recent developments and outstanding challenges in theory and modeling of liquid metals

www.elsevier.com/locate/jnoncrysol

Journal of Non-Crystalline Solids 353 (2007) 3444–3453

Recent developments and outstanding challenges in theoryand modeling of liquid metals

James R. Morris a,b,*, U. Dahlborg c,1, M. Calvo-Dahlborg c,1

a MS&T Division, Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831-6115, United Statesb Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN 37996-2200, United States

c LSG2M, CNRS, Ecole des Mines, Nancy, France

Available online 25 July 2007

Abstract

We review recent progress in accurately modeling properties related to liquid metals and alloys. In particular, we examine the abilityto provide true material specific models, and to use those models to examine real quantities of interest. This includes properties such asequilibrium melting lines for pure systems, alloy phase diagrams, and crystal-melt interfacial properties. We also discuss the role of suchcalculations in relation to understanding crystal nucleation and solidification. As an example of the type of phenomena that we wouldlike to be able to understand (and ultimately predict), we discuss recent experiments on the ‘simple’ eutectic system Al–Si, which has acomplex solidification behavior, due to the facetted interfaces that occur in this system. Recent neutron scattering experiments suggestthat there can be inhomogeneities in the melt, on the order of 10 nm in size, depending on the thermal history of the melt. Understandingthis type of behavior requires more accurate descriptions of alloy systems than have been previously available, as well as new techniquesand approaches similar to those developed in recent years for simpler systems.� 2007 Elsevier B.V. All rights reserved.

PACS: 61.20.Ja; 64.70.Dv; 68.08.�p; 61.25.Mv

Keywords: Liquid alloys and liquid metals; Nucleation; Diffraction and scattering measurements; Modeling and simulation; Phases and equilibria;Structure; Surfaces and interfaces

1. Introduction

The purpose of this paper is to review selected develop-ments in the theory and modeling of liquid metals. This hasbecome a popular topic for some time, and in the last fiveyears there has been a dramatic increase in interest in thearea. Thus, there is a vast amount of recent literature,and we do not attempt to provide a full summary of allof this. The subjects discussed in the paper below represent

0022-3093/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.jnoncrysol.2007.05.159

* Corresponding author. Address: MS&T Division, Oak Ridge NationalLaboratory, P. O. Box 2008, Oak Ridge, TN 37831-6115, United States.

E-mail address: [email protected] (J.R. Morris).1 Present address: GPM UMR6634, Universite de Rouen, 76801 St-

Etienne-du-Rouvray cedex, France.

some of what we feel is new and interesting, or what pre-sents specific challenges to the modeling area.

Simulations of liquids can provide much information tothe community; not only supplementing experimental data,but providing new tests of theories and ideas, making spe-cific predictions that require experimental tests, and ulti-mately helping to lead to the deeper understanding andbetter predictive behavior. The information can be very spe-cific, such as calculations of structure, diffusion or viscosityin specific systems at specific temperature [1–5]. On the otherhand, simulations can provide more ‘generic’ behavior: forexample, there has been significant interesting recent workin nucleation and crystal-melt interfaces in model systemssuch as hard-spheres and Lennard-Jones systems [6–20].

What, specifically, would we like to understand and pre-dict? While the liquids are interesting in their own right,

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J.R. Morris et al. / Journal of Non-Crystalline Solids 353 (2007) 3444–3453 3445

much of the interest is related to phase transitions: forexample, nucleation and growth of crystal phases, the glasstransition and glass formability, liquid–liquid phase transi-tions. (Again, there are many related, interesting questionsthat are not addressed here, including pattern formationduring solidification, devitrification behavior, and defor-mation of metallic glasses.) Much of this really requiresdetailed knowledge of non-equilibrium behavior. However,we stress that there is little predictive capability even inthe equilibrium behavior, and that without this, the non-equilibrium behavior will remain elusive.

More specifically, in equilibrium, we would like to beable to predict structure, viscosity, and diffusive behavior,in both pure metals and alloy systems. Phase diagrams,including melting curves [21–28], solidus and liquidus lines[29–35], are difficult to predict accurately without experi-mental input, even for simple binary systems [30–33]. Someof this review will specifically focus on equilibrium proper-ties of solid–liquid interfaces [3,9,12,16–18,35–54], includ-ing their free energies and mobilities. All of thesequantities affect the non-equilibrium properties we wouldlike to predict: nucleation rates (both homogeneous andheterogeneous), growth rates, and glass formability.

