Current Opinion in Solid State and Materials Science 20 (2016) 308–315

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Current Opinion in Solid State and Materials Science

journal homepage: www.elsevier .com/locate /cossms

Phase transformations at interfaces: Observations from atomisticmodeling

http://dx.doi.org/10.1016/j.cossms.2016.05.0031359-0286/Published by Elsevier Ltd.

⇑ Corresponding author.

T. Frolov a,⇑, M. Asta b, Y. Mishin c

a Lawrence Livermore National Laboratory, Livermore, CA 94550, USAbDepartment of Materials Science and Engineering, University of California, Berkeley, CA 94720, USAcDepartment of Physics and Astronomy, MSN 3F3, George Mason University, Fairfax, VA 22030, USA

a r t i c l e i n f o

Article history:Received 7 January 2016Revised 27 April 2016Accepted 1 May 2016Available online 16 June 2016

Keywords:Molecular dynamicsMonte Carlo modelingGrain boundary phasesSolute segregationGrain boundary migrationGrain boundary diffusion

a b s t r a c t

We review the recent progress in theoretical understanding and atomistic computer simulations of phasetransformations in materials interfaces, focusing on grain boundaries (GBs) in metallic systems. Recentlydeveloped simulation approaches enable the search and structural characterization of GB phases insingle-component metals and binary alloys, calculation of thermodynamic properties of individual GBphases, and modeling of the effect of the GB phase transformations on GB kinetics. Atomistic simulationsdemonstrate that the GB transformations can be induced by varying the temperature, loading the GB withpoint defects, or varying the amount of solute segregation. The atomic-level understanding obtained fromsuch simulations can provide input for further development of thermodynamics theories and continuousmodels of interface phase transformations while simultaneously serving as a testing ground for validationof theories and models. They can also help interpret and guide experimental work in this field.

Published by Elsevier Ltd.

1. Introduction

Recent years have seen a rapid growth of evidence suggestingthat materials interfaces are capable of first-order structuraltransformations in which the interface properties (such as solutesegregation, excess volume, mobility or sliding resistance) undergodiscontinuous changes [1]. Experiments have revealed a potentiallyimportant role of grain boundary (GB) phase transitions (sometimesreferred to as ‘‘complexion transitions” [1,2]) in abnormal graingrowth in ceramics [2], activated sintering [3] and liquid metalembrittlement [4]. Layering transitions associatedwith GB segrega-tionwere investigated using lattice gasmodels [5,6], first-principlescalculations [7] as well as advanced electron microscopy andspectroscopy methods [8–10]. Experimental investigation of thepotential impact of GB phase transitions on microstructure andother materials properties is currently a highly active area ofresearch [1,4,11–13]. The experimental studies have raised anumber of fundamental questions concerning the thermodynamicnature of the interface phases, their atomic structures and kineticproperties. A unified thermodynamic description of bulk andlow-dimensional phases has been recently proposed [14], phaserules for phases of any dimensionality have been formulated, andadsorption equations for interface phases and line defects

separating such phases have been derived [14]. Phase-field modelshavebeendeveloped [2,15,16], predicting a variety of possible inter-face transformations and mapping them onto bulk phase diagrams.

Although fundamentally important, the thermodynamic analysis[14,17–19] and phase-field models [2,15,16] do not provideatomic-level insights into interface phases and phase transforma-tions. The goal of this paper is to review the recent progress inatomistic modeling of interface transformations, focusing on GBsin metallic systems as an important practical case. Althoughmolec-ular dynamics (MD), Monte Carlo (MC) and other atomistic simula-tionmethodshavebeen successfully applied to study interfaces for along time [20,21], it has not been until very recently that it becamepossible to identify and structurally characterize individual GBphases, compute their thermodynamic properties, and observetransformations among them.Wedescribe the recentworkon struc-tural transformations in single-componentGBs (Section2), in binaryalloys (Section 3), and the effect of such transformations on GBmigration and short-circuit diffusivity (Section 5). In Section 5 wediscuss existing challenges in this field and outline future work.

