Low-temperature criticality of martensitic transformations of Cu nanoprecipitates in α-Fe
LLow-temperature criticality of martensitic transformations of Cu nanoprecipitates in α -Fe Paul Erhart
1, 2, ∗ and Babak Sadigh † Chalmers University of Technology, Department of Applied Physics, Gothenburg, Sweden Lawrence Livermore National Laboratory, Condensed Matter and Materials Division, Livermore, California, 94551, USA
Nanoprecipitates form during nucleation of multiphase equilibria in phase segregating multicom-ponent systems. In spite of their ubiquity, their size-dependent physical chemistry, in particularat the boundary between phases with incompatible topologies, is still rather arcane. Here we useextensive atomistic simulations to map out the size–temperature phase diagram of Cu nanoprecipi-tates in α -Fe. The growing precipitates undergo martensitic transformations from the body-centeredcubic (BCC) phase to multiply-twinned 9R structures. At high temperatures, the transitions ex-hibit strong first-order character and prominent hysteresis. Upon cooling the discontinuities becomeless pronounced and the transitions occur at ever smaller cluster sizes. Below 300 K the hysteresisvanishes while the transition remains discontinuous with a finite but diminishing latent heat. Thisunusual size-temperature phase diagram results from the entropy generated by the soft modes ofthe BCC-Cu phase, which are stabilized through confinement by the α -Fe lattice. PACS numbers: 64.70.kd, 64.70.Nd, 64.75.Jk, 81.30.Hd
The Fe–Cu system is a prototypical structurally mis-matched system with very small solubility of Cu in Fe.At low temperatures Fe adopts the body-centered cubic(BCC) phase, while Cu prefers the face-centered cubic(FCC) phase with BCC-Cu being mechanically unstable.Phase segregation in Fe–Cu alloys also has been and con-tinues to be the subject of considerable technological in-terest since Cu impurities contribute to high-temperatureembrittlement of ferritic steels [3]. Based on a series ofcareful experimental studies three stages in the transfor-mation of Cu precipitates have been identified [1, 2, 4–8]:( i ) Cu precipitates nucleate in the BCC structure of thehost lattice, ( ii ) as they grow they undergo a marten-sitic phase transition to a multiply-twinned 9R structure,which ( iii ) eventually transforms into FCC. In this Letterusing a novel combination of molecular dynamics (MD)and Monte Carlo (MC) simulations as well as empiricalinteratomic potentials, we develop a comprehensive pic-ture of the correlation between precipitate structure andsize as a function of temperature for the first two stagesof the transformation process.The Fe–Cu system was modeled using semi-empiricalinteratomic potentials [9–11] that provide an accurate de-scription of the alloy phase diagram and yield very goodagreement with density-functional theory calculations forthe energies of small Cu clusters in BCC-Fe. Simulationswere carried out using the hybrid MC/MD algorithm in-troduced in [12, 13] implemented in the massively par-allel MD code lammps [14]. Orthorhombic simulationcells were employed with cell vectors oriented along [111],[1¯10], and [11¯2]. Most simulations used simulation cellscontaining 42 × ×
45 unit cells (786,240 atoms) withadditional simulations of smaller precipitates at low tem-peratures using 31 × ×
33 unit cells (310,992 atoms).Periodic boundary conditions were applied in all direc-tions, and pressure and temperature were controlled us- ing Nos´e-Hoover thermostat and barostat [15]. Simula-tions were run for at least 1 . × and up to 4 × MD steps using a timestep of 2.5 fs. Every 20 steps theMD simulation was interrupted to carry out a numberof VC-SGC MC trial moves corresponding to 10% of afull MC sweep. Each simulation run thus comprises be-tween 8,000 and 20,000 attempts per particle to swap theatom type. The atomic structures were analyzed usingthe Ackland-Jones parameter [16], which uses bond angledistributions to classify local environments as BCC, FCC,or HCP. The accuracy of this analysis was enhanced byposition-averaging over 800 MD steps. Precipitate struc-tures have been rendered using
