NASA / TM--1998 -208665

Modeling of Substitutional Site Preference

in Ordered Intermetallic Alloys

Guillermo Bozzolo

Ohio Aerospace Institute, Cleveland, Ohio

Ronald D. Noebe and Frank Honecy

Lewis Research Center, Cleveland, Ohio

October 1998

https://ntrs.nasa.gov/search.jsp?R=19990009341 2018-05-11T17:16:23+00:00Z
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o
NASA/TM--1998-208665

Modeling of Substitutional Site Preference

in Ordered Intermetallic Alloys

Guillermo Bozzolo

Ohio Aerospace Institute, Cleveland, Ohio

Ronald D. Noebe and Frank Honecy

Lewis Research Center, Cleveland, Ohio

Prepared for the

"Interstitial and Substitutional Effects in Intermetallics" Fall Meeting

sponsored by the Minerals, Metals, and Materials Society

Rosemont, Illinois, October 11-15, 1998

National Aeronautics and

Space Administration

Lewis Research Center

October 1998
Acknowledgments

Fruitful discussions with N. Bozzolo are gratefully acknowledged. We would also like to thank C. Amador for

providing us with the first-principles input necessary for determining the BFS parameters used in this work.

NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076

Price Code: A03

Available from

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22100

Price Code: A03

II| TIV
MODELING OF SUBSTITU_ONAL SITE PREFERENCE IN ORDERED

INTERMETALLIC ALLOYS

Guillermo Bozzolo a, Ronald D. Noebe b and Frank Honecy b

aOhio Aerospace Institute, Cleveland, OH 44142, USA

b NASA Lewis Research Center, Cleveland, OH 44135, USA.

Abstract

We investigate the site substitution scheme of specific alloying elements in ordered compounds and the dependence of site occupancy on compound stoichiometry, alloy concentration.

This basic knowledge, and the interactions with other alloying additions are necessary in order to

predict and understand the effect of various alloying schemes on the physical properties of a material, its response to various temperature treatments, and the resulting mechanical properties. Many

theoretical methods can provide useful but limited insight in this area, since most techniques suffer from constraints in the type of elements and the crystallographic structures that can be modeled. With this in mind, the Bozzolo-Ferrante-Smith (BFS) method for alloys was designed to

overcome these limitations, with the intent of providing an useful tool for the theoretical prediction of fundamental properties and structure of complex systems. After a brief description of the

BFS method, its use for the determination of site substitution schemes for individual as well as

collective alloying additions to intermetallic systems is described, including results for the concentration dependence of the lattice parameter. Focusing on B2 NiA1, FeA1 and CoAl alloys, the

energetics of Si, Ti, V, Cr, Fe, Co, Ni, Cu, Zr, Nb, Mo, Ru, Hf, Ta and W alloying additions are

surveyed. The effect of single additions as well as the result of two simultaneous additions, discussing the interaction between additions and their influence on site preference schemes is considered. Finally, the BFS analysis is extended to ternary L12 (Heusler phase) alloys. A comparison

between experimental and theoretical results for the limited number of cases for which experimental data is available is also included.

\
Introduction

During thepastfew years,a convergenceof thecomputationalsciences,condensedmatter

physics,chemistryandmaterialssciencehasoccurredin suchawayasto providevaluabletools

andexpertisethat couldbecome of greatutility in thedesignof novel,technologicallyuseful

materials.In thepast,thedevelopmentof newstructuralalloysandevensmallimprovementsto

currentalloyshavemostlybeenperformedthroughextensiveexperimentaltrial anderror,which

is both costly and time consuming.This approachis slowly changing,due to the concurrent

effect of powerful computationaltechniquesfor differentlengthscales,which providesbasic

knowledgeneededateachstepof newmaterialdevelopment.

Of theseveralcomputer-basedtoolsnowavailableto the materials scientist, atomistic simulations based on sound theoretical methods have the potential to become useful at the early

stages of materials design, due to their ability to provide fundamental information at the atomic

level. Atomistic studies are broadly based on two types of approaches, first principles methods

and serniempirical techniques. These techniques are useful in separate but complementary

areas. First principles methods provide detailed electronic structure information, while semiempirical techniques are good for large scale simulations for the determination of bulk structural

properties. Ideally, first-principles approaches are best suited for providing the most accurate

and consistent framework for any study at the atomic level. However, the complexity of most

real life problems convoluted with the overwhelming computational requirements for even simple systems have, to date, prevented first principles methods from becoming common and economical predictive tools beyond their current use. In fact, very few calculations exist that go

beyond elementary binary alloys, and in some of these cases there are restrictions in scope and

accuracy. This is gradually changing, and it is expected that future growth in computer power

will correspondingly accelerate progress in this area. On the other hand, semiempiricai methods

based on quantum theory have enjoyed a great deal of attention in the last decade. This is mostly

due to their computational simplicity, resulting in substantial progress in areas previously inaccessible to theoretical treatment, such as alloy structure analysis and design. However, most of

the methods developed in the past have been severely restricted in the type and complexity of

the systems amenable to such studies.

IF| 1-
The purpose of these semiempirical methods is to provide an efficient but accurate way to

compute the total energy of arbitrary atomic systems in terms of the geometrical configuration of

the atoms, using approximations for the interactions between atoms. When applicable, the results

contribute a great deal to the general understanding of the system under investigation. However,

in most cases, the existing techniques are restricted to a limited number of systems or they require

specific (and therefore non-transferable) parameterizations or potentials. Naturally, it would be

desirable to overcome these limitations on the predictive power and applicability of these methods to increase their overall usefulness.

Recently, a new semiempirical method was developed with the goal of avoiding these limitations so that it could be adopted as a materials design tool. The Bozzolo-Ferrante-Smith (BFS) (1)

method for alloys, based on quantum perturbation theory, is particularly designed to deal with

complex systems and geometries. It has no constraints in its formulation that would limit the type

or number of elements under consideration or on the number or type of resulting phases. The two

distinguishing features of BFS are its novel way of modeling the alloy formation process - by

breaking the process into two coupled transformations - and its extremely simple mathematical

formulation, requiting the solution of a single transcendental equation for each transformation.

These features of the BFS method are particularly useful when dealing with multicomponent systems, as will be shown in the present work dealing with the role of alloying additions to high temperature ordered intermetallics.

It is now well recognized that alloying additions to structural intermetallics are essential for

the optimization of physical, chemical and mechanical properties. However, the role of these

additions in controlling properties is poorly understood due to the lack of detailed microstructural

information. As an example, in a NiAI+Ti system, Ti additions in the order of 2.5 to 3.0 at. % have

been shown to result in a 200 to 5000 fold reduction in creep rate as compared to that of the binary

NiA1 (2). However, from a microscopic point of view, this behavior is not clearly understood, thus

limiting our ability to modify and further improve the alloy (3).

In this work we take advantage of the versatility of the BFS method to tackle the problem of

site substitutions of alloying additions to ordered intermetallics. Concentrating on transition metal

rich B2 intermetallic alloys (NiA1, FeAI, CoAl), we studied the change in physical properties due

to solid solution alloying additions, determined the substitutional site preference schemes, as well

as interactions and changes due to multiple additives. To demonstrate the versatility of the BFS
methodforthistypeof applications,wealsoanalyzedthesubstitutionalbehaviorof alloyingadditions in a selectedgroupof Ni2(A1,X,Y)Heusleralloys.

The BFS Method

Since its inception a few years ago, the BFS method has been applied to a variety of problems

(1), ranging from bulk properties of solid solution fcc and bcc binary alloys to more specific applications like the energetics of bimetallic tip-sample interactions in an atomic force microscope

and the effect of stoichiometry on the defect structure of NiAI (4) and FeA1 alloys (5). The BFS

method has also been used to deal with alloy surfaces (1). These previous studies provide a foundation for the work presented in this paper.

An interesting consequence of the simple mathematical formulation of the BFS method is that

simple expressions can be derived for predicting the composition dependence of some bulk alloy

properties based solely on pure component properties (the BF rule) (6). As it will be shown later

in this work, these tools will become particularly useful when dealing with multiple additions to

binary or higher order alloys (e.g. Ni2A1Ti ).

In what follows, we provide a brief description of the operational equations of BFS in order to

introduce some concepts used later on. The reader is encouraged to seek further details in previous papers (1,4-6) where a detailed presentation of the foundation of the method, its basis in perturbation theory and a discussion of the approximations made are clearly explained.

The BFS method provides a simple algorithm for the calculation of the energy of formation of

an arbitrary alloy AH (the difference between the energy of the alloy and that of its individual constituents). In BFS, the energy of formation is written as the superposition of individual contributions of all the atoms in the alloy,

AH : _(e'i-ei)= _Ei,

i i

(l)

where ei is the energy of an atom i in a pure elemental crystal and e'i is the energy of the same

atom in the alloy being studied. For each atom, we partition the contribution e, to the energy of

formation AH into two parts: a strain energy and a chemical energy contribution. The first contri4

1[| _I!
bution takesinto accountthe atomicpositionsof theneighboringatomsto atom i, regardless of

their chemical identity. For its calculation, we use the actual geometrical distribution of the atoms

surrounding atom i, computed as if all of its neighbors were of the same species as atom i. In this

sense, the BFS strain energy differs from the commonly defined strain energy in that the actual

chemical environment is replaced by that of a single-element crystal. Its calculation is then

straightforward and even amenable to first-principles techniques. In our work, we use Equivalent

Crystal Theory (ECT) (7) for its computation, due to its proven ability to provide accurate and

computationally economical answers to most general situations.

The chemical environment of atom i is evaluated in the computation of the BFS chemical

energy term, where the surrounding atoms are forced to occupy equilibrium lattice sites corresponding to the reference atom i. Building on the concepts of ECT, BFS implements a straightforward approach for the calculation of the chemical energy, properly parameterizing the interaction

between dissimilar atoms.

Thus defined, the BFS strain and chemical energy contributions take into account different

factors, i.e. geometry and composition, computing them as isolated effects. A coupling function g

restores the relationship between the two terms. The coupling function is defined in such away as

to properly consider the asymptotic energy behavior where chemical effects are negligible for

large separations between dissimilar atoms. Consequently, the contribution ei to the energy of formation Zk/-/from an individual atom i is then given by

_'i = _S+g c. (2)

In what follows, we provide a brief description of the essential steps in the calculation of e,,

which should be complemented with additional details described at length in Ref. 1. The strain

energy contribution is obtained by solving the ECT perturbation equation (7)

NR_e_aR_ + MR_e-(a+ _)R_

= --.,)fr_je-(_+S(r ))rj (3)

]
where N and M are the number of nearest- and next-nearest neighbors respectively, and where p, l,

and _ are ECT parameters that describe element i (see Ref. 7 for definitions and details), r

denotes the actual distance between the reference atom and each of its neighbors and S(r)

describes a screening function (7). The ECT parameters, together with the LMTO results,

uniquely describe the physical properties of the element: p is related to the principal quantum

number n ( p = 2n-2), a parameterizes the electron density in the overlap region between two

atoms, 1 is a screening parameter that takes into account atoms located at distances greater than

nearest-neighbor distances, and 1 is a scaling parameter that ensures that the dependence of the

binding energy per atom as a function of lattice parameter follows satisfies the universal binding

energy relationship of Rose et al. (8). The sum runs over nearest and next nearest neighbors. Eq.

3 is solved for a s, the lattice parameter of the strain equivalent crystal, where the reference atom i

has the same energy as it has in the geometrical environment of the alloy. R 1 and R 2 denote the

nearest- and next-nearest neighbor distances in this equivalent crystal and are therefore related to

the unknown a S . Once the lattice parameter of the strain equivalent crystal a S is determined, the

BFS strain energy contribution is computed using the universal binding energy relation of Rose et

al. (8), which contains all the relevant information concerning a single-component system:

I_s = EcI1-(1 +as)e --as)

(4)

where EC is the cohesive energy of atom i and where the scaled lattice parameter a s is given by

• (a s- ae)

as = q I (5)

where q is the ratio between the equilibrium Wigner-Seitz radius and the equilibrium lattice

parameter a e. The values of the equilibrium lattice parameter a e, the cohesive energy E c, and the

bulk modulus B o for the bcc phase of all the elements used in this work are displayed in Table I.

These parameters are obtained from first principles, all-electron, density functional calculations of

the elemental constituents in the symmetry of the alloy (i.e., bcc-Ni, bcc-A1, etc.). The particular

6
LMTO results

Atom Lattice Cohesive Bulk

Parameter Energy Modulus

(/_) (eV) (GPa)

Ni 2.752 5.869 249.2

AI 3.192 3.942 77.3

Cr 2.837 4.981 286.1

Ti 3.213 6.27 121.0

Cu 2.8225 4.438 184.55

Co 2.7347 6.406 285.81

Zr 3.5525 7,353 93,77

Hf 3.4942 7.411 113.02

Ta 3.3091 9.310 205.06

Nb 3.3165 7.844 179.41

Si 3.0701 4.654 99.99

Fe 2.7423 6.071 311.53

Mo 3.1781 7.195 273.54

V 2.9770 7.512 199.01

W 3.1932 9.953 315.98

Ru 3.0700 7.664 338.32

ECT parameters

p ot _, 1

(A -1) (m -x) (/_)

3.067 0.763 0.2716

1.8756 1.038 0.3695

2.8580 0.646 0.2300

2.8211 1.0476 0.3728

3.0492 0.7615 0.2710

3.4779 0.7469 0.2658

3.4641 1,2257 0.4362

4.3132 1.1302 0.4022

4.4230 0.9664 0.3439

3.5099 0.9472 0.3371

6

4

6

6

6

6

8

10

10

8

4

6

8

6

10

8

1.9773 1.0158 0.3615

3.0939 0.6955 0.2475

3.5050 0.7506 0.2671

3.0050 0.9290 0.3306

4.4395 0.8194 0.2916

3.6116 0.7086 0.2522

Table 1. Linear-muffin tin orbital (LMTO) results for the lattice parameter, cohesive energy and

bulk modulus for the bcc phases of all elements listed. The last four columns display the resulting Equivalent Crystal Theory (ECT) (7) parameters determined from the LMTO results, p is

related to the principal quantum number n for the atomic species considered (p=2n-2), a parameterizes the electron density in the overlap region between two neighboring atoms, _, is a screening factor for atoms at distances greater than nearest-neighbor distance and l is a scaling length

needed to fit the lattice parameter dependence of the energy of formation with the universal

binding energy relationship of Rose et al. (8).
implementation used is the Linear-Muffin-Tin Orbitals (LMTO) method (9) in the Atomic Sphere

Approximation. The ensuing ECT parameters p,/, _,, and a e are also included in Table 1.

The BFS chemical energy is obtained by a similar procedure. As opposed to the strain energy

term, the surrounding atoms retain their chemical identity, but are forced to be in equilibrium lattice sites of an equilibrium (otherwise pure) crystal i. The BFS equation for the chemical energy

is given by

N_pi -tx,R, + MRP,e-( _' + _)R2

1( 1 e

= _k _lVikrlk e +Mikr2k e )

(6)

where Nik and Mik are the number of nearest- and next-nearest neighbors of species k of atom i.

The chemical environment surrounding atom i is reflected in the parameters O_ik, given by

O_ik = _i + Aki (7)

Ni

A1

Cr

Ti

Cu

Ni

.09160

.22482

.4958O

-.01708

AI

-.06078

-.01696

.23399

-.04993

Cr

-.03588

-.01524

.07343

-.01180

Ti

-.09062

-.08649

-.06107

-.07356

Cu

.01914

.05438

.02449

.20565

Co

-.05384

.17971

.09571

.67092

.13897

Zr

-.12210

-.06367

-.11502

-.05062

-.06767

Hf

-.12681

.25061

-.08881

.36712

.13561

Co .14364 -.06792 -.02624 -.09593 -.05734 -.06399 -.13074

Zr 1.00755 -.04861 .40891 -.01583 -.04778 -.05068 .14942

Hf .45749 -.07141 .01656 -.09376 -.07043 .54726 -.07487

Table 2 BFS interaction parameters axy (first subindex indicates the row, the second indicates the

column) and arx (in A'l)for X-Y alloys (X, Y = Ni, A1, Cr, Ti, Cu, Co, Zr, Hf), determined by fitring the lattice parameter and energy of formation of the corresponding B2 compounds via

LMTO.

8

ii[| _1!rr
wheretheBFSparametersAki (a perturbation on the single-element ECT parameter _ti used in Eq.

3) describe the changes of the wave function in the overlap region between atoms i and k. Once

Eq. 6 is solved for the equivalent chemical lattice parameter a c, the BFS chemical energy is computed using

Ec = i ( ,. "YiE¢ 1 - (1 + ac)e ) (8)

where Yi = 1 if a*c>O and _'i = -1 if a'c<0. The scaled chemical lattice parameter is given by

• (a c - ae)

a c = q. li (9)

Finally, as mentioned above, the BFS chemical and strain contributions are linked by a coupling function g which describes the influence of the geometrical distribution of the surrounding

atoms on the relevance of the chemical effects, and is given by

-a;

gi = e (10)

where the scaled lattice parameter a s is defined in Eq. (5).

In this work we use the BFS interaction parameters, Ava, determined following the procedure

outlined in Ref. 4. In order to provide parameters to the BFS method, we need to calculate the

equilibrium properties of the elemental solid for the same symmetry as the compound to be studied, since BFS is referenced to the ground state properties of the system in that symmetry. Once

these parameters are computed, they remain the same for any other calculation involving any of

these elements as related to the given crystal symmetry, requiring no further adjustment or

replacement. A partial list of the most relevant parameters is included in Table 2.

9
Results and Discussion

a. Single alloying additions to binary B2 compounds

While many aluminide-based intermetallic compounds have high melting temperatures,

excellent oxidation resistance and low density, they are limited in actual structural applications

due to the lack of sufficient room temperature ductility and elevated-temperature creep strength.

Ternary and higher order elemental additions can alter these basic properties and, therefore, a general understanding of the substitutional site preference schemes could be of great help in the alloy

design process. In this section, we perform BFS analyses of the site substitution patterns of ternary and higher order additions to NiAI, FeA1 and CoAl.

The results quoted in this paper are based on BFS calculations of the energetics of a 72-atom

cell in the AB B2 structure. This number of atoms is sufficient to study site substitutions of relevance to this work. A and B represent the two simple cubic sublattices of the B2 compound and X

represents a ternary alloying addition. From the basic 72-atom computational cell we build a catalogue of several atomic configurations, each describing a specific site substitutional scheme. We

introduce a convenient notation for such configurations: X(A) denotes an atom X substituting for

8- -_4- -.4_- - Q

4>'--,-4_ -I-e -,- O" i

!

4)

• Ni • A1 • X

Q- - 41- -_Q- - •

g'_l-O"_l-• -i- t1" i

I

Q

Q- - 4- -.4- - 4}

_ I

I

.I

X(A1) X(Ni)A1NN X(Ni)Alfar

(a) (b) (c)

Figure 1: (a) An X atom occupying an A1 site, (b) a Ni site with the Ni atom occupying the available A1 site, at nearest-neighbor distance from each other, and (c) same as (b), but with the substitutional and antistructure atoms separated by a distance greater than the equilibrium nearestneighbor distance. Ni and A1 atoms are denoted by black and grey circles, respectively. The alloying addition is denoted with a solid square.

10

It| ]:i
an atom A on the A sublattice, and X(A)B denotes the same case but with the displaced A atom

occupying a site in the B sublattice. In this second case, we distinguish between the pair of defects

X-A being nearest neighbors of each other (labeled NN in the figures) and the pair being separated by distances greater than that (labeled 'far' in the figures). Fig. 1 shows the simplest case,

when just one atom (representing the alloying addition) occupies either an A1 site (X(A1), Fig. 1.a)

or a Ni (or Fe, Co in FeAI or CoAl alloys) site (X(Ni)AI), with the Ni atom forming an antistructure defect by occupying an A1 site in a NisoA148.61XI.39 alloy. Figs. 1.b and 1.c distinguish

between the different relative locations of the substitutional atom and the antistructure atom. For

simplicity and also because most alloying additions are made at the expense of A1, we will hold

the transition metal composition constant at 50 at. % in most of the calculations unless otherwise

noted. We could just as easily do the same calculations for substoichiometric transition metal

compositions by taking into account vacancies in the calculations but space and actual interest in

transition metal poor compositions preclude a discussion of those alloys in the present paper.

-0.35

>

©

"_ -0.4:

0

ow,i

-0.55.

O

-0.65

Ni50(A148.61X 1.39)

x(m)

X(Ni)A1NN

X(Ni)AIf

u _ -- _ _ mm m

u

R

R

m

Fe Ru Co Cu Si Cr Zr Hf Ti v Nb Mo Ta W

X

Figure 2: Energy spectrum for alloying additions to Niso(A1,X)5 o. The thick solid line corresponds to X(A1) defects (see Fig. l.a), whereas thin solid lines and dashed lines correspond to

X(Ni)A1 defects as shown in Figs. 1.b and 1.c, respectively.

11
Thesitepreferenceof thealloyingadditionis determinedby computingtheenergyof formationof eachcell andcomparingthevaluesobtained.However,determiningtheconfigurationwith

the lowest defectenergymight not be sufficientfor understandingthe precisebehaviorof the

additive.While it is true that the configurationwith the lowestenergyis the most likely to be

found,it is importantto considerthedifferencein energybetweenconfigurationsdescribingdifferenttypesof substitutionpatterns.If thedifferencebetweenX(AI) andX(Ni)A1 is significant,it

is to beexpectedthatatomsX will alwaysoccupyA1sites.Experimentaldifficulties in determining thesitepreferenceof certainelementscouldberelated,amongotherfactors,to thesizeof this

energygap.Moreover,wewill seelaterthattheinterplaybetweenseveralsimultaneousadditions

candrasticallychangethebehaviorobservedwheneachadditionis consideredindividually, making theanalysisof higherenergyconfigurationsevenmoreimportant.

NiAI: In orderto highlight theenergydifferencesbetweenvariousatomic arrangements,we

presenttheresultsin theform of an'energyspectrum'.Fig. 2showsaspectrumof theenergiesof

formationof the configurationsdescribedin Fig. 1 (for oneatomof thealloying additionin the

72-atomcell), for severaldifferentelementaladditionsto NiA1. A wide rangeof behaviorsis

immediatelyapparent.In orderto categorizethedifference,wegroupedthevariouselementsconsideredby themagnitudeof theenergygap,G (definedasthedifferencein energybetweenthe

lowestenergylevelandits immediatelyhigherenergylevel,relativeto the energyof thelowestlying state).In doingso,wefindfour distinctgroups:(a)FeandRu,with agapof lessthan1%,a

negligibleenergydifferencethatis well within thefluctuationsintheparametersusedfor thiscalculation,(b) Co and Cu, showinga small but distinguishablegap,(c) Si,Cr, Zr, Hf with a well

definedgapand(d) Ti, V,Nb, Mo, TaandW with a gapthat is over20%of theenergyof formation of thedefect. Theselasttwo groupsarealsocharacterizedby the factthat all the additives

consideredmostlyoccupyA1sites.

In orderto considerthepossibility of changesin sitepreferencewith changesin concentration, we performedsimilar calculations,alsoin a 72-atomcell, for a numberof configurations,

correspondingto a Ni50A147.22X2.38alloy. It shouldbe notedthat evenif this concentrationis

greaterthanthe solubility limit for X in NiA1 alloys,it is feasibleto computetheenergeticsof X

in solidsolutionin NiAI, in spiteof thefactthatasecondphasemightform at thatconcentration.

The calculationis basedon a predefinedatomicdistribution,which may or may not reflectthe

correctgroundstatestructurefor thatcomposition.Thisissuehasbeenaddressedin ourprevious

12

II_I
work ontheeffectof Ti andCr additionsto NiA1(10).In thiswork,wedisregardthepossibilityof

precipitateformationfor thesakeof investigatinggeneraltrendsfor thissetof candidatealloying

additions.

Fig. 3 displayssomeof the possibleconfigurationsincludedin the catalogueof possible

structurescomposedof two soluteatomsin a72-atomcell, showingonly thosesectionsof thecell

affectedby thepresenceof thealloyingadditionsandthedifferentresultingdefectstructures.Fig.

4 showsthe energyspectrumcorrespondingto this set of configurations.Becauseof the larger

Y50A147.22X2.78

Y = Ni, Fe, Co X= Ti, Cr, Cu, Hf, Ta, Nb, Zr, Si, Mo, V, W, Ru, Ni, Fe, Co

4-- -.O- --O-- Q

O_ t- 0 "---r --Q-'-,- •"

_ I

I

8

8---•-- _O --- Q

.8- - -4- -;Q - - • •_l" 0-" "r "_ -i-I1" I

2X(A1)N N 2X(A1) f 2X(Y)A1NN

•- _-•- - _6 - _•

_ r"IB- "r-II_-I- II i

_ I

I

_" "" [""• " T "'_'--I -_" I

_ I

I

2 X(Y)AlNN 2X(Y)Alf

• - _-.•- -_-•- - •

• _ ¢-•- T -Q_ -m-•" i

[x(Al)+x(Y)Al3 a 

• "- r •: T ÷'-I -|" I

_ I

I

[X(A1)+X(Y)AINN]NN

•---41---•--11

_ I

I

[X(AI) +X(Y)AINN] f

• "---F-•_ T -I'_-,--•" j

[x(na)+x(Y)Alrlf

Figure 3: Configurations for two atoms in a 72-atom cell. A section of the cell containing the

defect is shown. Y atoms (Ni, Fe, Co in NiAI, FeAl and CoAl, respectively) are indicated with

solid black disks, A1 atoms with grey disks and the additive X with a solid square.

13
numberof possibledefectsdueto theirrelativelocationin thelattice,the spectrumnow includes

'bands' of states corresponding to the three main types of defects: the two solute atoms occupying

A1 sites (thick lines), the two solute atoms occupying Ni sites with the displaced Ni atoms in AI

sites (dashed lines), and a combination of both type of defects (thin solid lines). This energy spectrum can be seen as an extension of the one shown in Fig. 2. A comparison with the single atom

additions (lower concentration) provides information on the changes in the energy gaps due to the

©

O

t_

O

O

©

m

-0.15

n

-0.25

-0.45 m

-0.65

Ni50A147.22X2.78

_,_. 2X(A1)

X(A1) + X(Ni)A1

_._ 2X(Ni)A1

g. Bn

m . i m i,

m w . m w

__ . _m m

I . m

m . w m . w

! . _u - I

m_ m.w

m

m

m

m m

w - --

D - _ u .....

_m _ _im _-

m

w

l m

Fe Ru Co Cu Si Cr Zr Hf Ti V Nb Mo Ta W

X

Figure 4: Energy spectrum of the configurations shown in Fig. 3. For each element, the different

energy states are grouped depending on the type of defect: two X atoms in A1 sites (solid thick

line), one X atom in an A1 site and one in a Ni site (thin solid line) and both X atoms in Ni sites

(dashed lines). In those cases where the X atom occupies a Ni site, an antistructure defect is also

created. The different levels for the same type of defect correspond to the differences due to the

relative location of the defects with respect to each other.

14

II| l il
Ni50(AI+X)50

Ni

sublattice

No strong

preference

A1

sublattice

Co,V Ti,Cr,Fe,Cu Zr,Nb,Mo,Hf,Ta,W

Fe,Co,Cu Ti, V,Cr, Zr,Nb,

Hf, Ta,W

0.1 eVdifference Fe

betweenFe(A1),

Fe(Ni)A1

Co Cr,Fe,V.

(for Co

XAi >42.8at%) (for XAl< 42.8 at %)

Ti, V, Cr, Zr, Nb, Si

Hf, Ta, W ( both

sites when unfilled

with constituents)

Fe, Co, Cu

Fe

Ti, V, Cr, Zr, Nb,

Mo, Hf, Ta, W

Cr

Experiment/Theory

Theoretical model (11)

Allaverdoba et al.

Constitution diagrams (12)

First-principles calculations (13)

Fu and Zou

Theoretical Model (14) Kao et al.

Pseudo ground state model (15)

Hosoda et al.

ALCHEMI (16) Anderson et al.,

X-ray (EXAFS)(17)Chartier et al,

Atom probe field ion microscopy

(18) Duncan et al.

X-ray diffraction analysis (11)

AUaverdoba et al.

ALCHEMI (19)Field et al.,

ALCHEMI (20)Cotton et al.

Ti ALCHEMI (2) Kitabjian et al.

V ALCHEMI (21), Munrow et al.

Co Fe, Ru, Cu (0.01

eV difference

between X(AI)/

X(Ni)A1)

Fe, Ru, Cu, Si, Cr,

Zr, Hf, Ti, V, Nb,

Mo, Ta, W

BFS method (this work)

Table 3 Comparison between experimental (light shaded cells) and theoretical results for site preference of alloying additions in Ni(AI+X) alloys.

15
higher concentrationof the alloying additions,andits effecton the sitepreferencescheme.For

example,a largernumberof Ti atomsleavesthe energyof theTi(A1) level almostunchanged,

while theTi(Ni)AI levels(dashedlinesin Fig. 2, dashed/dottedlinesin Fig. 4) areclearlypushed

towardhigherenergies,increasingthepreferenceof Ti for A1sites.Fig. 4 alsoshowsthat,for the

atomicspeciesconsideredandtherangeof concentrationsstudied,thereis nochangein thebasic

sitepreferenceschemewith concentration.

A comparisonof theBFSresultswith experimentandothertypesof theoreticalcalculationsis

providedin Table3. In general,theBFSresultsarein excellentagreementwith experimentand

areconsistentwith othertheoreticalcalculations.Unfortunately,few experimentalresultsexist

andmost theoreticalcalculationshavebeenperformedfor a rathersmall numberof elements,

limiting thecomparisonto the fewcasesavailable.Still, theavailabledatais sufficientto put the

BFS techniqueintoproper perspective.

Thereis agreementbetweenall thetheoreticalmodelsandtheexperimentalresultswith BFS

predictionsfor thesitepreferenceof Co,with theexceptionof thepseudogroundstatemodelof

Hosodaet al. (15),wherea critical composition(XAI= 42.8at.% A1)is determinedfor achange

in sitepreferencebehavior.Abovethis critical concentration,Co is expectedto occupyNi sites_

The sitepreferenceof Fe in NiA1 hasbeenthesubjectof severaldetailedstudiesdueto the difficulty in establishingits behavior.In our work,Feis foundto haveavery smallvalueof G, with a

slight preferencefor A1sitesatzerotemperature.Thisis in agreementwith thedetailedfirstprinciples study of Fu and Zou (13), who found a small value(0.1 eV) for the energydifference

betweenthe two typesof defect.Severalexperimentalstudies(16-18)confirmthepreferenceof

Fe for A1sites.TheBFSresultsfor all theotheralloyingadditions(Si, Cr,Zr, Hf, Ti, V,Nb, Mo,

TaandW) arein agreementwith all theavailableexperimentalstudies: Cr (11,19-20),Ti (2, 11),

V (11, 21)andZr, Nb,Mo, Hf, Ta,W (11).

For completeness,we summarizethe numericalresultsfor all theNi(A1,X) alloy configurations in Table4. For eachentry,we list theenergyof formation(in eV/atom)andtheequilibrium

latticeparameterof theresultingalloy.This valueof theequilibrium latticeparameteris obtained

by minimizingtheenergyof formationof everyconfigurationwith respectto isotropicexpansions

or compressionsof theperiodiccell. Thechangesin latticeparameterwith elementaladditionsto

Ni(A1,X)areshownin Fig. 5. Theresultsobtainedcanbeusedto parameterizelinearexpressions

for thedependenceof thelatticeparameterwith concentrationa(x). In order to free these expres16

NiA1

2.846

Ta(0.40389)

(0.39379)

,Mo (0.36350)

Nb (0.32311)

1612)

.__V (0.02777)

Si (-0.03029)

(-0.04039)

(-0.04544)

(-0.07320)

_Fe (-0.10097)

I

1.39

Concentration of X (at. %)

Figure 5: Lattice parameter as a function of concentration of several alloying additions X ( X =

Si, Ti, V, Cr, Fe, Co, Cu, Zr, Nb, Mo, Ru, Hf, Ta,W) to NiA1. The values of _t (Eq. 11) are given in

parenthesis.

sions from their dependence on the end values (i.e. equilibrium lattice parameter of binary NiA1,

aNiAl determined by first principles methods (4)), we normalize a(x) by aNiAl, obtaining the general expression

a(x____)= 1 +_tx. (11)

aNiAl

where x is the concentration of the allowing addition. The values of _t are included in Fig. 5.

As mentioned before, the BFS method also allows for the derivation of simple expressions

that provide an approximate value of _ based only on the single-element properties (lattice

18

!_| 1 7
parameter,bulk modulusandcohesiveenergy,aslistedin Table1).TheBF rule (6) for thelattice

parameterof thealloy asafunctionof concentrationof eachelementis

ExiBia_

a(x) = i (12)

E xiB ia i

i

where a i and B i are the equilibrium lattice parameter and the bulk modulus, respectively, of element i in the symmetry of the alloy. In the particular case studied in this work, XAIso_xY x, with

X = Ni, Fe, Co, Eq. (12) becomes

1 2 2 2

_oxBx + (2- x)aAIBAI + XOyBy (13)

a(x) =

_axBx + (_- x)aalB al + XarBy

0

4....I

>

0

ow...l

O

-0.15_

-0.20

-0.30_

Fe50A148.61X1.39

._- X(A1)

X(Fe)AINN

X(Fe) Alf

m

m

m

m m

i

m

l

Ni Ru Co Cu Si Cr Zr Hf Ti V Nb Mo Ta W

X

Figure 6: Energy spectrum for Fe (AI+X) alloys with alloying additions of X. See Fig. 2 for

details concerning the type of defect structures modeled..

1-9
from which the approximatevalue of _t can be derived.Theseexpressionsdo not take into

accountspecificorderedstructures,thereforeleaving out the strongbonding characteristicsof

theseB2 compounds.However,in thecaseof severalalloying additions,it is obviouslyapractical wayof determininganapproximationfor thebehaviorof thelatticeparameterwith composition.

FeAI: Similar calculations were carried out for Fe(AI+X) alloys, following the same procedure used for NiA1. Fig. 6 displays the energy spectrum for a low concentration of the alloying

addition X. The numerical results for all possible configurations, both for low and high concentration, are listed in Table 5 As for NiAI, the different alloying additions are ordered according to the

relative magnitude of the energy gap between the two main site occupancy choices. Ni and Ru

2.920 (0.29191)

FeAI

Mo (0.22017)

(0.19296)

18801)

Zr (0.14101)

(0.03463)

Ru (0.00742)

v (-0.03448)

Cr, Cu (-0.10390)

Si (-0.11875)

Ni (-0.19296)

Co (-0.24491)

I

1.39

Concentration of X (at. %)

Figure 7: Lattice parameter as a function of concentration of several alloying additions X to

20
<_

•,"q C_ Z

e:_

.M.q

o

. ,...d

o_.._

Lt_

21
exhibit a negligible gap G <3 % while Co and Cu show a clear distinction between the two possible levels ( 3 < G < 10 %). All the other elements display a clear preference for A1 sites, with Si,

Cr, Zr and Hf having a distinct gap ( 10 < G < 30 %) and the rest (Ti, V, Nb, Mo, Ta, W) showing

preference for A1 sites with excessively large gaps (G > 70 %).

Completing the presentation of results for FeAI, Fig. 7 shows the change in lattice parameter

with concentration for all the alloying additions considered in this work, also showing the corresponding slopes _t of the linear dependence of the lattice parameter on the ternary addition with

concentration. Again, the slopes of the linear expressions for a(x) were normalized to the LMTOdetermined value for the lattice parameter of FeAI,

a(x) _ 1 + l.tx (14)

aFeA1

where x is the concentration of the alloying addition. The values of _t are included in Fig. 7.

Unfortunately, very few theoretical or experimental results exist for Fe(AI+X) alloys. The

preference of Ni and Co for Fe sites agree with the theoretical results of Allaverdova et al.

(11).They also predict the same site preference for Cu and Cr, although they admit that their prediction for Cr is questionable. The small energy gap found for Cu in this work could be an explanation of the ambiguity found in their work. That is not the case for Cr, where BFS predicts a

large gap favoring A1 sites, as found experimentally by synchrotron X-ray diffraction experiments

by Khosla (22), confirmed by the work of Munroe et al (21), who established the preference of Cr

for AI sites by means of ALCI-IEMI and X-ray methods. First principles calculations using Local

Density Functional theory by Fu and Zow (13) found Ti and Cr preference for A1 sites and Ni

preference with Fe sites, in complete agreement with our predictions. There is also excellent

agreement with the quasichemical model introduced by Kao et al. (14), which predicts V preference for AI sites and Co and Ni preference for Fe sites. The authors point out that the predictions

for other transition metal additions have large uncertainties and should not be used as basis of

comparison.

One of the most comprehensive studies of site distribution of transition metals is the recent

work of Anderson (23), which includes an ALCHEMI analysis of Ti, V, Cr, Mn, Co, Ni and Cu in

FeAI. The BFS results for Co, Ni and Cu exhibit a very small energy gap G. The slight preference

22
of CoandNi for Fe sitesis in agreementwith theALCHEMI predictions,butCu resultsarenot.

In thesesituations(i.e. small G), asnotedbeforefor Fe in NiAI, just the inclusionof entropic

effectsatfinite temperaturemayor maynotbesufficientto alterthepreferencescheme.

Themostmarkedpreferencefor AI sitesis thatof Ti, in agreementwith our work,with 85%

of thealloying elementoccupyingA1sites.TheresidualFe siteoccupancyis explainedby Andersonin termsof siteequilibrationkinetics,asdiffusionof substitutionaldefectsis required.Unfortunately,we havenot includedthe possibilityof vacanciesin our calculation,asNiAI - thebasis

of our study- doesnot havestructuralvacanciesfor Ni-rich alloys.A moregeneralcatalogue,

including vacancies,hasbeendefinedand testedin our previouswork on binary FeA1(5). The

defect structure (substitutions for Fe rich, triple defects for FeAI and Al-rich FeA1) was correctly

determined, in agreement with experiment and other theoretical approaches.

4-J

¢g

>

°_

Q

©

-0.80

-0.90

-1.00

-1.1(

-1.20

C°50A148.61X1.39

_. X(A1)

X(Co)A1NN

X(Co)Alf

/

m

I

m

Fe Ni Cu Ru Si Cr Zr Hf Ti V Nb Mo Ta W

X

Figure 8: Energy spectrum for Co(AI+X) alloys with alloying additions of X. See Fig. 2 for

details on the type of defect structures modeled.

23
II| I-
2.814 (0.58255)

*<

L,

ID

t_

t_

¢.2

t_

,.d

CoAl

2.78_

Mo (0.47429.)

Ta (0.39696)

Nb (0.32994)

Hf (0.27839)

Zr (0.25519)

Ru (0.15208)

Ti (0.10568)

V (0.04640)

Cr (-0.01289)

Si (-0.05413)

Cu (-0.05155)

Ni (-0.06444)

Fe (-0.07990)

I

1.39

Concentration of X (at. %)

Figure 9: Lattice parameter as a function of concentration of several alloying additions X to

CoAl. The values of _t (Eq. 14) are given in parenthesis.

CoAl: Having found agreement with available experimental and theoretical evidence for both

NiA1 and FeA1 raises confidence in the BFS method and the parameterization used in this work.

The parameters determined for the individual elements, as well as the BFS parameters aaB calculated for all pairs A-B are unique and transferable, making it trivial to use them for the calculation

of any other B2 alloy formed by the elements considered. It is therefore reasonable to extend the

application of this methodology to CoAl, a system for which we could find no experimental data

on the behavior of alloying additions in this compound in the current literature. Following the for25
matof ourpreviousexamples,wedisplaytheBFSpredictionsfor thesiteoccupancyin Co(AI+X)

alloysby showingtheenergyspectrumfor single-atomsubstitutionsin Fig. 8, andthe complete

setof resultsfor theenergyof formationandequilibrium latticeparameterasa function of compositionin Table6. Theelementsaregroupedbythevalueof theirenergygapG: Fe,Ni (G< 2 %)

; Cu, andRu(2 < G <5 %); Si, Cr, Zr andHf (5 <G<10%) andTi, V,Nb, Mo, TaandW (G>20

%) In addition,thechangein latticeparameteris displayedin Fig. 9 (seeEq. 11).

To concludetheanalysisof ternaryadditionstoXA1 (X= Ni, Fe,Co), wesummarizethesite

preferenceschemein Fig. 10,wherewe show,for eachintermetalliccompound,thesite preferencedistribution.It shouldbenotedthatwhilehelpful in obtainingaquick feelfor thebehaviorof

eachelement,thereis noexplicit considerationof thestrongorderingtendenciesthatcharacterize

thesecompounds.It is thereforeto beexpectedthat theBF rulepredictionsof theabsolutevalues

of thelatticeparametermight not beasreliableasthoseobtainedfrom full BFScalculations.As

with thelatticeparameterin Eq. 12,thebulk modulus,for arbitrarynumberor additionsandconA] s_

x site I

X - Ni, Fe, Co

Figure 10: Site preference for alloying additions to NiA1, FeAI and

CoAl. AI, X site denotes

those situations where

the alloying species X

II'I ii

NiA!

FeA!

CoAl

Ru

l

m

roll

I

..... _!;ii!iiil |

I'1_ _111_1I III I •

26
centrations,canbe computedfor anyof the alloys studiedor for those that can be constructed

from the elements parameterized in this work, by using the following expression

(_i xiBiai) 2

B(x) - XxiBi a2i (15)

i

where the single element values a i are listed in Table 1. The values of B i can be obtained from the

definition of the scaling length I,

Ec Ii = "12xqBiae

(16)

b. Double alloying additions to B2 compounds

Beyond examining all possible B2 alloys that can be composed of the 16 elements considered

in this work (i.e. CuA1, etc.), the BFS method allows for a straightforward generalization to higher

order additions, without any new determination of parameters and without any additional complexity in the calculations or the computer resources needed. A review of the description of the

BFS method should convince the reader that no particular provision should be taken when

increasing the number of elements in the system. The only additional effort consists of generating

a larger 'catalogue' of possible atomic configurations than that used for the simple ternary case.

We therefore constructed a suffici.ently large catalogue of atomic distributions contemplating all

the possible atomic arrangements and the ensuing defects for two or more simultaneous alloying

additions to binary B2 intermetallic compounds. One of the many goals of such effort is to investigate the changes, if any, in site preference of individual atomic species in the presence of other

alloying additions.

As can be seen in Fig. 1, some types of atoms might be susceptible to changing their substitutional behavior in the presence of other additions due to the small energy gap between different

27
configurations. Ti, for example,hasa definite preferencefor A1 sitesin NiAI, a fact clearly

provenexperimentallyandby anyothertheoreticalcalculation(2, 11, 12).The largeenergygap

betweenTi(A1) andTi(Ni)A1 substitutionsclearly supportthis fact.It is to be expectedthat the

addition of a fourth elementwould be, in general,irrelevantin changingthe behaviorof Ti in

NiA1. This canbe seenin Fig. 11, which showsthe energyspectrumof Ti in NiAI, and the

changesit undergoesin thepresenceof a fourth element.A detailedanalysisof this figure indicatesthat,in general,theTi(A1)choiceis not affectedby anyof theadditions.Thelower partof

the spectrumis completelydominatedby Ti(A1) defects(thick solid and dashedlines). Con-

-0.30

.-. -0.35

O

;>

= -0.40

O

Iw,4

-0.45

O

-0.55

-0.60

D

m

m

i

-Ti(Ni)A1

Ti(AI) + X(A])

==. ==Ti(AI) + X(Ni)A1

Ti(Ni)A1 + X(A1)

- - - Ti(Ni)AI + XONi)AI

m . --

m . i

u . w

w - --

film

Hi i

iB m

miD

BU g

Rim

it i

flUID

i

i

i

iU l

Ti(A1) _ "_"

Si Cu Fe Hf Cr Zr Co Ta Mo V W Ru

Ternary

Ni(A1,Ti)

Qautemary Ni(A1,Ti,X) alloys v

Figure 11: Energy spectrum of two simultaneous alloying additions to Ni(AI+X+Y) alloys. The

first column shows the ternary case, Ni(A1,Ti) The subsequent columns describe the energetics

of quaternary alloys with the defect structure indicated in the inset.

28
verseiy, the presence of Ti does not seem to affect the individual preference of the other elements.

Fe, for example, has a minuscule energy gap when acting alone, and it continues to do so in the

presence of Ti: Ti(A1) +Fe(AI) and Ti(AI)+Fe(Ni)AI defects are almost equivalent in energy.

The strong tendency of Ta, Mo, V and W for AI sites is emphasized in the presence of Ti,

making Ti(Ni)AI substitutions slightly more favorable than Ti(A1), although they are all very high

in energy to have any significance. The numerical results are displayed in Table 7, which in addition to the results plotted in Fig. 11 for Ni(AI+X) alloys, includes results for two-element additions to stoichiometric NiA1.

Co, on the other hand, benefits from the presence of Fe increasing its likelihood for choosing

Ni sites with respect to its isolatedbehavior. The same effect is seen in Ru, but in favor of A1 sites.

We could continue this analysis for all the possible combinations of alloying additions considered in this work, as well as extending it to include 3, 4 or more simultaneous additions, a calculation that would only entail the generation of the necessary catalogue of configurations. Such

massive presentation of results is naturally beyond the limitations of this paper, and it will be published elesewhere. The main goal of this paper is to simply introduce the BFS method for this type

of application with an appropriate illustration of typical results

c. Heusler structures

We conclude our analysis by considering single alloying additions to ternary Heusler phases.

Once again, the complexity of the calculation depends on our ability to define the proper catalogue of configurations which will adequately describe the system at hand. We continue to use the

same set of parameters that were used for the B2 compounds. While numerous Heusler phases

exist, there is very little known about these alloys in terms of their physical properties, except for

the lattice parameter of some alloys. No information on site preference of alloying additions

seems to be available. Fig. 12 shows the most important site substitution schemes consistent with

the structure of the Heusler phase (A atoms in one cubic sublattice, and B and C atoms in the

other sublattice, in alternating comers of the cube). Table 8 lists the corresponding energies of

formation and equilibrium lattice parameter for each one of these atomic arrangements, from

which the'site preference scheme can be extracted for a base Heusler compound of Ni2A1Ti.

29
" N_o(AI,Ti,X)5o -

] Si

Ti(Ai)+X(AI)

Ti(AI)+X(Ni)AI -.5340 -.5166

2.8668 2.8679

Ti(Ni)AI+X(AI) -.4140 -.4282

2.8794 2.8782

-.5709

2.8511

-.3619 -.5690 _[-.5767 -.5293

2.8860 2.8524 _[ 2.8511 2.8563

-.3709 -.4623 -.4543 -.4554 -.4403

2.8838 2.8616 2.86t0 2.8608 2.8643

Ti(Ni)AI -.4131 -.4198

+X(Ni)AI 2.8780 2.8768

-.5432

2.8575

-.4542

2.8653

-.3206 -.4570 -.4687 -.4520 -.4280 -.4503

2.8890 2.8621 2.8593 2.8618 2.8650 2.8656

-.4030 -.4176

2.8860 2.8849

Ni50.y/2Also-y/2(Ti,X)y

Ti(Al)+X(Ni)

Ti(AI)Ni+X(AI)

Ti(AI)Ni

+X(Ni)AI

-.3630 -.4515 -.4429 -.4293

2.8901 2.8681 2.8675 2.8672 2.8708

Ti(Ni)AI+X(Ni) ]

Ti_i)Al

+X(AI)Ni

i

-.4974

2.8787 2.8777

-.464 1 -.4520

2.8787 2.8793

-.3971 -.4i03

2.8849 2.8830

-.3971 -.3669

2.8849 2.8893

-.3831

2.8892

-.3181

2.8955

-.5187

2.8622

-.5024

2.8637

-.5019

2.8625

-.4081

2.8721

-.5249

2.8599

-.5033

2.8623

-.4027

2.8711

-.5059

2.8627

-.5046

2.8637

-.4021

2.8711

-.4673

2.8671

2.8719

-.3886

-.3242 2.8745

2.8941

-.3234

2.8939

-.5138

2.8657

-.4767

2.8690

-.4462 -.4577 -.4380 -.4181 -.4359

2.8686 2.8657 2.8687 2.8712 2.8724

-.3870

2.8773

Table 7 Energies of formation and equilibrium lattice parameter for Ni5o(A1,Ti,X)50 and for Ni5o_

y/2A150_y/2(Ti,X)y alloys (where the Ni and AI ratio is kept constant at 1:1), for X = Hf, Zr, Ta, Cu,

Co, Fe, Cr and Si. The shaded cells indicate the lowest energy state. The shaded cells indicate the

lowest energy state for each element. In all cases it is assumed that the Ti and X atoms are located

30

I[| 1!
(Ni'X)2AIT| Ni2(AI,X')Ti Ni2AI(Ti,X. )

- 0--.-_ -- -o-_

,e-rO"-'-e-%-O',e --O----o--.-o--e -0- -_--_o- •

' '_i ? -r_-,-e--,-b_ '

(Ti)

"- -__- _--

X(A1)Ni X(Ti)AI

_. o-.-.-Q- - -o - X(A1)Ti -'- x--,-*-'-_-b _" ,,-..o-_-_ _-._ _.-o .. _

X_ ' " ' ;'._._ -?-,,,--i-u-, e--VT6_-__--(>_.-•

_ A , n- : _u: l--_---1--1jr_;_ .

(TONi _.-o ___

.o---e--.o.__

? - r J¢--'-,- o--_-o-? X(AI,Ni)Ti

_.--_J ,.... ; 7-_Q-'-,-.:_-b'_*,

,'- --"--',-_-b'_: o-o--.-o--.-<>_ , _ X(Ni)Ti

_---0 _ - r 0-'-,- 0-'7 - 0 T" A- O-._.-e ---.-0 - -

_ ' , L ' V-l''V--l-g-,-1-¢'_ -0

X(AI,Ti)Ni _'-° _ "-_

X(Ni,Ti)AI X(Ni,A1)Ti

Figure 12: Atomic distributions of X additions to a Ni2A1T i alloy. Ni-poor, Al-poor and Ti-poor

alloys are indicated in each column. Ni, A1 and Ti atoms are indicated with solid black disks,

bol solidX. grey disks and open circles, respectively. The additional element is denoted with the sym31
INi2AITi÷X IHf lZr ICuICo

(Ni,X)2AITi

I '1''or I

XfN0

X(AI)Ni -0.3856

2.9391

X(Ti)Ni -0.2782

2.9488

X(Ti,AI)Ni -0.3810

2.9396

-0.3599 -0.2274

2.9398 2.9540

I-0.4102 [I-0_757 -0.4314 -0.4130 2.9373 2.9392 2.9226 2.9230

-0.3043 2.9467 ]-0.2845 2.9471

-0.3238 -0.3058

2.9322 2.9324

-0.4074

2.9235

X(AI,Ti)Ni -0.2782 I -0.3000 I -0.2731 -0.3238 -0.3067

2.9488 12.9472 [2.9483 2.9322 2.9324

Ni2(AI,X)Ti

.0.4,7,11-0.4°°_ -0.4366

X(AI) _] i_29298 2 9323 , _ 2.9164

X(Ti,Ni)AI -0.2842 -0.3104 I -0.2897 -0.3304

2.9426 2.9405 I 2.9411 2.9260

X(Ni)A! -0.4153 -0.3522 -0.2005 -0.4351

2.9294 2.9351 2.9512 2.9176

_2.-0.3124

9263

-0.4144 2.9227 -0.4135 2.9243 II

-0.3077 2.9321 -0.3016 2.9345 J

-0.4089 2.9232 [-0.4095 29246

-0.3084 2.9320 -0.3013 2.9345 [

-0.4,29 2.9154

-0.4384 2.9159 2.9178 -0.4348 ]'

-0.3143 -0.3081

2.9259 2.9284

-0.4101 2.9194

X(Ni,Ti)A! -0.4107 -0.3508 -0.2003 -0.4310 -0.4596 -0.4`80 -0.4063

2.9298 2.9352 2.9513 2.9180 Z9138 2.9154 2.9198

Ni2AI(Ti,X)

X(Ti) -0.4538 -0.4598 -o.4392 -0.4457 -0.4423

2.9258 2.9127 2.9133 2.9123 2.9143

X(AI)Ti -0.4552 -0.4089 -0.4435

2.9262 2.9288 2.9129

X(Ni)Ti

X(Ni,AI)Ti

Ti -0.3779

2.9317

-0.4162

2.9266

-0.4211

2.9261

-0.4040

2.9297

-0.3709

2.9306

-0.3745

2.9303

-0.3720

2.9314

-0.2282

2.9460

-0.2281

2.9460

-0.4245 -0.4058 -0.4073

2.9152 2.9156 2.9153

-0.4394

2.9144

-0.4435

2.9140

I-0.4658 I -0.4520

1129104/1291 

-0.4073

2.9167

-0.4127

2.9164

-0.4154

2.9162

Table 8 Hf, Zr, Ta, Cu, Co, Fe and Cr additions to Ni2AITi alloys. Energies of formation (in eV/

atom) and equilibrium lattice parameter for the atomic configurations displayed in Fig. 12. The

dark shaded cells indicate the lowest energy case. The light shaded cells indicate those states that

are both close in energy and lattice parameter to the lowest energy state.

32
Ni2AITi+X

X(Ni)

X(AI) Ni

X(Ti)Ni

X(Ti,AI)Ni

X(AI,Ti)Ni

INb IM°IV Iw

(Ni,X)2AITi

-0.2340 [ -0.2580 -0.3779 -0.2383

I

2.9524 [ 2.9482 2.9306 2.9516

-0.3915 2.9367 I[-0.3953 2.9359 [[t_ 1_ N[ 2.9379 -0.3892

I

-0.2981 [ -0.2962

2.9449 [ 2.9445

i

_!P_I!!_!r_!!!_!!N_]_ I

-0.3059 -0.2955

2.9370 2.9460

-0.4178

2.9269

Isi I

-0.3639

-0.4232

112.9287 I 2.9265

-0.3301

-0.2924 2.9425 2.9346

i -0.4286

2.9287

-0.2885 2.9466 [ -0.2930 2.9425

-0.2891 -0.2924 -0.3068 [ -0.3298

2.9457 2.9449 2.9369 I 2.9346

Ni2(AI,X)Ti

X(AI)

X(Ti)AI

X(Ti,Ni)A!

X(Ni)AI

X(Ni,Ti)A!

.o.4221 -0.4529

2.9304 2.9224

-0.3019

2.9385

-0.3123

2.9308

-0.3036

2.9388

-0.4516

2.9223

-0.3011 -0.2982

2.9400 2.9365

-0.2237 _".......

2.9480

-0.4614

2.9184

-0.3373

2.9283

-0.2260 -0.2434 -0.3825 -0.3561

2.9476 2.9449 2.9247 2.9277

-0.2254 -0.2417 -0.3776 -0.2227 -0.4576 -0.3555

2.9477 2.9451 2.9252 2.9481 2.9222 2.9277

X(Ti)

X(AI)Ti

i

X(AI,Ni)Ti

Ni2Ai(Ti,X)

I_J,_ _lq_t_lIlqi!_qll I

-0.4354 IR_N!l_q_q!_!qlllq_l_q: -0.4505 _ ,_ n

I__ _11_11_'-0.4594 -0.4689

2.9253 2.9167 _7_i:_1 ...... 2.9188 2.9148

-0.4289 -0.4289 _! -0.4292 ]1-0.4602 _'_'_

II

2.9259 2,9258 _°"°"_ 2.9270 /I 2.9189

|l

i_ -0.3904 -0.4169 -0.3854 -0.4243 -0.4172

2.9284 2.9189 2.9303 2.9213 2.9189

X(Ni)Ti -0.2511 -0.2602 -0.3733 -0.2465 -0.4616 -0.3809

i 2.9426 2.9407 2.9229 2.9433 2"9190 l 2.9226

X(Ni,AI)Ti -0.2509 -0.26000 -0.3741 -0.2459 t_'_[_'!!!_NI -0.3851

2.9426 2.9407 2.9228 2.9434 _ 2.9221

Table 8 (continued) Nb, Mo, V, W, Ru and Si additions to Ni2A1Ti alloys. Energies of formation

(in eV/atom) and equilibrium lattice parameter for the atomic configurations displayed in Fig. 12.

The shaded cells indicate the lowest energy case. The dotted cells indicate those states that are

both close in energy and lattice parameter to the ground state ones.

33
O

>

O

o_

O

-0.41

-0.42

-0.43

-0.44

-0.45

-0.40 __

m

-0.46

-0.47

u

m

Zr/AI/Ni

Hf/Ni) Zr(Ti,A1)Ni

_(m)

HffTi)A1 Zr(Ti)A1

HffA1)

Zr(A1)Ti

Hf(Ti)

Hf(AI)Ti - Zr_'T::-

r, (Ni,X)2A1Ti

Ni2(AI,X)Ti

-...... Ni2AI(Ti,X )

Fe(Ti)A1

Cu(AI) Ni Fe(Ni)A1

Cu(Ni) Fe(Ti)

Fe(Ni,Ti)AI

Fe_(.A1)Ti

Cu(Ti)A1 _Fe(N_i)_Ti_

Cu(A1) Fe(Ni)AI

_ _Cu_(Ti_)_ Co(Ni,Ti)AI __)_i

Cu(Al)Ti Co/Ni/A1

C6(Ni)Ti

Co(Ni,A1)Ti

Si/Al)Ni

Si(Ti,AI)Ni

Si(Ti)A1

Si(Al)

_ _si_(Tib _

Si(A1)Ti

Figure 13: Energy of formation (in eV/atom) for the Heusler alloys shown in Fig. 12 for Hf, Zr,

For such a complex System, the energy spectrum introduced in previous sections becomes perhaps the only way to investigate the trends exhibited by these systems. The energy spectrum corresponding to these numerical results is shown in Fig. 13. In spite of the similarities with the

ternary case, it should be noted that the bonding scheme in the Heusler phase is quite different

than that found in NiA1 alloys. Fig. 13 shows that while Zr, Hf, Cu and Si show a distinct preference for the sublatfice occupied by Ti and A1 atoms, there seems to be a low energy price to pay

34

I[_:1:
for substitutionsin AI or Ti sites,evenat theexpenseof anantistructuredefectcreatedwhenTi

goesto anA1sitein theX(Ti)A1cases.A cleardistinctionis seenfor CoandFe,wherebothshow

amarkedpreferencefor Ni sites.

A completepresentationof theBFSresultsfor all possibleadditionsto all theHeusleralloys

which can be obtainedfrom the 16elementsconsideredin this work is, asmentionedabove,

beyondthe scopeof this paper,but not thecomputationalmethod.We thereforelimit our commentson suchsystemsto theNi2A1Ti+X case(Table7, Figs. 12-13)andto providingapproximateexpressionsusingtheBF rule (6) for thechangesin latticeparameterfor alloying additions

in thedilutelimit for generalHeusleralloys(A2BCh_xXx,

with

= 1+ p,x (17)

a H

"_x - "_i _x - _i

- (18)

where "ti = 8ia_, z.,.= Bias, for i= A, B, C or X, and where TH and _H are given by

2_tA + '_'B + _'C (19)

Yn =

4

and

2_'a + _'B + _'c (20)

_'H "- 4

Several intermetallics of potential industrial interest form Heusler precipitates for small levels

of additions. Some alloying additions, like Hf, Zr, Ta or Nb in NiA1, have a low solubility limit

and result in the precipitation of Heusler particles. Such effect was studied in detail for the case of

Ti in NiAI (10), where BFS was applied to the determination of the solubility limit of Ti in NiA1

35
and secondphaseformation.It is thenimportantto examinethe role of otheradditionsandto

determine,beyondtheir sitepreferencein eitherphase(inthecaseof two phasealloys),theirpartitioning behavior.

Thecurrentformalismcanbeeasilyextendedto providesimilarinformationforthelargefamily of alloys that exhibit the Heusler structure(Ni2A1X, with X= Nb, Ta, Hf, Zr; C02A1X,

X=Nb,Ta,Hf,Zr and Ti; Cu2A1X,X= Hf, Zr; and FeeAIX,X= V, Cr, Co, Ni). In addition, as

notedfor thecaseof binarysystems,thereareno restrictions in the formalism for the number of

simultaneous alloying additions considered in the base alloy. Natural extensions of this work to

high order additions in binary and ternary ordered intermetallics as well as other base systems

(Ni3AI, Ni-base superalloys, etc. ) are also being performed.

Conclusions

The BFS method was used to determine the site preference scheme for a large number of

alloying additions to NiAI, FeA1 and CoAl. A detailed analysis of all the possible site substitutions was performed enhancing our understanding of the interrelationship between multiple alloying additions. Because the agreement with experiment was excellent in all cases, the analysis was

extended to ternary (Heusler) alloys where no data on the effect of alloying additions on structure

exist.

Acknowledgments

Frutiful discussions with N. Bozzolo are gratefully acknowledged. We would also like to

thank C. Amador for providing us with the first-principles input necessary for determining the

BFS parameters used in this work.

References

I. G. Bozzolo and J. Ferrante, J. Computer-Aided Mater. Design 2 (1995), 113, and references

therein; G. Bozzolo and J. Ferrante, Ultramicroscopy 42/44 (1992), 55; G. Bozzolo, R. IbanezMeier and J. Ferrante, Mat. Res. Soc. Symp. Proc. 355 (1995), 531.

36

Ii! 1 _-
2. E H. Kitabjian, A. Garg, R. D. Noebe and W. D. Nix, Creep and Fracture of Engineering Matedais and Structures, (The Minerals, Metals & Materials Society, Warrendale, PA 1997), pp. 667.

3. R. D. Noebe and W. S. Waiston, Structural Intermetailics 1997 (The Minerals, Metals & Matedais Society, Warrendale, PA, 1997), pp. 573.

4. G. Bozzolo, J. Fen'ante, R. D. Noebe and C. Amador, Scripta Mater. 33 (1995), 1907.

5. G. Bozzolo, J. Fen'ante, R. D. Noebe and C. Amador, Scripta Mater. 36 (1997), 813.

6. G. Bozzolo and J. Ferrante, Phys. Rev. B 50 (1994), 5971.

7. G. Bozzolo, J. Ferrante and A. M. Rodriguez, J. Computer-Aided Mater. Design 1 (1993), 285.

8. J. H. Rose, J. R. Smith and J. Ferrante, Phys. Rev. B 28 (1983) 1835.

9. O. K. Andersen, A. V. Postnikov and S. Y. Savrasov, Mat. Res. Soc. Symp. Proc. 253 (1992),

37.

10. G. Bozzolo, R. D. Noebe, J. Ferrante and A. Garg, Structural lntermetailics 1997 (The Minerals, Metals and Materials Society, 1997), pp. 665.

11..N.V. Allaverdoba, V. K. Portnoy, L. A. Kucherenko, A. V. Ruban and V. I. Bogdanov, J.

Less-Common Metals 141 (1988), 191.

12. D.J. Chakrabarti, Met. Trans. 8 (1977) 121; O. S. Zarechnyuk, N. V. German, T. I. Yanson,

R. M. Rykhal and A. A. Muravieva, Phase Equlibria in Metals and Alloys (Nauka, Moscow,

1981), pp. 69.

13. C. L. Fu and J. Zou, Acta Mater. 44 (1996), 1471.

14. C. R. Kao, L. M. Pike, S. -L. Chen and Y. A. Chang, Intermetallics 2 (1994), 235.

15. H. Hosoda, K. Inoue and Y. Mishima, Mat. Res. Soc. Symp. Proc. 364 (1995), 437.

16. I. 1VI.Anderson, A. J. Duncan and J. