Effects of vapor-liquid phase transitions on sound-wave propagation: A molecular dynamics study

To understand ultrasonic cavitation, it is imperative to analyze the effects of the vapor-liquid phase transitions on sound-wave propagation. Since current methods based on fluid dynamics offer limited information, it is imperative to carry out further research on this phenomenon. In this study, we investigated the effects of cavitation and near-critical fluid on sound waves using the molecular dynamics (MD) simulations of Lennard-Jones fluids. In the first-order liquid-to-vapor transition region (far from the critical point), the waveform does not continuously change with the temperature and source oscillation amplitude owing to the discontinuous change in the density due to the phase transition. Meanwhile, in the continuous transition region (crossing near the critical point), the waveform continuously varies with temperature regardless of the amplitudes because phase separation is not involved in this region. The density fluctuations increase as the amplitude increases; however, it does not affect the waveform. Thus, we clarified that the first-order and continuous transitions have different impacts on sound waves. Moreover, we determined the acoustic characteristics, such as attenuation and nonlinear parameters, by comparing the results of the numerical solution of Burgers' equation and MD simulation. Burgers' equation clearly describes the sound-wave phenomenon until phase separation or bubble formation occurs. In the continuous transition region, the attenuation parameters tend to diverge, reflecting a critical anomaly trend. We observed the bubbles move forward with the oscillation of their radii owing to their interaction with the sound waves. To the best of our knowledge, this is the first direct observation of the interaction using MD simulations.

Figure 1

Schematic of the computational domain. The positions of the oscillating and stationary walls are $x=0$ (red) and $x=L_x$ (gray), respectively. Two orange boxes on the stationary wall indicate the equilibration area by the Langevin thermostat. The gradation of the orange color in the left box schematically represents a linear increase in the friction constant of the thermostat.

Figure 2

Thermodynamic state of the LJ fluid in the simulation. The input temperature $T_0$ is varied with a constant mean density ${\rho }_0=0.6$ and 0.4 and is depicted as the circles along the blue and red lines, which represent the first-order transition region and continuous transition region, respectively. The black line denotes the vapor-liquid phase boundary of the LJ system, and the black filled circle is the critical point ($T_{\mathrm{c}}=0.94$, ${\rho }_{\mathrm{c}}=0.32$) [54].

Figure 3

Distribution of the loss function ${\chi }^2$ at temperature $T_0=0.95$ and density ${\rho }_0=0.6$ in the first-order transition region at the oscillation amplitude $A=10$. ${\chi }^2$ reaches a minimum at the attenuation parameter $b=2.0$ and the nonlinear parameter $\beta =5.5$.

Figure 4

Temperature $T_0$ dependence of the waveform in the first-order transition region (density ${\rho }_0=0.6$). (a) $T_0=1$, (b) $T_0=0.95$, (c) $T_0=0.9$, (d) $T_0=0.85$, and (e) $T_0=0.84$. The boiling temperature is $T_{\mathrm{b}}=0.85$, and the amplitude of the oscillating wall is $A=10$. As the temperature decreases, the wavelength decreases, and the attenuation rate of the sound wave increases. Although the change is continuous until $T_0=0.85$, that of $T_0=0.84$ is not continuous. The rapid change is attributed to the phase transition. The flow velocity is normalized by the maximum velocity $V_{\mathrm{w}}^{\max }$ of the oscillating wall, which was also applied to Figs. 5, 8, 9, and 12. The black dashed lines indicate the numerical solution of Burgers' equation by parameter fitting. The parameters are (a) $c_0=3.07$, $b=1.3$, $\beta =5.0$, (b) $c_0=2.90$, $b=1.8$, $\beta =5.7$, (c) $c_0=2.68$, $b=4.2$, $\beta =7.0$, and (d) $c_0=2.42$, $b=8.8$, $\beta =7.1$.