2. Grain boundary phase transitions in pure metals

2.1. The multiplicity of ground-state grain boundary structures

Atomistic computer simulations have proven to be an invalu-able tool for the study of GB properties [21]. Such simulations have

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been successfully applied to predict GB structures and calculatetheir thermodynamic and kinetic properties, such as GB free ener-gies, diffusivities and mobilities as functions of temperature andchemical composition [21–30]. However, despite the extensiveresearch, there has been no convincing simulation evidence of GBphase transformations until very recently (except for thedislocation-pairing transition in relatively low-angle GBs [31]). Ithas been demonstrated that the critical impediment to observa-tions of such transformations was rooted in inadequate simulationmethodology, namely, in fixing the total number of atoms andusing periodic boundary conditions. In order to sample all possibleatomic configurations, atoms need to be added to or removed fromthe GB region [32–36]. To overcome this shortcoming of theprevious simulations, a new simulation methodology has beendeveloped that allows for variations of the atomic density in theGB region as the latter approaches thermodynamic equilibrium[37]. Applications of this methodology have led to the discoveryof multiple GB phases and phase transitions by varying thetemperature and/or concentration of impurities or point defects[22,37,38].

The new simulation approach has been tested by studying theR5ð210Þ and R5ð310Þ symmetrical tilt GBs in several fcc metalsmodeled with embedded-atom method potentials. These arerepresentative high-angle, high-energy boundaries that have beenextensively investigated in the past. Their choice was additionallymotivated by the recent experimental measurements of Ag radio-tracer diffusion in Cu bicrystals with the R5ð310Þ GB [39], whichrevealed a peculiar temperature behavior of the GB diffusivity thatsuggested a possible structural transformation. To test this hypoth-esis, atomic structures of the two R5 GBs were optimized withrespect to both atomic density and rigid translations of the grains[37]. It was found that at 0 K, the GB energy as a function of the GBdensity (measured as the fraction of atoms in (210) and (310)planes) exhibits several local minima representing different GBphases (Fig. 1(a)). In each boundary, one minimum correspondsto the long-known structure composed of kite-shape structuralunits obtained without addition or removal of atoms(Fig. 1(b and d)). Other minima are new and represent the filled-kite and split-kite GB structures (Fig. 1(c, e, f)). These structuresare more complex and are either non-periodic or have a longperiod whose exact value remains unknown. These structures wereobtained in simulation blocks with a large GB area and should beconsidered as approximants of the actual ground-state structures.It is possible that structures with even lower energies could befound in still larger simulation blocks. Remarkably, and contraryto the common belief, none of the tested interatomic potentialspredicts the normal kite structure to be the lowest in energy forthe R5ð210Þ GB.

It should be noted that the phases found in the R5 GBs arecharacterized by different excess properties, such as the excessvolume and the interface stress [37]. These differences may resultin long-range elastic strain fields around interface phase junctionsin multi-phase states, which in turn can affect the coexistence ofdifferent GB phases and interact with dislocations and otherelements of the microstructure.

2.2. Temperature-induced grain boundary transitions

Atomistic simulations have shown that GB phases canreversibly transform to each other with temperature [37]. Sincesuch transformations require significant changes in atomic densityin the GB region, the conventional simulation methodology withperiodic boundary conditions and a fixed number of atoms cannotbe applied. To enable automatic adjustment of the atomic densityduring the transformation, a different methodology has beenproposed in which one or both edges of the GB terminates at an

open surface (Fig. 2(a)). In this setup, provided the temperatureis high enough, atoms can leave or enter the GB by diffusionto/from the surface. This effectively makes the GB an open thermo-dynamic system capable of exchanging atoms with a reservoir, acondition which naturally exists under experimental conditions.

