Downloaded by guest on September 29, 2021 necoigi civdb viightrgnu nucleation. heterogenous avoiding strong by where achieved is (10–14), undercooling techniques electromag- through levitation experiments netic/electrostatic containerless in studied been transitions growth-mode examined. unconventional a is about by of bring accompanied transformation can study interfacial phase first-order which a A growth. interfaces, crystal with multiphase special heterogeneous grow conclude at to kinetics We phase interfacial melt. bcc con- metastable the the of delimits from that grains amid system macroscopic fcc phase instabilities for model solidification–kinetic ditions the kinetic a for bases, drawn by is these the diagram undone On during stage. be made growth the demonstrated can selections is arbi- phase stage It into undercoolings, nucleation melt. large rates undercooled at the nucleation that from that coexist- developed of basins phase is calculations SCL methodology that trary practical rigorous of for (SCL) A clusters liquid liquid. allows in solid the of cluster with all solid multitude phase. ing contains a a liquid phase, which the crystal is with basin, every there coexist for can , define that We given phases any solid equilibria metastable at crystal–liquid of that thermo- its study of confirms extensive region the An of stability. outside dynamic well that grows show Large-scale and freezing nucleates . ultrarapid bcc close- of and hexagonal simulations the dynamics high as molecular well at as phases (bcc) transitions packed cubic phase model to body-centered undergoes a the lead that to to metal frame- that applied (fcc) and conditions general cubic developed face-centered solidification A is the growth phases. metastable-phase predicting crystal for metastable 2020) work 24, August kinetic of review to for lead (received stabilization can 2021 16, solidification January during approved processes and Nonequilibrium NJ, Princeton, University, Princeton Debenedetti, G. Pablo by Edited a Sadigh Babak kinetics solidification polymorphic from derived diagrams phase Metastable–solid PNAS of order the metals on mainly rates of past cooling the solidification at over rapid alloys, grown which has and macro- in literature to of decades grow body solid several large circumstances, metastable A suitable of sizes. nucleation under scopic of can that that is phases face- undercooling the outcome dramatic large into more A at place. takes final phase clusters (fcc) before (8, cubic phase centered occur particles (bcc) to colloidal cubic regions reported body-centered charged are bulk metastable block and as the where 7) well 9), to (6, as (5), solutions interface observed metals been copolymer the quenched has rapidly grow- of nucleation in the two-step experimentally ratio generalized so-called of the be This transformations to changes. as phase need nuclei structural (4), ing models for phase-field account that as to recognized well been theory as density-functional has and (3), It the (CNT) stages (2). nucle- theory with later phase nucleation of phase the classical bulk crystal stages and equilibrium the energy early the by free by the dominated interfacial that be solid–melt expected to smallest likely thus are is with ation energies It free different liquid. interfacial of respective the energies their free and bulk phases crystal the between correlation explicit T diagram phase arneLvroeNtoa aoaoy hscladLf cecsDrcoae iemr,C 94550 CA Livermore, Directorate, Sciences Life and Physical Laboratory, National Livermore Lawrence indtsbc oOtad() trslsfo akof lack a from results It (1). crystalliza- Ostwald during to nuclei back solid dates of polymorphism tion of idea he 01Vl 1 o e2017809118 9 No. 118 Vol. 2021 a,1 | iei stabilization kinetic usZepeda-Ruiz Luis , | metastability a n oahnL Belof L. Jonathan and , 10 | solidification 4 to 10 5 ,has K, | a,1 xrm odtos reigudrsc odtosi believed is at conditions solidification such of under kinetics Freezing unrav- the conditions. of into extreme forefront insights These the current tested. fundamental at the be eling currently of can are kinetics limits experiments transformation laboratory the phase therefore, of and theories experiments, techniques achieved these be cooling can in rapid undercoolings and extreme exceed Hence, melting can above. achieve described that to rates able at been solidification have droplet (15–23) the techniques sion at nucleation preferential cul- explained be likely the to be (13). surfaces Instead, suspected cannot energies. is free phases prit interfacial remain their metastable on controversies of based Nevertheless, nucleation experiments. derived when been have these phases crystal from different critical inter- of smallest energies solid–liquid nucle- free the for the facial with models in one Consequently, is place barrier. emerging takes nucleation phase selection the rationalized. phase stage; been that ation has conjectured is growth undercooling been It metastable-phase the has CNT, for liquids of necessary in undercooled application often sufficiently has Through phases, in quasicrystalline demonstrated. metastable structure, and crystalline of bcc icosahedral, emergence diverse the bcc, and of observed, fcc, solidification been as been result, such have a phases solidification during As the nucleation investigated. droplets, crystal suspended of freely the dynamics of diffraction X-ray Through doi:10.1073/pnas.2017809118/-/DCSupplemental at online information supporting contains article This 1 ulse eray2,2021. 22, February Published BY-NC-ND) (CC 4.0 NoDerivatives License distributed under is article access open This Submission.y Direct PNAS a is article This paper.y interest.y the competing wrote no J.L.B. declare and authors B.S. The and data; analyzed performed L.Z.-R. L.Z.-R. and and B.S. B.S. research; research; designed J.L.B. and L.Z.-R., B.S., contributions: Author owo orsodnemyb drse.Eal aih@llgvo belof1@ or [email protected] Email: addressed. be may llnl.gov.y correspondence whom To e,adterkntcsaiiisdrn h rwhsaeare stage growth identi- the are during stabilities characterized. nucleation kinetic their dominating and fied, undergo- phases materials the atomistic Through of principles. simulations, first for maps from solidification blueprint phase rapid a ing kinetic provides of work This development metastable these of lacking. of interpretation are physical experiments stabilization for necessary are kinetic that However, phases of pressures. and predictions temperatures accurate extreme understanding at unprecedented in bear character- of to situ advances in brought Recent as have infancy. well ization, their as nonequi- technologies, in power/laser produce still pulsed are that phases conditions for librium solidification experiments quantitatively the in can that predict established theories cooled been atomistic However, rapidly has decades. in alloys phases and metastable metals of stabilization Kinetic Significance eetpwru dacsi yai hc/apcompres- shock/ramp dynamic in advances powerful Recent https://doi.org/10.1073/pnas.2017809118 . y raieCmosAttribution-NonCommercial- Commons Creative . y https://www.pnas.org/lookup/suppl/ | f12 of 1

