at SciVerse ScienceDirect

Intermetallics 31 (2012) 292e320

Contents lists available

Intermetallics

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

Review

Long-range n-body potential and applied to atomistic modeling the formationof ternary metallic glasses

J.H. Li a,*, Y. Dai a,b, X.D. Dai a,c

aAdvanced Materials Laboratory, Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, People’s Republic of Chinab Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, People’s Republic of ChinacResearch Institute of Chemical Defense, Beijing 102205, People’s Republic of China

a r t i c l e i n f o

Article history:Received 28 March 2012Received in revised form29 May 2012Accepted 31 May 2012Available online 18 August 2012

Keywords:A. Ternary alloy systemsB. Glasses, metallicB. Alloy designE. Simulations, atomisticE. Phase stability, prediction

* Corresponding author.E-mail address: lijiahao@mail.tsinghua.edu.cn (J.H

0966-9795/$ e see front matter � 2012 Elsevier Ltd.http://dx.doi.org/10.1016/j.intermet.2012.05.018

a b s t r a c t

In this review article, a brief summary is firstly presented concerning the currently used n-bodypotentials for the metal systems, i.e. the second moment approximation of tight-binding, Finnis-Sinclairpotentials and embedded atom method. Secondly, a long-range n-body potential is proposed and provento be not only applicable for the three major crystalline structured, i.e. bcc, fcc and hcp, metals and theiralloys, but also be able to distinguish the energy differences between fcc and hcp structures of the metals.Furthermore, both the energy and force reproduced by the proposed potential could go smoothly to zeroat cutoff distance, without appearing some unphysical phenomena frequently observed in atomisticsimulations with other n-body potentials. Thirdly, for some selected ternary metal systems, the long-range n-body potentials are constructed with the aid of ab initio calculations and applied in moleculardynamics and Monte Carlo simulations to study the metallic glass formation. The simulation results notonly clarified that the underlying physics of the metallic glass formation is the crystalline latticecollapsing of the solid solution when the solute concentration exceeds the critical solid solubility, butalso predicted, for a ternary system, a quantitative composition region within which the metallic glassformation is energetically favored. Fourthly, the energy difference between the solid solution and theamorphous counterpart is defined as the driving force for the crystal-to-amorphous transition, and theamount of the driving force could thus be considered as a comparative measure of the glass-formingability of the amorphous alloy. It follows that in a ternary metal system, the largest driving forcecould be correlated to the optimized composition, of which the metallic glass is the most stable or easiestone to be produced in practice. It turns out that the predictions directly from the constructed potentialthrough atomistic simulations are well compatible with the experimental observations reported so far inthe literature, leading firm support to the relevance of the predicted glass-forming regions and theoptimized compositions of the respective ternary metal systems, as well as to the validity of the proposedlong-range n-body potential.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

In 1954, Buckel and Hilsch found that condensation of metalvapor onto a cooled substrate could result in the formation of anamorphous alloy [1,2]. A few years later, Duwez et al. obtained anAueSi amorphous alloy using a technique named liquid meltquenching [3,4]. These pioneering works explored a new categoryof metallic materials, i.e. amorphous alloys or later named metallicglasses. The metallic glasses, with a non-crystalline structure, differsignificantly from those widely used crystalline alloys, such as

. Li).

All rights reserved.

steels, aluminum alloys, titanium alloys [5e7]. Since the 1960s,liquid melt quenching has been extensively used to produce a largenumber of metallic glasses in binary, ternary as well as multicom-ponent metal systems [8e10]. In the early stage, metallic glassesobtained by liquid melt quenching were mostly thin films or foils.Since the late 1980s, researchers have found a number of ternaryand multicomponent metal systems, in which liquid meltquenching could readily produce the so-called bulk metallic glasseswith a geometrical size of up to a magnitude of centimeter [11e16].

In the field of metallic glasses, one of the fundamental scientificissues is the so-called glass-forming ability (GFA), which reflectsthe easiness or difficulty of the metallic glass formation. This issuehas been widely discussed generally in two ways [17e19]. First,consideration is given to the specific alloy and metallic glass

J.H. Li et al. / Intermetallics 31 (2012) 292e320 293

formation is considered as the frustration of the crystallization, orthe consequence of possibly enhanced stability of a liquid-like statedown to low temperatures. Second, consideration is given to, forexample, a specific binary metal system and in this situation, theglass-forming range (GFR) is used to indicate an alloy compositionrange within which the metallic glasses could be obtained in thesystem by certain non-equilibriummaterials producing techniques.In general, the issues of the GFA/GFR could be approached either byexperiment or atomistic simulation [13,20].

An interatomic potential of a system is used to describe theatomic interactions of the system and, in principle, if the potential isknown, then most of the properties or behaviors of the systemcould be obtained directly from the potential, e.g. throughcomputation and simulation. In this sense, the GFA/GFR of thesystem could surely be derived from the potential. However, theideal potential, which could commonly be valid, is not available. Infact, some approximations and assumptions have to be made to theinteratomic interaction. In the very early stage, Mie has proposeda phenomeno-logical model to describe the interatomic interac-tions [21]. In Mie’s model, the interatomic interactions could bedecomposed into two terms: one represents the repulsion and theother the attraction between the two atoms. Based on Mie’s work,Lennard and Jones proposed in 1925 a simple form of the pairpotential, i.e. the LennardeJones, or LeJ, 6e12 potential [22]. A fewyears later, some variants of the LeJ potential, such as the Morse[23] and BorneMayer potentials [24] were proposed. The uniquefeature of these potentials is their convenient mathematical formsto describe the properties of real materials. Because these pairpotentials do not incorporate the many-body effect, there are someintrinsic drawbacks, e.g. the dilemma of the cohesive energy andvacancy formation energy. For a pair potential, if the cohesiveenergy is correctly described, then the vacancy formation energy isnot, and vice versa. Besides, for cubic metals, the Cauchy pressuresderived from the pair potentials are always zero, yet the real Cauchypressures are usually non-zero [25,26].

In the 1980s, based on the concept of the local electron density,a significant progresswasmade by developing the so-called n-bodypotentials [27]. The main physical idea of the n-body potentials isthat the bonds would become weaker when the local environmentbecomes more crowded. Consequently, a plot of the cohesiveenergy as a function of coordination should not decrease linearly[28]. This means that the cohesive energy of an atom is largelygoverned by the local atomic configuration at the site where theatom is located. In the literature, several n-body potentials ofdifferent forms have been proposed for the transition metals andtheir alloys. The most widely used ones are the second-momentapproximation of tight-binding (TB-SMA) potential [29e35], theFinnis-Sinclair (F-S) potential [36], the embedded-atom method(EAM) potential and their various modifications [37e43]. In thepast decades, a great efforts have been made to improve theperformance of these n-body potentials. For example, Guellil andAdams proposed a polynomial truncation function for the electrondensity in the EAM potential [44]. Nonetheless, because a loga-rithmic function is adopted for the embedding function, thepotential energy goes to infinity instead of converging to zerowhenthe electron density approaches to zero, probably resulting in someunphysical phenomena in atomistic simulations [39,45]. Wadleyet al. proposed a segmented embedding function by introducingabout 20 parameters in the EAM potential [46]. Sutton and Chenproposed a Long-range F-S potential for fcc structured metals[47,48].

In recent years, by combining the unique features of the TB-SMAand F-S potentials, Li, Dai and Dai proposed a long-range n-bodypotential (abbreviated as the LDD potential for convenience). Theproposed LDD potential has been proven to be suitable for bcc, fcc

and hcp structured metals and their alloys [45,49e51]. The LDDpotential also solved the problem related to the appearance of someunphysical phenomena that occurred with other n-body potentials,because the energy and force derived from the LDD potential bothsmoothly converge to zero at the cutoff distance in the simulations.Furthermore, the LDD potential is also able to well distinguishenergy difference between the fcc and hcp structures of the tran-sition metals [52e54]. In addition, the LDD potential has also beenproven to be able to reproduce the static as well as dynamicphysical properties of the transition metals and alloys.

In the present review, a brief discussion is first presented con-cerning the currently available n-body potentials that are widelyused for the metal systems. Second, the newly proposed LDDpotential is introduced in detail, including its form, unique featureas well as the performance in reproducing the static and dynamicphysical properties of metals and alloys. Third, we applied theconstructed LDD potential to model the formation of ternarymetallic glasses, clarifying the underlying physics of metallic glassformation and to derive the related GFA/GFR for some represen-tative ternary transition metal systems through atomistic simula-tions. Finally, a few concluding remarks and prospects are given toend the present review.

2. Typical n-body potentials of the transition metal systems

For transition metal systems, several n-body potentials havebeen developed and are currently used in the field of computa-tional materials science. These potentials share a similar form, yetfrequently result in rather different parameterization for the samematerial. In many cases, the researchers could make a guess of thefunctions and fit the parameters to the experimental data, whereasfor some special cases, the researchers could derive the functionsand parameters by fitting the data acquired from the ab initiocalculations. In general, these n-body potentials still belong toempirical approaches and hence have some limitations and draw-backs. In this section, a brief discussion is firstly presented con-cerning the typical n-body potentials used in transition metalsystems, i.e. the EAM, TB-SMA and F-S potentials.

2.1. Embedded atom method

According to Daw and Bask [37,41,42], the basic principle of theEAM is that each atom can be viewed as an impurity embedded ina host created by its neighboring atoms and the energy of a systemconsisting of N atoms can be expressed by

Etotal ¼ 12

Xjsi

fij�rij�þX

i

FiðriÞ; (1)

where rij is the distance between atoms i and j and fij(rij) is the pairinteraction accounting for the electrostatic repulsion between theatoms. The local electron density, ri, is the electron density of thehost at the site of atom i. The embedding energy of atom i, Fi(ri),incorporating themany-body contributions, reflects the interactionbetween the embedded atom and the background electron gas. Thelocal electron density may vary from site to site, depending on thelocal atomic configurations, and can be approximated by thesuperposition of the contributions from the neighboring atoms, i.e.

ri ¼Xjsi

jj�rij�; (2)

where jj(rij) is the contribution of the neighboring atom j to thelocal electron density ri. It can also be considered as the electrondensity of the atom j at the site of the atom i. The embedding

Fig. 1. The pair interaction, atomic electron density, embedding function and thederivates of the EAM potential developed by Johnson for fcc Nickel [20,38].

Fig. 2. Total energy and the derivates of the EAM and F-S potentials developed byJohnson and Finnis et al. for bcc tantalum [36,39].

J.H. Li et al. / Intermetallics 31 (2012) 292e320294

energy defined here differs not only from theweak pseudopotential[28,55], in which the atomic volume is ambiguous e.g. at thesurface, but also from the quasiatom model and effective-mediumtheory [56,57], in which the average density may be invalid in thecalculation involving the relaxation, reconstructions of the defects.To apply the EAM to a real material, the functions j and F must bespecified for each atomic species, and f for each possible combi-nation of the atomic species. From the results of early ab initiocalculations, one can gain some important information concerningthe general behavior of these functions. The embedding energyshould converge zero when the electron density goes to zero, andshould have a negative slop and a positive curvature for the back-ground electron density. Assuming that only the nearest neighborscontribute to the pair interaction and the local electron density,Johnson, Banerjea and Smith et al. proposed a logarithmic functionfor the embedding energy [38,39,58]:

F�r� ¼ �EC

�1� gln

�r

re

���r

re

�g

: (3)

here, EC is the cohesive energy. g and re can be considered as twopotential parameters and they may be different for different solids.

In an AeB binary metal system, the pair interaction betweensimilar atoms, f, atomic electron density functions, j, and theembedding functions, F, can be considered as the same as those ofpure elements [59]. The pair interaction between dissimilar atoms,fAB or fBA, is the so-called cross potential and in general fAB ¼ fBA.The cross potential can be derived from the potentials of the pureelements. For example, Foiles et al. took the geometric average of

pure elements as the cross potential [60], i.e. fAB ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifAAfBB

q.

Alternatively, Johnson suggested that it is better to take the form forthe cross potential [61]:

fAB�r� ¼ 12

"jB�r�jA�r�fAA�r�þ jA�r�

jB�r�fBB�r�#: (4)

The cross potential can also be adopted a new function. Forinstance, Gao et al. proposed a new function for the pair interactionbetween the Ag and Pd atoms [62]:

fAB�r� ¼ �a

�1þ b

�r

r0;AB� 1

��exp

�b

�r

r0;AB� 1

��; (5)

where r0,AB is the nearest distance between the atoms A and B, andthe parameters a and b are determined by fitting to the physicalproperties of the related compounds. The cross potential can alsobe constructed by combining the two methods mentioned above.For example, Chen et al. used the following formula to construct thecross potential [63]:

fAB�r� ¼ AhfAA�ar þ b

�þ fBB�cr þ d�i; (6)

where A, a, b, c and d are the cross potential parameters to be fitted.One notes that a similar form was also proposed by Gong et al. forthe cross potential of CreW [64].

To date, several variant formulas of analytic EAM potentials havebeen proposed for metals and their alloys, for example, by Foiles[65], Oh [66], Johnson [39], Guellil [44], Pasianot [67], Zhang [68],Cai [69] and Pohlong [70]. It should be noted that these EAMs notonly take into account the contributions from the nearest neigh-bors, but also consider the effects of the second, third or even up toseventh neighbors. To give an intuitionistic impression, Fig.1 showsthe pair interaction, atomic electron density, embedding functionand the derivates of the EAM potential developed by Johnson for fccnickel [38]. Fig. 2 shows the total energy and the derivates of the

EAM and F-S potentials developed by Johnson and Finnis et al. forbcc tantalum [36,39]. It should be noted that, although the EAM hasbeenwidely applied to the calculation and simulation in the field ofmaterials science and condensed matter physics, it is still anempirical or semi-empirical method and therefore still has somelimitations. For example, since the embedding energy in the EAM

J.H. Li et al. / Intermetallics 31 (2012) 292e320 295

potential is calculated by a logarithmic function, the potentialenergy goes to infinity when the local electron density approacheszero, leading to unrealistic behavior in some cases. This indicatesthat the embedding function defined above might be invalid in thecase of small local electron densities [38]. To overcome thisproblem, Wadley, Minshi etc have proposed two segmentedembedding functions [71e73]. Besides, one notes from Eqs. (2) and(3) that, for a pure element, changing the atomic electron densityby a constant factor does not change the computed results, becausethis change simply results in a rescaling of the argument of theembedding function. For a multi-component system, however,changing the atomic electron density used for one of the compo-nents will strongly affect the mixing energies of the alloy. It istherefore essential for an alloy that a consistent choice should bemade for the atomic electron densities of all components.

One notes that the EAM discussed above and other empiricalpotentials mentioned later are all radially symmetric and thereforethey are also call as many-body pair-functional potentials. Thesepotentials do not reflect the directional nature of the bonding. Toimprove the performance of these potentials in this aspect, Baskes[37,40,42,74], Lee [43,75,76] and Ouyang et al. [77e79], forexample, have proposed the so-called modified EAM (MEAM)potentials, in which the local electron density is computed byconsidering the directionality of the bonding, i.e. taking intoaccount the angular contribution of the electron density to the localelectron density. In general, many-body pair-functional potentialsare still valuable both to study complex system that are intractablewith more rigorous methods, and to study generic properties thatdo not depend significantly on the details of the energetics.

2.2. Second moment approximation of tight-binding potential

In the second moment approximation of tight binding model[TB-SMA] [29,30,80e85], the bonding energy in a metal system isapproximatively proportional to the average bandwidth of the localelectron density of state. The bandwidth is determined by thesquare root of the second moment of the local electron density ofstate. In particular, the second moment of the electron density ofstate can be written as a sum of squares of hopping integralsbetween the atoms and their neighbors. Since the hopping integralis the function of the radial distance between the atoms i and j, theband energy can be written as

Ebond ¼ AffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXjsi

b2�rij�s: (7)

where A is a material dependent parameter and b(rij) is the hoppingintegral. The standard dependence of the hopping integral shouldbe r�4 or r�5, but the exponential form is widely used in practice. Toensure crystal stability, a repulsive interaction needs to be includedaside from the bonding contribution in Eq. (7), and this term isnormally assumed to be pairwise and described by the sum ofBorneMayer ioneion repulsions. Based on this argument, Tomaneket al. proposed an empirical formula to compute the potentialenergy of an atom [31,86,87],

Etotal ¼ pECp� q

Z1=20

�q

pZ1=2X

exp�� p

�rir0

� 1��

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX

exp�� 2q

�rir0

� 1��s �

; (8)

where the first term is the repulsive part for the electrostaticinteraction and the second term is the bonding energy accountingfor the many-body interaction of the system. EC and Z0 are the

cohesive energy and coordination number of an atom in the groundstate, respectively. Z is the coordination number of the atom. ri is thedistance of the atom to the neighboring atom i. r0 denotes thenearest neighbor distance in the ground state. Two empiricalparameters, p z 9 and q z 3 have been proposed for transitionmetals [87]. A few years later, a simple formula was proposed byRosato, Cleri et al. for the TB-SMApotential of hcp zirconium [33,35]:

Etotal ¼ 12

Xf�rij�� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX

j�rij�q; (9)

with

f�rij� ¼ Aabexp

�� pab

�rir0

� 1��

; (10)

and

j�rij� ¼ x2abexp

�� 2qab

�rir0

� 1��

: (11)

here, the four parameters, pab, qab, Aab and xab are assumed todepend only on the interacting atomic species a and b, and can bedetermined by fitting them to the experimental physical properties.In this way, the TB-SMBA potential was well established.

The form expressed by Eqs. (7)e(9) for the band term isgenerally equivalent to a functional expression of thetypeFðrÞ ¼ � ffiffiffi

rp

, with r¼ Sj(rij), representing a sum over the localelectronic charge density j(rij) induced at site i from atoms at sites j.In this sense the TB-SMA potential is formally analogous to the EAMpotential, although originating from a different physical perspec-tive, especially in its simplified version where the form of theelectron density function j(rij) is set to a simple exponential. Thetwo models have similar computational requirements and thenames are often used interchangeably. Nonetheless, there are somedifferences when dealing with a multi-components system. For anEAM potential, the pairwise term f(rij) depends on both species,while F(r) and j(rij) depend only on one specie. In contrast, for a TB-SMA potential, f(rij) and j(rij) depend on both species, while theFðrÞ ¼ � ffiffiffi

rp

is independent. This subtle distinction becomes rele-vant when they are applied to alloys. For the EAM potential, nofurther refitting of the many-body terms are required for eachspecies in some case, whereas for the TB-SMA potential the func-tions f(rij) and j(rij) need to be fitted for each pair of species.

Although the TB-SMA potential initially seemed to be applicableto a half-filled band metals, Ackland and Finnis have generalizedthe validity of the approach any band filling metals [32]. However,by rule of thumb, it has been shown that the TB-SMA potential ismore applicable to fcc and hcp metals. Since both the repulsiveterm and the electron density function are expressed by expo-nential functions, the interatomic interaction described by the TB-SMA exists in an infinite space. To truncate the long tail of theinteratomic interaction, Li et al. proposed a smoothly truncated TB-SMA potential [51,52], by incorporating a truncation function intothe pair term and local electronic charge density function:

f�r� ¼

8>>><>>>:A1exp

�� p1

�rr0

� 1��

; r � rm1

A2

�rc1r0

� rr0

�n1

exp�� p2

�rr0

� 1��

; rm1 < r � rc1

;

(12a)

j�r� ¼

8>>><>>>:B1exp

�� q1

�rr0

� 1��

; r � rm2

B2

�rc2r0

� rr0

�n2

exp�� q2

�rr0

� 1��

; rm2 < r � rc2

;

(12b)

J.H. Li et al. / Intermetallics 31 (2012) 292e320296

In the above expressions, both f(r) and j(r) are continuous andsmooth in the entire range of computation. It is found that thesmoothly truncated TB-SMA potential can also be applied to fcc andhcp transition metals and their alloys.

2.3. Finnis-Sinclair potential

According Finnis and Sinclair [36], the total energy of anassembly of N atoms can be empirically written as

Etotal ¼ 12

Xjsi

f�rij�þX

i

f ðriÞ; (13)

where rij is the distance between atoms i and j. f(rij) is the pairwiseterm, i.e. repulsive interaction, and described by a fourth-orderpolynomial

f�r� ¼ ðr � cÞ2

c0 þ c1r þ c2r

2: (14)

The second term in Eq. (13) is the n-body interaction describedby

f ðrÞ ¼ � ffiffiffir

p; (15)

and

r ¼ 12

Xjsi

j�rij�; (16)

with

j�r� ¼ ðr � dÞ2: (17)

The parameters c and d in Eqs. (14) and (17) are the cutoff radii ofthe pairwise and n-body interactions, respectively. c0, c1 and c2 arepotential parameters that can be determined by fitting to theexperimental data.

The Finnis and Sinclair (abbreviated as F-S) potential is somewhatequivalent to the TB-SMA, with the only difference that the spacedependence of the overlap integrals is assumed to be polynomialinstead of exponential. Some other physical interpretations could bemade. For example, r can be regarded as the local electron density atsite i and is constructed by a rigid superposition of atomic chargedensities j [36]. In this sense, the F-S potential can be considered asa variant of the EAM potential with two differences that theembedding function is always a negative square root instead ofa logarithmand the local density is dependent on the combination ofthe local atom and its neighbors instead of only on its neighbors.

Since both the repulsive term and the overlap integral in the F-Spotential are short-range and described by polynomial functions, itenables one to implement very efficiently simulation and compu-tation. The F-S potential has been used successfully to calculate thephysical properties or simulate the behaviors of bcc and fcc metals.It is found, however, that the force and energy described by the F-Spotential is too "soft" in some cases, especially when the atoms areforced to be close together, e.g. at high pressure [88,89]. Thecalculated energy is much lower than the experimental observationor theoretical prediction. To overcome this problem, severalmodified or extended F-S potentials have been proposed[47,48,88,90e92]. In these newly proposed potentials, additionalterms were inserted in or spindle functions were adopted for therepulsive term and overlap integral. To date, the modified,extended and long range F-S potentials have formed a loose family,including the Ackland potential [88,91], the Sutton potential [47].

In contrast to the original EAM, TB-SMA and F-S potentials thatdo not have a r�6 van der Waals tail and extend typically up to the

third neighbors in fcc and bcc crystals, Sutton et al. proposeda long-range F-S potential [47] whose, repulsive term and overlapintegral are expressed by

f�r� ¼ a

rm; (18a)

and

j�r� ¼ b

rn; (18b)

where the potential parameters a and b are positive real number.mand n are positive integers. Clearly, the pair potential is purelyrepulsive and the n-body term is purely cohesive. The furtheradvantages of the Sutton potential are analogous to the scalingproperties of the LennardeJones potentials. Since reciprocal func-tions are involved in Eqs. (18a) and (18b), the interaction describedby the Sutton potential is long-range and therefore some propertreatments at the cutoff distance are required. In this regard, newpower functions were proposed by the authors for the repulsiveterm and the overlap integral [93],

f�r� ¼ c1ðrc1 � rÞn1 ; (19a)

and

j�r� ¼ c2ðrc2 � rÞn2 ; (19b)

where n1 and n2 are two integers. rc1 and rc2 are the cutoff radii forrepulsive term and the overlap integral, respectively. c1 and c2 aretwo positive parameters. The power function potential proves to beable to describe the atomic interactions of Ag, Au, Cu, Ni, Pd, and Ptmetals. Due to its simple form and excellent performance near thetruncation point, it greatly simplifies the fitting procedure andsimulations, while maintaining adequate precision.

3. Long-range n-body potential proposed for the transitionmetal systems

As mentioned above, although sharing some similar physicalarguments, the TB-SMA, Finnis-Sinclair and EAM potentials andtheir various modified ones are all empirical approaches and hencehave some limitations or drawbacks in applications. For example, ithas been shown by rule of thumb that the EAM and F-S potentialsare applicable to fcc and bcc metals, whereas the TB-SMA potentialis somehow suitable to fcc and hcp metals. Therefore, it becomesa challenging problem to describe the interatomic interactions inhcpebcc binary metal systems. Concerning this problem, someresearchers have attempted to construct the EAM or TB-SMApotentials for hcpebcc metal systems. However, these constructedpotentials are not adequately validated and have some problems.Another problem is related to the cutoff radius. For an interatomicpotential, the proper treatment at the cutoff radius, i.e. incorpo-rating truncation functions is needed. Otherwise, the energy andforce described by the interatomic potential might jump whenatoms go into or out of the cutoff radius, and a large number ofthese events can spoil the energy conservation, leading tounphysical behaviors in simulations. Besides, the stackingsequences of closed-package plans in fcc and hcp structures areABCABC. and ABAB., respectively. The atomic configurations inthe first and second neighbors of fcc and hcp structures are verysimilar. consequently, the energy difference between them isconsiderably small and can not be identified by the short-rangepotentials. In this case, these short-range potentials might notgive the correct results, especially when applying them to study therelative stabilities of fcc and hcp structures. In recent years, by

J.H. Li et al. / Intermetallics 31 (2012) 292e320 297

combining the features of the TB-SMA and F-S potentials, Li, Daiand Dai developed a new long-range n-body potential [45,49,50],hereafter referred to as the LDD potential for convenience. It hasbeen shown that the newly developed LDD potential is capable ofsolving the inherent problems of other empirical potentialsmentioned above.

The physical argument of the LDD potential is also based on thesecond moment approximation of the tight-binding theory.Accordingly, the potential energy of atom i in a condensed systemconsisting of N atoms, Ei, can be expressed by,

Ei ¼ 12

Xjsi

f�rij�þ FðriÞ; (20)

where

FðriÞ ¼ � ffiffiffiffiri

p; (21)

and

ri ¼Xjsi

j�rij�: (22)

here, rij is the distance between atoms i and j of the system. For thesake of convenience, the physical argument can be ignored hereand therefore f and j are named as the pair term and local densityfunction, respectively. The pair term and local density function arecomputed by:

f�r� ¼ ðr� rc1Þm

c0þc1rþc2r

2þc3r3þc4r

4; 0< r< rc1;

(23a)

j�r� ¼ aðr � rc2Þnexp

�� b

�rr0

� 1��

; 0 < r < rc2; (23b)

where m and n are two adjustable parameters. rc1 and rc2 are thecutoff radii of the pair function and local density function, respec-tively. It can be shown that f, j and their first derivatives smoothlyconverge to zero at cutoff radii ifm and n are both greater than 3.m,n, c0, c1, c2, c3, c4, a and b are potential parameters that are deter-mined by fitting to the experimental or calculated results. Since thecutoff radius of this empirical potential is greater than 6 Å, which isgreater than the sixth-neighbor distance, it can be considereda long-range empirical potential.

The pair function in the LDD potential inherits the feature of theoriginal F-S potential, while the density function exhibits somecharacteristics of the traditional TB-SMA potential. Thus, it cantherefore be considered that the LDD potential combines theunique features of both the TB-SMA and F-S potentials. In addition,the LDD potential has been proven to be suitable for bcc, fcc and hcptransition metals and their alloys. To simplify the units of the

Table 1Potential parameters for Fe, Mo, W, V, Nb and Ta [45].

Fe Mo W

m 4 4 4n 6 6 6rc1 (Ǻ) 4.54 4.70 4.80rc2 (Ǻ) 6.60 6.60 6.80x0 (eV Ǻ�m) 6.6763 29.9727 28.57x1 (eV Ǻ�m�1) �10.2119 �41.9333 �39.8x2 (eV Ǻ�m�2) 5.8932 21.9665 20.77x3 (eV Ǻ�m�3) �1.5190 �5.0972 �4.80x4 (eV Ǻ�m�4) 0.1475 0.4414 0.415r0 (Ǻ) 2.4855 2.7280 2.736b �1.5958 �0.1903 �1.33a (eV2 Ǻ�n) 3.4793E-4 1.4774E-3 1.715

potential parameters involved in Eqs. (23a and 23b), the pair termand local density function can be rewritten as:

f�r� ¼A

�rr0

� rc1r0

�m

�1þ c1

rr0

þ c2

�rr0

�2

þc3

�rr0

�3

þc4

�rr0

�4�;

0 < r < rc1; ð24aÞ

j�r� ¼ B

�rr0

� rc2r0

�n

exp��b

�rr0

�1��

; 0< r< rc2: (24b)

In some cases, the LDD potential also includes the smoothlytruncated TB-SMA potential expressed by Eqs. (12a and 12b) sincethey originate from the same physical argument and exhibit similarbehavior.

3.1. Long-range n-body potentials for the bcc metals Fe, Mo, W, V,Nb and Ta

For bcc transition metals, the LDD potential parameters weredetermined by fitting to the basic physical properties, e.g. thecohesive energy, lattice constants, elastic constants, and vacancyformation energy that are obtained by experiment, Table 1 showsthe fitted potential parameters in Eqs. (24a) and (24b) for the bccmetals Fe, Mo, W, V, Nb and Ta [45]. Once the potential parameterswere determined, the structures of these metals were then opti-mized using the fitted LDD potentials, i.e. fully relaxing the bccstructures to their lowest energy states. The fully relaxed structuresand corresponding physical properties, i.e. the lattice constants,cohesive energies, and elastic constants are referred to as thereproduced physical properties. It turns out that, for these bccmetals, the lattice constants, cohesive energies and elasticconstants reproduced from the LDD potentials are totally identicalwith those obtained by experiment.

To evaluate the relevance of the fitted LDD potentials, the latticeconstants and energies of the possible metastable structures, i.e.,the fcc and ideal hcp structures of the six bcc metals at equilibrium,were calculated using the LDD potentials. Table 2 shows the energydifferences between the bcc, fcc, and ideal hcp structures repro-duced from the potential for the six bcc metals. For comparison, theresults derived from ab initio calculations (VASP [94], CASTEP [95])and from the F-S and EAM potentials are also listed in Table 2[36,39]. The values of DEfcc/bcc and DEhcp/bcc predicted by theconstructed LDD potentials agree well with those derived from theab initio calculations. In addition, the results derived from both LDDpotentials and ab initio calculations indicate that the bcc structurehas the lowest potential energy, reflecting the fact that the equi-librium states of the six studied metals are all bcc structures. As for

V Nb Ta

4 4 46 6 64.65 5.02 5.0796.70 6.70 6.700

57 7.3956 8.5031 6.6182051 �10.6019 �11.1037 �8.747515 5.7298 5.4619 4.369566 �1.3743 �1.1905 �0.97148 0.1228 0.09645 0.080816 2.6241 2.8579 2.857973 1.8017 1.6823 0.11751E-3 8.2946E-4 2.9256E-3 3.1530E-3

Table 2Energy differences between bcc, fcc and hcp structures for Fe, Mo, W, V, Nb and Ta.The first and second rows are derived from the LDD potentials and ab initio calcu-lations, respectively (The unit of energy is meV/atom. c/a ¼ 1.633 for hcp) [45].

Fe Mo W V Nb Ta

DEfcc/bcc LDD 48 486 364 144 209 188Ab initio 124 423 478 255 326 243

DEhcp/bcc LDD 75 491 425 151 214 199Ab initio 62 460 563 292 351 314

DEhcp/fcc LDD 27 6 61 7 5 5Ab initio �61 37 84 36 24 71F-Sa 0 0 0 0 0 0EAMb 0 0 0 0 0 0

a [36].b [39].

Fig. 3. The repulsive term, attractive term, and the total energy of bcc Ta calculatedfrom the LDD potential and the corresponding Rose equation [45].

J.H. Li et al. / Intermetallics 31 (2012) 292e320298

the relative stability of the metastable fcc and ideal hcp structures,the values of DEfcc/hcp predicted by the F-S and EAM potentials arezero, indicating that they are not able to distinguish the energydifference between fcc and hcp structures. The energy differencesDEhcp/fcc for Mo, W, V, Nb and Ta reproduced from the LDDpotentials qualitatively match with those obtained from the abinitio calculations. The only exception is that Fe shows an oppositesign to ab initio calculation. A possible reason could be due to theferromagnetic properties of Fe.

The surface energies for the six bcc transition metals were alsocalculated using the LDD potentials. According to the results re-ported by Foiles [96], relaxation does not greatly affect the calcu-lated surface energies. Consequently, the surface energies werecalculated only for the un-relaxed surfaces of three low-index faces,i.e., (110), (100), and (111). The results calculated using the LDD,EAM and MEAM potentials [44,97], as well as the experimentalvalues [98e100], are listed in Table 3. For all the six metals, theenergy sequence of these surfaces is completely consistent with theresults obtained by the three potentials, i.e., the close-packed (110)has the lowest energy, followed by the (100) surfaces and then the(111) surfaces, which is in agreement with the experimentalobservations [99,100]. In general, for the (110) surfaces, the resultsderived from the LDD and EAM potentials are closer to the exper-imental values than those obtained from the MEAM potential. Forthe (100) surfaces, the results calculated from the MEAM and thosefrom the LDD potentials are slightly larger and smaller, respectively,than the experimental values. In contrast, the surface energiescalculated from the EAM potentials are significantly different fromthe experimental values. In summary, the surface energies of thesix bcc metals calculated from the LDD potentials agree well withthe experimental results, showing a significant improvement overthe short-range empirical potentials.

Table 3Surface energies (J/m2) of three low-index faces for bcc metals Fe, Mo, W, V, Nb, andTa [45].

Fe Mo W V Nb Ta

(110) LDD 1.717 2.229 2.967 1.676 1.792 1.909EAMa 1.535 2.127 2.599 1.683 1.807 1.800MEAMb 2.356 2.885 3.427 2.636 2.490 2.778Experimentc 1.683 2.019 2.275 1.827 1.859 2.019

(100) LDD 1.899 2.459 3.223 1.929 2.101 2.223EAMa 1.685 2.284 2.809 1.831 1.968 1.990MEAMb 2.510 3.130 3.900 2.778 2.715 3.035Experimentc 2.388 2.868 3.221 2.596 2.628 2.868

(111) LDD 2.076 2.803 3.628 2.161 2.343 2.480MEAMb 2.668 3.373 4.341 2.931 2.923 3.247

a [44].b [97].c [98e100].

Another approach to evaluate the relevance of the LDD potentialis to derive the equation of state (EOS) and compare the deriveresult with the EOS obtained from theory or experiment. As anexample, the EOS of bcc Ta was calculated from the LDD potentialand is shown in Fig. 3, in which the Rose equation is also given[101,102]. One sees from Fig. 3 that the repulsive term, attractiveterm, and the total energy calculated from the LDD potential are allsmooth in the entire calculated range. The calculated total energyalso agrees well with the Rose equation. Therefore, it can beconcluded that the LDD potential can adequately describe theinteratomic interactions of bcc Ta even in the state far from equi-librium. In addition, the derivative of total energy with respect tothe lattice constant was also derived from the LDD potential and isshown in Fig. 3. Clearly, the derivative curve of the LDD potential issmooth in the entire calculated range, indicating that the LDDpotential can avoid the unphysical behaviors that may appears inthe simulations with other potentials. Furthermore, thepressureevolume relationships of the six bcc metals were alsocalculated using the respective LDD potentials. The calculatedpressureevolume relationships and experimental observations[103e106] are shown in Fig. 4. It is found that the agreementbetween the calculated results and experimental values is good,indicating that the LDD potential is able to adequately describe thePeV relationship in the bcc metals.

In summary, the above discussion proves that the constructedlong-range LDD potentials can satisfactorily reproduce the prop-erties of the bcc metals, and it can be concluded that they canadequately describe the interatomic interactions of the bcc metals.Furthermore, unlike the MEAM potential, the LDD potential hasa simple analytic form and can be widely applied to the metals.

3.2. Long-range n-body potentials for the fcc metals Cu, Ag, Au, Ni,Pd and Pt

The proposed LDD potentials were also successfully applied tothe fcc Cu, Ag, Au, Ni, Pd, and Pt metals [49]. Table 4 shows that thefitted potential parameters of Eqs. (24a) and (24b) for the six fccmetals. Interestingly, for these fcc metals, the lattice constants,cohesive energies and elastic constants reproduced from the LDDpotentials are also identical with those obtained by experiment[107,108]. From the LDD potentials, the energies of the metastablestructures, i.e., bcc and ideal hcp structures of the six fcc metalswere further calculated. The bcc and hcp structures were optimizedand then the energies of the optimized structures were calculatedusing the LDD potentials. Table 5 shows the calculated energy

Fig. 4. PressureeVolume relationships derived from the LDD potentials (lines) and observed in experiments (open circles) for bcc metals (reproduced from [45]).

J.H. Li et al. / Intermetallics 31 (2012) 292e320 299

differences between the metastable and fcc structures, i.e.,DEfcc/bcc, DEfcc/hcp, DEfcc/sc, and DEfcc/dia, where sc and diadenote the simple cubic and diamond structures, respectively. Forcomparison, the results obtained from Cai’s EAM potential [69],experimental observations, and ab initio calculations are also listedin Table 5. The energy differences predicted by the LDD potentialsquantitatively agree with those obtained from experiments or abinitio calculations. Furthermore, the calculated results indicate thatthe fcc structure has the lowest potential energy among the fcc, bcc,hcp, simple cubic, and diamond structures, matching well the factthat the equilibrium states of six studied metals are all fcc struc-tures at equilibrium. In contrast, the values of DEfcc/bcc andDEfcc/hcp predicted by Cai’s EAM potential [69], are systematicallyunderestimated compared with the experimental results, espe-cially for DEhcp/fcc, and the maximum relative error is greater than94%. Apparently, the constructed LDD potential is more relevantthan Cai’s potential for predicting the structural stability of the fcctransition metals.

For a realistic potential model, it should not only satisfactorilypredict the structural stability of the metals but also it shouldcorrectly calculate the properties of the metastable structures.Therefore, based on the constructed LDD potential, we all

Table 4Potential parameters for Cu, Ag, Au, Ni, Pd and Pt [49].

Cu Ag Au

m 4 4 4n 6 6 6rc1 (Ǻ) 6.100 6.375 6rc2 (Ǻ) 7.800 7.950 8x0 (eV Ǻ�m) 0.123554 0.235139 0x1 (eV Ǻ�m�1) �0.134361 �0.247471 �0x2 (eV Ǻ�m�2) 0.0543818 0.0983304 0x3 (10�2 eV Ǻ�m�3) �0.981194 �1.748544 �2x4 (10�3 eV Ǻ�m�4) 0.675816 1.174278 1A (10�4 eV2 Ǻ�n) 0.656618 0.805877 1B 1.836569 2.951121 5r0 (Ǻ) 2.552655 2.892067 2

calculated the elastic moduli, and elastic constants of the meta-stable bcc and hcp structures of Cu, Ag, Au, Ni, Pd, and Pt. and theresults are also listed in Table 6. For comparison, the results of abinitio calculations are also listed in the table. The results calculatedfrom the LDD potential for all the bcc and hcp structures of Cu, Ag,Ni, and Pd are in good agreement with those obtained from ab initiocalculations, with the largest relative error being 2.02% for thelattice constants. For the hcp structures of Au and Pt, the agreementbetween the results calculated from both methods are also in goodagreement, with the largest relative error being 4.35%. Concerningthe elastic modulus and elastic constants derived from the LDDpotentials, although they have some departures from thoseacquired by ab initio calculations, most of the relative errors are lessthan 30%, exhibiting a reasonable agreement between the twomethods. In summary, taking into account the computation error ofthe ab initio scheme in elastic calculations, it can be concluded thatthe results derived from the LDD potentials agree well with thosefrom the ab initio calculations, suggesting that the constructed LDDpotential is relevant in predicting the lattice constants and elasticproperties of these metastable structures for the transition metals.

The ability to correctly predict the phonon dispersion curves isalso considered as a useful criterion to validate the reliability of an

Ni Pd Pt

4 4 46 6 6

.400 6.000 6.300 6.440

.500 7.500 7.800 8.500

.346813 0.173210 0.311702 0.362085

.350701 �0.198386 �0.333143 �0.378121

.133662 0.0840575 0.134071 0.149113

.273741 �1.596247 �2.413259 �2.610984

.456262 1.172505 1.642024 1.706176

.011750 0.811753 1.560181 2.678379

.580155 0.376885 4.472177 4.354578

.884996 2.489016 2.750645 2.771859

Table 5Energy differences betweens structures, DE, reproduced from the EAM and LDDpotentials and obtained ab initio calculations and experimental observations. sc anddia denote the simple cubic structure and diamond structure, respectively.c/a ¼ 1.633 for hcp.