As an example, Fig. 2(a) shows a Cu bicrystal with the R5ð310Þboundary annealed at 800 K (0:6Tm; Tm ¼ 1327 K being themeltingtemperature predicted by this potential) for tens of nanoseconds.The zoomed-in views reveal the original normal-kite GB structureon the left and a new, split-kite structure on the right. The latternucleates at the surface and gradually propagates inside thebicrystal during the simulation. Its growth is enabled by the supplyof extra atoms from the surface, a process which is kinetically con-trolled by GB diffusion. At a certain temperature, the two GB phasescan coexist in thermodynamic equilibrium separated by a linedefect representing a one-dimensional phase boundary betweentwo two-dimensional phases. Knowledge of thermodynamic andtopological properties of these unusual defects is important forthe understanding of GB phase transitions (particularly, GB phasenucleation). These defects deserve a detailed study in the future.The observation of coexistence of GB phaseswith different densitiesseparated by an atomic-width line defect confirms that the phasetransformation is first order in character. Similar fully reversibletemperature-induced phase transitions were found in the R5ð210Þboundary [37].

2.3. Grain boundary transitions induced by point defects

During GB phase transformations, the boundary absorbs orrejects a large amount of atoms. This suggests that a phasetransformation can be induced by injection or removal of a suitablenumber of point defects at a fixed temperature. This was indeedconfirmed by simulations in which a GB phase was initially createdin a periodic supercell isolated from external sources or sinks ofatoms as in Fig. 3(a). A number of vacancies or interstitial atomswas then introduced into the GB region and the system wasre-equilibrated by an MD run. For example, 80 interstitial atoms(the number required for transformation to the split-kite phase)were introduced into the normal-kite structure of the R5ð310ÞGB followed by an anneal at 800 K. A possible outcome could bea uniform redistribution of the extra atoms in the initial GB struc-ture. Instead, a first-order transformation to the split-kite phasewas observed as illustrated in Fig. 3(c). This transformation wasfully reversible: when the same number of vacancies wassubsequently introduced into the split-kite phase, the lattertransformed back to normal kites (Fig. 3(e)). As a result of thisphase transformation cycle, the boundary returned to its initialstate having annihilated a large number of point defects.Analogous transformation cycles were observed upon loading ofpoint defects into the R5ð210Þ GB.

These simulations suggest that interfacial phase transformationscan greatly increase the capacity of GBs to absorb non-equilibriumpoint defects. This finding may have important practicalimplications for materials operating under extreme conditions.For example, the radiation tolerance of many nuclear materialsdepends on the efficiency of GBs as sinks of vacancies and intersti-tials created during irradiation by energetic particles [40–43]. GBphase transformations present a novel mechanism of radiationdamage healing in such materials.

3. Segregation-induced grain boundary transitions

In technological applications, dopants and impurities can play adominant role in microstructure evolution and properties. Recentexperimental studies of doped ceramics revealed discontinuous

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ΣΣ5(310)[001]5(310)[001]5(310)[001]

Fig. 1. (a) Energies of the R5ð210Þ½001� and R5ð310Þ½001� GBs in Cu calculated at 0 K as functions of the GB density (fraction of atomic plane) [37]. The local energy minimaare labeled (b) through (f) and the respective GB structures viewed along the [001] tilt axis are shown in panels (b)-(f). The structural units are outlined in red. (Forinterpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

(a)

)c()b(Fig. 2. (a) Structural phase transformation in the Cu R5ð310Þ½001� GB at 800 K in simulations with an open surface acting as a source of atoms. (b and c) Zoomed-in views ofthe original split-kite (b) and nucleated filled-kite (c) GB phases.

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changes in coarsening behavior that were attributed to theemergence of GB phases [1,2,13]. Such phases are characterizedby different segregated structures such as a monolayer, a bilayer,a trilayer, or a thick disordered layer [4,12]. Recent atomisticsimulations have provided insights in the thermodynamic originand physical mechanisms responsible for phase transitions insegregated GBs [22,38].