Figure 5

Dependence of the waveform on the amplitude $A$ of the oscillating wall at the first-order transition point (${\rho }_0=0.6$, $T_0=0.85$). (a) $A=10$, (b) $A=11$, (c) $A=12$, and (d) $A=13$. The black dashed lines show the numerical solution of Burgers' equation obtained by parameter fitting. The parameters are (a) $c_0=2.42$, $b=8.8$, $\beta =7.1$ and (b) $c_0=2.42$, $b=13$, $\beta =5.0$.

Figure 6

Typical snapshots of the density field in the first-order transition region (${\rho }_0=0.6$). The liquid and vapor phases are displayed in blue and red, respectively. (a) The liquid phase occupies the entire area at the temperature $T_0=1$ and oscillation amplitude $A=10$. (b) A few vapor bubbles appear at the boiling point $T_0=0.85$ and $A=10$. (c) $T_0=0.84$ and $A=10$. (d) $T_0=0.85$ and $A=13$. A further decreasing temperature or a strengthening of the oscillation generates a phase separation.

Figure 7

Temperature $T_0$ dependence of the maximum and minimum values of the time-averaged local density $\rho$ in the system at the oscillation amplitude $A=10$. The density and temperature are normalized by the initial density ${\rho }_0$ and boiling temperature $T_{\mathrm{b}}$, respectively. For the first-order transition region (blue), the minimum density shows a sharp drop in the vicinity of the boiling point. In contrast, the density changes continuously in the continuous transition region (red). The behavior of the change in density fields reflects the characteristics of the phase transition.

Figure 8

Temperature $T_0$ dependence of the waveform in the continuous transition region (${\rho }_0=0.4$) at the oscillation amplitude $A=10$. (a) $T_0=1.5$, (b) $T_0=1.4$, (c) $T_0=1.3$, (d) $T_0=1.2$, (e) $T_0=1.1$, (f) $T_0=1$, (g) $T_0=0.96$, (h) $T_0=0.95$, (i) $T_0=0.94$, and (j) $T_0=0.93$. The boiling temperature is $T_{\mathrm{b}}=0.94$. As the temperature decreases, the wavelength decreases, and the attenuation rate increases. Moreover, the rate of change increases with approaching the boiling point. The waveform changes continuously against the temperature, even in the phase-separation region. Black dashed lines show the numerical solution of Burgers' equation by the parameter fitting. The parameters are (a) $c_0=2.48$, $b=2.0$, $\beta =3.0$, (b) $c_0=2.33$, $b=2.2$, $\beta =3.0$, (c) $c_0=2.15$, $b=1.5$, $\beta =3.3$, (d) $c_0=1.96$, $b=2.0$, $\beta =3.5$, (e) $c_0=1.72$, $b=6.2$, $\beta =3.3$, (f) $c_0=1.34$, $b=6.5$, $\beta =4.8$, (g) $c_0=1.12$, $b=17$, $\beta =4.7$, (h) $c_0=1.02$, $b=22$, $\beta =5.8$, and (i) $c_0=0.91b=33$, $\beta =6.2$.

Figure 9

Dependence of the waveform near the boiling point (${\rho }_0=0.4$, $T_0=0.95$) in the continuous transition region on the amplitude $A$ of the oscillating wall. (a) The amplitude $A=10$, (b) $A=20$, (c) $A=30$, and (d) $A=40$. The waveform is independent of the amplitude, and no discontinuous changes are observed. Black dashed lines show the numerical solution of Burgers' equation by the parameter fitting. The parameters are (a) $c_0=1.02$, $b=33$, $\beta =6.2$, (b) $c_0=1.02$, $b=44$, $\beta =2.8$, (c) $c_0=1.02$, $b=49$, $\beta =1.8$, and (d) $c_0=1.02$, $b=39$, $\beta =2.0$.