APPLIED PHYSICAL SCIENCES 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 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 undercool- phase increases. It is thus imperative to identify under what con- ing, one could make the bcc precritical crystallites grow to large ditions the thus nucleated metastable-phase clusters can grow to sizes (35, 36). form micrometer-sized grains without transforming to the ther- In the following, we derive from atomistic first principles the modynamically stable phase. For this purpose, we define the kinetic maps of the crystal phases that are stabilized by nonequi- metastable solid cluster in liquid (MSCL) subspace, which is the librium processes during solidification. For this purpose, we gen- collection of microstates that are weakly connected to other SCL eralize the CNT and challenge the assumption that the precritical basins. More precisely, nuclei that belong to the MSCL subspace nucleation stage is solely responsible for crystal phase selection do not undergo solid–solid transformations during their growth during freezing. Hence, rather than studying the intricacies of stage, which for a micrometer-sized grain and a typical interface multistage nucleation and its effects on the rate of solidifica- velocity of 10 m/s, is on the submicrosecond timescale. tion (37–40), we focus on the product phases that result from It is important to note that while the definition of the SCL liquid–solid transformation kinetics and derive kinetic phase basin as a collection of microstates is quite straightforward, the boundaries. The latter specifies the undercooling conditions, in same is not true for the MSCL subspace. Metastability is the sta- the neighborhood of which multiple crystal phases are likely tistical property of the dynamic trajectories associated with the to grow from the melt. Near these boundaries, the postcritical microstates. This problem can be tackled by noting that large stage can play a big role in promoting or impeding the growth metastable-phase solid clusters transform to equilibrium through of competing crystal phases. Hence, in contrast to thermody- activated nucleation processes in their interiors similar to the namic phase boundaries that are straightforward consequences mechanism in the bulk. Hence, while the thermodynamic driv- of thermodynamic rules that can never be violated given enough ing force as well as the number of nucleation sites for solid–solid time, kinetic phase maps are mere guidelines for high-probability transformation increases with the size of the growing clusters, events and depend on the particular experimental context being the rate of nucleation of the equilibrium solid phase may in fact considered. Nevertheless, metastable-phase maps are crucial for decrease due to rising activation barrier. The latter comes about understanding solidification near the triple points of the thermo- because of increasing misfit strain energy of equilibrium-phase dynamic , where the liquid phase coexists with two inclusions inside growing metastable-phase clusters. The exis- solid phases. tence and extent of the region of long-lived metastable-phase CNT describes solidification in the language of canonical solid clusters in liquid that constitute the core of the MSCL sub- transition-state theory. It uses as the reaction coordinate the space can be determined effectively through the closed ensemble size of the solid cluster that nucleates inside the melt. It relies technique mentioned above, by which critical nuclei of different on separation of timescales involving two distinct processes: sizes are stabilized. activation of critical nuclei and growth of postcritical solid clus- With its core region identified, the boundary of an MSCL ters. The critical nucleus is the transition state that separates subspace can be delineated through direct canonical MD sim- the liquid and the solid basins of attraction. The rate of solid- ulations of small solid–cluster seeds. This procedure relies on phase nucleation is dictated by the flux of trajectories through subdivision of each SCL basin into a contiguous MSCL domain the critical nucleus. It is assumed that the solid-phase nuclei and a transient region that connects it to other basins. Tra- grow slowly enough that they can be considered near equilib- jectories that are initiated by nucleation of critical solid clus- rium at all times. A consequence of this is that the sizes of ters inside the MSCL subspace overwhelmingly grow to large the critical nuclei can be put in one to one correspondence sizes without structural transformation. Those initiated in the with the undercooling temperature. Furthermore, solid clusters transient regions have a finite chance of crossing over into can be stabilized within the melt by constraining volume and other SCL basins. This is analogous to transition path sam- energy fluctuations through change of statistical ensemble, from pling, where trajectories initiated near the saddle points of open (e.g., isothermal–isobaric) to closed (e.g., microcanoni- the potential are equally likely to decay into cal). This has been successfully utilized in computer simulations one or the other of the basins of attraction connected to it. and has allowed the study of equilibrium shapes and sizes of We find that phase transformations of the critical nuclei initi- solid clusters in liquid, as well as their coexistence tempera- ated in the transient regions proceed through far from equi- tures (41). Through the Gibbs–Thomson (GT) condition, this librium processes involving cross-nucleation of the new phase information has been used to extract solid–liquid interfacial free at the crystal–liquid interfaces. It is important to note that energies under the assumption of negligible anisotropy (41–43). the identification of the transient region boundaries does not Later in this paper, we will formulate a rigorous foundation for rely upon CNT (or any of its assumptions therein) and thus,

2 of 12 | PNAS Sadigh et al. https://doi.org/10.1073/pnas.2017809118 Metastable–solid phase diagrams derived from polymorphic solidification kinetics Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 eatbesldpaedarm eie rmplmrhcsldfiainkinetics solidification polymorphic from derived diagrams phase Metastable–solid al. et Sadigh yblqaiaieydpcstepeaec ftebct h close-packed the to each bcc of the content of blue prevalence 70 to phases. to the green 40 and depicts relative range bcc qualitatively The pressure observed. to symbol the In are equal hcp). phases blue simulations. and/or both with (fcc MD GPa, packed phase, direct close solid to in equal dominant observed green the was indicates melt color the which The from at temperature solid the of show circles nucleation filled colored respectively) The fcc–hcp, curves. and coexistence (bcc–hcp solid–solid and solid–liquid the delineate 1. Fig. shown diagram phase the in liquid appear and not bcc, does GPa it hcp, 79.5 coex- but at where liquid K, exists K, 3,478 and point 3,598 and coexistence hcp, and three-phase fcc, GPa third where A 85 coexist. K, at 3,320 one and and ist, GPa 71.6 at one for melt The the system. of model–Cu diagram (EAM) phase line method temperature–pressure the embedded shows 1 Fig. Diagram Phase Thermodynamic metastable-phase interface-driven for presented. is mechanism growth On a transitions. basis, growth-mode this by phase accompanied and study structures transformations the in-depth interface with unusual an coexistence their with in demonstrating melt, clusters paper (hcp) this close-packed metastable-phase conclude hexagonal of of delineated. We region be. are the freez- large can phase how which growth metastable–bcc noteworthy under quite the conditions is by the It dominated where phase is 5, kinetic follow- Fig. ing the the construct in In to shown line. methodology with map melt the 1, the develop Fig. along we in It ing, present shown (45). phases diagram, described solid potential phase system several complex interatomic Cu rather semiempirical a model constructed. exhibits a short-range be to a can applied diagrams by is phase methodology kinetic as This of well thermo- as combination which dynamic integrations, through a simulations, energy equilibrium on free multiphase under based and (44), melt simulations framework the MD practical nonequilibrium of follows, a what out outline In grow conditions. we that pressure and phases temperature solid different principal undercooling maps and the phase correspon- nuclei of solidification–kinetic one critical construct to the can one we of the temperature, sizes as the well between as dence liquid, kinetic undercooled the of of construction the for procedure maps. phase general a provides on htteeulbimsldpaeat phase (NPH) in P solid equilibrium described isobaric–isoenthalpic the that are found the Details (46–49). in ensemble simulations coexistence < sacneuneo h bv oooyo h hs space phase the of topology above the of consequence a As P 71.6 T > m hroyai n lrfs iei hs iga.Tebaklines black The diagram. phase kinetic ultrafast and Thermodynamic 85 (P P,hpi h interval the in hcp GPa, ) P.Tepaedarmcnan w rpepoints: triple two contains diagram phase The GPa. a enotie rmtopae(solid–liquid) two-phase from obtained been has 71.6 < T P m < sfcfrpressures for fcc is tis It Appendix. SI 85 P,adbcc and GPa, hs rnfraino hs lsesi lal nactivated that an indicates This clearly GPa. is 20 clusters to down these ns, of 10 durations to transformation with 5 simulations phase of MD range thousand in the stable several in found least be the at can that containing particles noted clusters be GPa. bcc should 85 above Nevertheless, It stable thermodynamically other. becomes phase each bcc clus- bulk to distinct hcp are coupled or basins weakly These fcc, nuclei. and bcc, fcc/hcp single-phase mixed 20 multiphase from range solid- to range pressure ters compact the They in GPa. of liquid 100 examples the solid–liquid to with NPH show coexist of that of 3 cores nuclei the the Fig. phase prevalence exploring subspaces. in for MSCL tool the the simulations effective an discuss constitute Multiphase ensemble we equilibria. section, metastable this In Metastable Multiphase Equilibria of Thermodynamics and Basins SCL experiments. metastable laboratory macroscopic-sized dis- in of observable and crystals, growth phenomenon bcc for this of conditions origin the follow- the physical cuss In the diagram. analyze phase we the of ing, point triple the solidification near polymorphic kinetics for close-packed proof the to mixed provide Hence, simulations observed. bcc GPa, be NEMD can as 80 nuclei well growing the below as of transformation pressures nucleation bcc At stabilizes and phase. close-packed kinetically bcc simula- process the metastable solidification fill nucleation to the the grow only Hence, nuclei Nev- not bcc box. the phase. above, but tion phase, bcc and bcc the GPa the in 80 occurs of pressures stability at thermodynamic ertheless, of region the grows. nucleus the as phase bcc inter- starts original which solid–liquid the occurs, bcc with phases competing the close-packed near form. to or nuclei transformation How- at bcc face, grow. small stage, first, growth and at their scenario, form During this do In 2). also nuclei (Fig. is growth observed bcc during phases pure close-packed to and transformation GPa, phases ever, 60 close-packed 40 to At transform At quickly vanish. noteworthy. nuclei quite bcc is formed, the are GPa nuclei but (fcc/hcp) close-packed 70 as well to as bcc 40 small GPa, from The increased simulations. is pres- as sure the kinetics GPa, evolution polymorphic in 70 observed the and place in difference 40 takes between region solidification the pressure polymorphic dominate the and In grow box. nuclei simula- simulation close-packed the the nucleation and dominate fcc observed, pure and GPa, is 40 grow below pressures nuclei at Similarly, bcc box. tion the nucleation and bcc close- pure observed, above, the and is depicting GPa phases. green 80 At and bcc phases. phase fcc/hcp the pressure, packed bcc each and the at observed fcc depicting phases blue the solid with the as color by well depicts as 1 Fig. hcp, the of for com- allows identification which cutoff method, evolution (50–52) adaptive (a-CNA) solid-phase the analysis the neighbor using mon rate tracked simulations have a NEMD We these at 1). during cooling (Fig. K/ns ultrarapid 100 of of simulations (NEMD) dynamics measured have We hsscintemxmlefc fkntc,b eemnn h start the determining temperature by kinetics, in of of effect explore limit maximal we the the section delineates variables, this in thermodynamic which phases of solid change line, and slow infinitely liquid melt the thermodynamic between boundary the the to contrast In Solidification Ultrarapid of and Map heats Phase latent the detailed in as A is well melting, points. as of volumes calculations, triple these respective of Clausius–Clapeyron the discussion the from by given starting slopes, relation, their hcp of the integration via of range higher stability slightly the the K. at melts 3,487 inside phase of hcp temperature falls the pressure, it this At because phase. 1 Fig. in tsol entdthat noted be should It h oi–oi oxsec ie hw nFg r drawn are 1 Fig. in shown lines coexistence solid–solid The T HS o reigwt aihn ulainbarrier. nucleation vanishing with freezing for T HS (P IAppendix. SI T ) HS hog oeulbimmolecular- nonequilibrium through talpesrsi lal outside clearly is pressures all at https://doi.org/10.1073/pnas.2017809118 PNAS | f12 of 3