3.1. Segregated grain boundary structures

The choice of the R5ð210Þ GB in Cu-rich Cu-Ag alloys as a modelsystem to study segregation effects was prompted by the recentexperimental measurements of Ag diffusion in this boundary [39]and the availability of a reliable interatomic potential reproducingthe Cu-Ag phase diagram [44]. The study combined MD with

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(c)

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(e)

Kites

Split kites

Kites

add interstitials

add vacancies

Fig. 3. GB phase transformations in the Cu R5ð310Þ½001� GB induced by interstitialsand vacancies at 800 K [37]. Panels (a) through (e) show selected GB structuresduring the cycle involving addition of interstitials and vacancies.

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semi-grand canonical MC simulations. While MC simulationscannot capture the correct dynamics of phase changes, they allowone to quickly obtain the equilibrium distribution of chemical com-ponents over the system and compute equilibrium thermodynamicproperties. This could not be achieved within the MD approach dueto the long time scales associated with solid-state diffusion. In thesemi-grand canonical ensemble, the atoms are allowed to moveaway from equilibrium positions and simultaneously switch theirchemical species while the temperature T, total number of atomsN and the diffusion potential M between Ag and Cu remain fixed.

Two different types of simulation block were employed. GBphase transitions were studied in a large bicrystal with a GB termi-nated at two open surfaces (Fig. 4(a)). Thermodynamic propertiesof individual GB phases were calculated in periodic simulationblocks containing a single GB phase of interest; due to the absenceof sinks and sources of atoms, phase transformations in such blocksdo not occur and properties of a single GB phase can be studiedover and wide range chemical compositions and/or temperatures.A GB phase is characterized by a set of excess properties such asthe excess energy, excess number of Ag atoms (segregation), inter-face stress and others. All these properties are readily accessible byatomistic simulations and served as input to thermodynamic inte-gration of the adsorption equation to obtain the GB free energy c[22,38]. As in the previous work [24], the excess quantities werecalculated at a fixed total number of atoms using Cahn’s formalismof generalized excesses [45].

Equilibrium segregated structures of the filled-kite and split-kite phases are illustrated in Fig. 4(b and c). The filled-kite phase

exhibits a monolayer segregation whereas the split-kite phase adouble layer segregation. The equilibrium segregation isothermsof the two phases at 900 K are shown in Fig. 4(d and f). It shouldbe emphasized that the segregated layers are only enriched inbut not fully packed with Ag atoms. In fact, the bilayer structurecontains less Ag atoms per unit GB area than does the monolayerstructure. Thus, caution should be exercised when interpretingexperimental images of segregated GBs: the number of segregationlayers is not a reliable indicator of segregation strength.

3.2. Segregation effect on grain boundary transformations

In pure Cu, the transition between the split-kite and filled-kitephases of the R5ð210Þ GB occurs at a temperature of about1050 K [37]. The segregation isotherms indicate that at the temper-ature of 900 K (when the split-kite phase in pure Cu is morestable), Ag segregation into filled kites increases with the graincomposition c faster than does the segregation into split kites(Fig. 4(d and f)). From the Gibbs adsorption equation, we canconclude that the free energy cFK of filled kites decreases withthe grain composition faster than does the free energy cSK of splitkites. One can expect, therefore, that at some composition c�, thetwo free energies become equal, cFK ¼ cSK � c�, and asegregation-induced phase transformation occurs.

To test this prediction, a simulation block with two opensurfaces was utilized (Fig. 4(a)). MC simulations were conductedat 900 K for a set of grain concentrations imposed by differentdiffusion potentials M. The boundary was initially composed oftwo phases separated by a phase junction. During the simulations,one of the phases always grew while the other shrunk, dependingon the chosen M. A value M ¼ M� was found at which the positionof the phase junction fluctuated in both directions without any dis-cernible growth of either phase. The respective grain composition,c� ¼ 0:02 at.%Ag, was identified with the GB phase equilibrium atthis temperature. The phase change occurring at this compositionis accompanied by a jump in Ag segregation marked by the verticaldashed line in Fig. 4(f). This calculation gives a point on thetemperature-composition phase diagram shown in Fig. 4(g).Similar calculations at other temperatures would recover theentire phase coexistence line on this diagram.