Cu Ag Au Ni Pd Pt

DEfcc/bcc (meV/atom) LDD 31.8 47.0 49.8 60.8 80.1 108EAMa 22.0 25.4 26.5 33.0 37.0 43.0Expt.b 40.0 30.0 40.0 70.0 100 150

DEfcc/hcp (meV/atom) LDD 8.1 3.1 7.0 19.6 18.5 21.7EAMa 1.2 1.2 5.0 1.0 1.6 1.1Expt.b 6.0 3.0 5.0 15.0 20.0 20.0

DEfcc/sc (eV/atom) LDD 0.49 0.37 0.34 0.70 0.45 0.48Ab initio 0.46 0.36 0.22 0.64 0.52 0.51

DEfcc/dia (eV/atom) LDD 1.24 0.88 0.69 1.96 1.05 0.71Ab initio 1.04 0.84 0.75 1.20 1.22 1.18

a [69].b [42].

J.H. Li et al. / Intermetallics 31 (2012) 292e320300

empirical potential. Therefore, the phonon spectra of the six fccmetals were calculated using the constructed LDD potentials andare shown in Fig. 5. For comparison, the experimental results arealso plotted in this figure [109e114]. One can see from Fig. 5 thatthe phonon spectra calculated from the LDD potential are in goodagreement with the experimental observations, especially in therange of the long wavelengths. The agreement is good for Cu, Ag,and Pd, and fair for Au, Ni, and Pt, indicating that the LDD potentialcan adequately model the lattice dynamics behavior of these fccmetals.

From the discussion above, one can conclude that the con-structed LDD potential has successfully overcome the structuralstability problem in traditional short-range potentials, and satis-fyingly resolved the energy and force jumps in Cai’s EAM [69] andCleri’s TB-SMA potentials [35]. Furthermore, it also proves that theconstructed LDD potential can well reproduce the lattice constant,cohesive energy, elastic constants of stable and metastable struc-tures of these fcc metals. The derived phonon spectra are also quiteagreeable with the experimental observations or the resultsderived from ab initio calculations [49].

3.3. Long-range n-body potentials for the hcp metals Co, Hf, Mg, Re,Ti and Zr

The fcc and hcp structures are similar and hence have a verysmall energy difference. In this respect, some frequently usedinteratomic potentials fail to identify the energy differencebetween the fcc and hcp structures [45,49]. In order to show the

Table 6Lattice constants, a, and c, bulk moduli and elastic constants, B0 and Cij, of bcc and hcppotentials and acquired by ab initio calculations [49].

Cu Ag Au

bcc hcp bcc hcp bcc hc

a (Ǻ) 2.865 2.553 3.271 2.887 3.250 2.82.866 2.538 3.274 2.892 3.317 2.9

c (Ǻ) 4.176 4.781 4.74.204 4.807 4.8

B0 (Mbar) 1.282 0.921 0.979 0.733 1.650 1.11.366 1.499 1.387 0.999 1.378 1.4

C11 (Mbar) 1.284 2.261 0.811 1.653 1.477 2.30.828 2.624 0.600 1.757 1.169 2.2

C12 (Mbar) 1.281 1.040 1.063 0.863 1.736 1.61.635 1.045 1.186 0.781 1.482 1.2

C13 (Mbar) 0.787 0.747 1.50.837 0.542 0.9

C33 (Mbar) 2.301 1.876 2.42.806 1.775 2.1

C44 (Mbar) 0.850 0.357 0.616 0.278 0.527 0.20.892 0.308 0.541 0.269 0.557 0.2

applicability of the proposed long-range n-body potential to thehcp metals, we tried to construct the LDD potential for the hcpmetals Co, Hf, Mg, Re, Ti, and Zr [50]. In fitting the potential, it wasinterestingly found that the parameter b in Eq. (23b) was nearlyzero for these hcp metals. To simplify the form of the potential andto reduce the computation time, the parameter bwas set to be zerofor these metals. Thus, the atomic density function can also bewritten as

f�r� ¼ aðrc2 � rÞn: (25)

Table 7 lists the fitted potential parameters for the six hcpmetals. Tables 8e10 list the reproduced physical properties of thebcc and fcc metastable structures and hcp stable structure of the sixmetals, respectively. One notes the physical properties of bcc andfcc metastable structures of the six metals were also derived by abinitio calculations. It can be seen that the lattice constants, cohesiveenergies, and bulk moduli calculated from the LDD potential agreewell with the results of experiments [107,108,115] and ab initiocalculations. The elastic constants derived by the LDD potentials aregenerally consistent with those obtained from experiment and abinitio calculations, although the agreement is not as good as that ofthe lattice constants and the cohesive energies. Considering thelimitations of the spherical symmetry of the LDD potential and theerrors involved in the ab initio calculations and experiments, thediscrepancy is acceptable. In addition, the LDD potentials preciselyreproduce the experimental c/a ratio for the six selected metals,indicating that the LDD potential can also be used to calculate someanisotropic properties of hcp metals.

For an empirical potential, it is important to ensure that nounphysical structural instabilities occur in the atomistic simula-tions. It is therefore necessary to derive the corresponding physicalproperties of metastable structures from the constructed LDDpotentials. Table 11 shows the calculated energy differencesbetween various possible metastable structures and the stable hcpstructure. For comparison, the results obtained from experimentalobservations and ab initio calculations are also listed in this table.The energy differences predicted by the LDD potential quantita-tively agree with those obtained by experiments and ab initiocalculations. Both the calculated and experimental results indicatethat the hcp structure has the lowest potential energy among thehcp, ideal hcp, fcc, bcc, sc and diamond structures, agreeing withthat the equilibrium states of the six metals are all hcp structures.The energy differences DEhcp/ideal hcp predicted by the LDDpotentials are in agreement with other works, especially with the

Cu, Ag, Au, Ni, Pd and Pt. The first and second rows are reproduced from the LDD

Ni Pd Pt

p bcc Hcp bcc hcp bcc hcp

74 2.795 2.485 3.102 2.742 3.110 2.76525 2.811 2.491 3.114 2.745 3.163 2.76571 4.075 4.555 4.55190 4.128 4.607 4.75888 1.505 1.193 1.648 1.271 2.516 1.80338 2.010 1.916 1.801 1.778 2.474 2.55586 1.527 3.149 1.361 2.791 2.251 3.80876 1.444 2.973 1.688 2.522 1.811 4.27442 2.251 1.256 1.792 1.642 2.648 2.44700 1.535 1.398 1.858 1.706 2.805 1.97914 0.823 1.414 2.21185 1.312 1.240 1.72893 3.077 2.762 3.47720 3.266 2.650 3.74844 1.188 0.513 0.772 0.345 0.908 0.44250 1.229 0.486 0.908 0.298 1.589 0.159

Fig. 5. Phonon spectra of the 6 fcc metals calculated from LDD potentials and observed in experiments [49].

J.H. Li et al. / Intermetallics 31 (2012) 292e320 301

AMEAM. The DEhcp/fcc and DEhcp/bcc predicted by LDD and MEAMpotentials [74] agree quantitatively with each other, while thevalues calculated from the AMEAM [116] are systematicallyunderestimated compared to the experimental results. Apparently,the LDD potential is more appropriate than the AMEAM potentialfor predicting the structural stability of the hcp transitionmetals. Insummary, these results indicate that the constructed LDD potentialcan reasonably predict the structural stability and can distinguishthe energy difference between the hcp and other structures.

To further evaluate the relevance of the constructed LDDpotential, the EOS of the six hcp metals were derived from thepotential and compared with the Rose equation [101,102]. The

Table 7Potential parameters for Co, Hf, Mg, Re, Ti, and Zr [50].

Co Hf Mg

m 4 4 4n 4 4 4rc1 (Å) 5.009038 6.249619 6.540rc2 (Å) 6.653700 7.288037 7.242c0 (eV) 2.092386 1.566590 0.326c1 (eV/Åmþ1) �2.809616 �1.719958 �0.349c2 (eV/Åmþ2) 1.407987 0.706121 0.139c3 (eV/Åmþ3) �0.310658 �0.127821 �0.024c4 (eV/Åmþ4) 0.025399 0.008576 0.001a1 (eV/Ån) 0.011414 0.030168 0.000

potential derived EOS and Rose equation are shown in Fig. 6. Onesees that there is no discontinuity of the curves in the entire range.In particular, at the cutoff radius, the energies and their derivativesconverge continuously and smoothly to zero. The total energyderived from the LDD potential is very close to that predicted by theRose equation in the vicinity of the equilibrium, indicating that theconstructed LDD potential can be applied to describe the inter-atomic interaction of the system only if the system is not very farfrom equilibrium.

The constructed LDD potentials were also applied to calculatethe formation energies of the structural defects, such as vacancies,stacking faults [50]. The calculated formation energies of vacancies

Re Ti Zr

4 4 44 4 4

678 5.429741 5.811177 6.437549303 7.345622 7.281491 7.222787862 4.036224 1.513990 0.788297058 �4.929920 �1.794443 �0.846083461 2.241564 0.795274 0.341507678 �0.448298 �0.155492 �0.061097626 0.033268 0.011279 0.004081876 0.028328 0.011887 0.036094

Table 8Lattice constants, a, cohesive energies EC, bulk moduli, B0, and elastic constants, Cij,obtained from potential and ab initio calculations (first rows and second rows) forbcc Co, Hf, Mg, Re, Ti, and Zr [50].

Co Hf Mg Re Ti Zr

a (Å) 2.51 3.19 3.21 2.76 2.95 3.232.51 3.19 3.21 2.76 2.95 3.23

c (Å) 4.07 5.05 5.21 4.46 4.68 5.154.07 5.05 5.21 4.46 4.68 5.15

EC (eV) 4.39 6.44 1.51 8.03 4.85 6.254.39 6.44 1.51 8.03 4.85 6.25

C11 (MBar) 2.813 1.387 0.484 5.747 1.312 1.2143.071 1.881 0.595 6.182 1.624 1.434

C12 (MBar) 1.149 0.650 0.191 2.223 0.593 0.5951.650 0.772 0.261 2.753 0.920 0.728

C13 (MBar) 1.288 1.017 0.250 2.428 0.940 0.7841.027 0.660 0.218 2.078 0.690 0.653

C33 (MBar) 4.100 2.757 0.955 7.606 2.700 2.1933.581 1.969 0.616 6.835 1.807 1.648

C44 (MBar) 0.970 0.734 0.205 1.965 0.705 0.4990.755 0.557 0.164 1.606 0.467 0.320

B0 (MBar) 1.909 1.211 0.367 3.695 1.141 0.9941.903 1.102 0.356 3.669 1.073 0.954

Table 10Lattice constants, a, cohesive energies EC, bulk moduli, B0, and elastic constants, Cij,obtained from potential and in experiments (first rows and second rows[107,108,115]) for hcp Co, Hf, Mg, Re, Ti, and Zr [50].

Co Hf Mg Re Ti Zr

a (Å) 3.542 4.471 4.521 3.895 4.136 4.5373.520 4.475 4.524 3.918 4.100 4.527

EC (eV) 4.376 6.406 1.496 8.003 4.820 6.2204.370 6.372 1.494 7.965 4.794 6.213

C11 (MBar) 2.544 1.394 0.434 5.141 1.278 1.0292.919 1.487 0.321 5.540 1.421 0.895

C12 (MBar) 1.269 0.708 0.230 2.486 0.657 0.6701.709 0.811 0.320 2.744 0.941 0.902

C44 (MBar) 0.951 0.428 0.185 2.024 0.424 0.3861.480 0.629 0.188 2.066 0.558 0.277

B0 (MBar) 1.694 0.937 0.298 3.371 0.864 0.7902.113 1.036 0.320 3.676 1.101 0.900

Table 11The structural energy differences DE (eV/atom) obtained by the n-body potentials,MEAM potential, and AMEAM potential, ab initio calculations and experimentalobservations [50].

J.H. Li et al. / Intermetallics 31 (2012) 292e320302

and divacancies and activation energies of self-diffusion byvacancies are all in good agreement with the values obtained inexperiments and other works. The calculated surface energies andstacking fault energies are also consistent with the experimentaldata and other theoretical works, showing that the constructedLDD potentials could well describe the interatomic interaction ofthese hcp metals.

From the discussion presented in this section, it can beconcluded that the proposed LDD potential is of relevance indescribing the interaction of bcc, fcc and hcp metals, thus providinga simple way to the construction the n-body potentials for thetransition metal systems with any combinations of the bcc, fcc andhcp structuredmetals. In particular, the LDD potential can deal withternary transition metal systems with three different structures.The following section will discuss the LDD potentials constructedfor some ternary metal systems.

3.4. Long-range n-body potentials of some ternary metal systems

The interatomic interactions in the AeB binarymetal system canbe classed into three types: AeA, BeB and AeB interactions. Thefirst two are the interactions between the same type of atom andcan be described by the interatomic potential constructed by fittingto the physical properties of the pure metals. The third is the

Table 9Lattice constants, a, cohesive energies EC, bulk moduli, B0, and elastic constants, Cij,obtained from potential and ab initio calculations (first rows and second rows) for fccCo, Hf, Mg, Re, Ti, and Zr [50].

Co Hf Mg Re Ti Zr

a (Å) 2.836 3.581 3.639 3.111 3.311 3.6152.809 3.540 3.580 3.117 3.248 3.570

EC (eV) 4.285 6.307 1.480 7.734 4.756 6.1884.284 6.266 1.478 7.715 4.744 6.179

C11 (MBar) 1.364 0.848 0.369 2.377 0.914 1.0621.538 0.663 0.216 3.639 0.841 1.326

C12 (MBar) 1.886 1.064 0.375 3.572 1.056 1.0112.186 1.201 0.382 3.557 1.228 0.799

C44 (MBar) 1.579 0.792 0.332 3.124 0.829 0.7311.259 0.513 0.322 1.800 0.330 0.635

B0 (MBar) 1.712 0.992 0.373 3.174 1.009 1.0281.970 1.021 0.327 3.585 1.099 0.974

interaction between different atoms, and it is called the crosspotential. The parameters of the cross potential can be determinedby fitting to the physical properties of the binary compounds. Ofcourse, the AeA and BeB potential are requisite in fitting the AeBcross potential since there are also AeA and BeB interactions in thebinary system. For a ternary metal system, there are three crosspotentials to describe the AeB, AeC and BeC interactions. Ingeneral, to simplify the complexity of the fitting procedure, somebinary compounds are selected for fitting the cross potentials ofa ternary system.

Differing from the case of a pure metal whose physical proper-ties are usually available, the properties of the compounds arefrequently scarce and thus it is a challenging task to construct thecross potential, especially for an immiscible system, since there isfrequently no any equilibrium compound existing in these system.In order to overcome this problem, ab initio calculations are carriedout to acquire some basic physical properties of equilibrium and/orhypothetical compounds such as the B2, L12, D019 structures. The abinitio calculated properties are then used in fitting the interatomicpotential. This method may be referred as the ab initio assistedconstructing interatomic potential [20]. Employing themethod, theproposed long rang LDD potentials were constructed for somebinary and ternary systems consisting of bcc, fcc and hcp transitionmetals.

Potentials Co Hf Mg Re Ti Zr

DEhcp/ideal hcp

(meV/atom)LDD 0.7 5.8 0.3 2.4 3.6 3.4MEAMa 0 5.9 0 0.5 7.1 10.5AMEAMb 0 6.8 0.1 1.8 4.1 2.9

DEhcp/fcc

(meV/atom)LDD 14.8 35.3 13.9 27.7 29.9 30.2ab initio 19.6 68.1 16.1 65.2 56.5 36.7MEAMa 0.5 53 4. 31.0 33. 17.AMEAMb 6.2 6.8 2.2 5.00 9.4 4.9Expt.a 100 26. 110. 60. 76.

DEhcp/bcc

(meV/atom)LDD 106 134 29.8 297 94.1 63.1ab initio 106 174 32.4 315 106 70.7MEAM 241 64 29 303 75 61AMEAM 16.4 30.1 17.6 164 14.3 17.9Expt.a 59 31 292 70 76

DEhcp/diamond

(eV/atom)LDD 1.07 1.67 0.64 1.89 1.27 1.62MEAMa 1.20 2.19 0.31 2.39 1.57 1.48

DEhcp/sc

(eV/atom)LDD 0.26 0.17 0.15 0.60 0.16 0.38MEAMa 0.59 0.51 0.13 0.94 0.41 0.45

a [74].b [116].

Fig. 6. LDD potentials and Rose equations for 6 hcp metals [50].

Table 13Lattice constants a, cohesive energies EC, elastic constants and bulk moduli B0derived from LDD potentials (first rows) and ab initio calculations (second rows).

Compounds a (Å) EC (eV) C11 (MBar) C12 (MBar) C44 (MBar) B0 (MBar)

L12 Ag3Cu 4.007 2.976 1.256 1.006 0.448 1.0894.009 3.009 1.274 0.992 0.603 1.086

L12 AgCu3 3.756 3.273 1.472 1.182 0.593 1.2793.755 3.264 1.500 1.170 0.676 1.280

B2 AgCu 3.083 3.111 1.207 1.114 0.478 1.1453.083 3.095 0.965 1.240 0.315 1.148

D03 Ag3Cu 6.399 2.925 0.856 1.081 0.536 1.0066.378 2.969 0.973 1.092 0.399 1.052

D03 AgCu3 5.987 3.221 1.072 1.231 0.637 1.1785.969 3.224 1.061 1.352 0.602 1.255

L12 Cu3Au 3.772 3.706 1.718 1.319 0.529 1.452

J.H. Li et al. / Intermetallics 31 (2012) 292e320 303

3.4.1. Long-range n-body potentials for the AgeAueCu systemWe take the AgeAueCu system as an example to show the LDD

potential constructed for a ternary metal system. The threeconstituent metals of the AgeAueCu system are all fcc structures inequilibrium state. The phase diagrams of the AgeAu and AgeCusystems are isomorphous and eutectic, respectively. No equilib-rium compound can be found in the two systems. The AueCusystem is also isomorphous but disordereorder phase transitionscan be observed at low temperatures. The disordereorder phasetransitions result in the three ordered intermetallics, L12 Au3Cu,AuCu3 and B2 AuCu compounds. Since there is a lack of availablephysical properties of compounds in the system, ab initio calcula-tion were carried out to acquire the lattice constants, cohesiveenergies, and elastic moduli of the L12 A3B and AB3, B2 ABcompounds, where A and B represent any two elements of Ag, Auand Cu. The calculated physical properties were then applied to fitthe AgeCu, CueAu, and AgeAu cross potentials. The fittedparameters of AgeCu, CueAu, and AgeAu cross potentials in Eqs.(23a) and (23b) are listed in Table 12.

After obtaining the interatomic potential of the AgeAueCusystem, the lattice constants, cohesive energies, elastic constantsand bulk moduli of some binary compounds in the system werereproduced from the constructed LDD potential. Table 13 shows thepotential reproduced and ab initio calculated properties. One cansee from the table that the reproduced lattice constants are in goodagreement with those acquired by ab initio calculations. The errors

Table 12Potential parameters for AgeCu, CueAu and AueAg interactions [49].

AgeCu CueAu AueAg

m 4 4 4n 6 6 6rc1 (Ǻ) 6.000000 6.000000 5.706344rc2 (Ǻ) 7.500000 7.800000 6.841334c0 (eV Ǻ�m) 0.233892 0.292061 0.468608c1 (eV Ǻ�m�1) �0.249618 �0.311393 �0.460405c2 (eV Ǻ�m�2) 0.101643 0.127695 0.175479c3 (10�2 eV Ǻ�m�3) �1.859480 �2.336064 �2.993292c4 (10�3 eV Ǻ�m�4) 1.284024 1.602712 1.885116a (10�3 eV2 Ǻ�n) 0.204474 0.379239 1.485145b 2.963432 2.915843 3.122075r0 (Ǻ) 2.646736 2.671407 2.816507

between the reproduced and ab initio calculated cohesive energiesof these compounds are all less than 6%, suggesting that theagreement between the two methods is also good. For the elasticconstants, the agreement of C12 is the best with a relative error lessthat 15%. The relative errors for C11 and C44 are less than 31% and59%, respectively. Considering the computational error of the abinitio method in calculating elastic properties, it can be concludedthat all the reproduced results are in good agreement with thosedetermined by ab initio calculations, suggesting that the con-structed LDD potential is appropriate for calculating the latticeconstants and cohesive energies of fccefcc alloys.

