metals

Article of FCC and BCC Metals Studied by Molecular Dynamics Simulation

Dmitri V. Louzguine-Luzgin 1,2,* and Andrey I. Bazlov 3,4

1 WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 2 Mathematics for Advanced Materials-OIL, National Institute of Advanced Industrial Science and Technology (AIST), Sendai 980-8577, Japan 3 Department of Physical Metallurgy of Non-Ferrous Metals, National University of Science and Technology “MISiS”, 119049 Moscow, Russia; [email protected] 4 Laboratory for Mechanics of Advanced Bulk Nanomaterials for Innovative Engineering Applications, St. Petersburg State University, 7/9 Universitetskaya nab., 199034 St. Petersburg, Russia * Correspondence: [email protected]; Tel.: +81-(22)217-5957; Fax: +81-(22)217-5957

 Received: 6 October 2020; Accepted: 13 November 2020; Published: 18 November 2020 

Abstract: The atomic structure variations on cooling, vitrification and crystallization processes in liquid metals face centered cubic (FCC) Cu are simulated in the present work in comparison with body centered cubic (BCC) Fe. The process is done on continuous cooling and isothermal annealing using a classical molecular-dynamics computer simulation procedure with an embedded- method potential at constant pressure. The structural changes are monitored with direct structure observation in the simulation cells containing from about 100 k to 1 M . The crystallization process is analyzed under isothermal conditions by monitoring density and energy variation as a function of time. A common-neighbor cluster analysis is performed. The results of thermodynamic calculations on estimating the energy barrier for nucleation and a critical nucleus size are compared with those obtained from simulation. The differences in crystallization of an FCC and a BCC metal are discussed.

Keywords: atomic structure; vitrification; crystallization; molecular-dynamics computer simulation

1. Introduction Crystallization of metals and alloys from the melt (liquid state) takes place by nucleation and growth [1]. This process has occupied the minds of scientists for quite some time owing to still unresolved issues, especially related to high nucleation and growth rates of pure substances. Pure metals are known to exhibit a high crystal nucleation rate at a low enough temperature [1] and high growth rates in a wide temperature range [2]. Crystallization of different substances [3,4] including liquid metals [5] and alloys [6] has been studied by molecular-dynamics (MD) simulation with various potentials. Furthermore, MD simulation of glass transition of pure metals, such as Fe [7,8], Ni [9,10], Cu [11], Al [12] and other metals [13] has been performed. In a recent work possible heterogeneous nucleation of Fe was studied at 0.67Tm and 0.58Tm (Tm is the melting point which in pure metals is equal to the liquidus temperature Tl)[14]. However, one should mention that Tm of 2400 K given by the potential used is significantly overestimated compared to the experimental value. Recent embedded atom potentials provide a better correspondence to the experimental values. The crystallization behavior of other metals was also studied via MD simulation [15] and compared with the experimental data [16]. The energy barriers for crystallization of pure metals were also compared with those obtained from computer simulation [17]. In the present work the atomic structure variation, vitrification and crystallization processes in liquid metals (FCC Cu and BCC Fe) are simulated on continuous cooling and isothermal

Metals 2020, 10, 1532; doi:10.3390/met10111532 www.mdpi.com/journal/metals Metals 2020, 10, x FOR PEER REVIEW 2 of 11

atomic structure variation, vitrification and crystallization processes in liquid metals (FCC Cu and

MetalsBCC2020 Fe), 10 are, 1532 simulated on continuous cooling and isothermal annealing using a classical2 ofMD 11 computer simulation procedure with an embedded-atom method potential at constant pressure. annealing2. Computational using a classicalProcedure MD computer simulation procedure with an embedded-atom method potential at constant pressure. Molecular dynamics (MD) computer simulation using a software package for classical 2.molecular Computational dynamics Procedure (LAMMPS) [18] was used to model the crystallization process of Fe and Cu metals at periodic boundary conditions. The simulation was performed at 1 fs time step using the embeddedMolecular atom dynamics potentials (MD) for computer Cu [19] and simulation Fe [20] usingat nearly a software constant package temperature for classical and pressure. molecular In dynamicsorder to let (LAMMPS) crystallization [18] was occur, used the to cubic model cell the size crystallization was typically process chosen of to Fe be and about Cu metals10 nm. at A periodictypical crystalline boundary conditions.cell of the atoms The simulation containing was108,000 performed for FCC at and 1 fs 128,000 time step atoms using for the BCC embedded structure atomwas potentialsheated to forof 2500 Cu [ 19K ]to and melt Fe [and20] atthen nearly cooled constant down temperature to a certain andtemperature. pressure. InIn ordersome tocases, let crystallizationlarger sized cells occur, were the used cubic as cell indicated. size was typicallyMelting is chosen confirmed to be aboutby the 10 radial nm. Adistribution typical crystalline function celland of stabilization the atoms containing of the 108,000density forvariation FCC and with 128,000 time. atoms Cooling for BCC was structure performed was heatedto different to of 2500temperatures K to melt andat about then 5 cooled × 1013 downK/s. Temperature to a certain temperature.upon simulation In some was cases,maintained larger at sized ±6 K cells (Figure were 1) usedwhile as pressure indicated. was Melting kept around is confirmed zero. by the radial distribution function and stabilization of the density variation with time. Cooling was performed to different temperatures at about 5 1013 K/s. × Temperature upon simulation was maintained at 6 K (Figure1) while pressure was kept around zero. ±

