Scaling Law for the Onset of Solidification at Extreme Undercooling.

Quasi-isentropic compression enables one to study the solidification of metastable liquid states that are inaccessible through other experimental means. The onset of this nonequilibrium solidification is known to depend on the compression rate and material-specific factors, but this complex interdependence has not been well characterized. In this study, we use a combination of experiments, theory, and computational simulations to derive a general scaling law that quantifies this dependence. One of its applications is a novel means to elucidate melt temperatures at high pressures.

Model for the Solid-Liquid Interfacial Free Energy at High Pressures.

The free energy involved in the formation of an interface between two phases (e.g., a solid-liquid interface) is referred to as the interfacial free energy. For the case of solidification, the interfacial free energy dictates the height of the energy barrier required to nucleate stable clusters of the newly forming solid phase and is essential for producing an accurate solidification kinetics model using classical nucleation theory (CNT)-based methods. While various methods have been proposed for modeling the interfacial free energy for solid-liquid interfaces in prior literature, many of these formulations involve making restrictive assumptions or approximations, such as the system being at or near equilibrium (i.e., the system temperature is approximately equal to the melt temperature) or that the system is at pressures close to atmospheric. However, these approximations and assumptions may break down in highly non-equilibrium situations, such as in dynamic-compression experiments where metastable liquids that are undercooled by hundreds of kelvin or overpressurized by several gigapascals or more are formed before eventually solidifying. We derive a solid-liquid interfacial free-energy model for such high-pressure conditions by considering the enthalpies of interactions between pairs of atoms or molecules. We also consider the contribution of interface roughness (disordering) by incorporating a multilayer interface model known as the Temkin n-layer model. Our formulation is applicable to a diverse variety of materials, and we demonstrate it by developing models specifically for two different materials: water and gallium. We apply our interfacial free-energy formulation to CNT-based kinetics simulations of several suites of dynamic-compression experiments that cause liquid water to solidify to the high-pressure solid polymorph ice VII and have found good agreement to the observed kinetics with only minor empirical fitting.

Measuring the melting curve of iron at super-Earth core conditions.

The discovery of more than 4500 extrasolar planets has created a need for modeling their interior structure and dynamics. Given the prominence of iron in planetary interiors, we require accurate and precise physical properties at extreme pressure and temperature. A first-order property of iron is its melting point, which is still debated for the conditions of Earth's interior. We used high-energy lasers at the National Ignition Facility and in situ x-ray diffraction to determine the melting point of iron up to 1000 gigapascals, three times the pressure of Earth's inner core. We used this melting curve to determine the length of dynamo action during core solidification to the hexagonal close-packed (hcp) structure. We find that terrestrial exoplanets with four to six times Earth's mass have the longest dynamos, which provide important shielding against cosmic radiation.

Minimization of Gibbs Energy in High-Pressure Multiphase, Multicomponent Mixtures through Particle Swarm Optimization.

We present a global optimization method to construct phase boundaries in multicomponent mixtures by minimizing the Gibbs energy. The minimization method is, in essence, an extension of the Maxwell construction procedure that is used in single-component systems. For a given temperature, pressure, and overall mixture composition, it reveals the mole fractions of the thermodynamically stable phases and the composition of these phases. Our approach is based on particle swarm optimization (PSO), which is a gradient-free, stochastic method. It is not reliant on good initial guesses for the phase fractions and compositions, which is an important requirement for the high-pressure applications considered in this study because data on phase boundaries at high pressures tend to be extremely limited. One practical use of this method is to create equation-of-state tables needed by continuum-scale, multiphysics codes that are ubiquitous in high-pressure science. Currently, there does not exist a method to generate such tables that rigorously account for changes in phase boundaries due to mixing. We have done extensive testing to demonstrate that PSO can reliably determine the Gibbs energy minimum and can capture nontrivial features like eutectic and peritectic temperatures to produce coherent phase diagrams. As part of our testing, we have developed a PSO-based Helmholtz-energy minimization procedure that we have used to cross-check the results of the Gibbs energy minimization. We conclude with a critique of our approach and provide suggestions for future work, including a PSO-based entropy-maximization method that would enable the aforementioned continuum codes to perform on-the-fly, phase-equilibria calculations of multicomponent mixtures.

Development of a Multiphase Beryllium Equation of State and Physics-based Variations.