APPLIED PHYSICAL SCIENCES we have not been able to stabilize mixed bcc/close-packed clusters in coexistence with the melt. The study of metastable-phase cluster–liquid equilibria (Fig. 3) reveals many surprising structural features. For instance, in contrast to previous works in Lennard–Jones systems (29, 30, 36), where the fcc clusters were found to interface the liq- uid through a bcc shell, negligible structural heterogeneities are found at the solid–liquid interfaces for either fcc or bcc clusters in this study. However, when examining the hcp clusters (Fig. 3C), we find that the hcp-(0001) facets interface the liquid through a shell of fcc-(111) layers. This amounts to a shift in the stack- ing of the close-packed layers of hcp clusters, in favor of the fcc symmetry, in the vicinity of the melt. It is shown in SI Appendix that the energy cost associated with the change in stacking of the solid layers is an order of magnitude smaller than the fcc–liquid interfacial free energy. Later in this paper, we discuss a kinetic mechanism that drives the growth of metastable–fcc phase at (0001)-hcp–liquid interfaces at all temperatures below the fcc melting point. This behavior is quite at odds with the other hcp– Fig. 2. A snapshot of an NEMD simulation during solidification at 60 GPa. liquid interface orientations. It lends credence to the unusual Particles have been classified as belonging to different crystal phases via hcp–liquid interface structure, seen in Fig. 3C, as well as the the a-CNA method (50–52): bcc (blue), fcc (green), and hcp (red). Particles existence of distinct MSCL basins of multiphase solid clusters identified as liquid have been removed. composed of mixed fcc/hcp stackings (see Fig. 3D). We conclude this section by pointing out that the existence of a multitude of MSCL basins demands that we quantify their process with a substantial barrier, which only increases with clus- relative thermodynamic stabilities or in other words, their phase ter size since nucleation of a new solid phase involves misfit strain space volumes. In the next section, we derive a rigorous and sur- that is easier to accommodate in smaller clusters interfacing prisingly general and convenient framework for extracting free the isotropic liquid. Hence, these solid–liquid equilibria consti- energies of solid–liquid metastable equilibria from macroscopic tute the core regions of their respective MSCL subspaces. It observables extracted from equilibrium two-phase simulations, should however be noted that most shallow SCL basins may lack utilizing the one to one correspondence between the sizes and metastable subspaces. As is laid out in some detail in SI Appendix, the coexistence temperatures of equilibrium solid clusters in

Fig. 3. A–D show four different crystal-phase clusters that can coexist in metastable equilibria with the liquid phase. Color coding is based on phase designation of each particle through the a-CNA method: fcc (green), bcc (blue), and hcp (red).

4 of 12 | PNAS Sadigh et al. https://doi.org/10.1073/pnas.2017809118 Metastable–solid phase diagrams derived from polymorphic solidification kinetics Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 nM iuain fsldlqi qiira swl edescribed be paper. will this as in equilibria, later solid–liquid size of cluster simulations average MD the in extracting for useful particularly are tions and respectively, where an be conditions way, contain- to the this and interface In volume particles. the associated no choose no ing to with convenient boundary parts thin is into infinitely It subdivision its unique. system, not physical real is a of property the is where reeeg nertdoe h nefc ra While area. interface the and over integrated respectively; energy energies, free free liquid and respectively; particles, G at as equilibrated written is sep- enthalpy system and liquid ble This temperature bulk region. uniform by interfacial surrounded a an solid by bulk of arated region a into ihteWlfcntuto sslto.Frorproe,the purposes, (53), our analytically For for solution. solved as be construction can Wulff limit the GT equilib- with the The clusters. in large shape of rium description physical accurate an is to sure eatbesldpaedarm eie rmplmrhcsldfiainkinetics solidification polymorphic from derived diagrams phase Metastable–solid al. et Sadigh this of shape equilibrium The directions. cluster all in extended were it if at ment closed a containing in equilibrium NPH) at (e.g., system ensemble solid–liquid two-phase a poten- of thermodynamic tial the constructing by directly real-world start or We computer experiments. for in observables macroscopic expecta- methodology of from values tion convenient parameters thermodynamic and interfacial extracting general a Thermodynamics. describe Systems Solid–Liquid discussed. kinetics is analyze phe- experiments to in standard is used solidification of commonly processes several are growth of that the rules the validity nomenological during and Con- the phases nucleation model. Furthermore, the these quantified. EAM–Cu of of the stages prevalence the of different relative of MSCLs simulations the bcc-phase computer sequently, the to and it crys- methodology applying fcc this their of by equilibrium of power demonstrated the complexity the paper, is the this regarding in or Later nuclei assumption structures. tal crystalline no the makes of shapes It melt. the Above, surface cluster as the coordinates spherical over in expressed integrated be can energy area free limit, this interfacial In approximation. the GT the GT beyond effects so-called curvature the size in particle) solid and one limit only containing one (i.e., where SL ti rfial oepesteitrailfe energy free interfacial the express to profitable is It R (T (Ω) P γ v , N n enthalpy and R γ P GT S n ˆ and Ω, GT S ne h osritta h lse oueis volume cluster the that constraint the under eq γ (Ω) κ(N and min = ) and I (Ω) steitrailfe nryo ciiu ntcluster unit fictitious a of energy free interfacial the is (T (N S stecytlsraenra fteitraeele- interface the of normal surface crystal the is H v , N S , sotie ymnmzto fEq. of minimization by obtained is P L N N , S γ T sys P ∞ = ) r h uklqi-adsldseicvolumes, solid-specific and liquid- bulk the are S L , V h hroyai oeta a hsbe thus can potential thermodynamic The . h v , P N (ˆ r h ubr ftesldadteliquid the and solid the of numbers the are S T sys n (T H Z ) S ; ( = ) G sys sitoue oicroaefiiecluster- finite incorporate to introduced is stesse’ oue h bv equa- above The volume. system’s the is T G , γ S b epooeasbiiino hssystem this of subdivision a propose We . P S , b ∞ (T T P + ) N and hc scnrle yteensem- the by controlled is which , (ˆ ) , S n N P ) sisitrailfe nrydensity energy free interfacial its is (Ω); N 2 3 S G + ) γ L and L b v GT N L r h uksnl-hs solid single-phase bulk the are N N T (T (T S L sys , N + G P , , L N P L P b )R atce toealpres- overall at particles a edtrie from determined be can ntefloig we following, the In (T L = ) )κ(N 2 = (Ω; , γ P V N I S + ) sys sys T , steinterfacial the is T , , , γ P , 5 I P )d ihrespect with (N ), G γ Ω. S I SL , v as T (T S This . , , P [3] [2] [1] [4] [5] P ) i ) , ymnmzn h hroyai potential thermodynamic the to minimizing by size cluster on on κ(N coefficient dependence dependence explicit curvature functional its the be of will gin there cases, ∇ ori- plane simplest crystal the the of functional entations nonlocal a becomes density pressure. view, change of can point shape tional equilibrium GT The when undercooled. require not is does system this on that depend note explicitly to not hence, does it and that size, property cluster of independent otipratfaueo Eq. of feature important most esoe ohl o ies olcino aeil (mainly materials of collection diverse a which for energies, hold free to interfacial showed solid–liquid he for (54) phenomeno- have rule Turnbull’s logical is to regard this essential most in the developments is of important One thermodynamics. it multiphase of thermodynamics, models accurate equilibrium that conditions predictions on to the contrast based in identifying selections, phase of kinetics-driven to purpose lead the for experiments Models. adequate. Phenomenological be will phase liquid the from for solid parameters the order distinguishing simplest the order. systems, bond-orientational most or for liq- zero energy Nonetheless, as of and such parameter, condition solid order ate the of noting (Eq. indistinguishable, volumes worth volume nearly specific is interface are it the phases Finally, where for. uid systems accounted the for be beyond also that solid can effects multiphase limit curvature as GT well finite-sized as Furthermore, multicomponent nuclei. can to framework the generalized and be made, are and approximations Eqs. uncontrolled temperature, using volume, by system’s exper- pressure from a atom of per measurements energy imental free of interfacial of effect extraction direct the sizes. cluster incorporating finite as limit, well interfacial as GT pressure the the and of temperature beyond determination energy for formalism free rigorous Eqs. a of tutes combination the Furthermore, clusters. relation the temperature used melting the from undercooling of where h bv mle that implies above The about N expansion in order first Eq. to correct is which lses stefloigTyo xaso a emade: be can expansion Taylor following the as clusters, coefficient curvature S n ˆ h qiiru/rtclcutrsize cluster equilibrium/critical The energy free interfacial the clusters, smaller for contrast, In nti eto,w aetu eie eea rmwr for framework general a devised thus have we section, this In N and S 6 S oacutfrcutrsraecraue hsi h ori- the is This curvature. surface cluster for account to : ∞, → γ salse ietrltosi ewe h observables the between relationship direct a establishes ∆ ∞ ∆ N H S T c n hne ihtmeaue eetees rmvaria- from Nevertheless, temperature. with changes ˆ m (T T hog h oa nefca reenergy free interfacial total the through ntenihoho fec ufc lmn.In element. surface each of neighborhood the in steltn etand heat latent the is , G , P P S b 1. = ) κ(N (T = ) , − S γ P h atrcnb elce o large for neglected be can latter The κ.  GT 2 3 ) a erpae yamr appropri- more a by replaced be can 3) ∂ − 2, ∆ ∂ )=1+ 1 = 1) log T N sol ucino eprtr and temperature of function a only is G H hnsuyn ai solidification rapid studying When and 3, S m κ L m b (T nEq. in ∆T 5 γ https://doi.org/10.1073/pnas.2017809118 , I sta t qiiru hp is shape equilibrium its that is P (N nrdcdi Eq. in introduced κ, R osvr supin or assumptions severe No 6. κ ∆ = )  eq 6 (1) S + 1 T (Ω) ) a engetdfrlarge for neglected be can N 1 3 ∆ = T S N c + γ H 3 2 T ob osre when conserved be to m S GT a o eobtained be now can N m . . . . o ag clusters, large For . − S c ∆ N a h desirable the has ∂ T T T S G ∂ ti important is It . log m m 6 N , T n ehave we and , PNAS and stedegree the is S ihrespect with m κ oethat Note . γ  I 8 , n the and | and 4, consti- f12 of 5 [6] [8] [7]

