Direct simulations of homogeneous bubble nucleation: Agreement with classical nucleation theory and no local hot spots

We present results from direct, large-scale molecular dynamics simulations of homogeneous bubble (liquid-to-vapor) nucleation. The simulations contain half a billion Lennard-Jones atoms and cover up to 56 million time steps. The unprecedented size of the simulated volumes allows us to resolve the nucleation and growth of many bubbles per run in simple direct micro-canonical simulations while the ambient pressure and temperature remain almost perfectly constant. We find bubble nucleation rates which are lower than in most of the previous, smaller simulations. It is widely believed that classical nucleation theory (CNT) generally underestimates bubble nucleation rates by very large factors. However, our measured rates are within two orders of magnitude of CNT predictions; only at very low temperatures does CNT underestimate the nucleation rate significantly. Introducing a small, positive Tolman length leads to very good agreement at all temperatures, as found in our recent vapor-to-liquid nucleation simulations. The critical bubbles sizes derived with the nucleation theorem agree well with the CNT predictions at all temperatures. Local hot spots reported in the literature are not seen: Regions where a bubble nucleation event will occur are not above the average temperature, and no correlation of temperature fluctuations with subsequent bubble formation is seen.

Figure 1

The temperature (dashed line) and pressure (solid line) differences from the targets for the setup and measurement phase of our shortest simulation, T6r1. In most runs the measurement phase is far longer. In this case, the rapid growth of only a few bubbles quickly leads to a significant pressure increase. Despite this, we are able to measure accurate nucleation rates, shown in the upper panel of Fig. 6. Size distributions and formation rates discussed in this paper are taken only during the simulations' “measurement” phase, after the setup, yet before the pressure increase.

Figure 2

Projection through run T8r1 at $t=230\tau$. Every pixel corresponds to a column of 319 cells of size ${\left(3\sigma \right)}^3$. A larger number of cells below the threshold density of 0.2 $m/{\sigma }^3$ results in a lighter color (gray scale). This run contains about 120 large, stable bubbles at this time; see Fig. 4. See Supplemental Material for animations of such projections [46].

Figure 3

Number of bubbles above various threshold volumes as a function of time for three runs at $T=0.855\epsilon /k$.

Figure 4

Number of bubbles above various threshold volumes as a function of time for two runs at $T=0.8\epsilon /k$.

Figure 5

Number of bubbles above various threshold volumes as a function of time for three runs at $T=0.7\epsilon /k$.

Figure 6

Number of bubbles above various threshold volumes as a function of time for three runs at $T=0.6\epsilon /k$.

Figure 7

Measured nucleation rates at $kT/\epsilon$ = 0.855 (a, b), 0.8 (c, d), 0.7 (e, f), and 0.6 (g, h) against ambient pressure (left column) and against the inverse of the pressure difference squared (right column). Our MD results (diamonds, down arrows for upper limits) are compared with the CNT (solid lines) and PCNT (dashed lines) predictions and to the MD FFS simulation results from Wang et al. [11] (circles) and Meadley and Escobedo [31] (crosses). The shaded areas around the PCNT model predictions illustrate the effect of the $1\sigma$ uncertainty in our planar surface tension estimates. The dotted lines are PCNT-like predictions with bubble size-dependent surface tensions: black (upper) dotted lines show the Tolman correction with ${\delta }_T=0.25\sigma$, red (lower) dotted lines use the surface tension from Eq. (11).

Figure 8

Surface tensions (solid lines) as a function of bubble radius at $T=0.7\epsilon /k$: The planar surface tension (thin solid), a small, positive Tolman length of ${\delta }_T=0.25\sigma$ (red solid line, lower one at large radii) and using Eq. (11) with ${\delta }_B=-0.1\sigma$ and $l$ = 1.0 ${\sigma }^2$ (gray solid line, upper one at large radii). Dashed lines show the corresponding free energy curves for run T7r3 (see axis label on the right): PCNT assumes the constant planar surface tension (thin). The other two $\Delta G$ curves use a small, positive Tolman length (red) and Eq. (11) (gray).

Figure 9

Measured nucleation rates divided by the PCNT estimate against $T/T_c$ for our simulations (green diamonds) and bubble nucleation results from the literature.

Figure 10

Free energy for bubble formation at $kT/\epsilon$ = 0.855. Reconstructed free energy curve from the MD bubble size distribution in terms of $r_{\mathrm{MD}}$ (circles) and in terms of the estimated equimolar radius ($r_{\mathrm{MD}}+0.56\sigma$) (crosses). Model predictions are shown with solid (CNT) and dashed (PCNT) lines.

Figure 11

Free energy for bubble formation at $kT/\epsilon$ = 0.8. Reconstructed free energy curve from the MD bubble size distribution in terms of $r_{\mathrm{MD}}$ (circles) and in terms of the estimated equimolar radius ($r_{\mathrm{MD}}+0.35\sigma$) (crosses). Model predictions are shown with solid (CNT) and dashed (PCNT) lines.

Figure 12

Average local temperature (top panel) and density (bottom panel) in bubble-forming cells of size ${\left(3\sigma \right)}^3$ as a function of time since the moment of bubble formation $t_{\mathrm{bf}}$ (thick solid lines). The upper and lower thick solid lines indicate the one sigma cell-to-cell scatter, dotted lines the averages over the entire simulation volume. The estimated effect of extra kinetic energy from growing and shrinking bubbles is shown with a red thin solid line; see text for details.

Figure 13

Same as Fig. 12 but using larger cells of size ${\left(6\sigma \right)}^3$. The values in the central cells of four bubble nucleation events are given with dotted lines (top panel) and thin solid lines (bottom panel); see text for details.