OVITO [17].Figure 1 exhibits precipitate structures observed insimulations at 700 K. Initially, Cu precipitates areisostructural with the surrounding BCC-Fe matrix, asshown exemplarily in Figure 1(a) by a precipitate con-taining approximately 15,000 atoms. Even larger pre-cipitates undergo a structural transformation from theBCC phase to a multiply-twinned 9R structure as il-lustrated in Fig. 1(b) where a cluster containing about36,000 Cu atoms is shown. The 9R phase is a closed-packed (CP) lattice that differs from FCC through itsstacking sequence of the CP planes, as shown on theright-hand side of Fig. 1(c). Using the Ackland-Jones pa-rameter for structure identification (see Method section),we obtain sequences of FCC and hexagonal close-packed(HCP) atoms in Fig. 1(b) that are in accord with theideal 9R structure, c.f. Fig. 1(c). In addition, we observestacking faults as well as several twins as indicated by ar-rows in Fig. 1(b). These results closely resemble the TEMmicrographs of 9R Cu precipitates in α -Fe [1, 2, 4]. Fur-ther analysis reveals that also parameters such as twinspacing and misorientation angles agree very well withexperiments.In order to map out the size–temperature phase dia- a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b [111] [100] [001] Aab BC2a/3a/3top views c a HCPHCPHCPHCPHCPHCPHCPFCCFCCFCCABCBCACABAside view(a) twin planes(b) (c)[112] [110]
BCC [112]
BCC [110]
FIG. 1. (a,b) Cross sectional (top) and perspective (bottom) views of (a) BCC and (b) twinned 9R Cu precipitates at 700 K.The precipitates contain approximately 15,000 and 36,000 atoms, respectively. The 9R precipitate in (b) exhibits a pronouncedBCC wetting layer. (c) Schematic of 9R structure [1, 2] which appears as a periodic sequence of two HCP atoms and one FCCatom along the [001] direction when using the Ackland-Jones parameter. gram of Cu nanoprecipitates, we record the number ofCu atoms in CP/BCC local environments as a functionof precipitate size at various temperatures. The opensymbols in Fig. 2(a) show the evolution of these popu-lations with increasing precipitate size at 700 K. Precipi-tates exceeding 28,000 atoms are found to undergo a dis-continuous structural transition as they transform fromBCC to multiply-twinned 9R structures. To study the re-verse transition, we start from a 9R precipitate obtainedfrom an earlier simulation and gradually reduce the Cucontent in the simulation. The thus obtained data areshown by the filled symbols in Fig. 2(a) demonstratingthat at approximately 18,000 atoms precipitates revert tothe BCC structure. There is thus a pronounced hysteresisbetween forward (BCC → → BCC)transitions at 700 K extending from about 18,000 atomsto 28,000 atoms.The above analysis was repeated for six temperaturesbetween 200 and 700 K, the results of which lead to thesize–temperature phase diagram depicted in Fig. 3. Itdemonstrates that the critical cluster size for the BCC–9R transition is strongly temperature dependent increas-ing from approximately 2,700 atoms (3.9 nm diameter) at200 K to about 22,000 atoms (7.9 nm) at 700 K. The tran-sition is decidedly first-order at high temperatures with apronounced hysteresis that vanishes below 300 K, where-upon the transition becomes nearly continuous . Thisis depicted in Fig. 2(b), where the numbers of BCCand CP atoms change around the critical size of 2,700atoms reversibly with negligible discontinuity. A moredetailed inspection of the precipitate structure at 200 K(see movies and snapshots in Supplementary Material)shows that precipitates below the transition point (2,700atoms) have completely adopted a BCC lattice structurewhile precipitates containing more than 3,000 atoms ex- N u m be r o f a t o m s i n d i ff e r en t en v i r on m en t s ( × ) Precipitate size (10 atoms) BCC−9R transitionBCCCPother close−packed coreBCC wetting layerA(b) 200 K 0102030 0 6 12 18 24 30 364 5 6 7 8 ∅ FIG. 2. Number of atoms in different local environments at(a) 700 K and (b) 200 K. Filled and open symbols in (a) showresults from forward (BCC → → BCC)simulations, respectively. The BCC–9R transition exhibits ahysteresis at higher temperatures that vanishes as tempera-ture is lowered. hibit multiply-twinned 9R structures as described above.In the intermediate size range [region A in Fig. 2(b)] theatoms that constitute the core of the precipitate rapidlyfluctuate between different CP motifs. These fluctuationsare absent at higher temperatures where the BCC–9Rtransition is sharp. T e m pe r a t u r e ( K ) Precipitate size (10 atoms) BCC → → BCC 200 300 400 500 600 700 0 5 10 15 20 25 304 5 6 7 ∅ BCC twinned−9R
FIG. 3. Diagram representing the structure of Cu precipi-tates as a function of temperature and size. The light coloredregions next to the phase transition line delineate the extentof the hysteresis. The hysteretic critical point is located alongthe dashed line between 200 and 300 K. −3.50−3.48−3.46−3.44−3.42 0 10 20 30 P o t en t i a l ene r g y ( e V / a t o m ) Precipitate size ( × )
700 K600 K500 K400 K300 K200 K(a)