APPLIED PHYSICAL SCIENCES metals). This rule can be generalized to an exact ansatz, at least phases, such as bcc and fcc, they are nearly constant over a wide in the GT limit, as follows: range of pressures in a typical metal. Furthermore, by explicit cal- culations of the interfacial energy α, we calculate the entropic ∆H m sphere contribution to the interfacial free energy and demonstrate the γI = (α − T ασ) 2/3 AI . [9] vS accuracy of the negentropic model. In order to calculate the Turnbull coefficients of the fcc and sphere Above, AI denotes the interfacial area of an equisized spher- the bcc phases, we have conducted careful calculations of fcc– ical cluster, ∆Hm is the latent heat of solidification, vS is the liquid and bcc–liquid equilibria in the pressure range from 60 to specific volume of the solid at temperature T , and α and ασ 100 GPa. For these studies, cubic simulation boxes, containing in are dimensionless quantities that incorporate all temperature excess of 13 million , have been utilized. At each pressure, dependence as well as shape effects beyond that of a sphere. The three two-phase systems at different Hsys are equilibrated for 8 ns conventional Turnbull coefficient is α(Tm ) = α − Tm ασ. each using NPH MD simulations with a time step of 2 fs, and Turnbull’s remarkable rule conjectures that the solid–liquid the coexistence temperatures Tc and the systems’ volumes Vsys interfacial tension scales as the difference in heat content of the are recorded. The coexistence temperatures Tc range from 30 to solid and the liquid phases at the melting point. As a result, 60 K undercooling below Tm . The measured values of Tc and the dimensionless Turnbull coefficient α(Tm ) is expected to be Vsys are inserted into Eqs. 2 and 3 to obtain NS, which in turn, nearly independent of material . However, Turnbull is used in Eq. 6 to obtain the interfacial free energy γI (Tc , P). found from experimentally inferred nucleation rates that for most The latter is further used in Eq. 9 for evaluation of α(Tc , P), metals, α(Tm ) ≈ 0.45, while for nonmetals, α(Tm ) ≈ 0.32 (54). sphere 1/3 2/3 with A = (36π) (NSvS) . At each pressure, the calcu- 20% I Furthermore, variation of about was predicted theoretically lated α(Tc , P) are fitted to a linear temperature dependence, between the Turnbull coefficients of the fcc and the bcc crystal which is used to estimate α(Tm , P). phases based on a polytetrahedral structural model of a liquid in Fig. 4A shows α(Tm ) in the pressure range from 60 to 100 contact with a rigid solid (55–57). It should be noted that extract- GPa for both the bcc and the fcc phases. To within the statisti- ing homogeneous nucleation rates from experiments is a daunting cal accuracy of our calculations, α(Tm , P) for both phases can task, as solidification in most cases is driven by heterogeneous pro- fcc be considered nearly pressure independent, with α (Tm , P) ≈ cesses, and error bars are usually very large. It is thus desirable to fcc 0.54 and α (Tm , P) ≈ 0.465. While this result confirms that the use computer experiments to quantitatively study this conjecture. Turnbull coefficient of the fcc phase exceeds that of the bcc The temperature dependence of the Turnbull coefficient, to phase, their ratio of about 1.16 is significantly smaller than was first order in deviation from the melting point, is determined found in ref. 59; Sun et al. (59) calculated the Turnbull coef- by the entropic contribution ασ. Nelson and Spaepen (58) have ficients of fcc and bcc iron using different empirical potential argued that the origin of the solid–liquid interfacial tension is fcc mainly entropic, whereupon they proposed the so-called negen- models and found α to be in the range from 0.5 to 0.55 in bcc tropic model according to which the interfacial tension is all agreement with this work, but they found α to be in the range entropic (i.e., α = 0). Since γI must necessarily be positive, from 0.29 to 0.36, which is significantly smaller, leading to a this entails a negative interfacial entropy, thereof the name ratio in excess of 1.5. Our result, however, is in closer agree- negentropic. The negentropic model in combination with Turn- ment with the calculated relative Turnbull coefficients of the fcc bull’s rule has been widely used to analyze rapid solidification and the bcc phases according to the polytetrahedral structural experiments (10). model of the liquid (55–57), as well as particle systems interact- ing via inverse power law potentials (60, 61). We thus conclude Validation by Computer Experiments. We now show that while that the value of the Turnbull coefficient for a crystal phase Turnbull coefficients α(Tm , P) have distinct values for different can be quite sensitive to particle interactions. Nevertheless,

AB

bcc fcc 0

-0.01 0.5 -0.02 Turnbull coefficients -0.03 bcc fcc

0.4 -0.04 60 70 80 90 100 ratio of interfacial energy to free 60 70 80 90 100 Pressure (GPa) Pressure (GPa) Fig. 4. (A) Calculated Turnbull coefficients for the C D fcc and the bcc phases as a function of pressure. (B) Ratio of the interfacial energy α to the inter- facial free energy α(Tm) of the fcc and the bcc 1.35 phases as a function of pressure. (C) Ratio of the latent heats of the fcc and the bcc phases as a 1.3 function of pressure. (D) Comparison of two differ- ent algorithms (gauges) for extracting cluster radii 1.25 from MD simulations. The filled circles are bcc-cluster 1.2 radii extracted from two-phase simulations at dif- ferent temperatures and pressures ranging from 1.15 60 to 100 GPa. The open squares are fcc-cluster radii extracted from two-phase simulations at dif-

Ratio of the latent heats fcc and bcc 1.1 20 40 60 80 100 ferent temperatures and pressures ranging from 60 Pressure (GPa) to 100 GPa.

