Nanosecond Freezing of Water at High Pressures: Nucleation and Growth near the Metastability Limit

The fundamental study of phase transition kinetics has motivated experimental methods toward achieving the largest degree of undercooling possible, more recently culminating in the technique of rapid, quasi-isentropic compression. This approach has been demonstrated to freeze water into the high-pressure ice VII phase on nanosecond timescales, with some experiments undergoing heterogeneous nucleation while others, in apparent contradiction, suggest a homogeneous nucleation mode. In this study, we show through a combination of theory, simulation, and analysis of experiments that these seemingly contradictory results are in agreement when viewed from the perspective of classical nucleation theory. We find that, perhaps surprisingly, classical nucleation theory is capable of accurately predicting the solidification kinetics of ice VII formation under an extremely high driving force ($|\mathrm{\Delta }\mu /k_{B}T|\approx 1$) but only if amended by two important considerations: (i) transient nucleation and (ii) separate liquid and solid temperatures. This is the first demonstration of a model that is able to reproduce the experimentally observed rapid freezing kinetics.

Figure 1

Representative experimental setup for multiple-shock compression and the phase diagram for water superimposed on an illustration of a hypothetical oceanic exoplanet. The quasi-isentropic loading path before the onset of freezing to ice VII may be approximated by the liquid principal isentrope. Ice VII has a body-centered cubic (bcc) lattice of oxygen. All the curves in the phase diagram are produced from our equation of state (EOS) for the two water phases [15]. Unlike single-shock compression (where the relevant curve is the Hugoniot), quasi-isentropic compression can probe deeply undercooled states, since the temperature rise along its loading path is far more attenuated. The two-phase isentropes for the oceanic super-Earths Gliese 581d (GJ 581d) and Gliese 1214b (GJ 1214b) are initiated at surface temperatures of 340 and 400 K, respectively, which are rough estimates taken from Refs. [16, 17].

Figure 2

(a) Ratio of exponential and preexponential factors for heterogeneous vs homogeneous nucleation as a function of contact angle $\theta$ for three different pressures along the isentrope in Fig. 1. (b) Comparison of the time evolution of the ice VII phase fraction $\varphi$ for these pressures, with the inset showing the behavior at early times. Freezing occurs primarily via homogeneous (heterogeneous) nucleation at higher (lower) pressures. Table 1 of Supplemental Material [27] shows the values of the quantities used to produce these plots.

Figure 3

Our simulation results for the ramp-compression experiment of Dolan et al. [10]. All of the results correspond to those from our dual-temperature model unless otherwise indicated. (a) compares the pressure wave profile for four cases, three of which are from our simulations run with different models and the fourth from Dolan et al. To aid in understanding, (a) also shows how the driving force $\mathrm{\Delta }\mu /k_{B}T$, the ice VII phase fraction $\varphi$, and the transient nucleation factor $I(t)$ evolve with time. (b) portrays the temperature in the two phases, (c) illustrates the induction time $\tau$ and number of ice molecules $n^{*}$ in critical size clusters, and (d) depicts the nucleation rate $J$ and growth rate $\gamma$, including insets that focus on the behavior near the onset of the transition.

Figure 4

Comparison between our simulation results and three multiple-shock experiments from Dolan et al. [10]. The peak pressure is high enough for the transition to occur in the 8 and 12 GPa experiments but not in the 5 GPa one. With the exception of the no transition curves (dashed), all of our results are produced with the dual-temperature model. The simulations include a plasticity model for the strength of ice VII that is active after solidification (solid blue and red curves) and results in a damping of the shock wave ringing after 400 ns (8 GPa) and 180 ns (12 GPa), in closer agreement with the experimental data than without strength (dotted curves).