300

250

200

150 Frequency 100

50

0 744 745 746 747 748 749 750 751 752 753 754 755 756 Temperature (K)

Figure 1. Frequency distribution of temperature for Cu held at 750 K fitted with the Gaussian function 2 withFigure R >1.0.99. Frequency The results distribution were similar of temperature for all other cases.for Cu held at 750 K fitted with the Gaussian function with R2 > 0.99. The results were similar for all other cases. Neither Fe nor Cu evaporated in the molten state as should have happened at zero pressure, likelyNeither because Fe there nor was Cu notevaporated enough timein the for molten a gaseous state phase as should to nucleate have happened from inside at the zero atomic pressure, cell (alikely gaseous because phase there has was to overcome not enough an time energy for barriera gaseou tos bephase formed to nucleate as well from as a crystallineinside the atomic one) and cell absence(a gaseous of the phase surfaces has to owing overcome to periodic an energy boundary barrier conditions. to be formed A thermostatas well as a was crystalline used to one) control and theabsence temperature of the surfaces [21,22] whileowing pressureto periodic was bounda maintainedry conditions. by a Barostat A thermostat [23]. A was software used to package control “OVITO”the temperature [24] was used[21,22] to visualizewhile pressure and analyze was mainta the simulationined by results.a Barostat Adaptive [23]. A Common software Neighbor package Analysis“OVITO” (CNA) [24] [was25] was used used to tovisualize analyze structuraland analyze changes the insimulation Fe and Cu. results. Adaptive Common 3.Neighbor Results andAnalysis Discussion (CNA) [25] was used to analyze structural changes in Fe and Cu.

3. ResultsThe density and Discussion changes on heating and cooling as a function of temperature are illustrated in Figure2a. Equilibrium liquid state was obtained at 2500 K in less than 1 ps. The holding time was also varied The density changes on heating and cooling as a function of temperature are illustrated in from 1 to 100 ps without visible changes in the liquid structure and density. Thus, the atomic cells have Figure 2a. Equilibrium liquid state was obtained at 2500 K in less than 1 ps. The holding time was been kept for at least 10 ps at 2500 K prior to cooling. also varied from 1 to 100 ps without visible changes in the liquid structure and density. Thus, the atomic cells have been kept for at least 10 ps at 2500 K prior to cooling. Metals 2020, 10, 1532 3 of 11 Metals 2020, 10, x FOR PEER REVIEW 3 of 11

9.0

8.5 Heating ) 3 Cooling 8.0

Heating Cu 7.5 Cooling Density (g/cm Density Fe 7.0 f Tg Cu Tl Fe Tl 6.5 300 600 900 1200 1500 1800 2100 2400 2700 Temperature (K) (a) 4

3

2

PDF(R) 300 K

1 2500 K

0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 R (nm) (b)

0.3 1.2

Cu 0.2 0.8 max min /PDF min PDF

PDF 0.1 0.4 Fe

0.0 0 400 800 1200 1600 2000 Temperature (K) (c)

Figure 2. 2. DensityDensity changes changes on on heating heating and and cooling cooling of Cu of Cuand andFe as Fe a asfunction a function temperature temperature (a). Pair (a). distributionPair distribution functions functions (PDFs) (PDFs) of Fe ofin Feliquid in liquid (2500 (2500 K) and K) glassy and glassy (300 (300K) state K) state(b). The (b). values The values of pair of • distributionpair distribution function function first first minimum minimum PDF PDFmin ( ) (and) and ratios ratios of of PDF PDFmin/PDF/PDFmaxmax (()N ()(c) cfor) for Cu Cu (violet) (violet) min • min and Fe (blue).

