1-\frac{\dot{R}}{C_l}\right)R\ddot{R}+\frac{3}{2}{\dot{R}}^2\left(1-\frac{\dot{R}}{3C_l}\right)=\left(1+\frac{\dot{R}}{C_l}\right)H+\frac{R}{C_l}\left(1-\frac{\dot{R}}{C_l}\right)\dot{H}$$ (1)

where R is the bubble radius and the dots represent the derivative with respect to time, H and C_l are the enthalpy and the sound speed at the bubble surface respectively, which are expressed as [30]:

$$H=\frac{{\left(B+P_0\right)}^{1/m}}{{\rho }_l}\frac{m}{m-1}\left[{\left(B+P_w\right)}^{\left(m-1\right)/m}-{\left(B+P_0+P_{\mathit{\text{ac}}}\right)}^{\left(m-1\right)/m}\right]$$ (2)

$$C_l={(C_0^2+(m-1)H)}^{1/2}$$ (3)

in which ρ_l and C₀ are the medium density and speed of sound in the medium, m and B are the specific constants whose values can be retrieved in the literature [31]. P₀ is the static pressure and $P_{\mathit{\text{ac}}}=-P_A\sin \left(2\pi ft\right)$ is the driving acoustic pressure where P_A is the ultrasound amplitude and f is the ultrasound frequency. According to pressure equilibrium conditions, the following relationship between the liquid pressure at the interface P_w and the internal pressure P_in of bubble can be obtained:

$$P_w=P_{\mathit{in}}-\frac{4\mu \dot{R}}{R}-\frac{2\sigma }{R}$$ (4)

in which μ and σ are the liquid viscosity and surface tension, respectively.

2.2. Peng-Robinson equation for interior gases

The internal pressure P_in is calculated from the IG law and the vdW EOS as presented in previous studies [32], [33]. In the present work, a more accurate PR EOS is used and its general form can be represented as [23]:

$$P_{\mathit{in}}=\frac{R_{g}T}{V_m-b_{\mathit{mix}}}-\frac{a_{\mathit{mix}}}{V_m^2+k_1{b}_{\mathit{mix}}{V}_m+k_2{b}_{\mathit{mix}}^2}$$ (5)

where V_m and T are the molar volume of the mixture and the temperature within the bubble, and R_g is the universal gas constant. k₁ and k₂ are specific constants for the EOS. Conveniently, the PR EOS (k₁ = 2, k₂ = −1) can readily reduce to the vdW EOS (k₁ = k₂ = 0). The coefficients a_mix and b_mix are calculated in Appendix A.

2.3. Mass transport during bubble expansion and collapse

During acoustic cavitation, water vapor will evaporate or condense due to the dis-equilibrium between its partial pressure within the cavitation bubble and the pressure in the surrounding medium. The time-dependent change rate of water vapor at the wall of bubble is calculated by the following formula [34]:

$${\dot{m}}_{\mathit{vap}}=\epsilon \overline{c}\frac{4\pi R^2}{M_{\mathit{vap}}}\frac{\left[{\rho }_{\mathit{vap}}^{\mathit{sat}}-{\rho }_{\mathit{vap}}\left(R,t\right)\right]}{4}$$ (6)

$$\overline{c}=\sqrt{\frac{8R_g{T}_s}{\pi M_{\mathit{vap}}}}$$ (7)

where ${\dot{m}}_{\mathit{vap}}$ is the evaporation–condensation molar rate of water vapor. $\epsilon$ is mass transport coefficient, $\overline{c}$ is the average velocity of molecules, ${\rho }_{\mathit{vap}}$ and $M_{\mathit{vap}}$ are the time-varying density and molar mass of water vapor. According to previous studies [35], [36], the temperature at the bubble wall is $T_{\text{s}}\text{=}{T}_0$, where $T_0$ is the ambient temperature of the surrounding medium. The saturated density of vapor ${\rho }_{\mathit{vap}}^{\mathit{sat}}$ is expressed as [37]:

$$\begin{array}{c}{\rho }_{\mathit{vap}}^{\mathit{sat}}=322\exp \left({ϛ}_1{\vartheta }^{2/6}+{ϛ}_2{\vartheta }^{4/6}+{ϛ}_3{\vartheta }^{8/6}+{ϛ}_4{\vartheta }^{18/6}\right)\\ \left(+{ϛ}_5{\vartheta }^{37/6}+{ϛ}_6{\vartheta }^{71/6}\right)\end{array}$$ (8)

with ${ϛ}_1$ = −2.03150240, ${ϛ}_2$ = −2.68302940, ${ϛ}_3$ = −5.38626492, ${ϛ}_4$ = −17.2991605, ${ϛ}_5$ = −44.7586581, ${ϛ}_6$ = −63.9201063 and ϑ = (1-T₀/T_c), where T_c = 647.37 K. Note that as the vapor partial pressure/temperature in the bubble reaches the water critical pressure/temperature, evaporation and condensation at the bubble wall are assumed to halt due to no clear distinction between liquid and gaseous phases [38].