The obtained phase coexistence point was used as a referencestate for thermodynamic integration. Fig. 4(e) shows the free ener-gies of the GB phases relative to this reference state, cFK � c� andcSK � c�, as functions of the grain composition. Note that the freeenergy difference between the phases, cFK � cSK , is very small. Evenwhen extrapolated to pure Cu at 900 K, it reaches only 2.2 mJ/m2.This is much smaller than typical GB energies in metals(e.g., 951 mJ/m2 for this boundary at 0 K). The free energy differencecFK � cSK is the driving force for the GB phase transformation perunit area. Its small magnitude demonstrates that, applying the pro-posed simulation method, thermodynamic properties of individualGB phases can be computed with a high precision, includingaccurate location of GB phase transformation points. The ability toevaluate the driving force for GB transitions is crucial for the under-standing of nucleation and growth of GB phases. For example,knowing the driving force and measuring the rate of GB phasegrowth, the mobility of the GB phase junction can be extracted.Kinetic properties of this novel line defect are presently unknown.

4. Effect of grain boundary transformations on kineticproperties

4.1. Effect on grain boundary motion

The recent experimental studies of the effect of GB transitionson grain growth kinetics [1,13] utilized polycrystalline samples

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Fig. 4. (a) A bicrystal with split-kite and filled-kite GB phases of the R5ð210Þ½001� GB in Cu(Ag) alloy connected to open surfaces [37]. Zoomed-in views of the split-kite (b)and filled-kite (c) structures reveal bilayer and monolayer segregation patterns, respectively. (d and f) Segregation isotherms calculated for the two phases at 900 K. (e) GBfree energies relative to the phase coexistence state with c� ¼ 0:02%. (g) Schematic of the phase coexistence line c�ðTÞ [38].

312 T. Frolov et al. / Current Opinion in Solid State and Materials Science 20 (2016) 308–315

and could only provide information related to collective behaviorand properties of multiple GBs. Atomistic simulations offer aneffective tool for gaining deeper insights by examining the effectof specific, fully characterized, GB phases on kinetic properties ofindividual GBs. One of the approaches to the evaluation of GBkinetics is to drive a GB by an applied shear stress and extract itsmobility from the velocity-force relation [46,47]. The stress-driven GB motion is caused by the shear-coupling effect, in whichnormal displacements of a boundary are coupled to paralleltranslations of the grains [48–53]. The degree of coupling ischaracterized by the coupling coefficient b ¼ v jj=v , where v jj is

the grain translation velocity and v the GB velocity. For idealcoupling, b is a geometric factor that can be predicted from theGB crystallography. In the particular case of [001] symmetrical tiltGBs, there are two coupling modes with coupling factorsb 1 0 0h i ¼ 2 tanðh=2Þ and b 1 1 0h i ¼ �2 tanðp=4� h=2Þ, where h is themisorientation angle. These two equations describe the 100h iand 110h i modes of coupling and were verified by both atomisticsimulations [49–51] and experiments on Al bicrystals [53,54].

The previous work on coupled GB motion of R5 GBs was per-formed on normal-kite structural units and did not investigateother GB phases. The latter has been done recently using the Cu

(a) (b)

(c)