We construct a family of beryllium (Be) multiphase equation of state (EOS) models that consists of a baseline ("optimal") EOS and variations on the baseline to account for physics-based uncertainties. The Be baseline EOS is constructed to reproduce a set of self-consistent data and theory including known phase boundaries, the principal Hugoniot, isobars, and isotherms from diamond-anvil cell experiments. Three phases are considered, including the known hexagonal closed-packed (hcp) phase, the liquid, and the theoretically predicted high-pressure body-centered cubic (bcc) phase. Since both the high-temperature liquid and high-pressure bcc phases lack any experimental data, we carry out ab initio density functional theory (DFT) calculations to obtain new information about the EOS properties for these two regions. At extremely high temperature conditions (>87 eV), DFT-based quantum molecular dynamics simulations are performed for multiple liquid densities using the state-of-the-art Spectral Quadrature methodology in order to validate our selected models for the ion- and electron-thermal free energies of the liquid. We have also performed DFT simulations of hcp and bcc with different exchange-correlation functionals to examine their impact on bcc compressibility, which bound the hcp-bcc transition pressure to within 4 ± 0.5 Mbar. Our baseline EOS predicts the first density maximum along the Hugoniot to be 4.4-fold in compression, while the hcp-bcc-liquid triple-point pressure is predicted to be at 2.25 Mbar. In addition to the baseline EOS, we have generated eight variations to accommodate multiple sources of potential uncertainties such as (1) the choice of free-energy models, (2) differences in theoretical treatments, (3) experimental uncertainties, and (4) lack of information. These variations are designed to provide a reasonable representation of nonstatistical uncertainties for the Be EOS and may be used to assess its sensitivity to different inertial-confinement fusion capsule designs.

Equation of state for a chemically dissociative, polyatomic system: Carbon dioxide.

A notorious challenge in high-pressure science is to develop an equation of state (EOS) that explicitly treats chemical reactions. For instance, many materials tend to dissociate at high pressures and temperatures where the chemical bonds that hold them together break down. We present an EOS for carbon dioxide (CO2) that allows for dissociation and captures the key material behavior in a wide range of pressure-temperature conditions. Carbon dioxide is an ideal prototype for the development of a wide-ranging EOS that allows for chemical-dissociation equilibria since it is one of the simplest polyatomic systems and because it is of great interest in planetary science and in the study of detonations. Here, we show that taking dissociation into account significantly improves the accuracy of the resulting EOS compared to other EOSs that either neglect chemistry completely or treat CO2 dissociation in a more rudimentary way.

Numerical modeling of solid-cluster evolution applied to the nanosecond solidification of water near the metastable limit.

Classical nucleation theory (CNT) is a promising way to predictively model the submicrosecond kinetics of phase transitions that occur under dynamic compression, such as the suite of experiments performed over the past two decades on the solidification of liquid water to the high-pressure ice VII phase. Myint et al. [Phys. Rev. Lett. 121, 155701 (2018)] presented the first CNT-based model for these types of rapid phase transitions, but relied on an empirical scaling parameter in their transient induction model to simulate the lag time that occurs prior to the onset of significant formation of ice VII clusters in the system. To build on that study, we model the liquid water-ice VII phase transformation using a numerical discretization scheme to solve the Zel'dovich-Frenkel partial differential equation, which is a fundamental CNT-based kinetic equation that describes the statistical time-dependent behavior of solid cluster formation. The Zel'dovich-Frenkel equation inherently accounts for transience in the nucleation kinetics and eliminates the need for the empirical scaling factor used by Myint et al. One major result of this research is that transience is found to play a relatively small role in the nucleation process for the dynamic-compression time scales of the liquid water-ice VII experiments being simulated. Instead, we show that it is possible to properly model the lag time using steady-state CNT by making small refinements to the interfacial free energy value. We have also developed a new dimensionless parameter that may be applied a priori to predict whether or not transient nucleation will be important in a given dynamic-compression experiment.

Two-phase equation of state for lithium fluoride.

We present an equation of state for the solid and liquid phases of lithium fluoride that covers a wide range of conditions from ambient pressure and temperature to the high pressures and temperatures exhibited in shock- and ramp-compression studies. The particular solid phase we have focused on in this work is the B1 phase. We have followed an approach where the pressure and heat-capacity functions of both phases are fit to experimental data and our own quantum molecular dynamics simulations and are then integrated in a thermodynamically consistent way to obtain the corresponding free-energy functions. This approach yields a two-phase equation of state that provides better overall agreement with experimental data than other equations of state for lithium fluoride, such as SESAME 7271v3, LEOS 2240, and the model presented by Smirnov. The last of these is a three-phase equation of state that predicts a B1-B2 transition along the shock Hugoniot at a pressure of about 140 GPa. This solid-solid transition has been a topic of speculation and debate in the literature for over 50 years, culminating in the work of Smirnov, who has developed the only potentially viable equation of state that allows for this transition. We explain why the proposed B1-B2 transition at 140 GPa is not consistent with recent velocimetry data.

