Cumulative distribution functions associated with bubble-nucleation processes in cavitation

Bubble-nucleation processes of a Lennard-Jones liquid are studied by molecular dynamics simulations. Waiting time, which is the lifetime of a superheated liquid, is determined for several system sizes, and the apparent finite-size effect of the nucleation rate is observed. From the cumulative distribution function of the nucleation events, the bubble-nucleation process is found to be not a simple Poisson process but a Poisson process with an additional relaxation time. The parameters of the exponential distribution associated with the process are determined by taking the relaxation time into account, and the apparent finite-size effect is removed. These results imply that the use of the arithmetic mean of the waiting time until a bubble grows to the critical size leads to an incorrect estimation of the nucleation rate.

Figure 1

Time evolutions of (a) temperature and (b) pressure. Only the data for $L=64$, 128, and 256 are shown for visibility. After thermalization at $T=0.9$, density suddenly decreases from 0.7 to 0.659 at $t=50$. Then the system relaxes to a metastable state with $T=0.868\left(2\right)$. The time evolutions of different system sizes are almost identical.

Figure 2

Phase diagram showing the vicinity of the phase boundary between the pure-liquid and gas-liquid coexistent regions. The solid line denotes the binodal line between the liquid and gas-liquid coexistent phases, and the dashed line denotes the spinodal line where spinodal decomposition takes place. The filled and open squares on the binodal and spinodal lines denote the observed point, which are used to draw the lines. The system is first thermalized in the pure-liquid phase, then is suddenly expanded. Then the system crosses the binodal line and becomes a metastable state (superheated liquid) in the vicinity of the spinodal line. The solid and open circles denote the states of the system before and after expansion, respectively. (Inset) Overall view of the gas-liquid phase diagram. The symbols “G,” “L,” and “GL” denote the pure gas, pure-liquid, and gas-liquid coexistent regions, respectively.

Figure 3

(Color online) Time evolution of bubble in the system with $L=128$. Left to right: snapshots at $t=100$, 150, and 550. We divide the system into small subcells and define them to be in the gas state when their densities are less than 0.2. Only the gas state subcells are shown. An embryo appears after a waiting time then grows slowly into a large bubble.

Figure 4

System-size dependence of the expectation value of the waiting time $⟨t_{\text{w}}⟩$. Decimal logarithms are taken for both axes. The error bars are smaller than the symbols. The solid line, which denotes $L^{-3}$, is a guide for the eyes.

Figure 5

Cumulative distribution functions of nucleation events. The values of $Y\left(t\right)=-\text{ln}\left(1-F\right)$ are shown as functions of time. The labels near lines denote system sizes.

Figure 6

System-size dependence of the characteristic time scale $\tau$ of the Poisson process. Decimal logarithms are taken for both axes. The error bars are smaller than the symbols. The solid line, which denotes $L^{-3}$, is a guide for the eyes.

Figure 7

Cumulative distribution function $F\left(t\right)$ plotted as a function of $X=1-\text{exp}\left[-\left(t-t_0\right)/\tau \right]$ for system sizes with the values listed in Table . If a nucleation process is perfectly described in Eq. (6), then this plot becomes a line connecting the origin to (1,1).