The mass flux of non-condensable gases ${\dot{m}}_{g,i}$ is given by the boundary layer theory and Fick’s law [39], [40], [41]:

$${\dot{m}}_{g,i}=4\pi R^2{D}_i\frac{\partial C}{\partial r}{|}_{r=R}=4\pi R^2{D}_i\frac{c_{\infty }-c_s}{L_{\mathit{diff}}}$$ (9)

$$L_{\mathit{diff}}=\min \left(\sqrt{\frac{R{D}_i}{|\dot{R}|},\frac{R}{\pi }}\right)$$ (10)

where D_i is the diffusion coefficient of gas species i (i.e., N₂, O₂ and Ar), $c_{\infty }=P_0{K}_H^{-1}$ is the concentration of dissolved gas, and $c_s=P_{\mathit{in}}\left(n_g/n_{\mathit{tot}}\right)K_H^{-1}$ is the transient concentration of molecules at the bubble wall. n_tot is the number of total moles inside the bubble, n_g is each non-condensable gas species mole inside the bubble and Henry’s constant K_H can be calculated as follows [42]:

$$K_{\mathit{\text{H}}\text{,}\mathit{\text{i}}}={\left[\frac{{\rho }_l}{P_0{M}_i}\exp \left(A_h+\frac{B_h}{{\tau }_h}+C_h\ln {\tau }_h\right)\right]}^{-1},{\tau }_h=\frac{T_0\left(\text{K}\right)}{100}$$ (11)

where M_i is the molar mass of each gas species i. The values of A_h, B_h, and C_h for gas species are given in Table 1 [43], [44], [45].

Table 1.

The alphabetical constants in Henry’s law.

Materials	Ah	Bh	Ch
Nitrogen[43]	−67.4	86.3	24.8
Oxygen[44]	−66.7	87.5	24.5
Argon[45]	−57.7	74.8	20.1

2.4. Heat transfer at the bubble wall and gas temperature

Similar to mass transfer, the heat exchange $\dot{Q}$ and thickness of thermal boundary layer L_th are expressed by [40]:

$$\begin{array}{c}\dot{Q}={\left(4\pi R^2{\lambda }_{\mathit{mix}}\frac{\partial T}{\partial r}\right|}_{\text{r}=R}=4\pi R^2{\lambda }_{\mathit{mix}}\left(\frac{T_0-T}{L_{\mathit{th}}}\right)\\ L_{\mathit{th}}=\min \left(\sqrt{\frac{R\kappa }{\left|\left(\dot{R}\right|\right)}},\frac{R}{\pi }\right)\end{array}$$ (12)

where λ_mix and $\kappa$ are the coefficients of thermal conductivity and diffusivity of an air-vapor mixture, respectively. The detailed calculating procedures are illustrated in Appendix B.

Furthermore, the overall equation governing the internal energy change of bubble is calculated by [39], [46], [47]:

$$\dot{E}\text{=}\dot{Q}-P_{\mathit{in}}\dot{V}+\sum _i^4(h_{w,i}-e_{\mathit{th},i}){\dot{N}}_i$$ (13)

$$h_{w,i}=\left(1+\frac{f_i}{2}\right)k_B{T}_0;e_{\mathit{th},i}=\frac{f_i}{2}{k}_{B}T+\sum _1^{\eta }\frac{k_B{\theta }_{\eta }}{\exp \left({\theta }_{\eta }/T\right)-1}$$ (14)

in which E is the energy inside the bubble, $\dot{V}=4\pi R^2\dot{R}$ is the change rate of bubble volume, h_w,i is the molecular enthalpy, e_th,i is the thermal energy of the molecule and ${\dot{N}}_i$ is the rate of the molecular number change caused by mass diffusion. η is the number of the characteristic vibrational temperatures θ_η. and f_i is the number rotational and translational degrees of freedom. Values of θ_η, and η, f_i for gases are given in Table 2.

Table 2.

The constants for each gas species [40].

Materials	θη (K)	η	fi
Water	2295,5255,5400	3	6
Nitrogen	3350	1	5
Oxygen	2273	1	5
Argon	/	/	3

By combining Eqs. (12), (13), (14), the temperature change with respect to time inside the bubble is calculated by [39], [46], [47]:

$$\dot{T}=\frac{\dot{E}}{C_{v,mix}}=\frac{\dot{Q}-P_{\mathit{in}}\dot{V}+{\sum }_i^4(h_{w,i}-e_{\mathit{th},i}){\dot{N}}_i}{C_{v,mix}}$$ (15)

$$C_{v,mix}=k_{\mathit{\text{B}}}\sum _{\text{i}=1}^4\left(\frac{f_{\mathit{\text{i}}}}{2}+\sum _1^{\eta }\left(\frac{{\left({\theta }_{\eta }/T\right)}^2\exp \left({\theta }_{\eta }/T\right)}{{\left(\exp \left({\theta }_{\eta }/T\right)-1\right)}^2}\right)\right)N_i$$ (16)

where $C_{v,mix}$ is the heat capacity of the air-vapor mixture.

In the simulation, the used physical parameters and their default values are given in Table 3 unless specified otherwise. The Runge-Kutta method is used for numerical integration of coupled ordinary differential equations.

Table 3.

Physical parameters.