We choose to focus primarily (though not exclusively)on atomistic modeling, primarily utilizing moleculardynamics or Monte Carlo techniques. We will do little todiscuss these techniques, though we will stress their calcu-lation. We will make some brief mention of theoreticaltechniques that are being used to address some of theissues, and how they can be compared with simulation orexperimental results.

The paper is organized as follows. In Section 2, we willdiscuss the methods used to calculate energies and forcesbetween atoms. Section 3 will describe developments inthe calculation of bulk equilibrium properties. Section 4will review recent developments in crystal-melt interfaces,which have received significant attention in the last fouryears, and also crystal nucleation from the melt. Section5 will demonstrate the challenges facing the community,using the ‘simple’ case of eutectic Al–Si. A brief final dis-cussion follows in Section 6.

2. Real materials: from pair potentials to first-principles

calculations

Ultimately, we would like to understand both generic

properties of liquid metals, as well as material specific prop-erties. Early work simulating liquids relied upon simplepotentials, in particular hard-sphere and Lennard-Jonesinteractions (for example, see [47–52,55,56]). To someextent, it may be surprising that 40 years later, these arestill of interest [7,14,16,40,57–60], and are still relevant tothe properties of liquid metals. For material specific prop-erties, pair potentials used to dominate, but these fail tocapture certain properties (such as elastic anisotropies).More recently, the literature has been dominated by theuse many-body potentials that may be written as a func-

tional of pair potentials [12,21,41–43,45,46,61,62],specifically:

EðfrigÞ ¼X

i

F ðqiÞ þX

<ij>

/ðrijÞ; ð1Þ

where

qi ¼X

j 6¼i

f ðrijÞ: ð2Þ

Thus, the energy of an atom depends only on the sets ofdistances of neighboring pairs, as in a pair potential. Poten-tials of this type include the embedded atom model (EAM)[63–66], Finnis–Sinclair (FS) models [67], ‘glue’ models[54,68], and effective medium models [69,70]. For conve-nience, we will simply refer to these as EAM models. Byincluding many-body terms, the potentials become moreaccurate, in particular under conditions where the atomicdensity or coordination changes significantly. However,the accuracy is limited, and for bulk liquid properties, itis not apparent that these add significantly to pair potentialdescriptions. It is useful to note that the energy can be ex-panded around the average density, to approximate this byan effective (density-dependent) potential [62,64]. Ofcourse, such an approximation will not work well in loca-tions where the density is significantly different from theaverage density.

Recently, the ready availability of sufficient computa-tional power has led to increased use of ab initio simula-tions of liquids [1–5,9,71–76], utilizing density functionaltheory (typically with the local-density approximation[LDA] or generalized gradient approximation [GGA]) tocalculate the total energies and forces including the elec-tronic structure. Such methods are, in principle, signifi-cantly more accurate than empirical potential, and do notrequire fitting potentials. However, they are more compu-tationally intensive, and thus the simulations tend to besmall (almost always less than 500 atoms, and often under100), with short simulation times. The use of such methodshas been encouraged by the availability of commercialcodes (for example, the ‘Vienna Ab initio Simulation Pack-age,’ a.k.a. VASP [4,5,77]) and also freely available codes(such as the ‘Spanish Initiative for Electronic Simulationswith Thousands of Atoms,’ a.k.a. SIESTA [78]). In allcases, the accuracy can be limited by the choice of energycut-offs and k-point sampling, but generally these are use-ful for examining material specific properties.

We note that there are approaches that bridge betweenthe classical potentials (such as EAM and pair potentials)and full electronic structure based methods. These includethe related tight-binding (TB) and bond-order potential(BOP) methods, which includes an empirical electronicstructure calculation. These schemes can be more accuratethan the classical potentials, by including the electronicstructure explicitly, while being significantly faster thanab initio calculations (by a factor of �100) by includingonly a limited basis. The use of these has been limited bythe difficulty of developing accurate potentials; however,

Fig. 1. Structure of liquid Al, as calculated from tight-binding moleculardynamics, and from neutron scattering results.

Fig. 2. Sample geometry from the (100) interface simulations. Atoms arecolored according to their order parameter, with the largest valuesindicating the crystalline region.

3446 J.R. Morris et al. / Journal of Non-Crystalline Solids 353 (2007) 3444–3453

they have been quite successful for quantitative predictions.As an example, we show the correlation function

GðrÞ � r gðrÞ � 1½ � ð3Þin comparison with experiment in Fig. 1, calculated using atight-binding potential currently in development for Al.The agreement is better than that of several EAM poten-tials [25,68,79], all of which have been explicitly developedto get liquid thermodynamics correct, even though the TBmodel has not had any liquid properties included in thefitting.