Vitrification of both Fe and Cu was attained at the cooling rate of about 5 × 1013 K/s (Figure 2). At the cooling rate of 5 × 1012 K/s and lower the studied metals start to crystallize directly Metals 2020, 10, 1532 4 of 11

Vitrification of both Fe and Cu liquids was attained at the cooling rate of about 5 1013 K/s × (Figure2). At the cooling rate of 5 1012 K/s and lower the studied metals start to crystallize directly on × cooling. Similarly, high critical cooling rate values indicating high instability of the supercooled liquid Metals 2020, 10, x FOR PEER REVIEW 4 of 11 were obtained in other works [13,26,27]. However, theoretical calculations [28] and the experimental 9 resultson cooling. [29] suggest Similarly, a lower high criticalcritical cooling rate rate values (of about indicating 10 K high/s) for instability glass-transition of the supercooled in pure metals, especiallyliquid were with BCCobtained lattice. in other The diworksfference [13,26,27]. could beHowever, connected theoretical with the calculations type of potentials [28] and used. the experimentalThe values ofresults glass-transition [29] suggest temperaturea lower critical (T coolingg) defined rate by (of the about change 109 K/s) of the for densityglass-transition as a function of temperaturein pure metals, slope especially are about with 1100 BCC K lattice. for Fe The and difference 800 K for could Cu (Figure be connected2a). Of with course, the thesetype of values correspondpotentials to used. this very high cooling rate used when viscosity is still low. However, the value for Fe correspondsThe values quite wellof glass-tr to theansition isochoric temperature conditions (T wheng) defined the volumeby the change of the liquidof the becomesdensity as similar a to function of temperature slope are about 1100 K for Fe and 800 K for Cu (Figure 2a). Of course, these that of the crystalline phase [30]. Two typical pair distribution functions (PDF) for Fe are shown in values correspond to this very high cooling rate used when viscosity is still low. However, the Figure2b. The values of T derived from PDF [31] and ratios of PDF /PDF [32] are shown value for Fe correspondsg quite well to the isochoricmin conditions when themin volumemax of the liquid in Figurebecomes2c. similar These to values that of are the about crystalline 100 Kphase lower [30]. than Two those typical obtained pair distribution from Figure functions2a. As (PDF) one can see for Fe Fe, are having shown a in BCC Figure lattice 2b. The at room values temperature of Tg derived has from much PDFmin lower [31] and PDF ratiosmin value of PDF comparedmin/PDFmax to Cu which[32] has are anshown FCC in lattice.Figure 2c. It resemblesThese values the are crystalline about 100 K structure: lower than therethose isobtained a deeper from gap Figure between 2a. the coordinationAs one can shells see for in BCCFe, having lattice. a SplittingBCC lattice of theat room first andtemperature second pairhas much distribution lower PDF functionmin value (PDF(R)) maximacompared intotwo to Cu peaks which is clearlyhas an observedFCC lattice. in It Figure resembles2b. It the indicates crystalline short- structure: and medium-range there is a deeper ordering of thegap liquid between and the glassy coordination phases shells on cooling in BCC and lattice. that Sp glassylitting Cuof the and first Fe and inherit second the pair short-range distribution order of the correspondingfunction (PDF(R)) . maxima into two peaks is clearly observed in Figure 2b. It indicates short- and medium-range ordering of the liquid and glassy phases on cooling and that glassy Cu and Fe Phase transformation kinetics for Fe and Cu are shown in Figure3; Figure4, respectively, inherit the short-range order of the corresponding crystals. by monitoring the density changes. Energy of the system changes in the way opposite to density. Phase transformation kinetics for Fe and Cu are shown in Figure 3; Figure 4, respectively, by Please,monitoring note that the the density final productchanges. densityEnergy isof dithefferent system from changes case toin case the owingway opposite to diff erentto density. fractions of the internalPlease, note defects. that the Full final system product equilibration density is differ uponent crystallization from case to case requires owing much to different longer fractions time. While at 1100of K the and internal above thedefects. density, Full volumesystem equilibration and energy valuesupon crystallization are visibly stable requires within much the incubationlonger time. period (relaxationWhile at proceeds 1100 K and in 10above ps forthe liquid density, Fe volume at 1100 and K) (Figureenergy 3valuesa), these are valuesvisibly graduallystable within change the at a lowerincubation temperature period owing (relaxation to structural proceeds relaxationin 10 ps for at liquid low atomicFe at 1100 mobility. K) (Figure The 3a), structure these values relaxation timegradually becomes change longer at than a lower the temperature simulation owing time below to structural 1100 Krelaxation for Fe which at low corresponds atomic mobility. quite The well to the equivolumestructure relaxation/isochoric time glass-transition becomes longer temperaturethan the simulation determined time beforebelow 1100 [30]. K It for also Fe takes which place at corresponds quite well to the equivolume/isochoric glass-transition temperature determined before about 800 K for Cu when the density and energy of the system gradually change with time prior [30]. It also takes place at about 800 K for Cu when the density and energy of the system gradually to crystallization.change with time Both prior BCC to crystallization. and FCC crystals Both BCC nucleate and FCC directly crystals from nucleate the melt directly and from no two-stagethe nucleationmelt and [33 no] two-stage is found. nucleation [33] is found.

