Thermally activated vapor bubble nucleation: The Landau-Lifshitz–Van der Waals approach

Vapor bubbles are formed in liquids by two mechanisms: evaporation (temperature above the boiling threshold) and cavitation (pressure below the vapor pressure). The liquid resists in these metastable (overheating and tensile, respectively) states for a long time since bubble nucleation is an activated process that needs to surmount the free energy barrier separating the liquid and the vapor states. The bubble nucleation rate is difficult to assess and, typically, only for extremely small systems treated at an atomistic level of detail. In this work a powerful approach, based on a continuum diffuse interface modeling of the two-phase fluid embedded with thermal fluctuations (fluctuating hydrodynamics), is exploited to study the nucleation process in homogeneous conditions, evaluating the bubble nucleation rates and following the long-term dynamics of the metastable system, up to the bubble coalescence and expansion stages. In comparison with more classical approaches, this methodology allows us on the one hand to deal with much larger systems observed for a much longer time than possible with even the most advanced atomistic models. On the other, it extends continuum formulations to thermally activated processes, impossible to deal with in a purely determinist setting.

Figure 1

Left panel: Comparison between the temperature dependence of the surface tension obtained through Eq. (2), when using the Lennard-Jones EoS [30], and the benchmark data provided at Ref. [31]. The value of the capillary coefficient is fixed to $\lambda m^2/\left({\sigma }^5\epsilon \right)=5.224$. Right panel: Density profiles of the critical nuclei, evaluated with the string method, at different thermodynamic conditions of the metastable liquid.

Figure 2

Left panel: Static structure factor comparison for a capillary fluid in a 3D system. We report the relative error $e=|S_f-S_t|/S_t$ between the theoretical prediction and numerical calculation for each wave numbers $k_x,k_y,k_z$ in the Fourier space. Right panel: Error of the density variance at different simulation time steps. As expected, the error follows a square power law $e\propto \mathrm{\Delta }{t}^2$.

Figure 3

Left panel: Bubble configurations along nucleation ($\rho =0.46,{\theta }_0=1.25$), from left to right $t=400$, $t=2000$, $t=25000$. Animation is available in the Supplemental Material [39]. Right panel: Bubble number evolution (red symbols) and number of coalescence events (blue line).

Figure 4

Same as Fig. 3 at $\rho =0.48,{\theta }_0=1.25$. Snapshots taken at $t=2000$, $t=11000$, $t=230000$. The bubble number vs time in the right panel is fitted by the dotted red line for better readability.

Figure 5

Left panel: Volume history of the bubbles surviving the entire simulation ($\rho =0.46$ and ${\theta }_0=1.25$). Intense coalescence events, characterized by a sudden volume jump, are identified in the red curves. The corresponding volumes are shown by the red dots in the inset providing the $V-V_{\mathrm{coll}}$ scatter plot, where $V_{\mathrm{coll}}$ is the volume acquired by coalescence. The largest bubbles experienced intense coalescence events. Right panel: Probability distribution function $f\left(V\right)$ of the bubble volumes during the nucleation, at different times (${\theta }_0=1.25$, $\rho =0.46$, critical volume $V_c=4445$).

Figure 6

Left panel: Comparison of the nucleation rate obtained via FH simulations (red squares at ${\theta }_0=1.25$ and blue squares at ${\theta }_0=1.20$) with respect to CNT predictions (green circles at ${\theta }_0=1.25$ and purple circles at ${\theta }_0=1.20$). The inset shows the comparison with other authors. Right panel: Time evolution of the number of supercritical bubbles $N_b$ and the total collapsed supercritical bubble $N_c$ up to time instant $t$. The number of bubbles is rescaled with the maximum number of bubble observed during the simulation, $N_{\max }$, which correspond to $N_{\max }=458$ in the thermodynamic condition with ${\rho }_L=0.46$ and $N_{\max }=52$ in the case with ${\rho }_L=0.48$. The time is shifted and rescaled in such a way that all the curves start when the first bubble appears, at $t=t_s$, and finish when $N_{\max }$ is reached, at $t=t_{\max }$.