Parameters	Definition	Value	Unit
R0	Initial bubble radius	6	μm
ρl	Ambient liquid density	998.2	kg·m−3
C0	Ambient liquid sound speed	1482	m·s−1
σ	Surface tension	0.073	N·m−1
μ	Viscosity	0.001	Pa·s
T0	Ambient temperature	293.15	K
f	Ultrasound frequency	45	kHz
P0	Static pressure	1.01325 × 105	Pa
PA	Ultrasound amplitude	1.42	atm
m	Empirical constant	7	/
B	Empirical constant	3.1309× 108	Pa
Rg	Universal gas constant	8.314472	J·mol−1·K
kB	Boltzmann constant	1.38065 × 10-23	J·K−1
$\epsilon$	Accommodation coefficient	0.4	/
Mvap	Molar mass of water vapor	18.015268 × 10-3	kg·mol−1
Mair	Molar mass of air	28.97 × 10-3	kg·mol−1
MN2	Molar mass of nitrogen	28 × 10-3	kg·mol−1
MO2	Molar mass of oxygen	31.9988 × 10-3	kg·mol−1
MAr	Molar mass of argon	39.95 × 10-3	kg·mol−1

3. Results and discussion

3.1. Pressure–temperature phase diagram for gases

The interior gases of the bubble are described by the EOS which directly determines the thermodynamic properties of the gases, consequently affecting the bubble dynamics as well as the resultant temperature, pressure and number of gas molecules within it [22]. As one of the commonly used equations to estimate the gases, the vdW EOS has a simple formula but it might result in the estimation with less accuracy. Comparatively, it is now well established that the PR EOS is one of the most potential and successful equations of state for liquid/gas thermodynamics and volumetric calculations for pure substances and mixtures [23], [24], [48], [49]. Each substance inside the air-vapor mixture bubble (i.e., water vapor, N₂, O₂, and Ar) has a different critical temperature, and the phase change is assumed to halt when the critical temperature is reached, according to previous studies [38], [50]. Therefore, we explore here how much the saturation pressures of the gases predicted by the PR and vdW EOS are different below the critical temperature of each gas, prior to investigating the acoustic cavitation characteristics of the air-vapor mixture bubble. The predicted pressure–temperature phase diagrams of the water vapor, N₂, O₂, and Ar by both PR and vdW EOS were compared, as shown in Fig. 1(a)−(d) respectively. The solid lines are the gas saturation curves for the PR (red) and vdW (black) EOS respectively, and the green point is the critical point. Compared to the vdW EOS, it is apparent that the saturation pressures of each gas predicted by the PR EOS are much closer to the experimental data of NIST (blue points) [51] within the whole temperature range. This evidence strongly suggests that the PR EOS can accurately predict the thermodynamic properties of the gases within the cavitation bubble. Moreover, the results also indicate that choosing the EOS could notably affect the predictions on the acoustic cavitation characteristics of the bubble due to the distinct differences in the thermodynamic properties of the interior gases estimated by the EOS. Therefore, the better PR EOS was selected to estimate the gases inside the bubble for a higher accuracy.

Fig. 1.

Pressure–temperature phase diagrams of (a) water vapor, (b) Nitrogen (N₂), (c) Oxygen (O₂), and (d) Argon (Ar) below the critical temperature of each gas, predicted by the Peng–Robinson (PR) (red line) and the Van der Waals (vdW) (black line) equations of state. The green point is the critical point. The experimental data of saturation pressure P_sat (blue points) were obtained from the NIST [51].

3.2. Comparison of acoustic cavitation between two equations of state

To examine the effects of the vdW and RP EOS on the acoustic cavitation of the air-vapor mixture bubble, the time-varying evolutions of the cavitation bubble radius, and the temperature, pressure, as well as number of gas molecules within it were first compared, as illustrated in Fig. 2. The simulation conditions were set as R₀ = 6 µm, f = 45 kHz, P_A = 1.42 atm, P₀ = 1 atm, T₀ = 293.15 K. As shown in Fig. 2(a), during the bubble expansion phase, no distinguishable difference can be observed between the two equations of state, resulting in the same maximum radius R_max (approximately 37 µm). This result is in line with that reported by Ahmed et al. [15], who obtained a similar value for the maximum radius (about 40 µm) at P_A = 1.42 atm and f = 40 kHz. More importantly, it means that choosing the EOS seems to exhibit little influence on the bubble dynamics at this stage. However, during the collapse stage, the minimum bubble radius obtained with the vdW EOS is larger than the one with the PR EOS, as shown in the inset of Fig. 2(a). This difference would lead to distinct estimations for the bubble collapse strength, which can be evaluated by several important indicators, such as the bubble wall velocity [52], compression ratio (R_max/R_min) [53], expansion ratio (R_max/R₀) [54], [55], relative expansion ratio (R_max − R₀)/R₀ [56] and ratio of maximum oscillation radius to the collapse time ($R_{max}^3/t_c$) [57]. The compression ratio (R_max/R_min) was chosen here to characterize the bubble collapse strength. Accordingly, there is a larger compression ratio (R_m = R_max/R_min), in other words, the collapse strength predicted by the PR EOS would be stronger. Taking the vdW EOS as a reference, the relative difference in the compression ratio is up to 12%, which can significantly affect the temperature and pressure within the violent collapsing bubble.

Fig. 2.