T. Frolov et al. / Current Opinion in Solid State and Materials Science 20 (2016) 308–315 313

R5ð210Þ½001� GB as a model. This boundary was previously foundto couple according to the 110h imode with a negative b [48–53].The split-kite and filled-kite phases of this boundary were createdin simulation blocks that did not contain sinks or sources of atoms,which precluded phase transformations. A shear stress was appliedto each boundary employing the previous methodology [49–51].As illustrated in Fig. 5(a and b), under the same shear rate, thetwo GB phases move in opposite directions and with differentspeeds. To quantify this difference, the inverse coupling factorscomputed for the two phases are plotted as functions of tempera-ture in Fig. 5(c). Until the temperature reaches about 0:83Tm, bothphases move with ideal coupling factors but in different couplingmodes: split kites in the 100h i mode (b > 0) while filled kites inthe 110h i mode (b < 0). This behavior demonstrates that GB phasetransformations can have a strong impact on GB migration. It sug-gests that in some cases, a phase transformation can even reversethe direction of GBmotion by triggering a switch between differentcoupling modes. Thus, a possible indirect way to discover GB phasetransitions is by measuring variations in the coupling factor atdifferent temperatures and impurity levels.

At temperatures above 0:83Tm, the coupling factors of both GBphases become non-ideal. The motion of the filled-kite phase isaccompanied by an increasing number of sliding events until theboundary completely pre-melts at 1300 K and its response to theapplied shear switches to pure sliding (b 1�!). The split-kite phasealso becomes increasingly disordered; its coupling factor becomesnon-ideal and at about 1200 K changes sign. The shear stressrequired for sustaining the GB motion with a given velocitydecreases with temperature (Fig. 5(d)). At high temperatures whenthe boundary becomes atomically disordered, the stressapproaches zero. It is interesting to note that the stress requiredto move the filled-kite phase is about 60% of that for thesplit-kite phase. This observation suggests that, under a givendriving force, a GB phase transition can significantly accelerate,slow down or even stop the GB motion.

(d)

Fig. 5. (a and b) Bicrystals with the split-kite (a) and filled-kite (b) phases of the CuR5ð210Þ½001� GB after 30 ns of stress-driven motion at 700 K [60]. The initiallyvertical red stripe is a marker line illustrating the shear deformation. The dashedwhite line shows the initial GB position and the black arrow the final position. (c)Inverse coupling factor as a function of temperature. The horizontal lines representthe ideal coupling factors for the h100i and h110i coupling modes. (d) Averageshear stress during the coupled GBmotion as a function of temperature. The verticaldashed line indicates the bulk melting point. (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.)

4.2. Effect on grain boundary diffusion

GB diffusion is a structure-sensitive property. It is natural toexpect that structural phase transformations in GBs can have a sig-nificant impact on GB diffusivity. Fig. 6 shows an Arrhenius plot,log P versus 1=T , of the calculated GB diffusion flux P ¼ Dgbd [37].Here, Dgb is the GB self-diffusion coefficient in the CuR5ð310Þ½001� GB and d is the GB width. The calculations wereperformed in a periodic simulation block prohibiting GB phasetransformations. The diffusion flux was computed from mean-square displacements of GB atoms during MD simulations [22].Note that the split-kite phase is characterized by a smalleractivation energy of diffusion in comparison with the normal-kitephase. The transition between the two phases occurs at approxi-mately 800 K, with the normal-kite phase being more stable belowthis temperature and the split-kite phase above. Fig. 6 suggests thatthe phase transition must be accompanied by a characteristic breakin the Arrhenius dependence of P at around 800 K, with a loweractivation energy at higher temperatures. This type of a break wasindeed observed experimentally at about 800 K [39,55]. Thisagreement strongly suggests that the unusual diffusion behaviorfound in experiments was indeed caused by the phasetransformation in this boundary.

In amore recent study, Divinski and co authors [39]measured Agimpurity diffusivity in the Cu R5ð310Þ GB. The GB diffusivity wasdescribed by the diffusion flux P ¼ Dgbsd, where Dgb is the GB diffu-sion coefficient of Ag and s is the impurity segregation factor. Similarto the self-diffusion case, a characteristic break in the Arrheniusdependence of P was found at about 800–850 K. To make a one to

one comparison with the experimental findings, the same diffusionflux was computed for Ag diffusion in the individual kite and split-kite phases of this boundary by combiningMDwithMC simulations.The latter was applied to create the equilibrium GB segregation ofAg in each phase and to extract the segregation factor s in the dilute

Fig. 6. Arrhenius diagrams of GB diffusion flux P for Cu self-diffusion (lower curves)and Ag impurity diffusion (upper curves) in the kite and split-kite phases of theR5ð310Þ½001� GB. For Ag diffusion, the experimental data [39] are shown forcomparison. Diffusion parallel and normal to the tilt axis is represented by filledand open symbols, respectively.