3.771 3.593 1.813 1.285 0.605 1.461L12 CuAu3 4.046 3.562 1.558 1.390 0.318 1.446

4.054 3.729 1.630 1.341 0.425 1.437B2 CuAu 3.103 3.693 1.747 1.255 0.297 1.419

3.103 3.680 1.336 1.471 0.439 1.426D03 Cu3Au 6.008 3.623 1.140 1.422 0.583 1.328

6.002 3.547 1.391 1.499 0.698 1.463D03 CuAu3 6.449 3.492 1.187 1.428 0.378 1.348

6.454 3.686 1.091 1.511 0.327 1.371L12 Ag3Au 4.125 3.284 1.268 1.035 0.216 1.112

4.123 3.217 1.297 1.029 0.525 1.118L12 AgAu3 4.145 3.541 1.444 1.325 0.194 1.365

4.149 3.642 1.610 1.230 0.472 1.357B2 AgAu 3.296 3.408 1.152 1.130 0.101 1.138

3.296 3.400 0.910 1.251 0.239 1.137D03 Ag3Au 6.599 3.210 0.837 1.119 0.286 1.025

6.568 3.177 1.048 1.078 0.508 1.068D03 AgAu3 6.616 3.478 1.123 1.338 0.232 1.267

6.608 3.601 1.010 1.397 0.524 1.268

J.H. Li et al. / Intermetallics 31 (2012) 292e320304

As mentioned previously, the EOS can be used to evaluatewhether a potential can correctly predict the properties of an alloyin the far from equilibrium state. We therefore calculated the totalenergies as a function of lattice constant for some AgeCu, CueAu,and AgeAu compounds based on the constructed LDD potentials,and the results are shown in Fig. 7. For comparison, the corre-sponding results derived from the Rose equation [101] are alsoshown in this figure. The total energies calculated from the LDDpotentials are in good agreement with the Rose equation in theentire range, indicating that the constructed LDD potentials cansatisfactorily describe the energy state of an fccefcc compoundseven in the far from equilibrium state. Moreover, the derivatives ofthe total energies calculated from the LDD potentials and the Roseequation are also shown in Fig. 7. The agreement between thederivatives derived from the LDD potential and Rose equation isexcellent, indicating the LDD potential can reasonably reflect theatomic interactions in these fccefcc compounds.

To further verify the relevance of the constructed potential, thelattice constants, cohesive energies and bulk moduli of four hypo-thetic AgeAueCu ternary structures, Ag2AuCu, AgAu2Cu, AgAuCu2and AgAuCu, were reproduced from the LDD potentials andcompared with those acquired by ab initio calculation (shown inTable 14). The physical properties obtained by the two methods arequite consistent with each other, suggesting that the LDD potentialconstructed by fitting to the properties of the binary compoundscould satisfyingly reflect the chemical complexity, i.e. it could stillreasonably describe the interaction of the ternary system.

In fact, apart from the AgeAueCu system, the proposed LDDpotentials were also successfully constructed for 15 binary systemsand further 3 ternary systems consisting of the fcc metals Ag, Au,Cu, Ni, Pd and Pt [118]. Appling these constructed LDD potentials,the vitrification temperature of the AgeCueNi melt and the re-crystallization temperature of the AgeCueNi amorphous alloywere predicted through molecular dynamics simulations. It isfound that the predicted results are in agreement with experi-mental observations. In addition, the LDD potentials formulized byEqs. (23a) and (25) were also successfully constructed for the sixbinary and four ternary systems consisting of the bcc metals Mo,Nb, Ta, and W [119]. Based on the successful application of the LDDpotentials to the fcc, bcc metals and their binary and ternary

Fig. 7. LDD potentials and their derivates and Rose equations for A

systems, it can be concluded that the LDD potential works well forbinary and ternary systems consisting of fcc or bcc metals.

3.4.2. Long-range n-body potential for the NieNbeTa systemIn this subsection, we will discuss the LDD potentials con-

structed for the NieNbeTa system, which consists of fcc and bccmetals. Nb and Ta are bcc metals and their phase diagram exhibitsan isomorphous behavior. There is no equilibrium compoundobserved in this system. Although there are two and five equilib-rium compounds observed in the NieNb and NieTa phase diagrams[120], respectively, the structure parameters and the physicalproperties of these equilibrium compounds are not adequate to fitthe NieNb and NieTa cross potential. We have therefore carried outab initio calculations to acquire some physical properties to fit theLDD potentials of the NieNbeTa system.

The potential parameters of the NieNbeTa system obtained byfitting the ab initio acquiredphysical properties are listed inTable 15.One notes that the LDD potentials expressed by Eqs. (23a) and (25)were also constructed for the Ni, Nb and Ta metal. The potentialreproduced and ab initio calculated physical properties are listed inTables 16e18 [122e127]. In fitting the NbeNi, NbeTa and NieTacross potentials, some physical properties of stable and hypothesiscompounds were used. These compounds include: B2 NiNb, NiTaand NbTa, D0a Ni3Nb and Ni3Ta, D019 Ni3Nb, NiNb3 and Ni3Ta, D022Ni3Nb and Ni3Ta, C11b Ni2Ta, C16 NiTa2, L12 Ni3Nb, NiNb3, Ni3Ta,Nb3Ta andNbTa3. They cover a large range of chemical compositionsand various structural characteristics. Comparing with experi-mental data and the ab initio calculations, themaximumerror of thepotential derived cohesive energies and lattice constants is less than2%, showing that the constructed LDD potentials reproduce wellthese physical properties of the metallic compounds in the system.Fig. 8 shows the potential energies as functions of the latticeconstants compared with the Rose equations for B2 NiNb, NiTa, andNbTa, L12 NiNb3, Ni3Nb, Ni3Ta, NiTa3, Nb3Ta and NbTa3 compounds.The energies and derivatives of the three metals and theircompounds derived by the LDD potentials are continuous andsmooth in the entire range, i.e. no discontinuities, avoidingunphysical behavior in atomistic simulations. From the abovediscussion, it can be concluded that the constructed LDD potentialsare appropriate for describing the atomic interactions in the system.

geCu, CueAu and AgAu compounds (reproduced from [49]).

Table 14Lattice constants, a, cohesive energies, EC, elastic constants, Cij, bulk moduli, B0, andshear modulus, C0 of the ternary AgeCueNi compounds obtained from LDDpotentials (first rows) and ab initio calculations (second rows) [117].

a (Å) Ec (eV/atom) B (Mbr)

Au2CuAg 6.441 3.509 1.196.414 3.497 1.26

Cu2AgAu 6.236 3.443 1.266.181 3.414 1.30

Ag2CuAu 6.426 3.327 1.106.413 3.226 1.13

AgCuAu 6.134 3.114 0.736.153 3.032 0.81

J.H. Li et al. / Intermetallics 31 (2012) 292e320 305

To further verify the relevance of the constructed LDD poten-tials, the properties of some ternary compounds were also repro-duced from the potentials and compared with the results of abinitio calculations. Table 19 shows the potential reproduced latticeconstants and cohesive energies of L21 Ni2NbTa, NiNbTa2 andNiNb2Ta, and C1b NiNbTa, NiTaNb and TaNbNi. The correspondingphysical properties obtained by ab initio calculations are also listedin the table. It can be seen that the maximum discrepancies of theproperties obtained by the two methods are all less than 5%. Thismeans that the constructed LDD potentials also predict well theenergies and the structures of the ternary compounds andreasonably describe the interatomic interaction in the NieNbeTaternary system.

3.4.3. Long-range n-body potentials for NieZreAg, NieZreAl andCueZreAl systems

The LDD potentials have been successfully constructed for abouta dozen of ternary systems among which are NieZreAg [128],NieZreAl [129], CueZreNi [130], CueZreAl [131] and CueHfeAl[132]. In this subsection, we will discuss the LDD potentials forthe AgeNieZr, NieZreAl and CueZreAl systems consisting of fccand hcp metals.

For the AgeNieZr ternary system, the LDD potential formulizedby Eqs. (23a) and (25) was constructed with the aid of ab initiocalculations and the fitted LDD potential parameters are shown inTable 20 [128]. To evaluate the relevance of the constructed LDDpotential, the equations of state of these metals and some hypo-thetical compounds were derived from the constructed LDDpotentials and compared with the Rose equations [101,102]. Fig. 9shows the reproduced equations of state and the correspondingRose equations of compounds B2 AgZr and NiZr, L12 Ag3Zr, AgZr3,Ni3Zr and NiZr3. It can be seen that the equations of state obtainedby the two methods are in good agreement with each other, sug-gesting that the constructed LDD potential is appropriate fordescribing the atomic interactions in the AgeNieZr system.

The next example is the NieZreAl system, which also consists offcc and hcp metals. In recent years, the NieZreAl system has

Table 15Potential parameters for the NieNbeTa system [121].

Nb Ni Ta

m 4 4 4n 8 5 5rc1 (Å) 4.8425 6.0607 4.8rc2 (Å) 6.8341 7.3000 6.1c0 (eV) 7.5899 0.1587 7.6c1 (eV/Åmþ1) �10.2173 �0.1845 �10.5c2 (eV/Åmþ2) 5.1983 0.0783 5.4c3 (eV/Åmþ3) �1.1710 �0.0147 �1.2c4 (eV/Åmþ4) 0.09787 0.00107 0.1a (eV/Ån) 0.0002591 0.001145 0.0

attracted considerable attention since the NieZreAl ternaryamorphous alloys were developed by Inoue et al. [133] Theoreti-cally, a realistic interatomic potential is a prerequisite to clarify, atatomistic scale, the formation, structure and related physicalproperties of the NieZreAl amorphous alloy. Consequently, theLDD potential formulized by Eqs. (12a) and (12b) were constructed[129]. It should be noted that the NieZr, NieAl and ZreAl crosspotentials and the Ni, Zr and Al potentials were all adopt the sameformula. Similarly, the potential parameters of the NieZreAlsystem were determined by fitting to the related physical proper-ties obtained by experiments or ab initio calculations. In the fittingprocess, the compoundswith different structures and compositionswere included to ensure that the constructed LDD potential candescribe the interatomic interactions in the entire compositionrange and in different chemical environments. The fitted potentialparameters of the NieZreAl system are shown in Table 21. Thecompounds involved in the fitting procedure are of differentsymmetries and shown in Tables 22e24. For example, D019, B82,and D88 are hexagonal, C16 and D023 are tetragonal, Bf and Pt5Ga3-type are orthorhombic, and C15 is cubic [124]. For the potentialreproduced cohesive energies and lattice constants of these NieZr,ZreAl and NieAl binary compounds, the maximum error is lessthan 3% compared with the experimental data and the ab initiocalculations. This means that the LDD potential is appropriate fordescribing the atomic interactions in these binary systems. To showthe relevance of the constructed NieZreAl potential, the repro-duced and ab initio calculated or experimentally obtained latticeconstants, cohesion energies, elastic constants of their compoundswith different structures are also listed in these tables. One notesthat the physical properties of three experimentally observedternary phases [146], i.e., s1-NiZrAl, s2-Ni2ZrAl and s3-Ni2ZrAl5 andtwo hypothetical ternary phases i.e. L21 NiZr2Al, and NiZrAl2 werederived and compared with the experimental and ab initio calcu-lated results. The maximum error of the cohesive energies andlattice constants between the potential reproduced and ab initiocalculated results is less than 3% [see Table 25]. This shows that theconstructed LDD potential predicts well the energy and the struc-ture of the ternary compounds and reasonably describes theinteratomic interactions of the ternary system.

The last ternary example is the CueZreAl system. The CueZreAlalloy is expected to be a promising candidate for mass-producedbulk metallic glasses. Wide and extensive studies have beencarried out for the CueZreAl system. Atomistic simulations can notonly provide guidance to help design the alloy with desiredstructures and properties, but can also clarify the physical originsand mechanism of the formation, as well as the behavior of theamorphous CueZreAl alloy. The LDD potential formulized by Eqs.(12a) and (12b) was therefore constructed for the CueZreAl system[131] with the aid of ab initio calculations. One notes that theequilibrium binary compounds observed in the diagrams wereused to fit the cross potentials of the system. The fitted potential

NbeNi NieTa NbeTa

4 4 46 6 6

458 5.2901 4.7830 5.3829078 5.6000 5.6751 6.5028678 1.2346 9.1014 2.2036138 �1.6148 �11.3792 �2.8240684 0.8130 5.2563 1.3794668 �0.1828 �1.0489 �0.30000991 0.01529 0.07527 0.024275430 0.02269 0.02625 0.007502

Table 18Lattice constants (a), cohesive energies (EC), and bulk modulus (B0) of NieTacompounds obtained from the LDD potentials (first rows), ab initio calculations(second rows), other works and experiments (third rows and fourth rows[122e127]) [121].

Ni3Ta Ni3Ta Ni2Ta NiTa NiTa2

L12 D0a D019 D022 C11b B2 C16

a, b, and/orc (Å)

3.66 5.14,4.27,4.52

5.16,4.27

3.63,7.48

3.13,7.97

3.08 6.12,4.83

3.70 3.083.70 5.20,

4.245.10,4.24,4.52

3.62,7.45

3.15,7.91

6.20,4.86

EC (eV) 5.733 5.756 5.707 5.773 6.065 6.246 7.1625.567 6.3605.597 5.672 5.745 5.734 6.060 7.210

B0 (MBar) 2.908 3.104 2.898 3.185 3.473 2.352 2.4322.048 2.1362.690 2.600

Table 16Lattice constants, a, cohesive energies, EC, and bulk moduli, B0, of NbeNi compoundsobtained from LDD potentials (first rows), ab initio calculations (second rows), otherpredictions and experiments (third rows and fourth rows [122e124]) [121].

Nb3Ni NbNi NbNi3

L12 D019 B2 L12 D0a D019 D022

a, b, and/orc (Å)

4.07 5.76,4.68

3.11 3.66 5.13,4.27,4.51

5.16,4.27

3.62,7.47

4.05 5.75,4.69

3.10 3.70 5.25,4.25

4.12 3.725.11,4.24,4.54

3.62,7.41

EC (eV) 6.704 6.695 6.133 5.523 5.541 5.523 5.5406.707 6.648 6.047 5.376 5.4786.724 6.711 5.439 5.586

B0 (MBar) 1.01 0.98 1.41 1.37 1.59 1.46 1.551.77 1.902.23 2.57 2.17 2.54 2.57

J.H. Li et al. / Intermetallics 31 (2012) 292e320306

parameters of the CueZreAl system are listed in Table 26. In orderto show the relevance of the constructed LDD potential, Table 27shows the fitted or reproduced and experimental or ab initiocalculated physical properties of the compounds [148,149]. Thepotential reproduced properties agree well with the experimentalproperties, indicating that the constructed LDD potential predictswell the energies and the structures of these ternary compounds. Itshould be mentioned that, the CueHfeAl [132], NieCoeZr [150],and NieZreTi [151] systems are also consisting of fcc and hcpmetals. The LDD potentials were also successfully constructed forthese systems, for which the reader can be referred to the literature.Based on the discussion above, it can be convincingly concludedthat the proposed LDD potentials can be applied to the binary andternary systems consisting of fcc and hcp metals.

As mentioned previously, it is once a challenge to constructa realistic potential for the hcp-bcc metal system. However, theproblem has been solved by using the proposed LDD potential. Takethe ZreNb system as an example, one may firstly construct the LDDpotential formulized either by Eqs. (12a and 12b) or Eqs. (23a and23b) for hcp Zr and then construct the LDD potential formulizedby Eqs. (23a and 23b) or Eq. (25) for the bcc Nb. The interatomicpotentials for some transition metals either bcc, fcc or hcp struc-tured have been constructed and are discussed in previous sections.For the cross potential of the ZreNb system, a similar fittingprocedure is carried out to fitting the potential parameter and thenthe LDD potential of the ZreNb system were obtained. If required,ab initio calculation was also performed to acquire the physicalproperties that were used in fitting process. In the same way, theLDD potential was constructed for the HfeTa system.

For the ZreNb and HfeTa systems, Table 28 shows the repro-duced and ab initio calculated physical properties of the

Table 17Lattice constants, a, cohesive energies, EC, and bulk moduli, B0, of NbeTa compoundsobtained from LDD potentials (first rows) and ab initio calculations (second rows)[54].

Nb3Ta NbTa NbTa3

L12 B2 L12

a (Å) 4.25 3.30 4.274.23 3.32 4.23

EC (eV) 7.531 7.838 7.8367.396 7.837 7.799

B0 (MBar) 1.076 1.773 1.2001.727 1.776 1.855

compounds that were involved in fitting process [52]. One seesfrom shows that the fitted physical properties agree well with abinitio calculations. Fig. 10 shows the potential derived EOS for theL12 Zr3Nb and HfTa3 compounds. For the potential energies as wellas their derivates, there are no discontinuities in the entire calcu-lated range. The results derived from the LDD potentials are smoothand agree well with those calculated using the Rose equations inentire range, indicating that the constructed LDD potential canreasonably describe the interatomic interaction of the system evenin far away from equilibrium state. Furthermore, to prove theapplicability and validation of the constructed LDD potentials,molecular dynamics and Monte Carlo simulations were carried outto investigate the interfacial stability between the two differentmetals and the solid-state amorphization in the metalemetalmultilayers. The simulation results are comparable with experi-mental observations. It again confirms that the proposed LDDpotentials are applicable to binary and ternary systems that consistof bcc, fcc and hcp metals. The LDD potentials are also relevant fordescribing the interatomic interactions of these systems and cantherefore be applied to study formation, structure, and otherphysical properties of metallic glasses.

4. Applying long-range n-body potentials to model theformation of ternary metallic glasses

As mentioned in Chapter 1, one of the fundamental scientificissues in the field of metallic glasses is to develop a theoreticalmodel to predict in which system, in what composition range orregion, and by using what producing technique, an amorphousalloy could be produced. This issue is often referred to the glass-forming ability, which relates to the readiness or difficulty of themetallic glass formation.

This problem is generally approached in two different ways[18,19]. First, consideration is given to a specific alloy, treating theglass formation as the consequence of a frustration of crystalliza-tion during freezing of the liquid melt [152]. Second, considerationis given to a specific metal system. This situation can be depicted asa competition between the crystalline and amorphous phases asa function of the alloy composition [19,127]. For the second situa-tion, the glass-forming range or region (GFR) is frequently used toindicate an exact alloy composition range or region within whichthe formation of amorphous alloys is energetically favored. There isa strong correlation between the two situations, but there are also

Fig. 8. LDD potential and Rose equations for NiNb, NbTa and NiTa compound (reproduced from [54]).

J.H. Li et al. / Intermetallics 31 (2012) 292e320 307

some subtle differences. Here, we will refer to the first and secondmeanings of the concept as the GFA of an alloy and the GFA ofa system, respectively.

Table 20Potential parameters for the AgeNieZr system [128].

AgeAg NieNi ZreZr AgeNi AgeZr NieZr

m 4 4 4 4 4 4n 10 6 4 6 8 5

4.1. Glass-forming range/region (GFR) and glass-forming ability(GFA)

In practice, technical parameters are frequently used to describethe GFA of an alloy or to compare the GFAs of two alloys. Forexample, when the liquid melt quenching technique is used toproduce the amorphous alloy, the critical quenching speed or themaximum size of the attainable metallic glass is used to describethe GFA. The lower the critical quenching speed or the larger theobtained metallic glass, the greater the GFA of the alloy. However,the critical quenching speed is difficult to measure experimentallyand the maximum attainable size strongly depends on the appliedglass-producing technique [16]. Therefore, several simple param-eters have been proposed based on the temperature or some otherphysical properties of the alloys or metal system. The most famousparameter is the reduced glass transition temperature, Trg, which isdefined as the ratio of the glass transition temperature, Tg, and the

Table 19Lattice constant, a, and cohesive energy, EC of compounds L21 NbTaNi2, L21 NbTa2Ni,L21 Nb2TaNi, C1b NbTaNi, C1b NiNbTa, and C1b TaNbNi obtained from LDD potentials(first rows) and ab initio calculations (second rows) [121].