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 (|Δµ/k_{B}T|≈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.

Rapid freezing of water under dynamic compression.

Understanding the behavior of materials at extreme pressures is a central issue in fields like aerodynamics, astronomy, and geology, as well as for advancing technological grand challenges such as inertial confinement fusion. Dynamic compression experiments to probe high-pressure states often encounter rapid phase transitions that may cause the materials to behave in unexpected ways, and understanding the kinetics of these phase transitions remains an area of great interest. In this review, we examine experimental and theoretical/computational efforts to study the freezing kinetics of water to a high-pressure solid phase known as ice VII. We first present a detailed analysis of dynamic compression experiments in which water has been observed to freeze on sub-microsecond time scales to ice VII. This is followed by a discussion of the limitations of currently available molecular and continuum simulation methods in modeling these experiments. We then describe how our phase transition kinetics models, which are based on classical nucleation theory, provide a more physics-based framework that overcomes some of these limitations. Finally, we give suggestions on future experimental and modeling work on the liquid-ice VII transition, including an outline of the development of a predictive multiscale model in which molecular and continuum simulations are intimately coupled.

Corrigendum: Rapid freezing of water under dynamic compression (2018 J. Phys.: Condens. Matter 30 233002).

Corrigendum.

Free energy models for ice VII and liquid water derived from pressure, entropy, and heat capacity relations.

We present equations of state relevant to conditions encountered in ramp and multiple-shock compression experiments of water. These experiments compress water from ambient conditions to pressures as high as about 14 GPa and temperatures of up to several hundreds of Kelvin. Water may freeze into ice VII during this process. Although there are several studies on the thermodynamic properties of ice VII, an accurate and analytic free energy model from which all other properties may be derived in a thermodynamically consistent manner has not been previously determined. We have developed such a free energy model for ice VII that is calibrated with pressure-volume-temperature measurements and melt curve data. Furthermore, we show that liquid water in the pressure and temperature range of interest is well-represented by a simple Mie-Grüneisen equation of state. Our liquid water and ice VII equations of state are validated by comparing to sound speed and Hugoniot data. Although they are targeted towards ramp and multiple-shock compression experiments, we demonstrate that our equations of state also behave reasonably well at pressures and temperatures that lie somewhat beyond those found in the experiments.

Arginine in membranes: the connection between molecular dynamics simulations and translocon-mediated insertion experiments.

Several laboratories have carried out molecular dynamics (MD) simulations of arginine interactions with lipid bilayers and found that the energetic cost of placing arginine in lipid bilayers is an order of magnitude greater than observed in molecular biology experiments in which Arg-containing transmembrane helices are inserted across the endoplasmic reticulum membrane by the Sec61 translocon. We attempt here to reconcile the results of the two approaches. We first present MD simulations of guanidinium groups alone in lipid bilayers, and then, to mimic the molecular biology experiments, we present simulations of hydrophobic helices containing single Arg residues at different positions along the helix. We discuss the simulation results in the context of molecular biology results and show that the energetic discrepancy is reduced, but not eliminated, by considering free energy differences between Arg at the interface and at the center of the model helices. The reduction occurs because Arg snorkeling to the interface prevents Arg from residing in the bilayer center where the energetic cost of desolvation is highest. We then show that the problem with MD simulations is that they measure water-to-bilayer free energies, whereas the molecular biology experiments measure the energetics of partitioning from translocon to bilayer, which raises the fundamental question of the relationship between water-to-bilayer and water-to-translocon partitioning. We present two thermodynamic scenarios as a foundation for reconciliation of the simulation and molecular biology results. The simplest scenario is that translocon-to-bilayer partitioning is independent of water-to-bilayer partitioning; there is no thermodynamic cycle connecting the two paths.

Epithelial tissue architecture protects against cancer.

We consider the design of colon crypts from the point of view of minimizing the likelihood of generation of cancerous mutations. A stochastic mathematical model (a finite branching process) is developed and fully analyzed. It is found that depending on the mutation rates, different designs are evolutionarily advantageous. If the mutation rates associated with stem cells are a lot higher than the mutation rates of daughter cells, then few stem cells per crypt is the evolutionarily optimal strategy. If the mutation rates of stem cells are of the same order of magnitude or lower than those for daughter cells, then having as many stem cells per crypt as possible is the desirable design. We also found that the optimal evolutionary strategy may work very well to protect the organism from cancer in the young age, but the same strategy becomes detrimental as the organism ages. It pushes the onset of cancer back in time, but it results in an elevated cancer initiation rates as the organism gets older. Our model quantifies the idea that cancer and aging are the two sides of one coin.