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Metastable–Solid Phase Diagrams Derived from Polymorphic Metastable–solid phase diagrams derived from polymorphic solidification kinetics Babak Sadigha,1 , Luis Zepeda-Ruiza, and Jonathan L. Belofa,1 aLawrence Livermore National Laboratory, Physical and Life Sciences Directorate, Livermore, CA 94550 Edited by Pablo G. Debenedetti, Princeton University, Princeton, NJ, and approved January 16, 2021 (received for review August 24, 2020) Nonequilibrium processes during solidification can lead to kinetic Through X-ray diffraction of the freely suspended droplets, the stabilization of metastable crystal phases. A general frame- dynamics of crystal nucleation during solidification have been work for predicting the solidification conditions that lead to investigated. As a result, solidification of diverse crystalline metastable-phase growth is developed and applied to a model phases such as fcc, bcc, icosahedral, and quasicrystalline has face-centered cubic (fcc) metal that undergoes phase transitions been observed, and emergence of metastable phases, often in to the body-centered cubic (bcc) as well as the hexagonal close- the bcc structure, in sufficiently undercooled liquids has been packed phases at high temperatures and pressures. Large-scale demonstrated. Through application of CNT, the undercooling molecular dynamics simulations of ultrarapid freezing show that necessary for metastable-phase growth has been rationalized. bcc nucleates and grows well outside of the region of its thermo- It is conjectured that phase selection takes place in the nucle- dynamic stability. An extensive study of crystal–liquid equilibria ation stage; the phase emerging is one with the smallest critical confirms that at any given pressure, there is a multitude of nucleation barrier. Consequently, models for solid–liquid inter- metastable solid phases that can coexist with the liquid phase. facial free energies of different crystal phases have been derived We define for every crystal phase, a solid cluster in liquid (SCL) from these experiments. Nevertheless, controversies remain basin, which contains all solid clusters of that phase coexist- when nucleation of metastable phases cannot be explained ing with the liquid. A rigorous methodology is developed that based on their interfacial free energies. Instead, the likely cul- allows for practical calculations of nucleation rates into arbi- prit is suspected to be preferential nucleation at the droplet trary SCL basins from the undercooled melt. It is demonstrated surfaces (13). that at large undercoolings, phase selections made during the Recent powerful advances in dynamic shock/ramp compres- nucleation stage can be undone by kinetic instabilities amid sion techniques (15–23) have been able to achieve melting and APPLIED PHYSICAL SCIENCES the growth stage. On these bases, a solidification–kinetic phase solidification at rates that can exceed rapid cooling techniques diagram is drawn for the model fcc system that delimits the con- described above. Hence, extreme undercoolings can be achieved ditions for macroscopic grains of metastable bcc phase to grow in these experiments, and therefore, the limits of the current from the melt. We conclude with a study of unconventional theories of phase transformation kinetics can be tested. These interfacial kinetics at special interfaces, which can bring about laboratory experiments are currently at the forefront of unrav- heterogeneous multiphase crystal growth. A first-order interfacial eling fundamental insights into the kinetics of solidification at phase transformation accompanied by a growth-mode transition extreme conditions. Freezing under such conditions is believed is examined. phase transitions j kinetic stabilization j metastability j solidification j Significance phase diagram Kinetic stabilization of metastable phases in rapidly cooled he idea of polymorphism of solid nuclei during crystalliza- metals and alloys has been established in experiments for Ttion dates back to Ostwald (1). It results from a lack of decades. However, atomistic theories that can quantitatively explicit correlation between the bulk free energies of different predict the solidification conditions that produce nonequi- crystal phases and their respective interfacial free energies with librium phases are still in their infancy. Recent advances in the liquid. It is thus expected that the early stages of nucle- pulsed power/laser technologies, as well as in situ character- ation are likely to be dominated by the crystal phase with the ization, have brought to bear unprecedented understanding smallest solid–melt interfacial free energy and the later stages of matter at extreme temperatures and pressures. However, by the equilibrium bulk phase (2). It has been recognized that accurate predictions of kinetic