L21NbTaNi2

L21NbTa2Ni

L21Nb2TaNi

C1bNbTaNi

C1bNiNbTa

C1bTaNbNi

a (Å) 6.21 6.45 6.59 6.37 6.10 6.206.17 6.41 6.42 6.11 5.96 6.10

EC (eV) 6.22 6.98 6.97 5.87 6.27 5.986.20 7.07 7.03 5.67 6.40 5.76

liquidus temperature of the alloy, Tl [17]. Later, the super-cooledliquid region, DTxg, the temperature difference between the crys-tallization onset temperature, Tx, and the glass transition, Tg, hasbeen used by Inoue et al. as a guide to search for the good glass-forming alloys [133,153].

Differing from the situation of a specific alloy, the GFA of a metalsystem is indicated by the GFR. The GFR not only indicates whetheror not the metallic glasses could be obtained in a system, but alsospecifies an alloy composition range/region, within which metallicglasses can be formed in the system. It follows that the wider theGFR, the greater the GFA of a metal system. If the GFR of a metalsystem is nil, no any metallic glass can be obtained in the system. Ametal system should have an intrinsic GFA, which is determined bythe internal characteristics of the system itself. The intrinsic GFAdenotes the maximum possible composition range energeticallyfavored for the metallic glass formation [154].

rc1 (Å) 5.29749 5.74953 6.43755 4.59037 5.06208 4.78334rc2 (Å) 7.92195 7.17191 7.22279 7.20221 6.50660 7.38540c0 (eV) 1.78642 0.29472 0.78830 2.67354 0.54529 0.63372c1 (eV/

Åmþ1)�2.17500 �0.35872 �0.84608 �4.05272 �0.42808 �0.82273

c2 (eV/Åmþ2)

1.00034 0.16319 0.34151 2.46597 0.12057 0.48530

c3 (eV/Åmþ3)

�0.20470 �0.03304 �0.06110 �0.69530 �0.01256 �0.14258

c4 (eV/Åmþ4)

0.01563 0.00253 0.00408 0.07708 0.000050 0.01681

a (eV/Ån) 0.0003558 0.01461 0.1501 0.03538 0.009587 0.05629

Fig. 9. LDD potential and Rose equations for AgZr and NiZr compound(reproduced from [128]).

J.H. Li et al. / Intermetallics 31 (2012) 292e320308

To quantitatively predict the GFA, i.e. to estimate the intrinsicGFR of a metal system, several empirical criteria or rules have beenproposed, based on some specific properties or intrinsic charac-teristics of the system. For example, based on the equilibrium phasediagram of a binary metal system, Turnbull has proposed the well-known deep eutectic criterion [155]. Egami and Waseda haveproposed a method to predict the metallic glass formation byconsidering the atomic size difference of the constituent metals ofthe binary metal system [156]. Based on extensive ion beammixingstudies, Liu et al. proposed a structural difference rule to predict theamorphous alloy formation by ion beam mixing [157], and furtherproposed that the total width of the two-phase regions observedfrom the equilibrium phase diagram is approximately the GFR ofthe binary metal system [158]. In addition, thermodynamicapproaches, which consider the formation enthalpy or formationenergy, have also been used to predict the metallic glass formation[159e161].

From a theoretical point of view, both the glass-forming rangeand glass-forming ability can be considered as the intrinsic char-acteristics of the alloy or metal system. They should be governed bythe interatomic interactions of the system. It is known that theinteratomic potential defines the atomic interactions in the system.Thus, once the potential is known, many properties of the system

Table 21Potential parameters for the NieZreAl system [129].

Ni Zr Al

A1 (eV/atom) 0.263517 0.5187 0.394A2 (eV/atom) 0.007499 3.800596 0.203B1 (eV2/atom) 4.868517 11.27019 4.316B2 (eV2/atom) 0.052075 0.660027 0.110rcm1 (Å) 3.500544 3.585809 3.500rc1 (Å) 8.438046 5.035498 6.159rcm2 (Å) 4.502313 4.003347 4.634rc2 (Å) 7.931756 8.689765 8.322r0 (Å) 2.489016 3.172329 2.863p1 13.07157 8.862382 7.625p2 11.05515 0.109254 3.318q1 3.659351 3.392549 4.236q2 0.030459 0.00795 0.354m 4 4 4n 5 5 5

could be calculated or derived. It is therefore believed that both theglass-forming range/region of a metal system and the glass-forming ability of an alloy could be derived from the interatomicpotential, e.g. through atomistic simulations. This principle hasproven to be feasible in binary metal systems, and the GFRs of morethan twenty binary metal systems have been predicted using thismethod [154,162]. In the following sections, it will be introduced anatomistic approach that is capable of predicting not only the glass-forming region but also the optimum chemical composition withwhich the alloy is of the greatest glass-forming ability in the ternarymetal system.

4.2. Predicting the glass-forming region of ternary metal systems

The formation of ametallic glass is always a far from equilibriumprocess, in which, due to the kinetic conditions being extremelyrestricted, complicated structured phases can not nucleate andgrow. It follows that the competing phase of the metallic glass, oramorphous phase, is the terminal solid solutions. Extensiveexperimental studies, e.g. ion beam mixing experiments [158,163],have also shown that, taking a binary metal system as an example,the metallic glasses can be obtained in a broad alloy compositionrange. The composition range is approximately equaling to the total

NieZr ZreAl NieAl

892 1.474505 1.012553 0.40198945 0.017217 0.067144 0.003002941 24.8934 16.1802 7.794842473 2.302459 5.54378 0.229068000 5.190125 3.148415 5.267137399 10.97635 8.90736 10.45212781 4.479109 3.55071 4.372795493 6.631971 6.662871 6.951622782 2.850088 2.99914 2.566223509 6.927372 7.613551 9.924476097 4.957115 5.530434 7.9447493 6.626752 4.975133 4.979426058 0.007449 0.156712 0.003866

4 4 45 5 5

Table 22Lattice constants, a, b and/or c, cohesive energies (EC), and bulk modulus, B0, of NieZr compounds obtained from LDD potentials (first rows), ab initio calculation andexperimental data (second rows and third rows [134e137]) [129].

Ni3Zr Ni3Zr Ni2Zr NiZr NiZr NiZr2 NiZr3

L12 D019 C15 Bf B2 C16 L12

a, b and/or c (Å) 3.768 5.318, 4.334 6.955 3.308, 10.012, 4.069 3.213 6.894, 4.813 4.2803.818 5.311, 4.298 7.081 3.312, 9.985, 4.078 3.238 6.798, 4.783 4.307

5.309, 4.303 6.915 3.271, 9.931, 4.107 6.483, 5.267EC (eV/atom) 5.414 5.438 5.520 5.875 5.753 5.772 5.786

5.398 5.412 5.348 5.831 5.816 5.791 5.751B0 (Mbar) 1.589 1.707 1.673 1.374 1.394 1.144 1.029

1.582 2.023 1.650 1.424 1.336 1.314 1.063

J.H. Li et al. / Intermetallics 31 (2012) 292e320 309

width of the two-phase regions, i.e. extending from the centralportion to near the edges of the two terminal solid solutionsobserved from the equilibrium phase diagram of the system. Inother words, metallic glasses can be experimentally obtained inalmost the entire alloy composition range, except in the regionsthat favor for the formation of the solid solutions. The experimentalobservations suggest that predicting the GFA/GFR of a metal systemcan be converted into an issue of determining the critical solidsolubilities, which splits the entire alloy composition range orregion into different parts that energetically favor the formation ofthe solid solutions or metallic glasses.

From the analysis above, predicting the GFR of a metal system issimply an issue of comparing the relative stability of the solidsolution and its amorphous counterpart as a function of soluteconcentration, enabling one to determine the critical points of solidsolubilities for a binary system or critical lines of solid solubilitiesfor a ternary system. The relative stability of the amorphous phaseand the solid solution are governed by the atomic interactions ofthe system, and hence can be determined through appropriateatomistic simulations. A natural choice is that, based on the inter-atomic interaction potentials, molecular dynamics or Monte Carlosimulations are carried out to compare the relative stability of thesolid solution versus its disordered counterpart.

4.2.1. The NieNbeTa systemThe GFR of the NieNbeTa ternary metal system were predicted

from the LDD potentials by carrying out the isothermaleisobaricmolecular dynamics simulation [54,121,164]. In moleculardynamics simulations, the stabilities of 129 solid solutions withdifferent compositions were investigated [121]. The typical pair-correlation functions, g(r), and projections of the atomic positionsin crystalline and amorphous states are shown in Fig. 11 [5]. Whenthe concentration of solute atoms is low, e.g. in the Ni85Nb5Ta10 alloy,the solid solutionmaintains its original crystalline state; whilewhenthe concentration is high, exceeding a critical value, e.g. in theNi75Nb5Ta20 alloy, the supersaturated solid solution collapses andturns into a disordered state. From the molecular dynamics simula-tion results, the critical solid solubilities of the NieNbeTa system

Table 23Lattice constants, a, b and/or c, cohesive energies (EC), and bulkmodulus, B0, of ZreAlcompounds obtained from LDD potentials (first rows), ab initio calculation andexperimental data (second rows and third rows [138e141]) [129].

Zr3Al Zr2Al Zr5Al3 ZrAl ZrAl3 ZrAl3

L12 B82 D88 B2 L12 D023

a, b and/orc (Å)

4.388 4.946, 5.959 8.395, 5.553 3.406 4.117 4.022, 17.4124.452 4.959, 5.942 8.400, 5.567 3.309 4.219 4.032, 17.4114.3917 4.894 5.928 8.184, 5.702 3.9993, 17.283

EC (eV/atom) 5.78 5.623 5.516 5.104 4.581 4.6075.774 5.611 5.438 5.124 4.559 4.471

B0 (Mbar) 0.906 0.859 0.83 1.592 0.951 0.9560.905 1.069 1.043 1.593 0.954 1.187

were derived, and the crystaleamorphous phase diagram of thesystem was obtained and is shown in Fig. 12. From thecrystaleamorphous phase diagram, one could conveniently obtainthe metallic glass-forming region in the composition triangle of theNieNbeTametal system. In fact, the two critical solid solubility lines,i.e., the dashed lines AB and CD, divide the composition triangle intothree regions. The alloy with composition locating on corners or thevicinity of the NbeTa side, the solid solution is stable. Whereas thealloy on the central area in the composition triangle, the solid solu-tion becomes unstable and turns into a disordered state. It followsthat the composition region thus defined by the two lines can beconsidered the GFR of the NieNbeTa ternary metal system.

To validate the LDD potential predicted GFR of the NieNbeTasystem, experimental and theoretical results were collected andare also shown in Fig. 12. For the NieNb alloys, Gallego et al.determined that the GFR of the NieNb system is 20e80 at.% Ni bycomparing the Gibbs free energies of the crystalline and amor-phous phases [165]. Tai et al. obtained the amorphous NieNb alloysin the range of 23e85 at.% Ni by ion beam mixing of metallicmultilayers [166]. Considering the approximation in simulationsand experiments [167], the predicted GFR of the NieNb system iscompatible with the composition range observed in experiments.The GFR of the NieTa system determined by Gallego et al. is 15e84at.% Ni [165]. In ion beam mixing of metallic multilayers conductedby Liu and Zhang [126], the GFRwas observed to be 25e75 at.% Ni. Itcan be seen that the LDD potential predicted GFR of the NieTasystem is in good agreement with the composition rangeobserved in experiments. For the binary NbeTa alloys, the equi-librium phase diagram [120] shows that Nb and Ta are completelymiscible and can form bcc solid solutions in the entire compositionrange, indicating that it is difficult to obtain amorphous alloys inthis system. The LDD potential predicted GFR is also consistent withthe conclusion. For the NieNbeTa ternary alloys, Lee et al. obtainedsome amorphous alloys with the compositions Ni60Nb40�xTax,x ¼ 0, 3, 5, 10, 20. These compositions are all in the predicted GFR[168]. Based on the above discussion, it can be concluded that theLDD predicted GFR is consistent with the experimental observa-tions and with other theoretical works.

Table 24Lattice constants, a, b and/or c, cohesive energies (EC), and bulkmodulus, B0, of NieAlcompounds obtained from LDD potentials (first rows), ab initio calculation andexperimental data (second rows and third rows [142e144]) [129].

Ni3Al Ni5Al3 NiAl NiAl3

L12 Pt5Ga3 type B2 L12

a, b and/or c (Å) 3.579 7.505, 6.709, 3.761 2.889 3.8523.570 7.504, 6.709, 3.761 2.90 3.7783.572 7.44, 6.68, 3.72 2.882

EC (eV/atom) 4.692 4.687 4.645 3.9284.700 4.656 4.615 3.985

B0 (Mbar) 1.793 1.636 1.609 1.1041.879 1.592 1.623 1.151

Table 25Lattice constants, a, b and/or c, cohesive energies (EC), and bulk modulus, B0, ofcompounds NiZrAl (s1), Ni2ZrAl (s2), Ni2ZrAl5 (s6), L21 NiZr2Al, and L21 NiZrAl2 ob-tained from LDD potentials (first rows), ab initio calculation and experimental data(second rows and third rows [145e147]) [129].

NiZrAl (s1) Ni2ZrAl(s2)

Ni2ZrAl5 (s6) NiZr2AlL21

NiZrAl2 L21

a, b and/orc (Å)

6.944, 3.489 6.144 4.032, 14.60 6.629 6.3366.953, 3.478 6.108 4.067, 14.60 6.587 6.3406.915, 3.466 6.123 4.023, 14.44

EC (eV/atom) 5.488 5.300 4.648 5.371 4.6565.340 5.440 4.508 5.479 4.693

B0 (Mbar) 1.55 1.437 1.143 1.123 1.0741.46 1.719 1.296 1.359 1.431

J.H. Li et al. / Intermetallics 31 (2012) 292e320310

By inspecting the line EF in Fig. 12, it can be seen that theNi20NbxTa80�x alloys, consisting of Ni20Nb80 alloy and the Ni20Ta80,remain their bcc structure. However, for the ternary metal alloys,e.g. Ni20NbxTa80�x with x is not equal to 0 or 80, the initial crys-talline bcc structure transformed into the amorphous state. Thissuggests that Nb and Ta atoms added into the NieTa and NieNballoys, respectively, help the formation of amorphous alloys. Inthis regard, Lee et al. also found that Ta could improve the stabilityof the Ni60Nb40 amorphous alloy [168,169]. For NixNb40-xTa60 alloys(line PQ in Fig. 12), when the composition changes from point P topoint Q, the crystal to amorphous transition takes place withincreasing the Ni concentration, indicating that Ni is more favorablethan Nb for the formation of a Ta-based NieNbeTa amorphousalloy. Similarly, Ni is more favorable than solute Ta for the formationof an Nb-based NieNbeTa amorphous alloy (line RS in Fig. 12). Insummary, Ni is an important glass-forming solute in the NieNbeTaternary metal system, while Nb and Ta play a secondary role.

According to Egami’s empirical rule [156], as Nb and Ta have thesame atomic radii, changing the ratio of Nb to Ta, and meanwhilekeeping the concentration of Ni constant, has no effect on the sizedifference of the constituent atoms. On the Nb or Ta side of thecomposition triangle, theonlyway to increase the sizedifferenceof theconstituent atoms is to increase the atomic pairs of NieNb and NieTa,which can help the formation of amorphous alloys in the NieNbeTasystem. Clearly, the GFR predicted in the present study is compatiblewith Egami’s empirical rule [156]. In addition, according to the struc-ture difference rule [157], an amorphous binary alloy consisting of twometal elements with different crystalline structure can be formed byion beammixing. It follows that, for the bccebcc NbeTa atomic pairs,increasing the fccebcc atomic pairs of NieNb and NieTa would befavorable for the formation of theNieNbeTa amorphous alloys,whichis in good agreement with the prediction by the crystaleamorphousphase diagram shown in the composition triangle.

Table 26Potential parameters for the CueZreAl system [131].

Cu Zr Al CueZr ZreAl CueAl

A1 (eV/atom) 8.049825 3.104994 2.917212 8.445146 6.428795 1.112537A2 (eV2/atom) 0.458708 13.997855 1.114067 14.501495 2.333929 1.38881B1 (eV/atom) 0.315491 0.751605 0.402184 0.607641 0.458326 0.977351B2 (eV2/atom) 3.854734 16.924785 4.738155 10.646367 7.985849 9.720919rc1 (Å) 3.634148 5.466890 4.607023 4.331234 4.237990 5.28644rm1 (Å) 2.124148 3.355556 2.764394 2.444527 2.067990 2.162239rc2 (Å) 6.215324 6.556265 6.515324 5.560000 6.568872 6.55rm2 (Å) 3.611361 3.218719 3.786874 2.390458 3.024899 3.618114r0 (Å) 2.553618 3.215897 2.864321 2.884757 2.883688 2.708969p1 9.625372 8.202755 8.77646 7.701745 6.895905 4.575749p2 2.860823 2.110120 2.558558 1.585780 1.580351 1.107382q1 4.903930 4.817954 5.249466 4.551226 4.068921 4.620459q2 0.000602 0.000198 0.000477 0.000478 0.000481 0.000619m 4 4 4 4 4 4n 5 5 5 5 5 5

In summary, two critical solid solubility lines in the compositiontriangle were determined, and the critical lines define a composi-tion region within which an amorphous phase is energeticallyfavored over its crystalline counterpart. This region can beconsidered as the GFR of the system. It is also found that thecalculated GFR agrees well with the experimental observationsreported in the literature.

4.2.2. The CueZreAl systemThe CueZreAl amorphous alloy is expected to be of very good

processability and mechanical performance. The CueZreAl systemhas therefore been extensively studied by experiments as well as bycomputer simulations [170e176]. In experimental studies, researchhas focused on the CueZr based systemwithminor addition of Al. Itis still unclear whether the CueAl based or the ZreAl basedmetallicglasses can be obtained and in which composition region thesemetallic glasses can be obtained. Thus, it is of interest to investigatethe GFR of the CueZreAl ternary system.

In a similar way, using the LDD potential constructed by Cui et al.[131], molecular dynamics simulationwere carried out to derive theGFRof the CueZreAl systembycomparing the stabilities of the solidsolution phase and its amorphous counterpart. The fcc Al based andCu based and hcp Zr based solid solutionwere annealed at 300 K bymolecular dynamics simulation for adequate molecular dynamicssteps, reaching a relatively stable state, i.e. the energy of the systemand the atomic configuration are almost unchanged. The structuresof the annealed CueZreAl alloys exhibit three different states:a crystal state (CS), a transition state (TS) and an amorphous state(AS). Take the Cu5Zr85Al10, Cu15Zr75Al10 and Cu25Zr65Al10 alloys asexamples, Fig.13 shows the projections of atomic positions andpair-correlation function for the three states. The projections of theatomic positions of the CS show a completely ordered state, whilethose of the AS show a completely disordered state. For the TS, theprojection exhibits amixed orderededisordered state. Furthermore,the g(r) curves of the CS exhibit a long-range ordered feature, whilein the g(r) curves of theAS, the crystal peaks beyond the secondpeakdisappeared, exhibiting a short-range ordered and long-rangedisordered feature. The g(r) curves of the TS showing mixturefeatures of amorphous and crystalline phases, indicating that it is anintermediate state between the fully ordered and disordered ones.The transition state can also considered as an inhomogeneouscomposite of amorphous and crystalline phases. One notes that thetransition region is also metastable, however, its formation mech-anism is not fully clear so far. A possible interpretation is related tothe fluctuation of chemical composition. One knows that thechemical composition distributes inhomogeneously at microscale.Compositions at somemicro-regions are somewhat higher than theothers and if it exceeds the critical solid solubility of the system, theamorphous phases may form at these micro-regions. By contraries,if the compositions of somemicro-regions are lower than the criticalsolid solubility, the crystalline phases may be observed at thesemicro-regions. Clearly, the fluctuation of chemical compositionresults in the formation of transition state1. Based on the obtainedstructures, we classify the CueZreAl alloys into the three states, i.e.the crystal state (CS), the transition state (TS) and the amorphousstate (AS), and then construct the CSeAS phase diagram as shown inFig. 14. One can see that the composition triangle are divided into

1 By the thermodynamics calculation, the Schwarz et al. argue there isa compositions region in which mixture of amorphous and crystalline phases canexist in a metastable state (Schwarz, and Johnson. Phys. Rev. Lett. 51, 415, see thenext paragraph). In fact, the mixture of amorphous and crystalline phases can beobserved frequently in experiment e.g. by Wang and Li et al. (App. Phys. Lett. 84,4029).