7.8 0.15 7.7 Kept at 1100 K 432k atoms 1024k at. Fe 128,000 atoms 1 Kept at 900 K 2

3 E ) ) 3 3 7.7 0.10 7.6 128k atoms Density (Mg/m Density 7.5 (Mg/m Density 7.6 0.05

7.4 7.5 0.00 100 200 300 400 100 200 300 400 500 Time (ps) Time (ps) (a) (b)

FigureFigure 3. Density 3. Density change change as as a a function function ofof timetime for pure Fe Fe at at 1100 1100 K K (different (different atomic atomic cell cell size size used used as indicated)as indicated) (a) and (a) 900and K900 (b K) (three(b) (three simulations, simulations, same same cell cell size size ofof 128,000128,000 atoms). atoms). Inset Inset in in(a) (showsa) shows an atomican atomic cell of cell 1.024 of million1.024 million atoms atoms at 250 at ps250 containing ps containing 5 growing 5 growing nuclei nuclei as as marked. marked. ( b(b)) also also shows shows the total energy difference (ΔE) for graph 1 for comparison. Initial 50 ps are related to heating the total energy difference (∆E) for graph 1 for comparison. Initial 50 ps are related to heating and and cooling time. cooling time. Metals 2020, 10, 1532 5 of 11 Metals 2020, 10, x FOR PEER REVIEW 5 of 11

8.6 0.10 8.5 0.07 108,000 atoms 108,000 atoms Kept at 850 K Kept at 750 K 0.06 ) )

131,072 at 0.05 ) )

3 8.5 3 ΔE eV/atom eV/atom ( ( ΔE 0.04 0.05 8.4 ence ence ence r r e e

f 0.03 f f f

256,000 at di di Density (Mg/m Density (Mg/m 8.4 y y g g

r 0.02 r Ene Ene 0.01

8.3 0.00 8.3 0.00 100 200 300 400 500 600 100 200 300 400 500 600 Time (ps) Time (ps) (a) (b)