3. Equilibrium behavior: melting curves and phase diagrams

interfaces

In utilizing empirical potentials, such as those describedabove, for studying liquids, the properties must be carefullyexamined. Often, potentials are used without evaluatingthe melting temperatures. For EAM potentials, it has beencommonly observed that the melting temperatures are sig-nificantly lower (30% or more) than experimental values.(A recent development [25] allows for the melting temper-ature to be adjusted using parameters in the potential, bytreating them as thermodynamical variables and using avariation of the Gibbs–Duhem relationship [28]. A similarapproach has been used to examine how the melting tem-perature changes as the potential changes [26,27].) Part ofthe difficulty is that the melting temperature is not triviallycalculated: the ‘brute force’ method of heating the crystal-line system produces large over-estimates of the meltingtemperature [80]. Calculating the free energies of the bulksolid and liquid phases by systematically perturbing thesystem and performing thermodynamic integration is onepossibility [57,74,81,82] that has been widely used. How-ever, all free energy techniques need accurate values forthe free energies of both phases, as errors in the meltingtemperature can be much higher than errors in the individ-ual free energies [62].

When sufficient sizes allow, a more direct method maybe used: performing a simulation of a coexisting system

(see [21,23,58] and references therein). The simulation isset up so that there are both liquid and solid systems indirect contact. A snapshot from such a coexisting simula-tion is shown in Fig. 2. In the simplest approach, a micro-canonical simulation (NVE) is performed, with the numberdensity N/V and total energy E chosen to be between thevalues of the solid and liquid phases. (Of course, some esti-mate must exist to satisfy these conditions.) Then, the sys-tem will evolve to a state where the time-averaged pressureP and time-averaged temperature T (from the kineticenergy) are a point on the coexistence line Pmelt(T). Wehave recently demonstrated the accuracy of this approachusing the Lennard-Jones system. There can be a difficultywith forming anisotropic stress in the system; however, thiscan be resolved by first performing a short equilibrationsimulation under NPT conditions [22]. Final simulationscannot be performed under this ensemble, however, as


the relationship between melting temperature and pressureis not known a priori.

The methods continue to be developed, and there ismuch recent literature on this topic [24,26–28,83]. Severalof the approaches utilize variations of the Gibbs–Duhemrelationship [24,26–28]. A more direct approach is to useperturbation from hard-sphere systems, utilizing theWeeks–Chandler–Anderson (WCA) [84,85] approach. Werecently utilized this [62] for an EAM model of Al, utilizingthe expansion of the EAM model into a density-dependentpair potential [64], and demonstrated that the free energiescan be calculated for both phases, in the same framework.This can be done to a high accuracy (less than 1% errors inthe free energies of the liquid and solid phases near themelting temperature).

For alloy systems, there are fewer approaches, and theseare significantly more computationally intensive. Bruteforce MD simulations of the solid and liquid phases areunlikely to produce equilibrium solute densities in boththe liquid and solid phases, due to the slow diffusion inthe crystalline phase. One approach is to use Monte Carloto accelerate the equilibration. However, direct insertion ofatoms into either the bulk liquid or solid phases are unli-kely to be accepted, due to the fact that both systems arequite dense. This can be alleviated using the ‘semi-GrandCanonical Monte Carlo’ method [29,34,35], in which theatoms can change their chemical species (i.e., from typeA to type B). An applied chemical potential at a given tem-perature allows the system to equilibrate. This has evenbeen used in coexisting simulations, to see the concentra-tion profile across the solid–liquid interface [35]. Anotherapproach is to use variations of the Gibbs–Duhemapproach, adapted for binary systems [29,34].