200 400 700 10 La t en t hea t ( e V ) / T r an s i t i on en t r op y ( m e V / K ) N u m be r o f a t o m s a t t r an s i t i on Temperature (K) ∆ S tr ∆ U tr N core N wetting layer (b) FIG. 4. (a) Potential energy of atoms in Cu precipitates asa function of precipitate size for different temperatures. Thevertical black bars represent the latent heat ∆ U tr associatedwith the transformation, which is plotted in (b) as a functionof temperature alongside the entropy change at the transition(∆ S tr = ∆ U tr /T tr ). The upward (downward) pointing tri-angles show the number of core (wetting layer) atoms at thetransition. Lines are guides to the eye. It is important to note that irrespective of tempera-ture even after the transition to 9R the precipitates stillcontain a substantial number of BCC atoms, which growswith precipitate size, albeit much slower than the numberof CP atoms. This observation reflects the existence of aBCC-Cu wetting layer between the BCC-Fe matrix andthe CP core of the Cu precipitate. It can be as thick as4 atomic layers at the transition point at 200 K, while itonly amounts to about 1 to 2 atomic layers for very largeprecipitates. The BCC-Cu wetting layer is also clearlyvisible in Fig. 1(b). In the following, we will see that thewetting layer plays a crucial role in the BCC–9R transi-tion, particularly at low temperatures. An important signature of first-order transitions is therelease of latent heat, which is absent in continuous tran-sitions. To conclusively resolve the character of the ob-served transitions we have extracted the internal energydensity of the Cu atoms in the precipitates as a func-tion of precipitate size at various temperatures as shownin Fig. 4(a). At high temperatures the data exhibitpronounced discontinuities when the higher-energy BCCphase transforms to the more stable 9R phase. As thetemperature is lowered the discontinuity in the poten-tial energy is reduced until it apparently vanishes be-tween 200 and 300 K, see Fig. 4(b). If true, this impliesthat a critical point exists in this temperature range, be-low which the BCC–9R transition occurs continuously.However, the observation of a continuous transition con-tradicts bulk thermodynamics, according to which thedistinct lattice symmetries require the BCC–9R trans-formation to be first order. Can this rule be violated forprecipitate sizes as astoundingly large as 2,500 to 3,000atoms?The key to answering this question lies in the het-erogeneous distribution of energy and entropy inside Cuprecipitates. For the purpose of this discussion we es-timate entropy by employing the Debye model for thevibrational spectrum, which provides a simple relationbetween entropy and mean square displacement (MSD).[18] Figure 5(a,b) shows the average atomic potential en-ergies (PE), and Fig. 5(c,d) depicts the MSD as a func-tion of distance to the centers of precipitates at 700 Kfor several BCC and 9R precipitates. The PE and MSDprofiles of the BCC precipitates have similar shapes witha constant region in the core that remains unchanged asthe precipitates grow. The wetting layer is identified asthe region outside the core where both the PE and theMSD of the Cu atoms drop steeply. This confirms thatthe vibrational entropy of the core region of BCC precip-itates is much larger than the wetting layer entropy. Itleads to the stabilization of BCC precipitates at elevatedtemperatures and derives from the vibrational modes ofbulk BCC-Cu with imaginary frequencies that are stabi-lized through confinement by the Fe matrix.Inspection of the PE and MSD profiles of the 9R pre-cipitates in Fig. 5(b,d) reveals a more complex behaviorthan in the case of the BCC precipitates. While the wet-ting layer seems to be quite similar in size, energy andentropy to the BCC precipitates, the core now consistsof two regions: ( i ) the inner core with uniform energyand entropy, and ( ii ) the outer core, which constitutesthe interface region between the twinned 9R inner coreand the wetting layer. It is strained to accommodate themisorientation between the inner-core 9R lattice and theBCC-Fe matrix. In this region, both energy and MSDincrease but never exceed the values of the core of theBCC precipitates. Hence the 9R core always contributespositively to the latent heat. However, the total latentheat ∆ U tr generated by the structural transformation is −3.50−3.45−3.40 0 20 40 P o t en t i a l ene r g y ( e V / a t o m ) Distance from center (Å) E int E core BCC