6 of 12 | PNAS Sadigh et al. https://doi.org/10.1073/pnas.2017809118 Metastable–solid phase diagrams derived from polymorphic solidification kinetics Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 oue hrfr,i sipratt s ossetgauge bcc- consistent and fcc- a of use energies free to the comparing important example, is for when, the and it pressure, are Therefore, temperature, observables chosen enthalpy, volume. the energy, physical to free abso- gauge-invariant system’s subject total the is only energy that The free reiterate interfacial the gauge. to the with of important compared value is clusters lute Furthermore, It fcc-phase ones. gauge. the underestimate fcc of zero-volume slightly the radii to for effective seems clus- the analysis than bcc neighbor the better common for clearly the methods is two radii the note ter between to interesting agreement however, the Overall is, that It temperatures. impressive. and quite pressures coexist- is different clusters agreement at fcc-phase liquid and with bcc- number ing of a from simulations extracted two-phase radii of cluster effective shows figure relation The the and gauge zero-volume the its using identifying radii by the an determined 4D between Fig. Finally, is analysis. cluster. Voronoi cluster solid through the surface the for applied to radius is belonging function effective not distribution atoms coor- radial a remove a Next, to on (50–52). based a-CNA the filter by dination determined gauge first zero-volume is the particle to described alternative method, an determining in as for such 41, liquid one ref. for used the in allow previously have within that We region systems. literature solid two-phase the the in characterization devised order accurate elaborate been error, statistical have the parameters reduce specific mul- to distinct any order having to In phases applicable volumes. coexisting and the use with to system infinitely convenient tiphase is is interface gauge the This assumes thin. which gauge, zero-volume the contribution energetic the calculate follows: we coefficient, bull to contribute phase heat fcc latent the larger γ of 1.12 the coefficient to Turnbull pressures larger pressure all the ambient at and Nevertheless, at GPa. 1.37 100 from at reduced is It dependent. eatbesldpaedarm eie rmplmrhcsldfiainkinetics solidification polymorphic from derived diagrams phase Metastable–solid al. et Sadigh size cluster the determining basis. case by should the case hence, a of and on validity dependent considered the be system Hence, be interface may (63). model densities between negentropic as significantly well vary as can orientations of the energy interfaces of solid–liquid content free entropic flat interfacial the fur- of that should study found system prior It Lennard–Jones a positive. the that stays As noted energy entropy. be free interfacial ther interfacial the than the magnitude result, in a smaller much as energy. inter- problems, is fcc–liquid conceptual bcc–liquid it no the poses the than energy that interfacial smaller negative and note The negative Also is metal. energy typical facial a for the Clearly, model GPa. 100 to of 60 part from energetic range pressure the in phases bcc where observed is it where fcc 4C, the and Fig. bcc in coeffi- the shown of Turnbull that are heats latent relative free latter relative The interfacial the the phases. the as on well by both as driven cients depends is which fcc–liquid undercoolings energy, flat large of at study phase recent a to with coefficient agreement (62). Turnbull interfaces in the of weak, dependence be pressure the find we i hss i.4B Fig. phases. uid fcc nodrt ics h eprtr eedneo h Turn- the of dependence temperature the discuss to order In ecnld hsscinb xlrn ifrn agsfor gauges different exploring by section this conclude We bcc the of nucleation preferential the that noted be should It γ > ∆ α H H bcc  S m b fcc ∆ v . and S 2/3 H > m ∆ H A H L N b I sphere α m r h uketape ftesldadteliq- the and solid the of enthalpies bulk the are R bcc S hw h ratio the shows nti ehd h hs hrce feach of character phase the method, this In . svr ml,wihvldtstenegentropic the validates which small, very is S hi ai s oee,srnl pressure strongly however, is, ratio Their . acltdi hswyadtoecalculated those and way this in calculated = H sys − N N S ehv ni o nyused only now until have We . S c H α S ( b α T −  m ) (N o ohtefcadthe and fcc the both for hw h comparison the shows − R N S S c = )H L b 4π 3 , N S v S α  [10]  1/3 as . h rcdr o eemnn h necoigtemperature undercooling the determining T for procedure the Boundaries. Solid–Nucleation-Phase from growth. promote hcp/fcc-phase that arising interfaces mixed intricacies hcp–liquid of of features discussion that structural a unusual small with so We paper improbable. this becomes become phase conclude hcp energies or fcc free bound- pure phase into stacking-fault solidification thermodynamic 1), hcp–fcc the (Fig. of noted ary vicinity be should the It in 1). where (Fig. that temperature GPa, thermodynamically melting is 85 the hcp) below below or stable (fcc pressures bcc- phases metastable close-packed at the of melt of problem one the the from it to apply growth sections we phase the two procedure, large following this basin, to of the grow MSCL to efficacy in likely an the are demonstrate of basin To this out sizes. to transition belong that of nuclei rate critical the timescale a than on we occurs shorter growth solidification result, whenever metastable-phase a that to As recognizing lead by undercooling. that conditions of the function identify a can as of basin rate MSCL partic- the any ular into predict can liquid can undercooled that we homogeneous sizes from way, basin cluster transition this MSCL solid-phase In the of metastable. of range considered the be extent undercooling words, the relative other of phase the in function each or quantify for a determine 1) as 2) nuclei to and critical need of solidi- we stabilities predicting during purpose, phase for phases this strategy crystal For general metastable fication. a of develop stabilization to kinetic ready now are particular We a Diagrams at Phase Metastable–Solid phase liquid the pressure. with and temperature coexisting nuclei phase t ytebcpae ihnteCT h aeo ulainof nucleation of rate the below CNT, temperature the and at Within clusters thereof) solid phase. combination critical bcc a close- the or by the hcp, it, by EAM–Cu. (fcc, dominated of phases phases is hcp packed nucleation the temperature, and this fcc the Above of those exceeds phase es 4ad6.W aefudti neesr ntepresent the in calcu- unnecessary be this found can have it We in desired, described 65. as If and clusters 64 critical liquid. the refs. the of fluctuations of size attachment from property of lated rate a the assump- mainly since two good is is making 1 Assumption by assumptions. equations above 1) the tions: solved have We where ntrso h uniisi Eq. in quantities the of terms fluctuations. in size cluster to respect with ture temperature at melt the with coexisting Above, n ie rsuei h mle ftetosolutions two the of smaller the T is pressure given any solid J equations: the two solving of by area determined unit a to liquid the and from cluster, attachment of sections. rate previous in determined numerically were S bcc hcp ∗ ∗ J Eq. (P G S (T (T fteequations the of SL cp eo hc h ulainrt ftemtsal bcc metastable the of rate nucleation the which below ), hcp T ∗ 11 (T c ∆G = ) cp ∗ τ = ) cp ∗ cp lostenceto-hs boundary nucleation-phase the allows ersnsete ftetotemperatures two the of either represents ) S s v ≈ − J L steecs reeeg faciia oi nucleus solid critical a of energy free excess the is S hcp τ ∆G G steseicvlm fteliquid. the of volume specific the is 2π bcc SL (T bcc k S 00 n 2) and (T B hcp ∗ (T T cp ec,tepaeboundary phase the Hence, ). ∗ c c = ) ) (N α hcp k S c B (T 2 T ≈ cp c https://doi.org/10.1073/pnas.2017809118 ∗ α )) 1: fcc log nti eto,w describe we section, this In 2/3 ∆G hs r ohreasonable both are These . T " c J v τ T S ∆H L is S bcc ∆ c = exp and , (T H m G bcc m ∆G cp fcc ∗ SL  α α = ) ∆G − cp bcc − S ∆G a edefined be can N  J τ T PNAS S 00 k S sys fcc T τ eoe the denotes ∗ B τ siscurva- its is S bcc (P G fcc (T cp T T ∗ (T L T b c ∗  or | ) fcc which , ∗ c (P fcc 2 ∗ ) obe to f12 of 7 ) # T  [12] [11] ) , and and hcp . ∗ at .