FigureFigure 4. Density 4. Density change change as aas function a function of timeof time for for pure pure Cu Cu at at 850 850 K (Ka )(a and) and 750 750 K K (b ()b for) for 108,000 108,000 atoms per cell.atoms (b )per also cell. indicates (b) also theindicates energy the di energyfference difference (∆E) in blue (ΔE) forin blue one plot.for one The plot. insert The indicatesinsert indicates the atomic the atomic cell at 270 ps with atoms belonging to FCC clusters only. The initial 50 ps are related to cell at 270 ps with atoms belonging to FCC clusters only. The initial 50 ps are related to heating, melting heating, melting and cooling time. and cooling time. As can be found for pure Fe in Figure 3, the incubation time (defined as an intersection of two As can be found for pure Fe in Figure3, the incubation time (defined as an intersection of two tangents to the plot before and after the onset time) changes irregularly with the atomic cell size tangentsowing to to the stochastic plot before character and afterof nucleation. the onset In time) a cell changescontaining irregularly 432,000 atoms, with two the atomicovercritical cell size owingnuclei to stochastic nucleated characterat about 120 of nucleation.ps. In a cell containing In a cell containing 1,024,000 atoms, 432,000 the atoms, first nucleus two overcritical was formed nuclei nucleatedat 150 atps, about the second 120 ps. at 170 In a ps, cell the containing third at 210 1,024,000 ps, the fourth atoms, at the230 firstps and nucleus the fifth was at 240 formed ps. If atone 150 ps, the secondassumes at total 170 ps,nucleation the third time at of 210 90 ps, ps thefor four fourth nu atclei 230 per ps a cell and of the 23 fifthnm size, at 240 then ps. the If oneaverage assumes totalhomogeneous nucleation time nucleation of 90 ps rate for is four 3.7 × nuclei1033 m− per3·s−1. aIt cell is a ofhigh 23 value nm size, but quit thene close the average to that reported homogeneous in nucleationan earlier rate work is 3.7 [8]. The1033 growingm 3 s 1nanocrystals. It is a high have value irregular but quite but closenearly tospherical that reported shape. In in a an cell earlier × − · − workcontaining [8]. The growing128,000 atoms, nanocrystals the first haveovercritical irregular stably but growing nearly nucleus spherical was shape. formed In at a 280 cell ps containing and consumed the entire volume at about 410 ps. Thus, at these temperatures (1100 and 900 K which are 128,000 atoms, the first overcritical stably growing nucleus was formed at 280 ps and consumed the 0.61 Tl and 0.50 Tl, respectively; when the liquidus temperature Tl = 1811 K) crystallization of Fe is entire volume at about 410 ps. Thus, at these temperatures (1100 and 900 K which are 0.61 T and 0.50 T , hardly statistically predictable, and it is not so sensitive to the cell size if its length exceeds aboutl 10 l respectively; when the liquidus temperature T = 1811 K) crystallization of Fe is hardly statistically nm. BCC Cu at 750 K (which is 0.55 Tl) has evenl higher homogeneous nucleation rate of 8 × 1034 predictable,m−3s−1. At and the it ishomological not so sensitive temperature to the cellhigher size than if its 0.65 length Tl crystallization exceeds about of 10 Cu nm. and BCC Fe Cuis not at750 K 34 3 1 (whichdetectable is 0.55 T withinl) has even3 ns of higher maximum homogeneous simulation nucleationtime. rate of 8 10 m− s− . At the homological × temperatureOwing higher to rather than high 0.65 nucleationTl crystallization rates, the of isothe Cu andrmal Fe crystallization is not detectable curves within can be 3 nsstatistically of maximum simulationwell reproduced time. below 0.55 Tl for Cu (Figure 4b) and below 0.4 Tl for Fe at the chosen atomic cells. AnOwing average to rather incubation high time nucleation is about rates,2 times the longer isothermal than that crystallization on pure Ni studied curves earlier can [34]. be statistically From this viewpoint Ni is a special liquid which is the least stable against crystallization among these well reproduced below 0.55 Tl for Cu (Figure4b) and below 0.4 Tl for Fe at the chosen atomic cells. metals. Furthermore, in spite of the fact that the nucleation barrier is quite high in pure Ni the An average incubation time is about 2 times longer than that on pure Ni studied earlier [34]. From this nucleation rate is also high attributed to the high atom attachment kinetics [35]. viewpoint Ni is a special liquid which is the least stable against crystallization among these metals. From the isothermal crystallization plots, the beginning, also called an incubation time, and Furthermore,finish time in of spite transformation of the fact can that be the calculated nucleation using barrier two corresponding is quite high intangents pure Nito the plot nucleation before rate is alsoand high after attributed the inflection to the point. high Note atom that attachment actual transformation kinetics [35 time]. is longer because the two tails of theFrom S-curve the isothermal are not taken crystallization into consideration plots, thewhen beginning, two tangents also are called applied. an incubation However, time,it becomes and finish timemore of transformation difficult to detect can the be incubation calculated period using for two each corresponding plot below 700 tangents K owing to to the the plot long before relaxation and after the inflectiontime manifested point. Note in thatconstant actual variation transformation of the timebaseline. is longer Nevertheless, because thea twotime-temperature- tails of the S-curve are nottransformation taken into consideration (TTT) diagram when was constructed, two tangents and are it applied.is shown However,in Figure 5. it The becomes simulation more results difficult to detectare the well incubation reproduced period (owing for to each high plot nucleation below 700 rate) K below owing 0.55 to the Tl for long Cu relaxation and below time 0.4 T manifestedl for Fe. in The nose of the TTT diagram is about 0.55 Tl for Fe and about 0.6 Tl for Cu. constant variation of the baseline. Nevertheless, a time-temperature-transformation (TTT) diagram was constructed, and it is shown in Figure5. The simulation results are well reproduced (owing to high nucleation rate) below 0.55 Tl for Cu and below 0.4 Tl for Fe. The nose of the TTT diagram is about 0.55 Tl for Fe and about 0.6 Tl for Cu. Metals 2020, 10, 1532 6 of 11 Metals 2020, 10, x FOR PEER REVIEW 6 of 11 Metals 2020, 10, x FOR PEER REVIEW 6 of 11 0.7 0.7

0.6 0.6 FCC Cu 108,000 atoms ) l FCC Cu 108,000 atoms T ) / l T T /

T 0.5 0.5

0.4 0.4 Temperature ( Temperature

Temperature ( Temperature BCC Fe 128,000 atoms 0.3 BCC Fe 128,000 atoms 0.3

0.2 0.2 2020 2002000 2000 Time (ps)

FigureFigure 5.5. 5. Time-temperature-transformationTime-temperature-transformation Time-temperature-transformation (TTT) (TTT)(TTT) diagram diagra form Feforfor (blue) FeFe (blue)(blue) and Cu andand (green). CuCu (green).(green). Circles Circles indicateCircles indicatebeginningindicate beginning beginning of crystallization of of crystallization crystallization while whilewhile diamondsdiamonds show end show of crystallization. endend ofof crystallization.crystallization.