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regime. The results have revealed a break in the slope of theArrhenius plot at about 800 K (Fig. 6), which is strikingly similarto the break observed in the experiment [39]. This close agreementprovides additional confirmation that the non-Arrhenius diffusionbehavior found by Divinski et al. [39] was caused by a GB phasetransformation. The computed segregation factorswere also in goodagreement with experimental measurements [39], lendingadditional credence to the simulation results.

5. Outlook

The concept of interface phase transformations has been knownsince many years [17,18,56–58]. In fact, the idea can be traced backto Gibbs [59]. The recent explosion of interest in the topic is pri-marily due to the appearance of new experimental evidence(although mostly indirect) for the existence of such transitionsand their possible impact on materials properties. Significant pro-gress has also been achieved in theoretical description of interfacetransitions [14,17] and their modeling by continuous methods(such phase-field approaches [15,16]) and atomistic simulations[22,37,38,60]. Many theoretical and computational challengesremain in this field. The current thermodynamic theory of interfacephase transformations [14] relies on simplifying assumptions thatmake it applicable, strictly speaking, only to isotropic systems suchas multicomponent fluids. The next challenge is to generalize thetheory to interfaces between solid phases. Such interfaces are oftenanisotropic and their properties may depend on the crystallo-graphic orientation of the interface plane. Furthermore, solid-solid interfaces possess two types of tension: the interface freeenergy and the interface stress, which are different quantities bothconceptually and numerically [45,59]. The theory also neglects thecurvature effects on interface transformations. Incorporation ofthese effects would enable applications of the theory to a widerrange of real materials.

One aspect that deserves special attention is related to theproperties and role of the one-dimensional ‘‘phase boundaries”,i.e., junctions between interface phases. Understanding of thesedefects is crucial for developing theories of phase nucleation atinterfaces and description of heterogeneous (multi-phase) statesthat may exist at interfaces under certain constraints. The recently

proposed thermodynamic theory of such junctions should beextended to solid-solid interfaces, such as GBs (especiallyasymmetric type). In particular, the role of the elastic strain fieldscreated by the phase junctions should be investigated and, ifnecessary, incorporated in the theory.

On the computational side, the search for interface phasesremains a difficult task even in the simplest case of single-component interfaceswith a high symmetry. Clearly, the traditionalapproach to creating interfaces by simply joining two crystals andapplying relative rigid translations and local atomic displacementsis insufficient for this purpose [32–36,61,62]. The recently proposedsimulation methodologies [22,37,38,60] create grand-canonicalenvironments in which the atomic density in the interface regionsbecomes an additional automatically adjustable parameter leadingto a deeper minimization of the interface free energy. However,such methodologies have certain limitations (e.g., the reliance onsufficiently fast GB diffusion) and are only implemented inspecifically-designed simulation software. An important task forthe future is the development of more robust grand-canonicalmethods and their implementation in commonly used MD andMC simulation packages. More efficient and accessible simulationapproaches would enable computational discovery of new interfacephases and deeper investigation of their junctions and multi-phaseinterface states. This would also provide a broader testing groundfor the existing theories and inspiration for future new theories ofinterface phase transformations.

Acknowledgements

This work was performed in part under the auspices of the U.S.Department of Energy by Lawrence Livermore National Laboratoryunder Contract DE-AC52-07NA27344. Y.M. was supported by theNational Science Foundation, Division of Materials Research, theMetals and Metallic Nanostructures Program. Use was made ofcomputational resources provided under the Extreme Scienceand Engineering Discovery Environment (XSEDE), which issupported by the National Science Foundation under Grant No.OCI-1053575.

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