Ideally, one could directly calculate the free energy ofthe solid and liquid phase as a function of solute concen-tration, by perturbing from a known system, and avoidingthe long simulations, as is done in the WCA calculationsfor pure systems. In that case, the system is treated as aperturbation from the hard-sphere case, as the hard-spherepair correlation functions are documented for both thecrystal and liquid phases. Thus, the free energies of bothphases may be calculated in a single framework [62,86].For alloy systems, however, the binary hard-sphere freeenergies and pair correlations are not generally known(even when the ratio of the radii are close to unity), sothere is no reference system (though there is some datafor particular values; see, for example, [30–33]). This isbeginning to be alleviated, by utilizing the ‘fundamentalmeasure density functional’ [87–90]. This allows the freeenergies and pair correlation functions for the hard-spheresystems to be calculated directly. Once this is done, aneffective perturbation theory may be constructed. Workon this, towards the binary phase diagram, is currently inprogress [86]. Of course, such calculations are limited topotentials that can be approximated by a pair potentialform (or, at least, the potential may be calculated knowingonly pair correlations, such as the EAM potentials). How-

ever, given the current limitations, such calculations wouldbe a significant advance to the field.

4. Coexistence, nucleation and solidification: the role of

crystal-melt interfacial properties

We now turn our attention to the properties of crystal-melt interfaces, in particular their free energies and mobil-ities. As indicated above, the interfacial free energy isimportant for determining the nucleation rate. However,it is also important for rapid solidification dynamics. Inparticular, for dendrite growth, the stability of the growthat the dendrite tip requires that the free energy be aniso-tropic with respect to the crystal interface. Such anisotro-pies are typically small for simple metals and alloys, onthe order of 1%. Yet they determine the dendrite growthdirection, and in the absence of the anisotropy, dendritegrowth is unstable to small fluctuations. At high growthrates, the growth morphology is likely determined also bythe anisotropic mobility of the interface. The interplaybetween continuum level descriptions and atomistic deter-mination of the interfacial properties is extremely rich,and rapidly developing [91].

4.1. Interfacial free energies

The theory of the crystal-melt interfacial free energycommonly refers to Spaepen [92,93] who estimated theentropy of the interface assuming that the liquid prefers atetrahedral arrangement close to the crystal interface. Thisordering of the liquid near the interface clearly accounts forthe dominant free energy loss for simple systems. However,there is also a relaxation of the crystal near the interface, asoftening that compensates somewhat. This has beenshown explicitly by [94], who calculated the free energyof the interface between a liquid and a static crystallineinterface, and obtained an interfacial free energy that wassignificantly higher than that of a true crystal-melt system[7,20]. (The methods used to do this are discussed below.)Commonly, the free energies of crystal-melt interfaces forsimple systems (hard-spheres, Lennard-Jones, and 1/rn

potentials) have been calculated using forms of the densityfunctional theory [36,37–39]. The results of these have beenwidely scattered, and in the absence of more direct calcula-tions, it is difficult to assess these.

Broughton et al. [49] did the first full simulationapproach to calculate the crystal-melt interfacial freeenergy, for a modified Lennard-Jones potential. Theirapproach was to attempt to adiabatically create the inter-faces from fully periodic bulk liquid and solid phases,and to thermodynamically integrate the work done in theprocess. This was done using a carefully constructed ‘cleav-ing’ potential that minimized the hysteresis in this process.Much more recently, Davidchack and Laird adapted andgeneralized this approach, avoiding the process of con-structing a cleaving potential, and applied it to hard-sphere[7] and Lennard-Jones [14] systems. For the latter case, the


potential was modified in precisely the same way as inBroughton and Gilmer, and they obtained the same values(within error bars).

An alternate approach has also been developed in thelast several years, that is more direct, and more sensitiveto the anisotropies in the free energy [41]. The approachis based on the observation the roughness of the interfaces.Such roughness can be seen in Fig. 2. When the interfacesare rough, then the magnitude of the fluctuations dependssensitively upon the interfacial free energy and its anisot-ropy. By measuring the height fluctuations, the interfacialfree energy may be accurately measured. This approach,along with experimental approaches to measuring boththe free energy and its anisotropy, was recently reviewed[18].

Recently, both of these techniques were used to examinethe interfacial free energy of the Lennard-Jones system,truncated as in the original cleaving work [49]. The resultswere in remarkable agreement: the two approaches gavevalues (10 0), (110) and (11 1) interfacial free energies thatwere all within the error bars of each other, and that weresufficiently accurate to resolve the ordering of the interfa-cial free energies. The original cleaving results [49] gave val-ues consistent with these, but with error bars too large toresolve the ordering of the energies.