Table 27The reproduced and experimental properties (first rows and second rows [148,149]) of CueZr, ZreAl and CueAl compounds [131].

Compounds Cu5Zr Cu8Zr3 Cu10Zr7 CuZr2 AlZr3 Al3Zr Al3Zr2 Al3Cu Al2Cu AlCu3

Space group F4 3m Pnma Aba2 I4/mmm Pm3 m I4/mmm Fdd2 Pm3 m I4/mcm Pm3 m

a, b, and/or c (Å) 6.92 7.88,8.15,10.16 9.37,9.39,12.78 3.24,11.20 4.30 4.00,16.82 9.37,13.57,5.37 4.01 6.00,5.08 3.706.87 7.87,8.15, 9.98 9.34,9.32,12.67 3.22,11.18 4.35 4.01,17.29 9.69,13.89,5.57 3.94 6.06,4.88 3.70

EC (eV/atom) 4.079 4.432 4.821 5.481 5.860 4.476 5.063 3.494 3.623 3.6604.080 4.431 4.820 5.481 5.857 4.614 5.069 3.493 3.623 3.658

B0 (Mbar) 1.286 1.268 1.162 1.126 1.219 1.125 1.297 0.590 0.681 0.8621.392 e e 1.121 1.014 1.025 1.086 0.611 0.792 1.253

J.H. Li et al. / Intermetallics 31 (2012) 292e320 311

seven regions by six critical solubility lines. TheAB, EFand IJ lines arethe CSeTS boundaries, suggesting that when an alloy compositionlocates on the lines andmoving toward one of the three corners, thecrystalline structures become stable. The CD, GH andKL lines are theTSeAS boundaries, suggesting that when an alloy compositionmoves from the lines toward the center of the hexagonal region,enclosed by CDKLHG, the crystalline structure becomes unstableand would collapse, turning into a disordered state. Between thecrystal and amorphous region, there are three transition stateregions, in which the initial solid solutions would turn into a mixedstate of the orderededisordered structures. Fromabove description,the CSeAS phase diagram and composition region favoring metallicglass formation are determined as shown in Fig. 14, in which thehexagonal composition region is defined as the quantitative glass-forming-ability (GFA) or glass-forming-region (GFR) of theCueZreAl system. To validate the predicted GFR of the CueZreAlsystem, experimental results [170e184] were collected and arealso shown in Fig. 14. Most the compositions of the CueZreAlamorphous alloys obtained in experiments fall within the pre-dicted hexagonal composition region, providing experimentalsupports to the CSeASphase diagramand theGFRderived fromLDDpotential of the CueZreAl system.

Summarily, the simulation results not only reveal that thephysical origin of the crystal to amorphous transition is the

Fig. 10. LDD potentials and their derivates and Rose equations for ZrNb and HfTacompounds (reproduced from [52]).

crystalline lattice collapsing when the solute atoms exceed thecritical value, but also predict a composition region favored for theformation of CueZreAl metallic glasses. Furthermore, the GFRs ofthe CueZreNi [130] and AleZreNi [129] ternary metal systemswere also predicted based on the constructed LDD potentialthrough molecular dynamics simulations and shown in Figs. 15 and16. It is found that the predicted GFRs of these ternary metalsystems all agree well with experimental observations.

4.3. Predicting the amorphous alloy of the largest glass-formingability in a metal system

In above section, an atomistic approach has been introducedand applied to derive the GFRs from the constructed LDD potentialsfor some typical ternary metals systems through moleculardynamics simulations. The derived GFRs indicate a compositionregion within which the ternary amorphous alloys can be obtainedby proper glass-producing techniques. In this section, the atomisticapproach will be further expanded so as to be capable of predictingthe ternary alloy with the largest GFA in a metal system. Theexpanded atomistic approach is of significance because it can helpone to design the chemical composition of an amorphous alloy thatis most easily produced or, the obtained amorphous alloy is moststable in a particular metal system.

As discussion before, in the process of metallic formation, theamorphous phase competes with the fcc, bcc or hcp simple struc-tural terminal solid solution. The energy difference between thetwo phases plays a vital role in determining whether the solidsolution or amorphous phase wins out in this process. Froma thermodynamic point of view, the energy difference can be

Table 28Lattice constants a, cohesive energies EC, bulk moduli B0 and elastic constants Cijreproduced from the LDD potentials (first rows) and acquired by ab initio calcula-tions (second rows) of compound L12 Zr3Nb, ZrNb3, Hf3Ta, HfTa3, Cu3Nb and CuNb3,and B2 ZrNb, HfTa and CuNb [52].

A (Å) EC (eV/atom) B0 (Mbar) C11 (Mbar) C12 (Mbar) C44 (Mbar)

L12 Zr3Nb 4.51 6.45 1.03 1.10 1.00 0.064.44 6.43 1.34 1.12 1.45 0.38

L12 Hf3Ta 4.55 6.30 1.15 1.40 1.03 0.494.40 6.66 1.27 1.10 1.35 0.81

L12 Cu3Nb 3.87 4.18 1.59 1.84 1.46 0.573.81 4.18 1.49 1.54 1.46 0.48

B2 ZrNb 3.57 6.77 1.38 1.78 1.18 0.163.43 6.79 1.17 1.19 1.15 0.15

B2 HfTa 3.47 6.65 1.63 2.12 1.38 0.523.40 7.00 1.66 1.65 1.65 0.39

B2 CuNb 3.26 4.87 1.55 1.69 1.48 0.453.12 5.35 1.69 1.66 1.70 0.48

L12 ZrNb3 4.44 6.92 1.09 0.84 1.21 0.054.29 7.52 1.35 1.15 1.45 0.24

L12 HfTa3 4.32 6.98 1.07 0.79 1.22 0.384.23 7.47 1.83 0.72 2.39 0.52

L12 CuNb3 4.04 5.94 1.36 0.83 1.63 0.494.05 6.40 1.60 0.94 1.92 0.44

Fig. 11. Projections of atomic positions and the corresponding pair-correlation functions of Ni85Nb5Ta10 crystalline and Ni75Nb5Ta20 amorphous alloys [121].

J.H. Li et al. / Intermetallics 31 (2012) 292e320312

considered as the driving force for the solid solution to amorphize.It is believed that the larger the driving force, the easier theamorphous alloy to form and the more stable the resulting amor-phous alloy [19,185,186]. Accordingly, once the interatomic

Fig. 12. Crystaleamorphous phase diagram of the NieNbeTa system derived from LDDpotential. The experimental results are marked by crosses and squares [121].

potential has been determined, the energies of the amorphousphase and solid solution can be calculated. Consequently, thedriving force for the solid solution to amorphize can be derived, andhence the chemical composition of the alloy with the largest GFA ina system can be predicted based on the driving force.

4.3.1. Calculating the driving force for the AleZr metallic glassformation

To demonstrate the procedure and verify the relevance of theproposed approach, we first take the AleZr binary metal system asan example to predict the chemical composition with which thealloy is of largest GFA in the metal system One notes that theformation of AleZr metallic glasses has been studied by experi-ments and atomistic simulations [181,187e189] and all the ob-tained results prove to be consist with each other. Therefore, onlythe Monte Carlo simulations are briefly presented below.

The LDD potential has been constructed for the AleZr systemand proven to be relevant in describing the interatomic interactionsof the system [189]. To acquire the GFR of the AleZr system,isothermaleisobaric Monte Carlo simulations were performed at300 K and 0 Pa both on fcc Al-based and hcp Zr-based solid solutionmodels with different chemical compositions [190e193]. Thesimulation results show that the solid solution retains its crystallinestructure when the solute concentration is low, but the crystallinelattice of the solid solution becomes severely distorted uponincreasing the solute concentration. Once the solute concentrationexceeds a critical value, the crystalline lattice collapses and thesolid solution turns into a disordered state. The pair-correlation

Fig. 14. Crystaleamorphous phase diagram of the CueZreAl system derived from LDDpotential [131].

Fig. 15. Crystaleamorphous phase diagram of the NieZreAl system derived from LDDpotential [129].

Fig. 13. Total pair-correlation functions and projections of atomic positions observed in the Cu5Zr85Al10 crystal state (a, d), Cu15Zr75Al10 transition state (b, e) and Cu25Zr65Al10amorphous state (c, f) [131].

J.H. Li et al. / Intermetallics 31 (2012) 292e320 313

Fig. 16. Crystaleamorphous phase diagram of the CueZreNi system derived from LDDpotential [129].

J.H. Li et al. / Intermetallics 31 (2012) 292e320314

functions were calculated for the MC annealed AleZr alloys withdifferent chemical compositions, and some typical pair-correlationfunctions are shown in Fig. 17. The calculated pair-correlationfunctions also show that, for either fcc Al-based solid solution orhcp Zr-based solid solution, when the solute concentration is low,the solid solution keeps its crystalline lattice; however, once thesolute concentration exceeds a critical value, the crystalline latticeof the solid solution collapses and becomes disordered. Forexample, for the Al90Zr10 alloy, all the fcc peaks can be

Fig. 17. Pair-correlation functions of AleZr alloys obtained by Monte Carlo simulations[193].

distinguished in almost the entire correlation distance. For theAl80Zr20, and Al77Zr23 alloys the fcc peaks can also be discerned,even at very large correlation distances, as shown in Fig. 17.However, when the concentration of solute Zr atoms reaches orexceeds 25 at.%, all the peaks except the first one are smeared outand become indistinct. To give more quantitative evidence on thecrystalline-to-amorphous transition of the AleZr system revealedby atomistic simulations, the structure factors were also calculatedfor the AleZr alloys with different chemical compositions, and thecalculated results are shown in Fig. 18. One sees that when theconcentration of solute atoms is relatively low, e.g., less than 23 at.%Zr for fcc Al-based solid solution and less than 33 at.% Al for hcp Zr-based solid solution, the calculated structure factors of both fcc Al-based and hcp Zr-based solid solutions are far greater than zero, i.e.,considerably closer to 1, suggesting that they can retain crystallinestructures. Once the concentration of solute atoms reaches orexceeds the critical values, i.e., 24 at.% Zr and 34 at.% Al for fcc Al-based solid solutions and hcp Zr-based solid solutions, respec-tively, the corresponding structure factors fall abruptly or becomevery close to zero, indicating that the supersaturated AleZr solidsolutions become unstable and, hence, become disordered. FromFigs.17 and 18, one can conclude that the glass-forming range (GFR)of the AleZr system is predicted to be 24e66 at.% Zr. The predictedresult is consistent both with the experimental observations andother theoretical prediction [181,187e189].

The formation energy of an AleZr alloy can be considered to bemade up three parts. The first part is the chemical contribution,DEchem, resulting from the shrinkage or expansion of atomicvolume, redistribution of valence electron when they approacheach other. The second part is the contribution from the latticedistortion, DEdistort. When solute atoms dissolve into a perfectcrystal lattice, atomic level stress would present. To relax theatomic level stress, atoms must adjust their equilibrium atomicpositions, and this adjustment results in lattice distortions. For anideal solid solution, the formation energy comes only from chem-ical contributions. For a real solid solution, the formation energyconsists of both parts. Apart from the two parts, the formationenergy of an amorphous alloy includes the contribution of latticecollapsing, DEcollps. The formation energy of an ideal solid solutioncan be directly computed by Monte Carlo or molecular statisticsmethods [53,193]. The formation energies of the real solid solutionand amorphous alloys can be calculated by Monte Carlo ormolecular dynamics simulations.

To give a thermodynamics picture concerning the structuretransition observed in the AleZr system by Monte Carlo simula-tions, and to reveal the correlation between GFA/GFR and the

Fig. 18. Planar structure factors of MC annealed AleZr alloys with different chemicalcompositions [193].

J.H. Li et al. / Intermetallics 31 (2012) 292e320 315

thermodynamics factor, the formation energies of the ideal solidsolutions, relaxed solid solutions, and the amorphous phases in theAleZr system were all calculated based on the results of MonteCarlo simulations. Fig. 19 shows the calculated formation energiesof these AleZr alloys with the different structures. In general, thelattice distortion depends on the concentration of the solute atoms,and the lattice distortion increases with the concentration of soluteatoms. When the concentration exceeds a critical value, the latticedistortion is so large that the solid solution totally collapses intoa disordered state. For the AleZr solid solution, with the soluteatoms increasing, the contribution of lattice distortion to theformation energy becomes greater and greater (absolute value)[193].

From a thermodynamics viewpoint, the energy of latticecollapse, DEcollps, i.e. the energy difference between the amorphousphase and the real solid solution, is the driving force for the solidsolution to amorphize. According to Xia et al. and Wang et al.[185,186], the driving force is correlated to the glass-forming abilityof the alloy, i.e. the larger the driving force the greater the glass-forming ability of the corresponding alloy in the same system. Inother word, the alloy with the largest driving force should be theeasiest to be produced in the system. According to the argument,one can predict that the Al56Zr44 alloy has the largest driving forceand hence the alloy has the glass-forming ability in the AleZrsystem, indicating that the Al56Zr44 amorphous alloy is easier tobe obtained than other alloys in this system. It also means that themaximum attainable size of the Al56Zr44 metallic glass is the largestor this amorphous alloy is the most stable in the AleZr system[189,193].

Fig. 19. Formation energies (a) and driving force (b) of AleZr alloys. The dot lineshown in (a) stands for the corresponding properties of the unstable solid solutions[193].

To indicate the GFA of an alloy, some thermodynamics param-eters have been proposed previously. For example, Xia et al. arguedthat the GFA is directly proportional to the parameter g ¼ DEamor/(DEamor � DEcomp), where DEamor is the formation of the amorphousalloy and DEcomp is the reference state determined by thecompounds [185]. Based on some different arguments, Wang et al.also proposed a similar parameter to evaluate the GFA of a specificalloy [186]. Although these parameters give different values of GFA,the compositions of the alloys with the largest GFAs determined byboth methods are similar and close to that obtained by simplycomparing the formation energies of solid solutions and amor-phous alloys. In practice, it is also difficult to distinctly differentiateenergies of lattice collapse and lattice distortion, especially fora ternary amorphous alloy, since they intermingle with each other.Consequently, the formation energy of the amorphous alloy, con-sisting of energies of lattice collapsing and lattice distortion, can beused to search for the composition of the alloy with the largestglass-forming ability in a ternary metal system.

4.3.2. Optimized compositions for the CueZreNi and CueHfeAlmetallic glass formation

Cui et al. have constructed the LDD potential for the CueZreNisystem and applied the LDD potential to derive the GFR of thesystem [130]. It predicted that the amorphous alloys with compo-sitions locating in the predicted region could be produced.However, it is not clear which alloy in the composition region is theeasiest to obtain by a specific glass-producing technique. This is theproblem to search for a composition with which the CueZreNiternary alloy is of the largest GFA in the system. In general, it canbe solved by calculating and comparing the GFAs of CueZreNialloys with different compositions in the system. As mentionedabove, the driving force is correlated with the GFA of the alloy, i.e.the larger the driving force the greater the glass-forming ability ofthe alloy in the system.

The alloy with the composition locating in the glass-formingregion, the crystalline lattice of the solid solution collapses spon-taneously into a disordered state and forms a metallic glass,resulting in the failure of molecular dynamics simulation to calcu-late the formation energy of the solid solution. In this regard, themolecular statics calculations proposed by Dai is an efficient andrelevant scheme to compute the formation energy of ternarymetallic solid solutions in entire composition region [53]. Themolecular statics calculation differs significantly from themoleculardynamics simulation. The objective of the molecular dynamicssimulation is to give the information about the crystal-to-amorphous transition to reveal the mechanism of metallic glassformation,whereas theobjective of themolecular statics calculationis to optimize the solid solution structure and find the minimumenergy of the solid solution at a specific composition. Themotions ofatoms in molecular dynamics simulation are governed only by theforce acted on the atom. The symmetry of the initial structure inmolecular dynamics simulation might be spoiled in some cases, e.g.the crystalline lattice collapsing when the solute concentrationexceeds critical value. However, in the molecular statics calculation,the symmetry of the initial structure keeps unchanging and only thestructure parameters such as the lattice constants are relaxed so asthat the energy of a solid solution reaches the lowest state. Usingthese atomistic calculations and simulations, the formation energiesof the amorphous phase and the solid solution can be obtained.Accordingly, the energy difference between them, i.e. the drivingforce for the solid solution to amorphize, can be computed byDEDriving Force ¼ DEAmorphous � DESolid Solution.

Based on the results of the molecular dynamic simulations andmolecular statics calculations, the driving forces for the CueZreNisolid solutions to amorphize, with compositions locating in the

Fig. 21. GFR of the CueHfeAl system derived from the LDD potentials [132].

J.H. Li et al. / Intermetallics 31 (2012) 292e320316

glass-forming region, were derived and are shown in Fig. 20 [130].The driving forces, DEDriving Force, are negative over the entire pre-dicted GFR, indicating that the formation enthalpy of the amor-phous phase is less than that of the solid solution. It shoes that theformation of an amorphous phase in the GFR is energeticallyfavored. One can see that from Fig. 20 that the alloys withcompositions marked by red dots have lower DEDriving Force than theother alloys, and the Cu16Zr60Ni24 alloys marked by a purplepentagram has the lowest DEDriving Force. As mentioned above, thelarger the formation energy difference the larger the driving forcefor amorphization. The optimized composition for metallic glassformation can therefore be predicted to be around the Cu16Zr60Ni24.It is interesting to compare the calculated results with those re-ported from experiments [194e196]. The experimental data inFig. 20 are densely distributed around the optimized compositionregion, suggesting that the alloys around the optimized regionwerereadily obtained. Yang et al. have obtained the Cu40�xZr60Nix andCuyZr82�yNi18 metallic glasses with the compositions x ¼ 14, 16, 18,20, 22, 24 and 26 and y ¼ 14, 18, 20, 22, 24, 26 and 30, respectively.It is found that the alloy Cu20Zr62Ni18 has the largest reduced glass-transition temperature [195]. Hu et al. also reported that the largesuper-cooled liquid region is 34 K for Cu10Zr60Ni30 and 48 K forCu20Zr60Ni20 [196]. The results obtained from atomistic simulationsare consistent with those from experiments. Clearly, these experi-mental observations provide a firm support to the validity of theconstructed LDD potential as well as the predicted GFR of theCueZreNi system and the GFA of CueZreNi alloys [132].

For the CueHfeAl ternary system, the GFR was derived from theconstructed LDD potential through molecular dynamic simulationsand the derived GFR is shown in Fig. 21 [132]. The CueHfeAlcomposition triangle is spilt into seven regions by six criticalsolubility lines. The three corners are the crystalline regions withinwhich the CueHfeAl solid solution is more stable than its amor-phous counterparts. The central hexagonal region enclosed byCDKLHG is the GFR of the CueHfeAl ternary system within whichthe crystalline structure of a solid solution becomes unstable andwould collapse into a disordered state. Between the crystalline andthe amorphous regions, there are orderededisordered mixedregions, within which the initial solid solutions would transforminto a state featuring with an orderededisordered coexistingstructure. The experimental results were collected and are marked

Fig. 20. Diving forces for the CueZreNi solid solutions to amorphize, with composi-tions locating in the glass-forming region, derived from LDD potential [130].

by red solid circles in Fig. 21. One can see that the experimentalcompositions mostly fall within the hexagonal region [197e204],suggesting that the LDD predicted GFR the CueHfeAl system is ofrelevance.

Fig. 21 shows that the compositions of the experimentallyobserved CueHfeAl metallic glasses are densely clustered ina small composition area: CuxHfyAl100�x�y, x ¼ 45e65 andy ¼ 35e55. These experimental results suggest that the mostfavored composition for the CueHfeAl metallic glass formation iswithin this composition area. Accordingly, for the CueHfeAl solidsolutions locating in this area, the driving force to amorphize werecalculated and the calculated results are shown in Fig. 22. Thecalculated driving force is negative over the entire calculated area,suggesting that the formation of metallic glasses is energeticallyfavored within this area. Moreover, the compositions marked byred dots have lower driving forces than the other points, and theCu48Hf41Al11 composition marked by a black circle has the lowestdriving force, suggesting that the most favored compositions ofmetallic glass are around Cu48Hf41Al11. Using the same method, theGFR of NieZreAl, NieZreTi ternary metal systems and GFA of theseternary alloys were derived from the constructed LDD potential

Fig. 22. Diving forces for the CueZreAl solid solutions to amorphize derived from LDDpotential [132].