stabilization of metastable classical nucleation theory (CNT) and density-functional theory phases that are necessary for physical interpretation of these (3), as well as phase-field models (4), need to be generalized experiments are lacking. This work provides a blueprint for to account for structural phase transformations of the grow- development of kinetic phase maps of materials undergo- ing nuclei as the ratio of the interface to the bulk regions ing rapid solidification from first principles. Through atomistic changes. This so-called two-step nucleation has been observed simulations, the phases dominating nucleation are identi- experimentally in rapidly quenched metals (5), as well as block fied, and their kinetic stabilities during the growth stage are copolymer solutions (6, 7) and charged colloidal particles (8, characterized. 9), where metastable body-centered cubic (bcc) phase clusters Author contributions: B.S., L.Z.-R., and J.L.B. designed research; B.S. and L.Z.-R. performed are reported to occur before final crystallization into the face- research; B.S. and L.Z.-R. analyzed data; and B.S. and J.L.B. wrote the paper.y centered cubic (fcc) phase takes place. A more dramatic outcome The authors declare no competing interest.y at large undercooling is that of nucleation of metastable solid This article is a PNAS Direct Submission.y phases that can under suitable circumstances, grow to macro- This open access article is distributed under Creative Commons Attribution-NonCommercial- scopic sizes. A large body of literature has grown over the past NoDerivatives License 4.0 (CC BY-NC-ND).y several decades in which rapid solidification of mainly metals 1 4 5 To whom correspondence may be addressed. Email: [email protected] or belof1@ and alloys, at cooling rates on the order of 10 to 10 K, has llnl.gov.y been studied in containerless experiments through electromag- This article contains supporting information online at https://www.pnas.org/lookup/suppl/ netic/electrostatic levitation techniques (10–14), where strong doi:10.1073/pnas.2017809118/-/DCSupplemental.y undercooling is achieved by avoiding heterogenous nucleation. Published February 22, 2021. PNAS 2021 Vol. 118 No. 9 e2017809118 https://doi.org/10.1073/pnas.2017809118 j 1 of 12 Downloaded by guest on September 29, 2021 to play a major role in determining the structure of the planetary this approach and as a result, present a general algorithm for cores, which in turn, is responsible for many important properties conveniently calculating accurate interfacial free energies that (e.g., whether a planet can support a magnetic dynamo). It is thus incorporate interfacial anisotropy as well as finite cluster-size essential to develop capabilities for predicting metastable-phase curvature effects. formation under these nonequilibrium conditions. Let us now examine the phase space of the undercooled liq- From a theoretical standpoint, the preference for nucleation uid by carving it into distinct subspaces, each containing all of the bcc and the icosahedral phases from the undercooled liq- of the configurations of solid clusters of a particular phase φ, uid was first predicted decades ago through Landau expansion in coexisting with the liquid phase. We refer to each such region density fluctuations near the liquid phase (24). The observation as the φ-SCL (solid cluster in liquid) basin. The relative rates of two-step nucleation in molecular dynamics (MD) simulations of nucleation of solids belonging to two different basins φ1 was initially inconclusive (25–28). The majority of attempts failed and φ2 are proportional to exp(−∆G=kB T ), where ∆G is the to demonstrate any convincing signature of intermediate phases Gibbs free energy difference between the critical nuclei of the during nucleation. The breakthrough came through the work two phases at the undercooling temperature T . ∆G has con- of ten Wolde et al. (29, 30), where they performed umbrella tributions from free energies of both the bulk solids as well sampling (31) simulations using the bond-orientational order as the solid–liquid interfaces. Hence, it is easy to see that at parameter Q6 (32, 33) to induce crystallization at moderate large-enough undercooling, when the critical nuclei are rela- undercoolings. They thus observed nucleation of predominantly tively small, the interfacial free energy may favor nucleation of bcc crystallites that would transform to fcc in their cores while the phase that is not the thermodynamically stable bulk phase. the interface regions remained bcc. This finding was later repro- However, during the growth stage, the thermodynamic driv- duced by classical density-functional theory (34). Subsequently, ing force for transformation to the thermodynamic equilibrium it was observed that when varying both pressure and
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