Figure 10

Typical snapshots of the density field in the continuous transition region (${\rho }_0=0.4$). (a) The temperature $T_0=1.2$ and oscillation amplitude $A=10$. (b) $T_0=1$ and $A=10$. (c) $T_0=0.95$ and $A=10$. (d) $T_0=0.93$ and $A=10$. (e) $T_0=0.95$ and $A=20$. (f) $T_0=0.95$ and $A=40$. Low-density parts appear as the temperature decreases. In the phase-separation region ($T_0<T_{\mathrm{b}}=0.94$), a slight tendency of phase separation is observed. However, no clear phase separation occurs. The density fluctuations increase with increasing amplitude.

Figure 11

Temperature dependence of (a) attenuation parameter $b$ and (b) nonlinear parameter $\beta$ at the oscillation amplitude $A=10$. The temperature $T_0$ is normalized by the boiling temperature $T_{\mathrm{b}}$.

Figure 12

Dependence of the waveform at the first-order transition point (${\rho }_0=0.6$, $T_0=0.85$) on the amplitude $A$ of the oscillating wall for a larger system. (a) $A=5$, (b) $A=10$, (c) $A=15$, (d) $A=20$, and (e) $A=40$. The waveform changes are almost negligible in the range of $5\le A\le 10$. However, for $A\ge 15$, the attenuation rate of the wave increases with increasing amplitude $A$. Moreover, the shape of the wave changes. Although such a change occurs, the wavelength remains unchanged (approximately three waves exist in all conditions). Black dashed lines show the numerical solution of Burgers' equation by the parameter fitting. The parameters are (a) $c_0=2.42$, $b=3.0$, $\beta =9.5$, (b) $c_0=2.42$, $b=3.0$, $\beta =9.0$, and (c) $c_0=2.42$, $b=34$, $\beta =4.0$.

Figure 13

Typical snapshots of the density field in the first-order transition point (${\rho }_0=0.6$, $T_0=0.85$). (a) The oscillation amplitude $A=5$, (b) $A=10$, (c) $A=15$, (d) $A=20$, and (e) $A=40$. The liquid and vapor phases are displayed in blue and red, respectively. At $A=5$, the entire area is in the liquid phase. At $A=10$, small bubbles appear instantaneously in various locations and immediately disappear. When the amplitude reaches $A=15$, a large spherical bubble appears for a long lifetime. As the amplitude increases, a cylindrical bubble or phase separation occurs. Simulations with a larger system revealed that bubbles emerge before phase separation occurs.

Figure 14

Bubble dynamics from the inception to the collapse in the first-order transition point (${\rho }_0=0.6$, $T_0=0.85$) at the oscillation amplitude $A=15$. (a) A bubble generation near the oscillating wall, (b) bubble traveling toward a wave propagation direction and bubble nucleation, (c) bubble collapse, and new bubble generation and growth. The bubbles repeat a similar cycle from generation to collapse.

Figure 15

Time evolution of the gyration radius $R_{\mathrm{g}}$ of the largest bubble generated in the first-order transition point (${\rho }_0=0.6$, $T_0=0.85$) at the oscillation amplitude $A=15$. The amplitude of the pressure wave $p_{\mathrm{in}}$ that reached the bubble position is also shown. The phase difference of $R_{\mathrm{g}}$ and $p_{\mathrm{in}}$ is about $0.6\pi$.

Figure 16

Time evolution of (a) the pressure $p$ near the oscillating wall and (b) the total number of bubbles $N_{\mathrm{b}}$ in the first-order transition point (${\rho }_0=0.6$, $T_0=0.85$). The amplitudes of the oscillating wall are $A=10$ and 15. The pressure is estimated by applying the virial theorem to the slice near the wall, where the slice volume $V_{\mathrm{slice}}=25\times 100\times 100=250000$. For $A=10$, the pressure oscillates at a constant amplitude. Moreover, the frequency of bubble generation is almost unchanged. In contrast, for $A=15$, an increase and decrease in the pressure wave amplitude occur. As the amplitude increases, the frequency of bubble generation also increases. The dynamics of nucleation and growth during bubble generation affect the pressure wave.