Temporal evolutions of (a) bubble radius, (b) temperature, (c) pressure, and (d) number of gas molecules inside the bubble as predicted by the Peng–Robinson (PR) and the Van der Waals (vdW) EOS. Simulation conditions were set as R₀ = 6 µm, f = 45 kHz and P_A = 1.42 atm. Gas molecules include water vapor molecules and air molecules. Inset in each figure denotes the locally magnified part of bubble evolution around the end of collapse. The blue point represents the water critical temperature, and the position pointed by the blue dotted arrow corresponds to the time when the water critical temperature is located.

For the maximum temperature and pressure inside the collapsing bubble, the theoretical measurements are within the range of 6000 K and several thousand MPa under 1.5 atm of ultrasound amplitude [22]. As shown in Fig. 2(b) and (c), the maximum temperature and pressure predicted by the two equations of state are about 5000 K and 5000 MPa respectively, which are consistent with the theoretical measurement data [22]. Compared to the results using the vdW EOS, the maximum temperature predicted by the PR EOS exhibits an obvious increase (approximately 741 K), with a relative difference of 16%, as presented in Fig. 2(b). Similarly, it can be observed in Fig. 2(c) that the increment in the maximum pressure predicted by the PR EOS is up to 144 MPa (a relative difference of about 3%). Taken together, the predicted maxima of both temperature and pressure with the vdW EOS are smaller than those with the PR EOS, which are consistent with the results of the compression ratio. The larger the compression ratio is, the stronger the bubble collapse is and consequently the higher the peak values of temperature and pressure of collapsing bubble can get.

Furthermore, the most striking result to emerge from the Fig. 2(d) is that the number of water vapor molecules within the bubble greatly changes over time, taking the non-condensable air gas as a reference. These results strongly indicate that the effects of evaporation and condensation of water vapor at the wall of bubble are more significant than the diffusions of non-condensable gases. In addition, it has been demonstrated that water vapor has an important impact on the temperature and pressure at the moment of collapsing bubble [35], [58]. As a result, the present study mainly discusses the impacts of water molecules rather than the air molecules. Noticeably, the number of water vapor molecules predicted by the vdW EOS is apparently different from that predicted by the PR EOS. In particular, at the end of the collapse phase, the number of water vapor molecules trapped in the collapsing bubble is distinctly higher for the PR EOS compared to the vdW EOS. It might be explained that below the water critical temperature (647.37 K), more water vapor molecules are generated within the bubble via evaporation for the case of PR EOS relative to the vdW EOS, as illustrated by the inset of Fig. 2(b) and (d). Overall, these results confirm that choosing the EOS has an obvious influence on the simulation results, showing distinct differences in the acoustic cavitation characteristics, especially for the minimum bubble radius, the peak temperature and pressure, as well as the number of water vapor molecules within the bubble at the moment of transient collapse of the bubble.

3.3. Effects of mass and heat transfers

In addition to the mass transport mechanism (i.e., evaporation and condensation of water vapor), it has been found that the heat transfer on the bubble interface would also play a critical role in the acoustic cavitation process [59], [60]. Therefore, the effects of both mass and heat transfers on the acoustic cavitation characteristics of the air-vapor mixture bubble were elucidated based on the PR EOS. Specifically, the effects of mass and heat transfers on the evolution of the bubble radius, temperature, pressure and number of water vapor molecules were presented as a function of time, as shown in Fig. 3(a)–(d) respectively. The model only considering the heat transfer is denoted as Q, the model only considering the non-equilibrium evaporation and condensation is denoted as m, and the model considering both of them is denoted as m + Q (i.e., normal model).

Fig. 3.

(a) Bubble radius, (b) temperature, (c) pressure, and (d) number of water vapor molecules inside the bubble as a function of time with three models (with mass transfer, denoted as m (light yellow lines); with heat transfer (blue lines), denoted as Q; with mass and heat transfers (light grey lines), denoted as m + Q) based on the Peng–Robinson (PR) EOS. Simulation conditions were set as R₀ = 6 µm, f = 45 kHz and P_A = 1.42 atm. Inset in the figure denotes the magnified part of pressure inside the bubble evolution around the end of collapse.

As shown in Fig. 3(a), the maximum radius with the normal model is found to be largest. The decrease of the maximum bubble expansion in the other two models is due to the energy deficit. During the expansion, the bubble loses some work, which should be recovered by heat conduction and water evaporation towards the bubble interior. Therefore, once the heat transfer or mass transport is ignored for the sake of simplification, the less energy absorbed by the expanding bubble is not able to reach the maximum radius with the normal model. Moreover, according to the Ref. [59], heat conduction has a higher energetic influence on the energy inside the bubble compared to water evaporation during the bubble expansion. Consequently, R_max in the model with heat transfer is greater than the model with mass transfer. To evaluate the bubble collapse strength in different cases, the values of the compression ratios are classified in an increasing order as follows: 40.92 for the model without heat transfer, 50.39 for the normal model and 51.31 for the model without mass transfer. These results evidently confirm that the bubble collapse predicted by the model without mass transfer is much stronger than those for the other two cases [59].