APPLIED PHYSICAL SCIENCES context because we have found that neglecting the right-hand The line connecting this point in Fig. 5 to the bulk thermody- ∗ side of Eq. 12 altogether only leads to a slight rise in T (P). namic fcc–hcp–liquid triple point at P = 71.6 GPa and Tm = This increase is largest at low pressures. At 20 GPa, it amounts 3,320 K is the boundary that separates the temperature–pressure to 0.7 K, while at 40 GPa, it is reduced to only 0.25 K. Assump- regions where nucleation (according to the CNT) is dominated tion 2 is reasonable due to the similarity of the two close-packed by the fcc phase from those where the hcp phase is most favored. phases both structurally and energetically. In the pressure range However, we expect kinetic factors beyond those considered of interest (around 70 GPa), we find that the latent heats of melt- in this section to promote mixed fcc/hcp solidification at T > ing of the hcp and the fcc phases differ by only 2%, while that of T ∗(P) within at least a 10-GPa-wide pressure window about the the bcc phase is 20% smaller. Furthermore, their respective equi- thermodynamic triple point at P = 71.6 GPa, due to diminish- librium shapes shown in Fig. 3 A and C suggest that αhcp > αfcc. ing energy cost of stacking faults. At first sight, this may seem This assertion is based on two observations: 1) the equilibrium only marginally relevant to solidification kinetics and have lit- shapes of fcc clusters in the melt are nearly spherical, from which tle effect on the liquid–solid interfacial kinetics. However, we it can be concluded that the fcc–liquid interfacial free energy will show in a later section that in regions where the CNT pre- only depends weakly on fcc crystal plane orientations, and 2) the dicts hcp-dominated nucleation and growth, the liquid interface equilibrium shapes of hcp clusters in the melt exhibit hcp-(0001) with special hcp plane orientations can in fact promote fcc-phase facets that interface the liquid through a shell of fcc-(111) layers. growth by a kinetic mechanism. This implies that the liquid–fcc interfaces are favored relative to liquid–hcp ones and thus, have lower free energies. Kinetic Phase Stabilities of Postcritical Nuclei. In the previous sec- ∗ Eq. 12 and its solutions are discussed in detail in SI tion, we computed the phase line T (P), below which critical Appendix. The resulting phase boundary T ∗(P), separating the nucleation is dominated by the bcc phase. In this section, we temperature–pressure regions where bcc nucleation dominates determine the conditions for the bcc postcritical nuclei to grow to [T < T ∗(P)] from those where fcc or hcp nucleation prevails large sizes without transforming to other solid phases. Hence, we [T > T ∗(P)], is shown in Fig. 5. At pressures P < 71.9 GPa, quantify the temperature–pressure region within which critical T ∗(P) (green line) is the nucleation phase boundary between nucleation occurs inside the bcc–MSCL subspace. The weak cou- the bcc and the fcc phases, and at P > 71.9 GPa (red line), it con- pling of the MSCL subspace to other basins guarantees growth stitutes the boundary between the bcc- and the hcp-phase critical of postcritical nuclei without solid–solid transformation. In con- nuclei. Whenever a critical nucleus belongs to an MSCL basin, it trast, the transient regions of the SCL basins contain postcritical is expected to be long lived. It can thus grow to large sizes and clusters that may undergo structural instabilities. The extent be observed during solidification and likely even after the phase of the transient subspace can be studied straightforwardly via transformation has completed. In the next section, we determine NEMD simulations in the isobaric–isothermal (NPT) ensemble the extent of the bcc–MSCL basin upon very strong undercooling by 1) preparing atomistic configurations of critical solid-phase by studying the kinetic stability of postcritical bcc nuclei. clusters embedded in the melt at different undercooling tem- Before ending this section, we remark on the fcc–hcp–bcc crit- peratures, 2) initiating dynamic trajectories from these config- ical nucleation triple point at P = 71.9 GPa and T ∗ = 3,263 K. urations, and 3) characterizing the evolution of their solid-phase content as the solid clusters grow. The approach we have cho- sen to pursue for accomplishing the above tasks is detailed in SI Appendix. We have carefully applied this procedure to examining the kinetic stability of postcritical bcc nuclei at pressures 20, 30, 40, and 60 GPa. Our aim has been to identify the undercooling temperatures Tu (P), below which the growing bcc nuclei trans- form to fcc/hcp phases. In other words, the critical nuclei smaller than Nu do not belong to the bcc–MSCL basin, as they readily transform to fcc/hcp when growing. The result is shown in Fig. 5, where the Tu (P) line is green below 40 GPa and brown below 80 GPa. This coloring is intended to convey the fact that at tem- peratures below Tu , in the green pressure region the unstable bcc nuclei are observed to transform to close-packed structures dom- inated by the fcc phase (SI Appendix, Fig. S9), while in the brown region, multiphase fcc/hcp clusters are observed (SI Appendix, Fig. S7). Of course, there is no sharp boundary between the two regions, and the hcp content of the multiphase region rises with increasing pressure. At 80 GPa, Tu and THS coincide, and solid- ification at all undercoolings is found to be dominated by the bcc phase. This has been proven by direct NEMD simulations, which consistently show bcc nucleation and stable growth at P ≥ 80 Fig. 5. A solidification–kinetic phase map. The top and bottom solid lines bound the region where solidification can take place. The four lines in order GPa under ultrarapid cooling conditions. from top to bottom are 1) the thermodynamic melt line Tm, with the green It is beneficial here to discuss in some detail the physical part depicting liquid/fcc-phase boundary and the red part depicting liq- meaning of the temperatures Tu (P). They designate the low- uid/hcp; 2) the boundary T∗, where the CNT rate of bcc nucleation equals est undercooling at which all of the computationally prepared the fcc rate (green part) or the hcp rate (red part) with a critical nucleation bcc nuclei that could grow under this condition grow without bcc–hcp–fcc triple point at P = 71.9 GPa and T = 3,263 K; 3) the boundary transforming to other solid phases. This does not exclude that Tu, below which the postcritical bcc nuclei become unstable and transform at lower temperatures, there may exist bcc clusters that can grow to fcc (green part) and mixed fcc/hcp (brown part); and 4) the boundary THS, without transformation to close-packed phases. This rather con- below which the liquid phase becomes unstable and crystallization occurs servative measure of metastability is justified if we seek to predict with vanishing nucleation barrier and thus, can be observed in MD sim- ulations. The colors depict the observed crystal phase in the simulations: the conditions for experimental observations of macroscopic- fcc: green; mixed fcc/hcp: brown; and bcc: blue. The solid parts of the T∗ sized solid phases. For this purpose, it should be recognized and Tu lines bound the region, where the bcc phase is nucleated inside the that our computer experiments are quite limited in their bcc–MSCL subspace. representation of the perturbations that can occur in realistic