Fig. 23. Diving forces for the NieZreAl solid solutions to amorphize derived from LDDpotential [205].

J.H. Li et al. / Intermetallics 31 (2012) 292e320 317

through atomistic simulations [129,151] and the derived results aresummarized in Fig. 23. According to the prediction, the largestdriving force for NieZreAl solid solutions to amorphization isobserved at the composition of Ni25Zr60Al15 around. It means thatthe predicted Ni25Zr60Al15 alloy has the best glass-forming ability inthe system. In other words the Ni25Zr60Al15 amorphous alloy iseasier to be obtained or more stable than other alloys in thissystem. Indeed, these predictions are generally compatible withexperimental observations [205], providing solid support to therelevance of the constructed LDD potential as well as the atomisticapproaches applied.

5. Conclusions and remarks

1) We have proposed a long range n-body potential (LDD) thatcombines the major features of the second-moment approxi-mation of tight-binding and Finnis-Sinclair potentials. The LDDpotential is applicable to the three major crystalline structures,i.e. bcc, fcc and hcp transition metals and their alloys, and iscapable of distinguishing the energy difference between fccand hcp structures. In addition, when applying the LDDpotential to atomistic simulations, the reproduced energy andforce both converge smoothly to zero at the cutoff distance,without the appearance of unphysical phenomena frequentlyobserved with other empirical potentials.

2) With the aid of ab initio calculations, over thirty LDD potentialshave been constructed for bcc, fcc and hcp structured transitionmetals and their binary and ternary alloys. The constructedpotentials well reproduce some important static properties, e.g.the lattice constant, cohesive energy, elastic constant and bulkmodulus, as well as important dynamics properties, e.g. thedefects and phonon behavior, providing support to the rele-vance of the proposed LDD potential in describing the atomisticinteractions of the respective metals and alloys.

3) For some selected ternary metal systems, applying the con-structed LDD potentials, molecular dynamics and Monte Carlosimulations not only clarify that the underlying physics of themetallic glass formation is the crystalline lattice collapse of thesolid solution when the solute concentration exceeds a criticalsolid solubility, but also predict the compositional triangle,a glass-forming region (GFR), for each ternary system withinwhich formation of metallic glasses is energetically favored,providing important guidance for designing favored alloycompositions to produce the desired ternary metallic glasses.The energy difference between a solid solution and its amor-phous phase is proposed to be the driving force for the crystal

to amorphous transition, and the amount of the driving forcecan be considered as a comparative measure of the glass-forming ability (GFA) of a metallic glass. The driving forces toamorphize of about half dozen ternary metallic solid solutionswere derived from atomistic simulations. Based on the deriveddriving forces, some ternary amorphous alloys with the largestglass-forming abilities in their respective metal systems werepinpointed and predicted to be the most stable or the easiestone to be produced in practice by specific glass-producingtechniques.

4) In the atomistic simulations of the present work, the initialstate is an ideal and perfect crystalline solid solution. Thesimulations have not taken into account some other possiblefactors that would affect metallic glass formation, e.g. impuri-ties, defects and lattice distortion. The simulations have norelation to any practically applied glass-producing technique,and hence the potential predicted GFA/GFR through simula-tions can be considered as an intrinsic property of a metalsystem. Ab initiomethods have been established for a long timeand are widely applied to study some important scientificissues in many fields. Nonetheless, due to the inherent limita-tions, e.g. size limitations and time scale, the method is notsuitable or is difficult to be used in studying some systems. Asa consequence, atomistic simulations are still of vital impor-tance at present and for the foreseeable future. In this regard,the proposed LDD potential could be used for atomistic simu-lations in various fields.

Acknowledgments

J. H. Li, one of the authors, wishes to express his great thanks tohis PhD advisor, professor B. X. Liu, who provided the inspirationthat got the work started and the enthusiasm to keep him goingwhen frustration set in and his interests waned. This work wouldhave been much leaner without Professor Liu’s ongoing help,encouragement and useful conversation, for all of which Li is verygrateful. Sincere thanks also give to the members of Prof. Liu’sresearch group, especially Dr. S. H. Liang, Ms. S. Z. Zhao andMs. Y. Y.Cui, for their contributions to the present work. Financial supportfrom theMinistry of Science and Technology of China (973 Program2011CB606301, 2012CB825700), the National Natural ScienceFoundation of China (50971072, 51131003) and the Administrationof Tsinghua University are also gratefully acknowledged.

References

[1] Buckel W, Hilsch R. Einfluss der kondensation bei tiefen temperaturen aufden elektrischen widerstand und die supraleitung fur verschiedene metalle.Zeitschrift Fur Physik 1954;138:109e20.

[2] Buckel W. Elektronenbeugungs-aufnahmen von dunnen metallschichten beitiefen temperaturen. Zeitschrift Fur Physik 1954;138:136e50.

[3] Klement W, Willens RH, Duwez P. Non-crystalline structure in solidifiedgoldesilicon alloys. Nature 1960;187:869e70.

[4] Duwez P, Willens RH, Klement W. Continuous series of metastable solidsolutions in silverecopper alloys. Journal of Applied Physics 1960;31:1136e7.

[5] ZallenW. The physics of amorphous solids. NewYork: JohnWiley & Sons; 1983.[6] Elliott SR. Physics of amorphous materials. London: Longman; 1984.[7] Greer AL. Metallic glasses. Science 1995;267:1947e53.[8] Güntherodt HJ, Beck H. Glassy metals. New York: Springer-Verlag; 1981.[9] Anantharaman TR. Metallic glasses: production, properties and applications.

Switzerland: Trans Tech Publications; 1984.[10] Luborsky FE, editor. Amorphous metallic alloys. London: Butterworths; 1983.[11] Kawazoe Y, Suzuki A-P, Yu J-Z, Masumoto T. Nonequilibrium phase diagrams

of ternary amorphous alloys. Sendai: Springer-Verlag; 1997.[12] Inoue A. Bulk amorphous alloys: preparation and fundamental characteris-

tics. Uetikon-Zuerich, Switzerland: Trans Tech Publications; 1998.[13] Inoue A. Bulk amorphous alloys: practical characteristics and applications.

Zurich, Switzerland: Trans Tech Publications; 1999.[14] Wang WH, Dong C, Shek CH. Bulk metallic glasses. Materials Science and

Engineering R e Reoport 2004;44:45e89.

J.H. Li et al. / Intermetallics 31 (2012) 292e320318

[15] Wang WH. The elastic properties, elastic models and elastic perspectives ofmetallic glasses. Progress in Materials Science 2012;57:487e656.

[16] Miller M, Liaw P. Bulk metallic glasses: an overview. Boston: Springer; 2007.[17] Turnbull D. Under what conditions can a glass be formed. Contemporary

Physics 1969;10:473e88.[18] Massalski TB. On the glass forming ability of metalic alloys. In:

Hassen P, Jaffee RI, editors. Amorphous metals and semiconductors:proceeding of an international workshop. Coronado, California: Perga-mon Press; 1986. p. 178.

[19] Lu ZP, Bei H, Liu CT. Recent progress in quantifying glass-forming ability ofbulk metallic glasses. Intermetallics 2007;15:618e24.

[20] Li JH, Dai XD, Liang SH, Tai KP, Kong Y, Liu BX. Interatomic potentials of thebinary transition metal systems and some applications in materials physics.Physics Reports-Review Section of Physics Letters 2008;455:1e134.

[21] Mie K. Kinetic theory of monatomic bodies. Annales de Physique Leipzig1903;11:657e97.

[22] Lennard-Jones E. On the forces between atoms and ions. Proceedings of theRoyal Society of London 1925;109:584e97.

[23] Morse PM. Diatomic molecules according to the wave mechanics II, vibra-tional levels. Physical Review 1929;34:57e64.

[24] Born M, Mayer JE. Zur gittertheorie der ionenkrstalle. Zeitschrift Fur Physik A1932;75:1e18.

[25] Cousins CSG, Martin JW. Extended Cauchy discrepancies: measures of non-central, non-isotropic interactions in crystals. Journal of Physics F-MetalPhysics 1978;8:2279e91.

[26] Rathore RPS, Agarwal RM. Cohesion and Cauchy discrepancy in stable simplecubic metals. Physica Status Solidi B 1980;97:597e600.

[27] Voter AF. Interatomic potentials for atomistic simulations. Mrs Bulletin 1996;21:17e9.

[28] Raabe D. Computational materials science. New York: Wiley-VCH; 1998.[29] Ducastelle F. Elastic moduli of transition metal. Journal of Physics (Pairs)

1970;31:1055e62.[30] Ducastelle F, Cryrot-Lackmann F. Moments developments and their appli-

cation to the electronic charge distribution of d bands. Journal of Physics andChemistry of Solids 1971;31:1295e306.

[31] Tomanek D, Aligia AA, Balseiro CA. Calculation of elastic strain and electroniceffects on surface segregation. Physical Review B 1985;32:5051e6.

[32] Ackland GJ, Finnis M, Vitek V. Validity of the second moment tight-bindingmodel. Journal of Physics F-Metal Physics 1988;18:L153e7.

[33] Rosato V, Massimo C, Legrand B. Thermodynamical and structural propertiesof fcc transition metals using a simple tight-binding model. PhilosophicalMagazine A 1989;59:321e36.

[34] Willaime F, Massobrio C. Development of an n-body interatomic potential forhcp and bcc zirconium. Physical Review B 1991;43:11653e65.

[35] Cleri F, Rosato V. Tight-binding potentials for transition-metals and alloys.Physical Review B 1993;48:22e33.

[36] Finnis MW, Sinclair JE. A simple empirical n-body potential for transition-metals. Philosophical Magazine A 1984;50:45e55.

[37] Baskes MI. Application of the embedded-atom method to covalent materials:a semi-empirical potential for silicon. Physical Review Letters 1987;59:2666e9.

[38] Johnson RA. Analytic nearest-neighbor model for fcc metals. Physical ReviewB 1988;37:3924e31.

[39] Johnson RA, Oh DJ. Analytic embedded atom method model for bcc metals.Journal of Materials Research 1989;4:1195e201.

[40] Baskes MI, Nelson JS, Wright AF. Semiempirical modified embedded-atompotentials for silicon and germanium. Physical Review B 1989;40:6085e100.

[41] Daw MS. Model of metallic cohesion: the embedded-atom method. PhysicalReview B 1989;39:7441e52.

[42] Baskes MI. Modified embedded-atom potentials for cubic materials andimpurities. Physical Review B 1992;46:2727e42.

[43] Lee BJ, Baskes MI. Second nearest-neighbor modified-atom-method poten-tial. Physical Review B 2000;62:8564e7.

[44] Guellil AM, Adams JB. The application of the analytic embedded atommethod to bcc metals and alloys. Journal of Materials Research 1992;7:639e52.

[45] Dai XD, Li JH, Kong Y. Long-range empirical potential for the bcc structuredtransition metals. Physical Review B 2007;75:052102.

[46] Wadley HNG, Zhou X, Johnson RA, Neurock M. Mechanisms, models andmethods of vapor deposition. Progress in Materials Science 2001;46:329e77.

[47] Sutton AP, Chen J. Long-range Finnis-Sinclair potentials. PhilosophicalMagazine Letters 1990;61:139e46.

[48] Kaszkur Z. Modification of Finnis-Sinclair potential to account for surfaceeffect. In: the XVII congress and general assembly of the International unionof crystallography seattle. Washington: 1996.

[49] Dai XD, Kong Y, Li JH. Long-range empirical potential model: application tofcc transition metals and alloys. Physical Review B 2007;75:104101.

[50] Dai Y, Li JH, Liu BX. Long-range empirical potential model: extension tohexagonal close-packed metals. Journal of Physics-Condensed Matter 2009;21:385402.

[51] Li JH, Dai XD, Wang TL, Liu BX. A binomial truncation function proposed forthe second-moment approximation of tight-binding potential and applica-tion in the ternary NieHfeTi system. Journal of Physics-Condensed Matter2007;19:086228.

[52] Li JH, Dai Y, Dai XD, Wang TL, Liu BX. Development of n-body potentials forhcpebcc and fccebcc binary transition metal systems. ComputationalMaterials Science 2008;43:1207e15.

[53] Dai XD, Li JH, Liu BX. Molecular statics calculation of the formation enthalpyfor ternary metal systems based on the long-range empirical interatomicpotentials. Applied Physics Letters 2007;90:131904.

[54] Dai Y, Li JH, Che XL, Liu BX. Proposed long-range empirical potential to studythe metallic glasses in the NieNbeTa system. Journal of Physical Chemistry B2009;113:7282e90.

[55] Johnson RA. Relationship between two-body interatomic potentials ina lattice model and elastic constants. Physical Review B 1972;6:2094e100.

[56] Stott MJ, Zaremba E. Quasiatoms: an approach to atoms in nonuniformelectronic systems. Physical Review B 1980;22:1564e83.

[57] Norskov JK, Lang ND. Effective-medium theory of chemical binding: appli-cation to chemisorption. Physical Review B 1980;21:2131e6.

[58] Banerjea A, Smith JR. Origins of the universal binding-energy relation.Physical Review B 1988;37:6632e45.

[59] Johnson RA. Alloy models with the embedding-atom method. PhysicalReview B 1989;39:12554e9.

[60] Foiles SM. Calculation of the surface segregation of NieCu alloys with the useof the embedded atom method. Physical Review B 1985;32:7685e93.

[61] Johnson RA. Phase stability of fcc alloys with the embedded-atom method.Physical Review B 1990;41:9717e20.

[62] Gao N, Lai WS. Calculation of the elastic constants of Ag/Pd superlattice thinfilms by molecular dynamics with many-body potentials. Chinese PhysicsLetters 2006;23:2913e6.

[63] Chen JK, Farkas D, Reynolds WT. Atomistic simulation of an fcc/bcc interfacein NieCr alloy. Acta Materialla 1997;45:4415e21.

[64] Gong HR, Kong LT, Lai WS, Liu BX. Glass-forming ability determined by an n-body potential in a highly immiscible CueW system through moleculardynamics simulations. Physical Review B 2003;68:144201.

[65] Foiles SM, Daw MS. Application of the embedded atom method to Ni3Al.Journal of Materials Research 1987;2:5e15.

[66] Oh DJ, Johnson RA. Simple embedded atom for fcc and hcp metals. Journal ofMaterials Research 1988;3:471e8.

[67] Pasianot R, Savino EJ. Embedded-atom-method interatomic potentials forhcp metals. Physical Review B 1992;45:12704e10.

[68] Zhang BW. Theoretical calculation of thermodynamic data for bcc binaryalloys with the embedded-atom method. Physical Review B 1993;48:3022e9.

[69] Cai J, Ye YY. Simple analytical embedded-atom-potential model includinga long-range force for fcc metals and their alloys. Physical Review B 1996;54:8398e410.

[70] Pohlong SS, Rama PN. Analytic embedded atom method potentials for faced-centered cubic metals. Journal of Materials Research 1998;13:1919e27.

[71] Mishin Y, Farkas D, Mehl MJ, Papaconstantopoulos DA. Interatomic poten-tials for monoatomic metals from experimental data and ab initio calcula-tions. Physical Review B 1999;59:3393e407.

[72] Williams PL, Mishin Y, Hamilton JC. An embedded-atom potential for theCueAg system. Modelling and Simulation in Materials Science and Engi-neering 2006;14:817e33.

[73] Mitev P, Evangelakis GA, Kaxiras E. Embedded atom method potentialsemploying a faithful density representation. Modelling and Simulation inMaterials Science and Engineering 2006;14:721e31.

[74] Baskes MI, Johnson RA. Modified embedded atom potentials for hcp metals.Modelling and Simulation in Materials Science and Engineering 1994;2:147e63.

[75] Lee BJ, Shim JH, Baskes MI. Semiempirical atomic potentials for the fcc metalsCu, Ag, Au, Ni, Pd, Pt, Al, and Pb based on first and second nearest-neighbormodified embedded atom method. Physical Review B 2003;68:144112.

[76] Kin YM, Lee BJ, Baskes MI. Modified embedded-atom method interatomicpotential for Ti and Zr. Physical Review B 2006;74:014101.

[77] Ouyang YF, Zhang BW, JIn ZP, Liao SZ. The formation enthalpies of rareearthealuminum alloys and intermetallic compounds. Jom-Journal of theMinerals Metals & Materials Society 1996;87:802e5.

[78] Hu WY, Shu XL, Zhang BW. Point-defect properties in body centered cubictransition metals with analytic EAM interatomic potentials. ComputationalMaterials Science 2002;23:175e89.

[79] Hu WY, Deng HQ, Yuan XJ, Massahiro F. Point-defect properties in hcp rareearth metals with analytic modified embedded atom potentials. EuropeanPhysical Journal B 2003;34:429e40.

[80] Cyrot-Lackmann F. On the calculation of surface tension in transition metals.Surface Science 1968;15:535e48.

[81] Ziman JM. The physics of metals I electrons. Cambridge: CambridgeUniversity Press; 1969.

[82] Friedel J. On the band structure of transition metals. Journal of Physics F-Metal Physics 1973;3:785e94.

[83] Allan G, Lannoo M. Vacancies in transition metals: formation energy andformation volume. Journal of Physics and Chemistry of Solids 1976;37:699e709.

[84] Sutton AP, Finnis M, Pettifor WD, Ohta GY. The tight-binding bond model.Journal of Physics C: Solid State Physics 1988;21:35e66.

[85] Foulkes WMC, Haydockt R. Tight-binding models and density-functionaltheory. Physical Review B 1989;39:12520e36.

J.H. Li et al. / Intermetallics 31 (2012) 292e320 319

[86] Gupta RP. Lattice relaxation at a metal surface. Physical Review B 1981;23:6265e70.

[87] Tomanek D, Mukherjee S, Bennemann KH. Simple theory for the electronicand atomic structure of small clusters. Physical Review B 1983;28:665e73.

[88] Ackland GJ, Thetford R. An improved n-body semi-empirical model for body-centered cubic transition metals. Philosophical Magazine A 1987;56:15e30.

[89] Rebonato R, Welch DO, Hatcher RD, Bilello JC. Modification of the Finnis-Sinclair potentials for highly deformed and defective transition metals.Philosophical Magazine A 1987;55:655e67.

[90] Ackland GJ, Bacon DJ, Calder AF, Harry T. Computer simulation of point defectproperties in dilute FeeCu alloy using a many-body interatomic potential.Philosophical Magazine A 1997;75:713e32.

[91] Ackland GJ, Tichy G, Vitek V, Finnis MW. Simple n-body potentials for thenoble-metals and nickel. Philosophical Magazine A 1987;56:735e56.

[92] Dai XD, Kong Y, Li JH, Liu BX. Extended Finnis-Sinclair potential for bcc andfcc metals and alloys. Journal of Physics-Condensed Matter 2006;18:4527e42.

[93] Li JH, Kong Y, Guo HB, Liang SH, Liu BX. Proposed power-function n-bodypotential for the fcc structured metals Ag, Au, Cu, Ni, Pd, and Pt. PhysicalReview B 2007;76:104101.

[94] Kresse G. http://cms.mpi.univie.ac.at/vasp/vasp/vasp.html.[95] Segall MD, Lindan PJD, Probert MJ, Pickard CJ, Hasnip PJ, Clark SJ, et al. First-

principles simulation: ideas, illustrations and the CASTEP code. Journal ofPhysics-Condensed Matter 2002;14:2717e44.

[96] Foiles SM. Reconstruction of fcc (110) surfaces. Surface Science 1987;191:L779e86.

[97] Lee BJ, Baskes MI, Kim H, Cho YK. Second nearest-neighbor modifiedembedded atom method potentials for bcc transition metals. PhysicalReview B 2001;64:184102.

[98] Tyson WR, Miller WA. Surface free energies of solid metals: estimation fromliquid surface tension measurements. Surface Science 1977;62:267e76.

[99] Sundquist BE. A direct determination of the anisotropy of the surface freeenergy of solid gold, silver, copper, nickel, and alpha and gamma iron. ActaMetallurgica 1964;12:67e86.