In Fig. 3(b)–(d), compared to the normal model, the model without mass transfer (blue lines) gives a higher peak temperature and pressure, while the number of water vapor molecules is obviously lower. In this case, less water vapor molecules within the bubble would decrease the heat capacity of the air-vapor mixture and results in an increase in temperature. Meanwhile, the bubble is easy to compress, and the pressure inside the bubble is substantially increased. On the other hand, when ignoring heat conduction through the bubble wall (light yellow lines), the peak temperature has the largest value since a huge amount of energy is prevented from being released when the bubble collapses without the heat transfer. Moreover, the corresponding peak pressure and the number of water vapor molecules inside the bubble have lower values as compared with the normal model (light grey lines). The bubble collapse strength in this case is much weaker than that predicted by the normal model due to a smaller compression ratio (40.92 vs. 50.39), resulting in a decrease in the internal pressure and less vapor molecules trapped inside the bubble.

Bearing in mind all these arguments, bubble cavitation characteristics are obviously different with or without considering mass and/or heat transfers. This indicates that the mass and heat transfers play a vital role in the acoustic cavitation. Furthermore, these results also highlight the necessity to consider the effects of mass and heat transfers while investigating the bubble acoustic cavitation, otherwise the simulated results might have a large deviation from the realistic results.

3.4. Parameters affecting the acoustic cavitation

3.4.1. Effects of the ultrasound amplitude

This section mainly investigated the effects of the ultrasound amplitude P_A on the acoustic cavitation characteristics and the influences of the two equations of state were also compared. As indicated in Fig. 4(a), the compression ratio (R_m = R_max/R_min) for both equations of state increases with an increase in P_A, which is consistent with the results of Merouani et al. [61]. They found that higher R_max and lower R_min (i.e., higher R_m) values were obtained when higher ultrasound amplitudes were applied. By contrast, the value of R_m is greater for the PR EOS compared to the vdW EOS. It can be explained that the two equations predict the same R_max regardless of the variation of ultrasound amplitude, and the R_min predicted by the PR EOS is always smaller than the one predicted by the vdW EOS (Fig. 2(a)). Moreover, the difference of R_m between the two equations of state gradually increases as P_A increases and the relative difference in R_m is increased to 12% at 2 atm. For the maximum temperature T_max inside the bubble, it shows that as P_A increases, the T_max predicted by the vdW EOS continues to increase, while it first increases and then slightly decreases as predicted by the PR EOS (Fig. 4(b)). In addition, it is apparent from Fig. 4(c) and (d) that both the maximum pressure P_max and number of water vapor molecules inside the bubble at the end of collapse N_vap increase with the increase of P_A using either the PR or vdW EOS. These increasing trends may be explained by the growth of R_m with an increase in P_A, resulting in a more intense bubble collapse. This is in line with the results of the previous literatures [22], [52], [62], which found that as P_A increases, the bubble collapse strength will be stronger.

Fig. 4.

Variations of (a) compression ratio R_m, (b) maximum temperature T_max, (c) maximum pressure P_max, and (d) number of water vapor molecules N_vap inside the bubble at the end of collapse with respect to the ultrasound amplitude P_A. The effects of the Peng–Robinson (PR) and the Van der Waals (vdW) EOS on the acoustic cavitation characteristics at various P_A were compared. Simulation conditions were set as R₀ = 6 µm and f = 45 kHz.

It should be pointed out that the amount of water vapor trapped within the bubble predicted by the PR EOS is much larger at a higher P_A than that predicted by the vdW EOS (Fig. 4(d)). At a higher P_A, there are more water vapor molecules within the bubble for the PR EOS, which reduces the compression heating [38], [63]. This provides a possible explanation for the slight decrease of T_max with the continuous increase of P_A in the case of the PR EOS shown in Fig. 4(b).

Compared to the vdW EOS, the values of the T_max, P_max and N_vap in the PR EOS are considerably larger, probably due to a more intense bubble collapse predicted by the PR EOS (Fig. 4(a)). Meanwhile, what stands out in Fig. 4 is that the differences between the two equations of state increase as P_A increases. Similar results have been reported in the work of Kerboua [22] who compared the effects of the IG and vdW EOS on the acoustic cavitation characteristics and found that the differences between the two models increase with the ultrasound amplitude increasing. Overall, the larger P_A is, the stronger the acoustic cavitation effect is, and the greater the differences between the equations of state become.

3.4.2. Effects of the initial bubble radius and the ultrasound frequency

Fig. 5 shows the contour plots of the differences of the acoustic cavitation characteristics predicted by the PR and vdW EOS as a function of the initial bubble radius R₀ and ultrasound frequency f. It can be found in Fig. 5(a) that the difference of the compression ratios ΔR_m increases with a decrease in the R₀, f, or both of them. Moreover, when the R₀ and f are smaller, it can be seen from Fig. 5(b) that the difference of the maximum temperatures ΔT_max is larger. Similarly, decreasing the f can also increase the difference of the maximum pressure ΔP_max but only exhibits a distinct effect at smaller values of the R₀ and f (Fig. 5(c)). For the number of internal water vapor molecules at the final moment of bubble collapse, its difference ΔN_vap between the PR and vdW EOS decreases with an increase in the f but increases with R₀ increasing, and the effects were most pronounced only at larger R₀ and lower f, as displayed in Fig. 5(d). Previous researches have established that the smaller R₀ and/or f are beneficial to get the larger collapse pressure and stronger acoustic cavitation effects [22], [59]. Therefore, the results suggest that the increased differences of T_max and P_max between the two equations of state at lower R₀, f, or both of them are closely and positively correlated with the stronger acoustic cavitation effects.