8 of 12 | PNAS Sadigh et al. https://doi.org/10.1073/pnas.2017809118 Metastable–solid phase diagrams derived from polymorphic solidification kinetics Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 hsi h xetdrgm ae nOtadsse ue which rule, step Ostwald’s on based regime stabilization undercooling. expected kinetic significant the at discussed is solid This have bcc single-phase we metastable paper, of this in Hitherto Transitions Growth-Mode to Coupled Transformations Phase Interfacial Layers: Interface Extraneous-Phase bulk the of properties cannot structural alone. it phase dynamical crystal and the mechanism driven, by This interface explained phonons. be and bulk nonequilibrium stable explicitly with is clusters growing transformations phase in solid–solid drive can kinetics show interface fast we However, conditions. these under instability in phase vibrational bcc the bulk to the due of be to instability This expected undercoolings. conventionally large is at EAM– fcc in to undercooled clusters transform bcc which within growing Cu, the growing upon touched clusters pertaining have solid phenomenology We liquids. of vast stability a kinetic of to surface the scratched only at kinetics slower the the for undercoolings. within suited small described better is be which cannot compact framework, that kinetics from CNT far their noted repro- are and be undercoolings spherical, is should large and result at It This clusters subspace. critical bcc. bcc–MSCL the the as the demonstrating sizes of configurations, large boundary initial to different grows from while it ducible phase, at fcc K, cluster the to 1,863 this transitions it growing at ensemble, When NPT the ensemble. in K NPH 1,860 the metastable about in a contains shrinking cluster slowly It bcc by obtained stability. was cluster dynamic This of metry. verge the on N region. GPa transient 30 the at S9 and Fig. domain Appendix, bcc–MSCL the between growth, of the early some time. confirms solidification, during with increasing of 2, phases content close-packed start close-packed Fig. the with the to in at transform shown nuclei which GPa, bcc 60 of formation at solid- ultrarapid snapshot the ification of examination Nevertheless, experiments. rate. steady at melts slab hcp The circles): 3) filled clusters. fcc fluctuating of film a liq- through the uid interfaces substrate hcp The observed. is motion interface steady-state and le ) h iudpaetasom otefcpae hc rw nthe on 3,489 grows 2) which substrate. phase, hcp fcc the to transforms phase liquid The circles): filled 1) regimes. temperature three into titioned in specified dimensions with (11 and (0001) orientations with interfaces planar 6. Fig. eatbesldpaedarm eie rmplmrhcsldfiainkinetics solidification polymorphic from derived diagrams phase Metastable–solid al. et Sadigh u ecnld hsscinb onigotta eew have we here that out pointing by section this conclude We boundary the at states the discuss briefly to instructive is It IAppendix SI ≈ T m 200 = eprtr eedneo h oiiyo w ifrn hcp–liquid different two of mobility the of dependence Temperature ,9 .Tecluain r efre nproi lbgeometries, slab periodic in performed are calculations The K. 3,497 atce dnie yaCAt anyhv c sym- bcc have mainly to a-CNA by identified particles htfrfo qiiru odtoscue by caused conditions equilibrium from far that ≤ hw neapeo oi rtclcluster critical solid a of example an shows T ≤ IAppendix SI ,0 gendse iewt pncrls:No circles): open with line dashed (green K 3,503 h 00)itraekntc spar- is kinetics interface (0001) The . T < T ,8 gensldln with line solid (green K 3,489 ≥ 0 tpressure at 20) ,0 rdsldln with line solid (red K 3,503 P = 0GPa 80 SI etcnrbtn usatal otersaiiy sa example, an As representing the stability. line with their the interfaces to their substantially to contributing due melt phase, other phases equilibrium prefer bulk can the liquid than in clusters solid small that suggests peree ntefltitraelmt ntefloig we hcp- orientations, following, hcp-(1,1 interface the two and simulations for In two-phase (0,001) geometries by limit. limit slab this periodic interface of of flat study in-depth the an should structure present in shell the even case, the appear for not ramifications If kinetics. deep undoubtably growth possible is crystal This with nuclei. phenomenon solid noteworthy growing a the of expected interfaces be the to are at transformations structural of so, interfa- question If the curvature. on the cial dependent forth are bring arrangements atomic inevitably such 3C) whether Fig. (e.g., nuclei sized features. unusual the its with study planes to the hcp-(0,001) opportunity fcc–bcc an of However, offers interface of systems. melt the our cost at any in structure energy in shell high the fcc structures relatively that shell be suggests bcc must we which interfaces such earlier, systems, observe mentioned our to As of able literature. been the not in have explored this have of been solidification ramifications of not kinetics the the knowledge, for inter- structure our their interfacial at To unusual layers melt. bcc by the coated with be faces to found were nuclei system, fcc Lennard–Jones the the of (29) studies computational earliest energy free of interfacial order solid–liquid Appendix). by an (SI the is rationalized energy than be free can smaller fault This magnitude stacking shell. the that fcc above observation an the the via liquid via of the explained interface existence 3C be the Fig. particular, cannot of In approach. suggestive that is phenomenology 3C) such (Fig. structures the interface the of structure in internal immersed the neglected. nuclei as be solid long can interface as of solid–liquid reasonable energy is free This contribut- overall surface melt. only dividing the particles, a no to containing as ing and interface space the this analysis, no in consider this earlier occupying to described In suffices As study. it role. this passive paper, of a system from assumes Cu interface phase model the the metastable–bcc for of melt nucleation the for required cooling c-00)itraewt h iudsol ei h c phase, fcc the in hcp-(11 be the the should to 3C, liquid contrary Fig. the of discussions with previous interface our hcp-(0001) on Based 1). (Fig. phase point iertmeauedpnec.I atclr nteinterval the in particular, In ∆ dependence. temperature so-called linear sites, the of all of (67). large, 1/4 sites step are about repeatable interfaces be The rough to (66). the assumed heat on commonly latent attachment diffu- the for liquid and sites by velocity, available controlled thermal are of only that detachment often surfaces and sivity, crystal attachment the of to rates atoms crystal latent by and dominated small rough, atomically is relatively is growth interface the the metals, to simple of due (i.e., cal system that EAM–Cu the Note of 6. heat Fig. in mobility stud- about temperatures 6, of Fig. range in narrow ied the In implies sign growth. negative of and solid-phase phase, vicinity liquid func- the growing a indicates as in velocity interface each temperature result of velocity of the remarkable tion depicts It the 6. Fig. reveals in shown small at and the configurations of solid–liquid undercooling study these systematic of A S11). properties Fig. dynamic Appendix, (SI phase bulk the xetd hsi lal bevdfrtehcp-(11 the for observed clearly is This expected. a nsa nenlsrcue fsldlqi nefcsi finite- in interfaces solid–liquid of structures internal Unusual the in fact, In new. not is structure shell a of observation The heterogeneous with nuclei hcp of observation the However, ncnrs,te(01 nefc eoiyhssrnl non- strongly has velocity interface (0001) the contrast, In 349 3,503) (3,489, = T m 3,497 = t8 P,weehpi h ukequilibrium bulk the is hcp where GPa, 80 at K 1% T 20) ,itraemblt aihso the on vanishes mobility interface K, ∗ 20) (P ntenihoho ftemelting the of neighborhood the in , eito from deviation ) nefc htsasiotutrlwith isostructural stays that interface nFg eit h iia under- minimal the depicts 5 Fig. in hw htte(01-c planes (0001)-hcp the that shows https://doi.org/10.1073/pnas.2017809118 ∆ H /k T B m T T oiiesg fthe of sign positive ; m m ierrsos is response linear , ≈ ,wihi typi- is which 1), PNAS 20) interface | f12 of 9

APPLIED PHYSICAL SCIENCES A 20 6 variations. Above 3,503 K, the hcp phase shrinks smoothly at a speed proportional to the superheating, as shown in Fig. 6 as interfacial fcc content well as in SI Appendix, Fig. S13. Below 3,489 K, the interface hcp-liquid interface thickness region is solid fcc and grows steadily at a speed proportional to 15 the undercooling from the fcc melting point at 3,489 K (Fig. 6 and SI Appendix, Fig. S14). 4 Fig. 7A explains the sudden dynamical arrest of the (0001) 10 interface, observed at 3,503 K in Fig. 6, by invoking a first- order structural-phase transformation at the interface. The order parameter for this transformation is the fcc content of the inter- 5 face. Above 3,503 K, the concentration of the fcc particles at 2 the interface is dilute, which we designate as phase I. It under- = Interface width (no. of layers)

Phase II Phase I int goes a discontinuous jump at 3,503 K, transitioning to phase Number of fcc particles ( x 1000 )