We note that the ordering from the fluctuation approachhas been the same for all FCC systems studied[12,14,16,20,41–43,46]. Initial results from cleaving forthe hard-sphere system gave different results [7]; however,subsequent calculations for the 1/rn systems, includingresults extrapolated to the n !1 limit [19], are consistentwith these results. More recently, the extrapolated valuesfor the interfacial free energy and anisotropy in hard-sphere limit were confirmed with the fluctuation method[20]. However, surprisingly, both of these latter approachesgive lower average interfacial free energies than were foundin the original cleaving results for hard-spheres [7], whichwere quite similar to values extrapolated from calculationsof the free energy of nuclei [10,13]. This is problematic forcomparisons that utilize these values for predicting nucle-ation properties [95], as the lower value implies a muchlower nucleation rate.

How well do these approaches work for real materials?This is currently being tested for Al, which has a weakanisotropy [96]. In the original calculation for Al usingthe fluctuation technique [12,18], an EAM model of Alwith an accurate melting temperature [68] was used, andproduced values reasonably close to those from grain-boundary groove measurements [97]. However, valuesfrom nucleation data [98,99] are significantly lower. Sinceour original calculation, we have demonstrated that theliquid structure from the Ercolessi and Adams potentialis significantly different than experiments. Calculationswith other Al potentials that have better liquid structuresand similar melting temperatures [25,79] have significantlylower free energies, close to the values from nucleationdata.

The origins of the discrepancies between the potentials,and between the experiments, are unclear. For the simula-tion results, we note that the pair correlation function ofthe original potential has hallmarks of having moreshort-range order (possibly icosahedral) than the others(which are close to hard-sphere like [62], and this may affectthe interfacial free energy. On the experimental side,grain-boundary groove experiments provide a more directmeasurement of the interfacial free energy. However, inpractice, some alloying is required for the metallography,and the measurements [97] use small amounts of Cu. Ifthe Cu affects the liquid structure, then it could also affectthe interfacial free energy. Given the fact that a number ofternary Al–Cu compounds form icosahedral quasicrystals,it is interesting to speculate that the Cu additions have asignificant ordering effect on the liquid Al. Experimentsand ab initio simulations examining the structure of AlCuare currently underway.

All of the above work has primarily dealt with FCC lat-tice structures with rough interfaces. Of course, there isalso broad interest in other systems. BCC systems are ofparticular interest, not only because of metals (such asiron) that solidify into the BCC phase, but also becauseof suggestions that the BCC phase may be preferentiallynucleated [100–103]. Calculations for iron potentials [46]and 1/rn potentials [19] show that the interfacial free ener-gies are significantly lower than in FCC systems. The aniso-tropies are also much smaller. Several authors have alsoindicated that BCC ordering may take place near thesolid–liquid interface in pure Lennard-Jones systems[6,40]. This is rather surprising, given the large free energydifference between the BCC and FCC phases near the melt-ing temperature. In our simulations, we have seen no evi-dence of BCC ordering near the equilibrium interfaces. Acloser examination of this is worthwhile, especially forthe 1/r6 potential where the FCC and BCC phases arenearly degenerate in their free energies at the melting tem-perature (with the BCC phase being the thermodynami-cally stable phase). We have performed a number ofsimulations of the FCC–liquid interfaces, including six dif-ferent interface orientations, and only in one case did thestable BCC phase nucleate. If BCC ordering at the inter-face was significant, we would expect more formation ofthe stable BCC phase.

4.2. Crystal nucleation

As indicated above, the crystal-melt interfacial freeenergy plays an important role in classical nucleation the-ory. While the above discussion deals with the interfacein equilibrium, there has been extensive recent work onnucleation in systems, both theoretical and experimental.This topic is important enough to warrant a separatereview, and we make only a few points here (aside from rel-evant comments above).

It has long been recognized that in simple systems,nucleation may be observed on molecular dynamic time


scales. Simulations with fully periodic boundary conditionsprovide perfectly homogeneous conditions, making themideal for testing ideas. Recently, nucleation in the Len-nard-Jones system was carefully studied, including both acareful examination of the thermodynamic driving forces,and also a statistical exploration of the nucleation rate[8]. This work emphasizes the difficulty in such simulations,essentially demonstrating that steady-state nucleation ratescannot be achieved in molecular dynamics time scales.They analyze the system in terms of time-dependent nucle-ation, and report a crystal-melt interfacial free energy thatis comparable to more direct calculations [14,16,51]. Thework also emphasizes the importance of finite-size effects:for systems of less than 1000 atoms, they see a significantlyhigher nucleation rate than in larger systems. This is notsurprising, given that a sphere with a radius of the correla-tion length will typically contain on the order of 500 atoms,near a melting point.