Fig. 5.

Contour plots of the differences between the Peng–Robinson (PR) and the Van der Waals (vdW) EOS with respect to various initial bubble radius R₀ and frequencies f at P_A = 1.42 atm, including the differences in (a) compression ratio ΔR_m, (b) maximum temperature ΔT_max, (c) maximum pressure ΔP_max, and (d) number of water molecules ΔN_vap inside the bubble at the end of collapse.

3.4.3. Effects of the surface tension and viscosity of the surrounding liquid

Contour plots of the differences of the acoustic cavitation characteristics predicted by the PR and vdW EOS as a function of the surface tension σ and viscosity μ of the surrounding liquid were plotted in Fig. 6. It has been demonstrated that increasing the σ or μ can lead to a weaker bubble collapse with reducing the acoustic cavitation effects [64], [65], [66]. Similarly, Fig. 6 shows that with increasing the σ, μ, or both of them, the differences between the two equations of state decrease due to the abated bubble collapse. Overall, these results suggest that the better PR EOS should be used to predict the acoustic cavitation characteristics of the bubble and can obtain stronger bubble collapse, higher collapse temperature and pressure, as well as more water molecules within the collapsing bubble as compared to the vdW EOS, especially for the cases at a lower σ and/or μ where the acoustic cavitation effects are much stronger.

Fig. 6.

Contour plots of the differences between the Peng–Robinson (PR) and the Van der Waals (vdW) EOS for the (a) compression ratio ΔR_m, (b) maximum temperature, ΔT_max, (c) maximum pressure, ΔP_max, and (d) number of water molecules inside the bubble at the end of collapse ΔN_vap with respect to various surface tension σ and the liquid viscosity μ. Simulation conditions were set as R₀ = 6 µm, f = 45 kHz and P_A = 1.42 atm.

3.4.4. Effects of the liquid temperature

Fig. 7 displays the variations of the acoustic cavitation characteristics under different liquid temperatures T_l. From the obtained results, it is apparent that the liquid temperatures strongly affect the collapse strength of the cavitation bubble. Specifically, as shown in Fig. 7(a), when the liquid temperature T_l is small, the R_m predicted by the PR and vdW EOS both increase slightly and then decrease dramatically with increasing T_l. This is due to the growth rate of the predicted minimum bubble radius being larger than that of the maximum bubble radius at high liquid temperatures [67]. The resultant T_max, P_max and N_vap as function of T_l are presented in Fig. 7(b)–(d), respectively. It is evident that the P_max exhibits the same variation trend as that of the R_m, the T_max decreases while the N_vap increases with T_l increasing, which agree well with the variation trends reported in Refs. [38], [67]. The R_m is lower at higher T_l, leading to less violence of the bubble collapse, and consequently both T_max and P_max inside the bubble become smaller. The possible explanation for the increase of N_vap with T_l increasing is that the saturated water vapor pressure increases when the T_l increases (Fig. 1(a)), resulting in an increase in the evaporation rate of vapor at the expansion stage, and a decrease in the mass condensation rate at the collapse phase [67]. Consequently, the number of water vapor molecules trapped within the collapsing bubble increases.

Fig. 7.

Variations of (a) compression ratio R_m, (b) maximum temperature T_max, (c) maximum pressure P_max, and (d) number of water molecules N_vap inside the bubble at the end of collapse with respect to the liquid temperature T_l. The effects of the Peng–Robinson (PR) and the Van der Waals (vdW) EOS on the acoustic cavitation characteristics were compared at various T_l. Simulation conditions were set as R₀ = 6 µm, f = 45 kHz and P_A = 1.42 atm.

From Fig. 7(b) and (d), within the whole range of the T_l, the T_max and N_vap calculated by the PR EOS are larger than those calculated by the vdW EOS. Nevertheless, it is noteworthy that when exceeding the liquid temperatures indicated by the blue arrows (R_m: T_l > 353 K, P_max: T_l > 308 K), the R_m and P_max predicted by the PR EOS are lower than those of the vdW EOS (Fig. 7(a) and (c)), which is well explained in Fig. 8. The minimum bubble radius R_min predicted by the PR EOS is larger at higher liquid temperatures indicated by the blue arrow (T_l > 353 K), whereas the EOS has a negligible effect on the maximum bubble radius R_max, as illustrated in Fig. 8(a). Accordingly, the PR EOS predicts smaller compression ratios (i.e., R_max/R_min) at higher liquid temperatures (T_l > 353 K) as compared to the vdW EOS, as shown in Fig. 7(a). On the other hand, ΔP_in is the difference of the gas pressures inside the bubble at the time of R(t) = R_min and R_max, i.e., ΔP_in = P_max − P_Rmax, where P_Rmax is the pressure at R(t) = R_max. It is obvious that both equations of state predict the same values of the P_Rmax for all T_l, whereas at higher liquid temperatures indicated by the blue arrow (T_l > 308 K), the values of ΔP_in in the case of the PR EOS are lower than those in the case of the vdW EOS. In accordance with the expression of ΔP_in, a lower P_max will consequently be obtained with the PR EOS at a higher temperature as compared to the vdW EOS (Fig. 7(c)).