W II, in which larger fcc clusters form and dissolve (SI Appendix, 0 3492 3498 3504 3510 Fig. S12). While the rate of attachment to the fcc sites on the Temperature (K) underlying hcp substrate is high, the clusters cannot become crit- ical as long as the temperature is above the fcc melting point of B 5000 30 3,489 K. Further cooling toward the latter temperature causes Interfacial fcc population the fcc content at the interface to steadily rise. As shown in SI Appendix, Fig. S11, the thicknesses of fcc-containing films at 4000 T = 3505 K the interface are smallest near the transition temperature (≈ 20 3,503 K) and increase upon cooling. Below 3,489 K, metastable 3000 fcc phase grows on the (0001) interface. An important feature of first-order phase transformations is the presence of , which implies that metastable phases 2000 Interface displacement 10 exist in the vicinity of the transformation temperature. This is Phase II confirmed by NPT MD simulations at T = 3,503 K, which can Number of fcc particles 1000 equilibrate in either phase I or II depending on initiation. In con- Phase I Interfacial displacement (Å) 0 trast, Fig. 7B clearly demonstrates that phase II is unstable at 0 3,505 K. It shows the time evolution of the fcc population at the 0 2000 4000 6000 8000 10000 interface in an NPT MD simulation starting from a configuration time (ps) in phase II (large fcc content) generated at 3,500 K. During the course of the simulation, the fcc clusters dissolve, and the inter- C facial structure transforms to phase I (SI Appendix, Fig. S15). 1 T = 3505 K (Initial state) Similarly, we find that phase I is unstable at 3,501.5 K. Hence, T = 3505 K (final state) T = 3510 K the hysteresis window is at most 2 GPa. T = 3500 K Fig. 7B also records the motion of the interface with time. The intimate coupling of the interfacial structure to its dynam- ics is clearly evidenced in this figure. The fcc shell at the interface seems to constitute a protective shield for the hcp sub- 0.5 strate, which starts dissolving at around 7,000 ps, when the fcc population has dwindled. In order to explain the coupling mechanism between interfa- cial structure and its dynamics, we study the atomic roughness of the hcp–liquid interfaces at different temperatures. It corre-

Distribution density function (D) lates strongly with the width of an interface, which in turn, can be 0 quantified by the extent of the region, where an order parameter 0.3 0.6 0.9 field (OPF) exhibits large gradients. In the following, we outline a Normalized OPF gradient rigorous method for measuring interfacial thickness from atom- istic simulations. As a result, we prove that hcp–liquid interfaces Fig. 7. (A) The blue curve depicts the number of fcc particles in the sys- undergo an atomically diffuse to sharp transition concurrent with tem that mainly reside near the (0001)-hcp–liquid interfaces, and the black the structural transformation from phase I to phase II. Con- curve shows the interface widths Wint as a function of temperature. (B) Time evolution of an NPT MD simulation at T = 3,505 K, initiated from an (0001)- sequently, the growth mode changes abruptly from continuous hcp–liquid equilibrium slab configuration at T = 3,500 K. The blue line shows (fast) to layer by layer with high nucleation barrier. the evolution of the number of fcc particles, while the black line represents We choose as the order parameter the fraction of the particles for the same simulation the displacements of the (0001)-hcp–liquid interfaces in a volume element that are designated via the common- as a function of time, derived from the size fluctuations of the simulation box. neighbor analysis as hcp. Let η(z) be the OPF along the hcp- The symmetry character of the particles have been identified via the common- (0001) plane normal, and define the OPF gradient η˙(z) as the neighbor analysis using a cut-off radius of 2.875 A.˚ (C) The distribution density derivative of this function. It is now straightforward to define function D(η˙). The blue curves depict the distribution functions of the phase I and calculate the distribution of OPF gradients from the simula- atomically rough interfaces with relatively small OPF gradients, and the black tions. For this purpose, it is convenient to work with normalized curves show the sharp phase II interfaces with much higher OPF gradients. The ˙ dashed lines have been obtained by averaging over the first and the last 2,000 OPF gradients: η(z) =η ˙(z)/η˙max, where η˙max = ηhcp/∆zhcp, with time steps of the simulation shown in B. ηhcp being the value of the order parameter in bulk hcp solid and ∆zhcp being the interlayer spacing of the hcp-(0001) planes. The physical representation of η˙max is a maximally sharp inter- simulation timescale exceeding 10 ns (dashed green line in face, with a bulk solid layer next to a liquid layer with negligible Fig. 6). Outside of this interval, interface motion is observed to order. We can now define the distribution density function D(η˙ ) be normal, diffusive, and linearly changing with temperature in the interval 0 < η˙ < 1, normalized by the relation

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W nthe in int [13] [15] [14] is zto n nlsso iuainsasoswr are u sn h OVITO the using out carried were snapshots simulation of Visual- analysis step. and time ization 1-fs a for with performed model were simulations not reasonable When specified, pressure. otherwise a under metal constitutes close-packed generic it a in data, not interactions is atomic initio the it ab While represents condition. high-pressure model ambient to This at potential well fitted (45). quite interatomic al. Cu sim- et of EAM MD properties Mishin physical the by large-scale bulk constructed and of as (44) series metal, a code Cu for of LAMMPS results the presented using have ulations we paper, this In Methods and Materials sur- be dividing must generalized. boundary thin and phase infinitely abandoned thermodynamic the an with notion as analogy fundamental in boundary face the phase result, kinetic a the As of phases. crystal growth structure disparate promote interfacial may of orientations of crystal role different dynamics, explicit their and in this phases can to crystal and Due competing of neighborhoods. clusters kinetics growth solid tran- the sharp the in induce sitions turn, of in complex which bulk transformations, have phase the undergo can from radi- interfaces distinct are These uniform. behaviors structures others. as whose from this treated interfaces, different In be special cally considered. are cannot is there interface hcp Rather, solid–liquid as such the with becomes competition phase case, in shortcoming crystal fcc anisotropic metastable This of an excluding subspaces. growth by the MSCL after aside apparent the complexities too real from is all them it sweeps diagrams, and phase restrictive solidification–kinetic of construction that those instabilities all growths. kinetic early of undergoing their clusters exclusion amid of MSCL number by finite the a identified to into easily lead nucleation be ther- to can Hence, lead subspace subspace. that MSCL grows the conditions inevitably to nucleus modynamic transition the to instabili- after, kinetic enough time unless short large occurs basin, a SCL within nucleation the occur critical of ties if region transient clusters Hence, the large real- size. in the cluster the only with of in transformations solid–solid increases all of lies nucleation contain to idea barrier the subspaces this since MSCL of that power ization phase real subspace further without MSCL The evolve particular to transformations. nucleation guaranteed a is Hence, certainty into near basins. liquid with SCL and undercooled other the basins to from SCL that connected former this of the weakly For of concept are subdomains diagram. as phase the defined subspaces, a introduced MSCL of restric- their have notion a the in we preserve kinetics purpose, to including paper, as by this inter- way idea In be tive this and diagrams. generalized drawn phase have be thermodynamic we can critical like diagrams just a phase fail. preted no kinetic sizes of way, large to phase this grows In CNT, the the picture, to this according govern which In nucleus, during forces CNT. selection the driving phase within of thermodynamic being described result stage is a nucleation as that the metastable discussed mechanism been of has kinetic stabilization phases the kinetic Conventionally, on considered. depend They versal. (68). of nuclei domain near-equilibrium the of determine assumption the to kinet- of necessary validity nonequilibrium are the nucleation No during of made. ics studies are simplifying sophisticated structures more no or doubt, However, shapes their investigated. regarding well are assumes assumptions as stabilities paper thermodynamic kinetic this whose as in nuclei, adopted critical procedure near-equilibrium The unstable. ically xeto h eatbergo.I sbuddfo bv by above from significant bounded the boundary is observe by phase It to region. solid-nucleation sobering metastable the quite the is of this phase It extent of 5). metastable diagram (Fig. a phase system thermodynamic construct the thus complementing map We potential. interatomic hl h bv sqiearaoal udn rnil for principle guiding reasonable a quite is above the While uni- not are maps phase kinetic that noted be should It T u (P ne hc h rwn c ulibcm kinet- become nuclei bcc growing the which under ), https://doi.org/10.1073/pnas.2017809118 T ∗ (P ) n rmbelow from and PNAS | 1o 12 of 11

APPLIED PHYSICAL SCIENCES program package (69). Solid-phase designation of each particle has been ACKNOWLEDGMENTS. B.S. and J.L.B. are grateful for the guidance, insights, done via a-CNA (50–52). Particles are colored according to their phase and support from Prof. Russell Hemley. We acknowledge Alex Chernov, Lorin designation: fcc phase (green), bcc phase (blue), and the hcp phase (red). Benedict, Sebastien Hamel, and Amit Samanta for helpful discussions. We also thank A. Arsenlis, D. P. McNabb, B. K. Wallin, and R. S. Maxwell for project support. This work was performed under the auspices of the US Data Availability. All study data are included in the article and/or SI Department of Energy by Lawrence Livermore National Laboratory Contract Appendix. DE-AC52-07NA27344.

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