An interesting development has been the direct examina-tion of the free energy required to form a crystal nucleus, asa function of nucleus size [10,13] using Monte Carlo simu-lations of hard-sphere systems. While classical nucleationtheory is found to be inadequate, they examine the scalingof the size-dependent free energy, and find an interfacialfree energy that is quite close to the value originally calcu-lated using the cleaving method [7]. However, as indicatedabove, more recent calculations [17,19] suggest a lowervalue for the hard-sphere system.

Part of the difficulty in classical nucleation theory is thatthe interface is not sharp, but extends over several latticespacings, including significant relaxation of the bulk prop-erties. This width can be comparable to the critical radius.Thus, the crystalline portion of the nucleus does not havethe bulk free energy. A phase-field approach has been usedrecently, to demonstrate that this can account for the dis-crepancies between nucleation rates in simulations andthose predicted by classical nucleation theory [11,15].

We have recently compared simulations with classicalnucleation theory, in a model Al system where are relevantproperties have been calculated, including the temperature-dependent driving force, diffusion rate, and the interfacialfree energies at melting [104]. The results for the tempera-ture-dependent time until nucleation are consistent withclassical nucleation theory at deep undercoolings, oncethe following factors are taken into account. First, a tem-perature dependence of the solid–liquid interface isassumed, following the Turnbull relation [98,99] with thetemperature-independent Turnbull constant determined atthe melting temperature, but using the temperature-depen-dent latent heat. Secondly, transient nucleation must beproperly accounted for, allowing for the time for nucleito develop. Finally, for smaller systems, the scatter in thetimes is large; in order to reduce this, large simulations pro-duced narrow distributions. For all system sizes, the short-est time until nucleation (taken over multiple simulations)is essentially constant (assuming the system size is largerthan the expected critical nucleus size).

4.3. Interfacial mobilities

Similarly, there has been much recent work examiningthe mobility of the crystal-melt interfaces using bothnon-equilibrium molecular dynamics and equilibrium fluc-tuations [44,45,49,52–54,59,60,105,106]. We will not exten-sively review this work, in part due to recent discussions inthe literature [44,45]. Most of these approaches directlyexamine the velocity of the interface under conditions ofoverheating or undercooling. One of the difficulties is toachieve the linear regime, i.e., where the velocity is a linearfunction of driving force, in molecular dynamic time scales.Some researchers have reported an asymmetry in themobility [54]: the velocity at a given undercooling is differ-ent than when overheated by the same amount. In thesmall driving force limit, this cannot be true, as it wouldviolate microscopic equilibrium – fluctuations would drivethe interface preferentially in one direction even at thermo-dynamic equilibrium. Other simulations have checked thisand found symmetrical behavior [45,105]. Asymmetricbehavior is possible far away from the transition. Oneimportant goal is to understand how microscopic mecha-nisms lead to the mobility of the interface [107–109].

For pure systems, the anisotropy of the interface hasbeen examined from simulations for some time [49,52].As in the case of the interfacial free energy, the anisotropyis important for understanding the solidification dynamics(for a recent discussion, see [110]). Early molecular dynam-ics simulations showed that the anisotropy can be signifi-cant, and proposed a model for this [49,52]. The modelfailed to explain the slow growth of the (111). This wasexplained in terms of the difficulty of consistently growingthe FCC crystal structure: the system can easily form stack-ing faults during the growth, and the process of ‘correcting’these could slow down the growth. The model had otherdifficulties: the explicit dependence of the growth rate onplanar spacings would predict unphysical velocities forhigh-index planes. An alternate model for the anisotropyfrom density functional theory [107], however, appears tocapture the anisotropy, at least for FCC metals, and rea-sonable corrections give more quantitative agreement aswell [45].

In the case of alloy systems, there are significantly fewerresults [53,105,106]. This is an important case, not onlybecause of the wide interest in alloy systems, but alsobecause even dilute solutes can dramatically affect theinterfacial dynamics. In [53,105], the systems were chosento represent idealized cases: [53] explored the limit thatthe minority atoms have no solubility in the crystal, andcan only be found there due to solute trapping. In[53,105], model parameters were chosen to examine thecases where atom types A and B either preferentiallybonded with each other, or preferentially bonded withthemselves. Also, a case where the bonding strength wasneutral, but the minority atoms were smaller, was exam-ined. In all cases, the mobility was decreased in comparisonto the pure system. Experimentally, studying mobility in

ig. 3. Small-angle neutron scattering results for eutectic Al–Si, indicatingat the scattering at 950 �C depends upon the thermal history. Theraight line indicates the Q�4 behavior associated with Porod’s law.


alloyed systems is difficult, as the planar interface can easilybecome unstable, especially under driving conditions rele-vant to rapid solidification.