AccordingAccording to toto adaptive adaptiveadaptive common commoncommon neighbor neighborneighbor analysis analysis (Figure (Figure(Figure6), crystallization 6),6), crystallizationcrystallization becomes detectable becomesbecomes detectablebydetectable volume by andby volume volume energy and and changes energy energy when changeschanges the fractionwhen the of fraction atoms in ofof crystalline atomsatoms inin crystallinecrystalline clusters attains clusters clusters about attains attains 2%. aboutTheabout increase 2%. 2%. The The in increase theincrease numbers in in the the of numbersnumbers atoms with ofof atomsatoms the icosahedral-like with the icosahedral-likeicosahedral-like configuration configuration configuration prior to the prior prior nucleation to to the the nucleationofnucleation crystalline of of crystalline phases crystalline found phases phases in earlier foundfound works inin earlierearlier [36 ,wo37]rks is not[36,37] seen isis in notnot the seenseen present in in the the study present present (see study study Figure (see (see6). FigureAlthoughFigure 6). 6). Although onlyAlthough BCC only crystalsonly BCC BCC form crystalscrystals and formform grow and in grow Fe FCC, in Fe BCC FCC,FCC, and BCCBCC hexagonal andand hexagonal hexagonal close packedclose close packed packed (HCP) (HCP)atomic(HCP) atomic arrangements atomic arrangements arrangements are found are are in foundfound Cu owing inin CuCu to owing low stacking to lowlow stacking faultstacking energy faultfault of energy energy this metal of of this this (well metal metal estimated (well (well estimatedexperimentally)estimated experimentally) experimentally) as seen in Figureas as seen seen7 . inin FigureFigure 7.

7.9 7.9 8.5 CommonCommon Neighbour Neighbour Analysis Analysis for Fe atat 750 750 K K 8.5Common Neighbour Analysis ρ 7.8 7.8 for CuCommon at 750 NeighbourK, 108,000 Analysisatoms ρ 128,000 atoms 128,000 atoms 100100000,000 for Cu at 750 K, 108,000 atoms 100,000 7.7 7.7 ρρ 100000 8.4 s ) 7.6 r 8.4 3 7.6) 3 ) )

3 10 000 3 7.5 10,000 , 7.5 10000 10000 FCC 1st 1st FCC 7.4 8.3 1st nucleus nucleus 1st 7.4 FCC clusters HCP FCC 1st overcritical overcritical of cluste of HCP f 8.3 nucleus r

7.3 nucleus 1000 HCP 1000 overcritical overcritical BCC 7.3 BCCHCP 7.2 1000 Density (Mg/m BCC 1000 BCC ICO Density (Mg/m Density ICO clusters of Number

8.2Density (Mg/m 7.2 All ICO

7.1 Numbe All clusters of Number Density (Mg/m ICO 8.2 100 All 100 Common Neighbour Analysis Analysis Neighbour Common 7.1 7.0 Number o All 100 Common Neighbour Analysis NeighbourCommon Analysis 100 Analysis Neighbour Common 7.0 6.9 8.1 Common Neighbour Analysis Analysis Common Neighbour 6.9 6.8 10 508.1 100 150 200 250 300 350 400 50 100 150 200 250 300 350 400 450 500 550 600 650 700 50 100 150Time 200 (ps) 250 300 350 400 6.8 Time (ps) 10 Time (ps) 50 100 150 200 250 300 350 400 450 500 550 600 650 700

Time ((ps)a) (b) FigureFigure 6. 6.Adaptive Adaptive common common(a) neighbor neighbor cluster cluster analysis analysis for for Fe Fe (a )(a and) and Cu Cu (b )( cellsb) cells at(b 750)at 750 K together K together with ρ Figurethewith density the 6. Adaptivedensity (ρ) curve ( )common curve (blue) (blue) as neighbor a function as a function cluster of time. ofanalysis time. for Fe (a) and Cu (b) cells at 750 K together with the density (ρ) curve (blue) as a function of time. Metals 2020, 10, 1532 7 of 11 Metals 2020, 10, x FOR PEER REVIEW 7 of 11

(a) (b)

FigureFigure 7. 7.Growing Growing crystalline crystalline particles particles (a )( ofa) BCCof BCC Fe atFe 900 at 900 K (blue) K (blue) and and FCC FCC Cu (green) Cu (green) at 850 at K 850 (b). K HCP(b). stackingHCP stacking faults faults are shown are shown in red. in red.