[100] Grenga HE, Kumar R. Surface energy anisotropy of iron. Surface Science1976;61:283e90.

[101] Rose JH, Smith JR, Guinea F, Ferrante J. Universal features of the equation ofstate of metals. Physical Review B 1984;29:2963e9.

[102] Li JH, Liang SH, Guo HB, Liu BX. Four-parameter equation of state of solids.Applied Physics Letters 2005;87:194111.

[103] Kinslow R. High-velocity Impact phenomena. New York and London:Academic Press; 1970.

[104] Gray DE. American institute of physics handbook. New York: McGraw-Hill;1972.

[105] Hixson RS, Fritz JN. Shock compression of tungsten and molybdenum.Journal of Applied Physics 1992;71:1721e8.

[106] Cynn H, Yoo C-S. Equation of state of tantalum to 174 GPa. Physical Review B1999;59:8526e9.

[107] Kittel C. Introduction to solid state physics. 7th ed. New York: John Wily &Sons; 1996.

[108] David RL. CRC handbook of chemistry and physics: a ready-reference book ofchemical and physical data. Boca Raton: CRC Press; 2008.

[109] Nicklow RM, Gilat G, Smith HG, Raubenheimer LJ, Wilkins MK. Phononfrequencies in copper at 49 and 298 K. Physical Review B 1967;164:922e8.

[110] Kamitakahara WA, Brockhouse BN. Crystal dynamics of silver. Physics Letters1969;A29:639e40.

[111] Lynn LW, Smith HG, Nicklow RM. Lattice dynamics of gold. Physical Review B1973;8:3493e9.

[112] Birgeneau RJ, Cords J, Dolling G, Woods AD. Normal modes of vibration innickel. Physical Review 1964;136:A1359e65.

[113] Miller AP, Brockhouse BN. Crystal dynamics and electronic specific heats ofpalladium and copper. Canadian Journal of Physics 1971;49:704e23.

[114] Dutton DH, Brockhouse BN, Miller AP. Crystal dynamics of platinum byinelastic neutron scattering. Canadian Journal of Physics 1972;50:2915e27.

[115] Simmons G, Wang H. Single crystal elastic constants and calculated aggre-gate properties: a handbook. 2nd ed. Cambridge: MIT Press; 1971.

[116] Hu WY, Zhang BW, Huang BY, Gao F, Bacon DJ. Analytic modified embeddedatom potentials for HCP metals. Journal of Physics-Condensed Matter 2001;13:1193e213.

[117] Dai XD. Construction and application of N-body potential in binary andternary transition metal systems. Thesis: Materials Science and Engineering,Tsinghua University. Beijing: 2007.

[118] Dai XD, Zhao LF, Chen CS, Li JH. Molecular statics calculations of the phasestability for binary alloys based on the long-range empirical interatomicpotential. Journal of the Physical Society of Japan 2011;80:34602.

[119] Dai Y, Li JH, Che XL, Liu BX. Analytic calculation of formation enthalpiesdirectly from interatomic potentials for binary and ternary refractory metalsystems. Science China Technological Sciences 2010;53:2015e9.

[120] Massalski TB, Okamoto H, Subramanian PR, Kacprzak L. Binary alloy phasediagrams. Materials Park, OH: The Materials Information Society; 1990.

[121] Dai Y, Li JH, Che XL, Liu BX. Glass-forming region of the NieNbeTaternary metal system determined directly from n-body potential throughmolecular dynamics simulations. Journal of Materials Research 2009;24:1815e9.

[122] Zhang ZJ, Liu BX. Thermodynamics and crystallographic mechanism ofmetastable phase-formation in NbeFe, NbeCo, and NbeNi systems by ionmixing. Journal of Applied Physics 1994;75:4948e52.

[123] Zhang ZJ, Huang XY, Zhang ZX. Hexagonal metastable phase formation inNi3RM (RM ¼ Mo, Nb, Ta) multilayered films by solid-state reaction. ActaMaterialia 1998;46:4189e94.

[124] Villars P. Pearson’s handbook: crystallographic data for intermetallic phases.Materials Park, OH: ASM International; 1997.

[125] Zhang Q, Lai WS, Liu BX. Interfacial reaction, amorphization transition, andassociated elastic instability studied by molecular dynamics simulations inthe NieTa system. Physical Review B 2000;61:9345e55.

[126] Liu BX, Zhang ZJ. Formation of nonequilibrium solid-phases by ion irradia-tion in the NieTa system and their thermodynamic and growth-kineticsinterpretations. Physical Review B 1994;49:12519e27.

[127] Liu BX, Lai WS, Zhang Q. Irradiation induced amorphization in metallicmultilayers and calculation of glass-forming ability from atomistic potentialin the binary metal systems. Materials Science and Engineering R e Reoport2000;29:1e48.

[128] Dai Y, Li JH, Liu BX. Glass-forming ability and atomic-level structure of theternary AgeNieZr metallic glasses studied by molecular dynamics simula-tions. Journal of Applied Physics 2011;109:053505.

[129] Zhao SZ, Li JH, Liu BX. Formation of the NieZreAl ternary metallic glassesinvestigated by interatomic potential through molecular dynamic simula-tion. Journal of the Physical Society of Japan 2010;76:064607.

[130] Cui YY, Li JH, Dai Y, Liu BX. Prediction of favored and optimized compositionsfor CueZreNi metallic glasses by interatomic potential. The Journal ofPhysical Chemistry B 2011;115:4703e8.

[131] Cui YY, Wang TL, Li JH, Dai Y, Liu BX. Thermodynamic calculation andinteratomic potential to predict the favored composition region for theCueZreAl metallic glass formation. Physical Chemistry Chemical Physics2011;13:4103e8.

[132] Cui YY, Li JH, Dai Y, Liu BX. Interatomic potential to calculate the drivingforce, optimized composition, and atomic structure of the CueHfeAl metallicglasses. Applied Physics Letters 2011;99:011911.

[133] Inoue A, Zhang T, Masumoto T. ZreAleNi amorphous-alloys with high glass-transition temperature and significant supercooled liquid region. MaterialsTransactions Jim 1990;31:177e83.

[134] Becle C, Bourniquel B, Develey G, Saillard M. The intermetallic compoundNi3Zr. Journal of the Less-Common Metals 1979;66:59e65.

[135] Brochado Oliveira JMDC, Harris IR. Valency compensation in the Lavessystem, Ce(Co1�xNix)2. Journal of Materials Science 1983;18:3649e60.

[136] Bououdina M, Lambert-Andron B, Ouladdiaf B, Pairis S, Fruchart D. Structuralinvestigations by neutron diffraction of equi-atomic ZreTi(V)eNi(Co)compounds and their related hydrides. Journal of Alloys and Compounds2003;356e357:54e8.

[137] Havinga EE, Damsma H, Hokkeling P. Compounds and pseudo-binary alloyswith the CuAl2(C16)-type structure I. Preparation and X-ray results. Journal ofthe Less-Common Metals 1972;27:169e86.

[138] Meng WJ, Faber J, Okamoto PR, Rehn LE, Kestel BJ, Hitterman RL. Neutrondiffraction and transmission electron microscopy study of hydrogen inducedphase transformations in Zr3Al. Journal of Applied Physics 1990;67:1312e9.

[139] Hansen RC, Raman A. Zum aufbau einiger zu T4-B3 homologer und quasi-homologer systeme II. die systeme TitaneAluminium, ZirkoniumeAlumi-nium, HafniumeAluminium, MolydaneAluminium und einige ternaresysteme. Zeitschrift Fur Metallkunde 1962;53:548e61.

[140] Nandedkar RV, Delavignette P. On the formation of a new superstructure inthe zirconiumealuminium system. Physica Status Solidi (a) 1982;73:K157e60.

[141] Ma Y, Rømming C, Lebech B, Gjønnes J, Taftø J. Structure refinement of Al3Zrusing single-crystal X-ray diffraction, powder neutron diffraction and CBED.Acta Crystallographica Section B 1992;48:11e6.

[142] Rao PVM, Satyanarayana MK, Suryanarayana SV, Naidu SVN. Effect of ternaryadditions on the room temperature lattice parameter of Ni3Al. Physica StatusSolidi (a) 1992;133:231e5.

[143] Enami K, Nenno S. A new ordered phase in tempered 63.8Ni-1Co-Almartensite. Materials Transactions Jim 1978;19:571e80.

[144] Dutchak YI, Chekh VG. High-temperature X-ray diffraction study of thelattice dynamics of the compounds AlCo and AlNi. Russian Journal of PhysicalChemistry 1981;55:1326e8.

[145] Zumdick MF, Hoffmann R-D, Pöttgen R. The Intermetallic zirconiumcompounds ZrNiAl, ZrRhSn, and ZrPtGa e structural distortions and metal-emetal bonding in Fe2P related compounds. Zeitschrift Fur NaturforschungSection B 1999;54:45.

[146] Markiv VY, Matushevskaya NF, Rozum SN, Kuzma Yu B. Uber den aufbaueiniger zu TiAl3 verwandter legierungsreihen III. untersuchungen in einigenTeNieAle and TeCu-Al-systemen. Zeitschrift Fur Metallkunde 1965;56:99e109.

[147] Markiv VY, Krypyakevich PI, Belyavina NM. Dopovidi Akademii NaukUkrainskoi RSR Seriya A 1982;44:78.

[148] Ghosh G. First-principles calculations of structural energetics of CueTM(TM ¼ Ti, Zr, Hf) intermetallics. Acta Materialia 2007;55:3347e74.

[149] Ghosh G, Asta M. First-principles calculation of structural energetics ofAleTm (Tm ¼ Ti, Zr, Hf) intermetallics. Acta Materialia 2005;53:3225e52.

J.H. Li et al. / Intermetallics 31 (2012) 292e320320

[150] Ye D, Jiahoo L, Baixin L. Prediction of glass-forming ability and character-ization of atomic structure of the CoeNieZr Metallic Glasses by a proposedlong range empirical potential. Journal of Applied Physics 2012;111:033521.

[151] Zhao SZ, Li JH, Liu BX. Favored composition region for metallic glassformation and atomic configurations in the ternary NieZreTi system derivedfrom n-body potential through molecular dynamics simulations. Journal ofMaterials Research 2011;26:2050e64.

[152] Chen HS, Turnbull D. Formation, stability and structure of palladiumesiliconbased alloy glasses. Acta Metallurgica 1969;17:1021e31.

[153] Inoue A, Zhang T, Masumoto T. Glass-forming ability of alloys. Journal ofNon-Crystalline Solids 1993;156:473e80.

[154] Li JH, Dai Y, Cui YY, Liu BX. Atomistic theory for predicting the binarymetallic glass formation. Materials Science and Engineering R e Reoport2011;72:1e28.

[155] Turnbull D. Amorphous solid formation and interstitial solution behavior inmetallic alloy systems. Journal de Physique 1974;35:C4.1e4.10.

[156] Egami T, Waseda Y. Atomic size effect on the formability of metallic glasses.Journal of Non-Crystalline Solids 1984;64:113e34.

[157] Liu BX, Johnson WL, Nicolet MA, Lau SS. Structural difference rule for amor-phous alloy formation by ion mixing. Applied Physics Letters 1983;42:45e7.

[158] Liu BX. Ion mixing and metallic alloy phase formation. Physica Status Solidi(a) 1986;94:11e34.

[159] Weeber AW, Bakker H. Extension of the glass-forming range of NieZr bymechanical alloying. Journal of Physics F-Metal Physics 1988;18:1359e69.

[160] Bormann R, Gartner F, Haasen P. Determination of the free-energy ofmetallic glasses. Zeitschrift Fur Physikalische Chemie Neue Folge 1988;157:29e34.

[161] Bormann R, Gartner F, Zoltzer K. Application of the Calphad method for theprediction of amorphous phase formation. Journal of the Less-CommonMetals 1988;145:19e29.

[162] Zhang Q, Lai WS, Liu BX. Glass-forming ability determined by the atomicinteraction potential for the NieMo system. Physical Review B 1999;59:13521e4.

[163] Liu BX. Prediction of metallic-glass formation by ion mixing. MaterialsLetters 1987;5:322e7.

[164] Parrinello M, Rahman A. Polymorphic transitions in single-crystals e a newmolecular-dynamics method. Journal of Applied Physics 1981;52:7182e90.

[165] Gallego LJ, Somoza JA, Alonso JA, Lopez JM. Prediction of the glass-formationrange of transition-metal alloys. Journal of Physics F-Metal Physics 1988;18:2149e57.

[166] Tai KP, Wang LT, Liu BX. Distinct atomic structures of the NieNb metallicglasses formed by ion beam mixing. Journal of Applied Physics 2007;102:124902.

[167] Lai WS, Zhang Q, Liu BX. Solubility criterion for sequential disordering inmetalemetal multilayers upon solid-state reaction. Philosophical MagazineLetters 2001;81:45e53.

[168] Lee MH, Bae DH, Kim WT, Kim DH. Ni-based refractory bulk amorphousalloys with high thermal stability. Materials Transactions 2003;44:2084e7.

[169] Lee MH, Kim JH, Park JS, Kim JC, Kim WT, Kim DH. Fabrication of NieNbeTametallic glass reinforced Al-based alloy matrix composites by infiltrationcasting process. Scripta Materialia 2004;50:1367e71.

[170] Schroers J. Processing of bulk metallic glass. Advanced Materials 2010;22:1566e97.

[171] Inoue A, Zhang W. Formation, thermal stability and mechanical properties ofCueZreAl bulk glassy alloys. Materials Transactions 2002;43:2921e5.

[172] Wang XD, Jiang QK, Cao QP, Bednarcik J, Franz H, Jiang JZ. Atomic structureand glass forming ability of Cu46Zr46Al8 bulk metallic glass. Journal ofApplied Physics 2008;104:093519.

[173] Cheung TL, Shek CH. Thermal and mechanical properties of CueZreAl bulkmetallic glasses. Journal of Alloys and Compounds 2007;434e435:71e4.

[174] Kumar G, Ohkubo T, Mukai T, Hono K. Plasticity and microstructure ofZreCueAl bulk metallic glasses. Scripta Materialia 2007;57:173e6.

[175] Lu ZB, Li JG, Shao H, Gleiter H, Ni X. The correlation between shear elasticmodulus and glass transition temperature of bulk metallic glasses. AppliedPhysics Letters 2009;94:091907.

[176] Inoue A, Zhang T, Masumoto T. The structural relaxation and glass-transitionof LaeAleNi and ZreAleCu amorphous-alloys with a significant supercooledliquid region. Journal of Non-Crystalline Solids 1992;150:396e400.

[177] Zhang BW, Jesser WA. Formation energy of ternary alloy systems calculatedby an extended Miedema model. Physica B 2002;315:123e32.

[178] Wang D, Tan H, Li YY. Multiple maxima of glass-forming ability in threeadjacent eutectics in ZreCueAl alloy system e a metallographic way topinpoint the best glass forming alloys. Acta Materialia 2005;53:2969e79.

[179] Georgarakis K, Yavari AR, Louzguine-Luzgin DV, Antonowicz J. Atomicstructure of ZreCu glassy alloys and detection of deviations from idealsolution behavior with Al addition by X-ray diffraction using synchrotronlight in transmission. Applied Physics Letters 2009;94:191912.

[180] Cheng YQ, Ma E, Sheng HW. Atomic level structure in multicomponent bulkmetallic glass. Physical Review Letters 2009;102:245501.

[181] Yang JJ, Yang Y, Wu K, Chang YA. Amorphous alloy thin films as precursormetals of oxide tunnel barrier used in magnetic tunnel junctions. Journal ofApplied Physics 2005;98:74508.

[182] Yan HB, X. GF, Huang DQ. Evaporated CueAl amorphous alloys and theirphase transition. Journal of Non-Crystalline Solids 1989;112:221e7.

[183] Xu DH, Lohwongwatana B, Duan G, Johnson WL, Garland C. Bulk metallicglass formation in binary Cu-rich alloy series-Cu100�xZrx (x ¼ 34, 36 38.2, 40at.%) and mechanical properties of bulk Cu64Zr36 glass. Acta Materialia 2004;52:2621e4.

[184] Buschow KHJ. Thermal-stability of amorphous ZreCu alloys. Journal ofApplied Physics 1981;52:3319e23.

[185] Xia L, Fang SS, Wang Q, Dong YD, Liu CT. Thermodynamic modeling of glassformation in metallic glasses. Applied Physics Letters 2006;88:171905.

[186] Wang TL, Li JH, Liu BX. Proposed thermodynamic method to predict the glassformation of the ternary transition metal systems. Physical ChemistryChemical Physics 2009;11:2371e3.

[187] Gudzenko VN, Polesya AF. Structure of splat cooled from liquid-state zirco-niumealuminium alloys. Fizika Metallov i Metallovedenie 1975;39:1313e5.

[188] Yoshioka H, Habazaki H, Kawahima A, Asami K, Hashimoto K. Anodicpolarization behavior of sputter-deposited AleZr alloys in a neutral chloride-containing buffer solution. Electrochimica Acta 1991;36:1227e33.

[189] Zhao SZ, Li JH, Liu BX. Local structure of the ZreAl metallic glasses studied byproposed n-body potential through molecular dynamics simulation. Journalof Materials Research 2010;25:1679e88.

[190] Farahani RN. Monte Carlo simulation of structural and mechanical propertiesof crystal and bicrystal systems at finite temperature. Thesis: Department ofNuclear Engineering, Massachusetts Institute of Technology; 1980.

[191] Frenkel D, Simit B. Understanding molecular simulation: from algorithms toapplications. San Diego: Academic Press; 2002.

[192] Allen MP, Tildesley DJ. Computer simulation of liquids. Oxford: Clarendon;1987.

[193] Li JH, Zhao SZ, Dai Y, Cui YY, Liu BX. Formation and structure of AleZrmetallic glasses studied by Monte Carlo simulations. Journal of AppliedPhysics 2011;109:113538.

[194] Zhang T, Inoue A, Masumoto T. Amorphous (Ti, Zr, Hf)eNieCu ternary alloyswith a wide supercooled liquid region. Materials Science and Engineering A-Structural Materials Properties Microstructure and Processing 1994;182:1423e6.

[195] Yang H, Wang JQ, Li YY. Glass formation in the ternary Zr-Zr2Cu-Zr2Nisystem. Journal of Non-Crystalline Solids 2006;352:832e6.

[196] Hu C-J, Lee P-Y. Formation of CueZreNi amorphous powders with significantsupercooled liquid region by mechanical alloying technique. MaterialsChemistry and Physics 2002;74:13e8.

[197] Inoue A, Zhang W. Formation and mechanical properties of CueHfeAl bulkglassy alloys with a large supercooled liquid region of over 90 K. Journal ofMaterials Research 2003;18:1435e40.

[198] Jia P, Guo H, Li Y, Xu J, Ma E. A new CueHfeAl ternary bulk metallic glasswith high glass forming ability and ductility. Scripta Materialia 2006;54:2165e8.

[199] Li GQ, Borisenko KB, Cockayne DJH. Local structures of two metallic glasseswith good plasticity. Journal of Physics: Conference Series 2010;241:012066.

[200] Jia P, Xu J. Comparison of bulk metallic glass formation between CueHfbinary and CueHfeAl ternary alloys. Journal of Materials Research 2009;24:96e106.

[201] Haein C-Y, Park J-Y, An J-H. Development of CueHfeAl ternary systems andtungsten wire/particle reinforced Cu48Hf43Al9 bulk metallic glass compositesfor strengthening. Journal of the Korean Physical Society 2010;56:666e71.

[202] Nagy E, Rontó V, Sólyom J, Roósz A. Investigation of new type CueHfeAl bulkglassy alloys. Journal of Physics: Conference Series 2009;144:012035.

[203] Haein C-Y. Study on glass formation and properties of the CueHfeAl (Ti)alloy system. Journal of the Korean Physical Society 2009;54:200e3.

[204] El-Eskandarany MS. Solid-state amorphization reaction in mechanicallydeformed AlxHf100�x multilayered composite powders and the effect ofannealing. Journal of Alloys and Compounds 1999;284:295e307.

[205] Zhao SZ. Metallic glass formation and atomic configurations in some binaryand ternary metal systems studied by atomistic simulations. Thesis: Mate-rials Science and Engineering, Tsinghua University. Beijing: 2007.