5. Challenges: the ‘simple’ case of Al–Si

As indicated above, there have been dramatic develop-ments not only in the computational power available, butalso in the methods that can be applied. Ab initio calcula-tions reveal detailed information about real materials;new methods allow for the examination of not only struc-ture, but detailed thermodynamic and non-equilibriumproperties. We are now in a position to start examiningmore complex behavior of real materials.

As an example of the complexity that we may now beginto address, we consider Al–Si, which demonstrates somevery interesting and challenging scientific issues (aside fromits technological importance). From the point of view of itsphase diagram, this is a very simple system. It is a classiceutectic (at 12 at.% Si), with no ordered phases. There issome solubility of Si in FCC Al, but almost no solubilityof Al in diamond-structured Si. However, the solidificationprocess is much different, due to the formation of facetedinterfaces between crystalline Si and the liquid Al–Si. Thisresults in very novel solidification microstructures [111].Small amounts of ternary additions can also dramaticallyaffect the microstructure by inducing fine-scale twinning;we will not discuss this here.

With no other information, this provides a fresh set ofchallenges. Solidification microstructures in alloys are com-monly studied using phase-field models, which use acoarse-grained free energy, and are capable of modelingon the length scale of microns [112,113]. However, in orderto do this, they utilize a non-physical width of the solid–liquid interface, much larger than actual widths. For roughinterfaces, this can be a reasonable approximation, but itdoes not apply in a straightforward way for facetted inter-faces. Recent approaches have begun to address this [114].

Modeling the materials at the atomistic length scale isalso challenging. Ab initio simulations are sufficient to sim-ulate small systems [75,76]. However, they are not sufficientto examine yet the interfaces between the Si and the alloyliquid. Moreover, the system is not ideal for using classicalpotentials: even for pure Si, accurately describing both theliquid and solid phases with classical potentials is difficult.However, both Al and Si are good candidates for tight-binding modeling, due to the limited number of orbitalsneeded to describe the bonding. Excellent potentials forSi already exist, and we have recently developed (and arecurrently testing) an Al potential as well. Thus, accuratemodeling of Al–Si systems, for systems on the order of1000–2000 atoms, may be available soon.

However, there are fresh challenges in the liquid Al–Sisystem, aside from solidification issues. There are now sev-eral measurements [115] for that indicate hysteresis in theliquid: density and diffraction measurements give differentresults, depending upon the thermal history of the liquid.

Fthst

We demonstrate this in Fig. 3, where we show the results ofsmall-angle neutron scattering for eutectic Al–Si. The datais plotted on a log–log plot, to compare with Porod’s lawfor scattering off of sharp, homogeneous particles of areaAp and volume Vp:

SðQÞ ¼ 2pAp

1

V 2pQ4¼ KP

Q4: ð4Þ

In the figure, it is apparent that the scattering at 950 �C onheating is similar to that at 700 �C. At 1200 �C, the scatter-ing for Q < 0.01 A�1 has decreased significantly, nearly bya factor of 10. On cooling back to 950 �C, the scattering re-mains low. Note that the measurements at each tempera-ture take several hours.

This dramatic change indicates that there are structuralchanges occurring at scales of �100 A, with particles thatscatter neutrons at significantly different strengths thanthe surrounding liquid. One possibility is that there areclusters of Si in the system, which remain metastable untilthere is a significant overheat. However, the implied stabil-ity is remarkable: the system apparently cannot reach equi-librium at nearly 200 �C above the eutectic temperatureover the course of several hours. Aside from the remark-able scientific issues, this can also have important effectson the casting of such materials.

What can modeling add to this? Phase-field simulationsof the growth can add information about the solidificationmicrostructure, if the faceted interfaces can be handledaccurately. (To correctly model the observed microstruc-ture, it will also be necessary to describe twin formationduring solidification [111].) However, little is known aboutthe mobilities of the interface. The mobilities are key inunderstanding if Si crystallites can be stable in the Al–Simelt as well: if stable, faceted Si clusters exist, it may explainthe observation of the slow approach to equilibrium in the


liquid. We also note that growth of crystal Si from the meltwill be very different from the processes described earlier,not only because of the faceted interfaces (which also existin pure Si) but also because the Si atoms are a minority inthe liquid. Thus, the dynamics may be more related toattachment kinetics at steps, similar to processes occurringin vapor deposition, but dramatically different due to thepresence of the liquid phase at the interface.