Fig. 8.

Variation curves of the (a) bubble radius and (b) pressure inside the bubble as a function of liquid temperature T_l, predicted by the Van der Waals (vdW) and the Peng–Robinson (PR) EOS. R_max,: maximum bubble radius, R_min: minimum bubble radius, P_Rmax: pressure inside the bubble at R_max, ΔP_in: the difference of the P_max and P_Rmax. Simulation conditions were set as R₀ = 6 µm, f = 45 kHz and P_A = 1.42 atm.

4. Conclusion

This study set out to numerically investigate the acoustic cavitation characteristics of the air-vapor mixture bubble, by combining the Gilmore-Akulichev equation and the PR EOS (i.e., Gilmore-PR model). As compared to the experimental measurements, it can be found that the PR EOS can predict the gas thermodynamics for both air and water vapor more accurately than the vdW EOS. Moreover, the simulation results obtained with the more accurate PR EOS were also compared with those obtained using the vdW EOS to highlight the effects of the EOS. The results obtained with the two equations of state exhibit similar variation trends for the bubble compression ratio, the maximum temperature and pressure, as well as the number of water vapor molecules within the bubble, while varying the acoustic parameters (e.g., ultrasound amplitude and ultrasound frequency), the initial bubble radius and the liquid parameters (e.g., surface tension, viscosity and ambient temperature). However, it demonstrates that the differences between the results from the two equations of state increase at a higher ultrasound amplitude or a lower ultrasound frequency. By contrast, with the increase of the initial bubble radius and the liquid parameters, the differences between the two equations of state decrease. Compared to the vdW EOS, the PR EOS can predict a stronger bubble collapse, thus achieving higher internal temperature and pressure, as well as more water vapor molecules within the bubble.

The current study provides valuable insights into the impacts of the EOS on the acoustic cavitation characteristics of the bubble and might be contributing to further optimizing its utilization in sonochemistry and biomedicine. For instance, the coated microbubbles consist of C₃F₈ or C₄F₁₀ gases are commonly used in the biomedical applications, and the effects of the EOS for the interior gases as well as the influences of the mass and heat transfers on the bubble dynamics are usually ignored for simplification [68], [69], [70]. The current results suggest that choosing a proper equation describing the state of the C₃F₈ or C₄F₁₀ gases inside the cavitation bubble, and meanwhile considering the mass and heat transfers at the bubble wall are of great importance for accurately simulating the bubble dynamics. Furthermore, the comprehensive Gilmore-PR model developed in this work can also extend the prediction of the recently developed models for acoustic waves propagation properties in a bubbly liquid to higher Mach numbers [71], [72]. Specifically, at high driving acoustic pressures, the pressure-dependent attenuation and sound speed in a bubbly medium as well as the influences of the EOS on them can be accurately evaluated, which will gain better insights on the attenuation and sound speed phenomena of bubbly media. Nevertheless, the chemical reactions inside the bubble and the heat of reactions [59], as well as the interactions between bubbles [55], [73], [74] and the non-spherical bubble oscillations [26] are ignored, in the future, their effects on the acoustic cavitation characteristics of bubbles need to be investigated in detail.

CRediT authorship contribution statement

Dui Qin: Conceptualization, Methodology, Visualization, Writing – original draft, Writing – review & editing, Funding acquisition. Shuang Lei: Methodology, Software, Visualization, Writing – original draft, Writing – review & editing. Bo Chen: Visualization, Validation. Zhangyong Li: Writing – review & editing. Wei Wang: Writing – review & editing, Project administration, Resources. Xiaojuan Ji: Conceptualization, Project administration, Resources.

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.

Appendix A. The coefficients of the vapor and air mixture

The values of coefficients a_mix and b_mix of the mixtures are calculated by [75]:

$$a_{m\text{ix}}=\sum _{\text{u}}\sum _v\left(1-k_{\mathit{uv}}\right)x_u{x}_v{\left(a_u{a}_v\right)}^{0.5}$$ (A1)

$$b_{\mathit{mix}}=\sum _u{x}_u{b}_u$$ (A2)

in which x_u (x_v) and k_uv are the mole fraction and interaction parameters, respectively. When the components have the same structure, the coefficient k_uv is zero. In contrast, the coefficient k_uv is non-zero in the calculations of phase-equilibrium for mixtures consisting of compounds with great differences in molecular shape and size [75]. In this Section, subscript u and v indicate that the water vapor and air components (i.e., N₂, O₂, Ar). Therefore, the coefficient k_uv is zero and expressions (A.1), (A.2) are as follows:

$$a_{\mathit{mix}}={\left(\frac{N_{\text{air}}}{N_{\mathit{tot}}}\right)}^2{a}_{\mathit{air}}\text{+}{\left(\frac{N_{\mathit{vap}}}{N_{\mathit{tot}}}\right)}^2{a}_{\mathit{vap}}\text{+ 2}\sqrt{a_{\mathit{air}}{a}_{\mathit{vap}}}\left(\frac{N_{\text{air}}}{N_{\mathit{tot}}}\right)\left(\frac{N_{\mathit{vap}}}{N_{\mathit{tot}}}\right)$$ (A3)

$$b_{\mathit{mix}}=\left(\frac{N_{\text{air}}}{N_{\mathit{tot}}}\right)b_{\mathit{air}}+\left(\frac{N_{\mathit{vap}}}{N_{\mathit{tot}}}\right)b_{\mathit{vap}}$$ (A4)

where $N_{\mathit{tot}}$ is the sum of the number of water vapor and air molecules within the bubble, i.e., $N_{\mathit{tot}}=N_{\mathit{vap}}+N_{\text{air}}$.