0 20 40 E int ∆ E E core M ean s qua r e d i s p l a c e m en t ( Å ) Distance from center (Å) −10 0 10 20 200 400 600 E ne r g y d i ff e r en c e ( m e V / a t o m ) Temperature (K) ∆ U core ∆ U tr γ wl (e) FIG. 5. (a,b) Potential energy and (c,d) mean square displacement as a function of distance from precipitate center for (a,c)BCC and (b,d) 9R precipitates at 700 K. (e) Latent heat ∆ U tr [from Fig. 4(b)] as well as core ∆ U core and wetting layer γ wl energy differences between BCC and 9R precipitates obtained by analyzing PE profiles. always smaller than the core contribution since the wet-ting layer energy is always slightly lower in BCC than in9R precipitates. The total latent heat and its associatedentropy ∆ S tr = ∆ U tr /T are shown in Fig. 4(b). Bothdecrease dramatically upon cooling mainly because atlower temperatures transitions involve much fewer atomsas they occur at much smaller cluster sizes. Furthermoreat lower temperatures the latent heat per atom drops sig-nificantly [see Fig. 5(e)] due to the fact that in smallerprecipitates the core region, which provides the positivecontribution to the latent heat, shrinks while the com-pensating wetting layer thickens. This is supported bythe number of atoms in the core and the wetting layerregions at the transition as depicted in Fig. 4(b).Nevertheless based on the reported temperature de-pendence of the entropy of transformation in Fig. 4(b),one expects the BCC–9R transformation to involve a di-minishing but finite latent heat at temperatures clearlybelow the critical temperature ( ≈
300 K) at which thehysteresis vanishes. Note that at this temperature, asmany as 500 Cu atoms form the core of the 9R precipitateand thus directly participate in the transformation, seeFig. 4(b). These observations thus support the existenceof a dynamic critical point in the BCC–9R transitionof Cu nanoprecipitates. At this temperature, the nucle-ation barrier between the two phases disappears. Themicroscopic origin of this behavior lies in the mechani-cal instability of bulk BCC-Cu on the one hand and thestabilization of a BCC wetting layer with stiff phononmodes at the interface with the BCC-Fe host lattice onthe other.It is important to note that while our observations can-not prove the existence of a thermodynamic critical point(TCP) in the BCC–9R transition of Cu nanoprecipitates,they suggest that the phase diagrams of nanoprecipitatesmay contain TCPs that have no counterpart in the corre-sponding bulk thermodynamic limit. It is of course well-known that the phase diagrams of nanoparticular systems deviate from the bulk due to confinement. In particu-lar the loci of phase boundaries become size dependentand thus phase transition temperatures and pressures aremodified for liquid-gas [19], liquid–solid [20–24] as wellas solid–solid transformations [25–34]. Furthermore, dy-namic critical points leading to vanishing hysteresis havebeen observed to precede the TCP in liquid-gas transi-tions [19, 35]. In contrast, one might expect the first-order character of transitions between phases with dis-tinct symmetries to be conserved. The presence of a TCPin BCC–9R transformations of Cu nanoprecipitates vio-lates this expectation. It is caused by the heterogeneousnature of nanoprecipitates, where the total latent heatcan vanish with the core and wetting layer regions ex-hibiting compensating discontinuities. This heterogene-ity is best illustrated in Fig. 5(e), which shows the totallatent heat per atom, as well as the inner core and wet-ting layer energy differences between BCC and 9R pre-cipitates. For temperatures above 500 K, the contribu-tion to the latent heat from the inner core region reachesits asymptotic limit. Below this temperature, this con-tribution is slowly reduced as the inner core region isgradually shrinking and becoming strained. Figure 5(e)demonstrates that the latent heat per atom can nearlyvanish while significant differences in the energetics ofBCC and 9R precipitates remain. This points to the factthat the balance between wetting layer and core ener-gies is fundamental to the low-temperature criticality inthis system. Furthermore, if a TCP existed, higher orderderivatives of the free energy would become discontinu-ous rather than diverge.The phenomenology described in this work not onlyhas direct consequences for our understanding of the Fe–Cu system but transpires to many more materials. Basedon the analysis presented here one can anticipate the oc-currence of similar effects in other immiscible systems,whenever the minority phase is unstable in the latticestructure of the host.
ACKNOWLEDGMENTS
Parts of this work were performed under the auspices ofthe U.S. Department of Energy by LLNL under ContractDE-AC52-07NA27344. P.E. acknowledges funding fromthe Swedish Research Council in the form of a Young Re-searcher grant and the
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