TypicalTypical growth growth rates rates (crystal (crystal radius radius change change as a functionas a function of time) of aretime) shown are inshown Table 1in. TheTable growth 1. The ratesgrowth are lowerrates are than lower those than found those earlier found in simulations earlier in simulations [15]. The growth [15]. The rate growth of FCC rate Cu crystalsof FCC isCu lesscrystals temperature is less temperature dependent thandependent that of than BCC that Fe. of Furthermore, BCC Fe. Furthermore, one should one note should that some note growingthat some particlesgrowing have particles much have lower much growth lower rate growth owing torate interaction owing to with interaction other nuclei with (evenother under-critical)nuclei (even under- [38]. Atcritical) the same [38]. time At the the observed same time growth the rateobserved values gr forowth Fe arerate quite values close for to theFe are experimental quite close ones to ofthe 40experimental m/s [39]. The ones observed of 40 m/s growth [39]. rateThe ofobserved the order grow of tensth rate meters of the per order second of tens are lowermeters than per thosesecond obtainedare lower for than flat interfaces.those obtained for flat interfaces.

Table 1. Typical growth rates (U ) values at some temperatures for the studied metals estimated from Table 1. Typical growth ratesN (UN) values at some temperatures for the studied metals estimated anfrom average an average single radiuscrystal measuredradius measur in twoed mutuallyin two mutually perpendicular perpendicular directions. directions. Metal T (K) U (m/s) Metal T (K) NUN (m/s) 11001100 48 48 FeFe 900900 20 20 850 18 Cu 850 18 Cu 750 16 750 16

2 AA liquid liquid//solid interfacial interfacial energy energy (σ ()σ value) value of of 0.37 0.37 J/ mJ/mis2 is obtained obtained in in the the present present work work for for Fe Fe at at 11001100 K K as as an an excess excess energy energy per surfaceper surface area area of a singleof a single particle particle of 1.8 nmof 1.8 radius. nm radius. It is calculated It is calculated taking intotaking account into theaccount potential the energypotential di ffenergyerence difference of 23.7 eV atof 17023.7 ps eV of at the 170 simulation ps of the time simulation between time its valuebetween for the its entirevalue atomicfor the cellentire and atomic a sum cell of that and energy a sum values of that for energy the atoms values belonging for the atoms to a crystalline belonging particleto a crystalline of about particle 3.6 nm diameterof about 3.6 formed nm diameter at that stage formed (1657 at that atoms) stage and (1657 the potentialatoms) and energy the potential of the other atoms in the cell belonging to liquid phase (128,000 1657 = 126,343 atoms). The potential energy energy of the other atoms in the cell belonging to liquid− phase (128,000−1657 = 126,343 atoms). The forpotential the liquid energy part for of the the atomic liquid part cell wasof the obtained atomic atcell the was 50 obtained ps stage whenat the all50 atomsps stage belonged when all to atoms the supercooledbelonged to liquid the supercooled phase while liquid that ofphase a crystal while was that taken of a crystal for fully was crystalline taken for statefully oncrystalline heating thestate 2 initialon heating perfect the crystal initial to perfect 1100 K. crysta The valuel to 1100 of 0.33 K. JThe/m forvalue the of Cu 0.33 crystal J/m2 atfor 850 the K Cu is found crystal in at a similar850 K is 2 way.found These in a values similar are way. not These too far values from theare experimental not too far from values theof experimental 0.28 and 0.18 values J/m obtained of 0.28 and for 0.18 Fe andJ/m Cu,2 obtained respectively for Fe [1 ].and Please Cu, noterespectively that although [1]. Please the calculated note that values although aresomewhat the calculated overestimated values are owingsomewhat to the overestimated internal defects owing in the to growing the internal crystals, defects the valuein the for growing Fe is still crystals, higher the than value that for Cu.Fe is Thestill earlier higher simulation than thatresults for Cu. indicated The earlier that simulation the crystal–melt results interfacial indicated energy that the calculated crystal–melt from interfacial a planar interfaceenergy calculated needs to be from adjusted a planar to a smallerinterface value needs in to order be adjusted to predict to the a smaller correct value nucleation in order rate to at predict deep supercoolingthe correct nucleation [40]. rate at deep [40]. Calculation of the critical nucleus size (Rc) and work for the formation of critical nucleus (W*) was done according to classical nucleation theory: Metals 2020, 10, 1532 8 of 11

Calculation of the critical nucleus size (Rc) and work for the formation of critical nucleus (W*) was done according to classical nucleation theory:

Rc = 2σ/∆Gv (1)

3 2 W* = 16πσ /(3∆Gv ) (2)