For the coefficients of water vapor:

$$a_{\mathit{vap}}={\mathrm{\Omega }}_a\frac{{(R_g{T}_c)}^2}{P_c}\alpha b_{\mathit{vap}}={\mathrm{\Omega }}_b\frac{R_g{T}_c}{P_c}$$ (A5)

For the coefficients of air:

$$a_{\mathit{air}}={(\sum x_v{a}_v^{0.5})}^2{b}_{\mathit{air}}=\sum x_v{b}_v$$ (A6)

$$a_v={\mathrm{\Omega }}_a\frac{{(R_g{T}_c)}^2}{P_c}\alpha b_v={\mathrm{\Omega }}_b\frac{R_g{T}_c}{P_c}$$ (A7)

where R_g is the universal gas constant and α is a function of reduced temperature for each component. The subscript c represents the critical condition. From the data in Ref. [23], the Van der Waal equation of state (vdW-EOS) has Ω_a = 0.421875, Ω_b = 0.1250, and the Ω_a = 0.457240, Ω_b = 0.0778 is set for the Peng–Robinson equation of state (PR-EOS). α = 1 used in the vdW-EOS, and the expression for α in the PR-EOS is written as:

$$\alpha ={\left(1+\chi (1-T_r^{0.5})\right)}^2$$ (A8)

where $T_r=T/T_c$ is reduced temperature and $\chi$ is a function of the acentric factor w, which has the following form:

$$\chi =0.37464+1.54226w-0.26992w^2$$ (A9)

The values of P_c, T_c, and w of each component are listed in Table A1 [76].

Table A1.

Values of critical and acentric factor for each of materials [76].

Materials	Pc (bar)	Tc (K)	w
Water	221.20	647.37	0.343
Nitrogen	33.90	126.20	0.037
Oxygen	50.43	154.58	0.020
Argon	48.70	150.58	0.000

Appendix B. The thermal diffusivity and conductivity of the vapor and air mixture

The thermal diffusivity κ of the vapor and air mixture can be expressed as [40]:

$$\kappa =\frac{{\lambda }_{\mathit{mix}}}{c_p}$$ (B1)

$$c_p=\sum \frac{f_i+2}{2}{k}_B\frac{N_i}{V}$$ (B2)

where c_p, f_i and k_B are the heat capacity concentration of an air-vapor mixture, number of degrees of freedom for gas species i and Boltzmann constant, respectively. N_i/V is the molecular concentration, V = 4πR³ / 3 is the bubble volume and λ_mix is the thermal conductivity of the air-vapor mixture, which is given as follows [77]:

$$\begin{array}{c}{\lambda }_{\mathit{mix}}=\frac{b_{m\text{ix}}}{V_m}\left(\frac{1}{y}+1.2+0.755y\right){\lambda }_i\\ y=\frac{b_{m\text{ix}}}{V_m}+0.6250{\left(\frac{b_{m\text{ix}}}{V_m}\right)}^2+0.2896{\left(\frac{b_{m\text{ix}}}{V_m}\right)}^3+0.1150{\left(\frac{b_{m\text{ix}}}{V_m}\right)}^4\end{array}$$ (B3)

where b_mix is the coefficients of the air-vapor mixture and the ${\lambda }_i$ is expressed as [78]:

$${\lambda }_i={\left(\frac{n_{\mathit{vap}}}{n_{\mathit{tot}}}\sqrt{{\lambda }_{\mathit{vap}}}+\frac{n_{\mathit{air}}}{n_{\mathit{tot}}}\sqrt{{\lambda }_{\mathit{air}}}\right)}^2$$ (B4)

where $n_{\mathit{tot}}\text{=}{n}_{\mathit{vap}}+n_{\mathit{air}}$ is the number of total mole inside the bubble, $n_{\mathit{vap}}$ and $n_{\mathit{air}}$ are the number of water vapor and air mole. The temperature-dependent of thermal conductivities of vapor ${\lambda }_{\mathit{vap}}$ and of air ${\lambda }_{\mathit{air}}$ are described as follows [79]:

$$\begin{array}{c}{\lambda }_{\mathit{vap}}={\alpha }_{\mathit{vap}}T+{\beta }_{\mathit{vap}}\\ {\lambda }_{\mathit{air}}={\alpha }_{\mathit{air}}T+{\beta }_{\mathit{air}}\end{array}$$ (B5)

where ${\alpha }_{\mathit{vap}}$ = 9.98 × 10^-5 [W m⁻¹ K⁻²], ${\alpha }_{\mathit{air}}$= 5.39 × 10^-5 [W m⁻¹ K⁻²], ${\beta }_{\mathit{vap}}$= −0.0119 [W m⁻¹ K⁻¹], ${\beta }_{\mathit{air}}$= 0.0108 [W m⁻¹ K⁻¹].

Data availability

No data was used for the research described in the article.