6. Final discussion

The purpose of this review has been to describe themany issues associated with determining the thermody-namic and dynamic processes associated with liquid metalsand their associated crystalline phases. Clearly, there aremany challenges to be met, but the community is poisedto making great strides to narrowing the gap between prac-tical experimental issues on real materials, and resultsbased on simulations. This is due in part to the develop-ment of sufficient computer power and availability of elec-tronic structure methods capable of performing accuratesimulations of liquid metals, alloys and other systems. Onthe other hand, there have been significant developmentsin approaches to calculating properties of liquids, includingcrystal-melt interfacial properties and nucleation behavior,which have been carefully studied for model systems. Thefuture of the field clearly is both further fundamental devel-opments, and the application of the approaches to morerealistic simulations of real materials.

The most significant challenge is in the area of alloys,where both models and techniques are lacking. As indi-cated in this paper, there are beginnings of importantresults in this area, including new developments that allowefficient determination of phase diagrams, improved poten-tials, and better understanding of the properties. As anexample of a challenge in this area, we have discussed theproperties of liquid Al–Si, its interfaces with the Al andSi crystalline phases, and the impacts on the directionalsolidification behavior. This is simply one area of interest.It is interesting to compare this with questions (not dis-cussed in detail here) on ‘bulk metallic glasses’ (BMG):these complex alloys (typically three or more components)have been studied widely in the past 10 years. Clearly, whenthe liquid behavior for ‘simple’ systems such as Al–Si is stilla challenge, the behavior of BMGs will raise questions formany years.

Of course, the converse question of glass formation isthe prediction of nucleation rate. Given that there havebeen important developments examining the crystal-meltinterfaces and the nucleation barriers in monatomic sys-tems, we can now envision understanding how smallamounts of solutes affect the nucleation rate. Preliminarywork has recently begun in this area [116] showing thatsmall amounts of solute can change the critical cooling rateof simple systems by several orders of magnitude. Morestudies along these lines, combined with crystal-melt simu-lations, would provide clearer understanding of nucleation

in the presence of solute. This information may also beincorporated into longer length-scale descriptions, as inRef. [11]. Such work may provide the proper scientificfoundations not only for nucleation and growth of alloyphases, but also for understanding the BMG behavior.

As we have focused on relatively simple systems, espe-cially FCC and BCC systems, the other generalizations willalso consider more complex crystal structures. We havealready indicated the formation of alloy phases above;however, we also look towards studying complex crystal-line structures. Diamond-structured phases are an interest-ing beginning: in Si, it is already known that certaininterfaces are facetted [117,22]. The free energies of suchinterfaces cannot be determined by the fluctuation tech-nique described above, but are ideal for the cleavingmethod. The free energies of orientations near these willbe determined by step free energies, which will also dependupon the fluctuations that occur at the steps. Thus, map-ping out the interfacial free energies and nucleation barri-ers [118] for this system is a very interesting challenge.Again, we point to the Al–Si system as a key model whereall of these issues come into play. Also of interest are thenucleation barriers and free energies of icosahedral anddecagonal systems; we have discussed this recently in Ref.[18].

Acknowledgements

The authors acknowledge the Institut Laue Langevin inGrenoble, France, for allowing beam time and providingfunding and experimental support (C. Dewhurst) duringthe measurements on D11. J.R.M. would like to thankmany useful discussions with R.S. Aga, M. Asta, R.Davidchack, T. Egami, L. Granasy, K.M. Ho, J.J. Hoyt,A. Karma, K. Kelton, M. Kramer, B. Laird, S. Liu, M.Mendelev, R.E. Napolitano, X. Song, D. Sordelet, D.J.Srolovitz, R. Trivedi, C.Z. Wang, J. Warren, Y.Y. Yeand X.C. Zeng. This research has been sponsored by theDivision of Materials Sciences and Engineering, Office ofBasic Energy Sciences, US Department of Energy undercontract DE-AC05-00OR-22725 with UT-Battelle and con-tract W-7405-ENG-82 with Iowa State University of Sci-ence and Technology. We also acknowledge partialfunding from the Department of Energy’s ComputationalMaterials Science Network project, ‘Microstructural Evo-lution Based on Fundamental Interfacial Properties,’ andcomputer time from the Department of Energy’s NationalEnergy Research Scientific Computing Center.

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