2 2 Using the Gibbs free energy difference (∆Gv). σ values of 0.277 J/m for Fe and 0.178 J/m for Cu were obtained from Ref. [1]. The temperature dependences of enthalpies of liquid and crystalline phases were used to estimate the enthalpy of melting at about 1809 K. The enthalpy difference of 0.164148 eV/at or 15.837 kJ/mol found for Fe is not far from the experimental value of 15.2 kJ/mol [1]. The value of 13 kJ/mol was used for Cu [1]. ∆Gv can be estimated from the liquid crystallization enthalpy (∆Hc) and the liquidus temperatures Tl using the following equation [41]:

∆Gv = (∆Hc ∆T/T ) (7T/(T + 6T)) (3) · l · l According to thermodynamic calculations the critical nucleus radius in the studied temperature range is about 0.4–0.6 nm for Cu and for 0.5–0.6 nm Fe. Steadily growing particles found in this simulation were larger than 0.8 nm in size. At such small size the energy barrier W* to the thermal energy RT ratio is about 20 for Cu and about 30 for Fe. A higher energy barrier found for Fe is rather responsible for a lower number of nuclei found for Fe at the same homological temperature. It is interesting to note that even at such high energy barriers homogeneous nucleation takes place after hundreds of picoseconds. The number of crystal nuclei formed within the typical cells of about 10 nm length in all metals 24 3 ranges from 1 to 10, leading to the number density of precipitates of the order of 10 m− . The highest number density of nuclei is observed in Cu which leads to better reproducibility of the crystallization traces at different simulation runs. Such high number densities of precipitates are more or less typical for crystallization of pure metals at extremely low homological temperatures of about 0.55 Tl or lower, which are unattainable in the experiments on undercooling but quite common in various metallic glasses which nanocrystallize in a similar temperature range [42,43].

4. Conclusions Liquid structure variation of cooling and glass transition indicated that glassy Cu and Fe inherit the short-range order of the corresponding crystals. Nevertheless, a nucleation and growth type (with an energy barrier) phase transformation is observed. The crystallization kinetics of FCC Cu and BCC Fe were analyzed in isothermal conditions by monitoring the density and energy variation as a function of time, and a TTT diagram was constructed. BCC Fe shows a lower number density of precipitates than FCC Cu. A higher energy barrier for nucleation found for Fe is rather responsible for a lower number of nuclei found for Fe at the same homological temperature. On the other hand, BCC Fe, in general, shows a shorter incubation period than FCC Cu and generally a higher growth rate though the nucleation rate of Cu crystals at a higher rate. The nose of the TTT diagram is about 0.55 Tl for Fe and about 0.6 Tl for Cu. Homogeneous nucleation of both metals becomes difficult to detect above about 0.65 Tl within a reasonable simulation time scale, while below ~0.6 Tl a high population density of crystalline precipitates is obtained. The growing nanocrystals have an irregular but nearly spherical shape. The observed growth rates of the order of tens meters per second are lower than those obtained for flat interfaces. Some of the crystalline particles displayed a lower growth rate than the others owing to the lattice defects within the growing crystals and existing order in the supercooled liquid. The crystal growth rate was scattered from crystal to crystal likely owing to the local order in surrounding environment and the lattice defects (mostly HCP staking faults) within the growing crystals in Cu. According to thermodynamic calculations the critical nucleus radius in the studied Metals 2020, 10, 1532 9 of 11 temperature range is about 0.4–0.6 nm for Cu and for 0.5–0.6 nm Fe. The energy barrier to the thermal energy ratio is about 20 for Cu and about 30 for Fe.

Author Contributions: Conceptualization, D.V.L.-L. and A.I.B.; methodology, D.V.L.-L.; software, D.V.L.-L.; validation, D.V.L.-L. and A.I.B.; formal analysis, D.V.L.-L. and A.I.B.; investigation, D.V.L.-L. and A.I.B.; resources, D.V.L.-L. and A.I.B.; data curation, D.V.L.-L. and A.I.B.; writing—original draft preparation, D.V.L.-L. and A.I.B.; writing—review and editing, D.V.L.-L. and A.I.B.; visualization, D.V.L.-L. and A.I.B.; supervision, D.V.L.-L. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the World Premier International Research Center Initiative (WPI), MEXT, without a particular number, Program to Increase the Competitiveness of NUST “MISiS”, grant number K2-2020-006 and partly by RFBR, project number 19-33-60078. Acknowledgments: The authors gratefully acknowledge the financial support received from the World Premier International Research Center Initiative (WPI), MEXT, Japan Science of the Russian Federation in the framework of the Program to Increase the Competitiveness of NUST “MISiS” (K2-2014-013 and K2-2020-006) and partly by RFBR, project number 19-33-60078. Conflicts of Interest: The authors declare no conflict of interest.

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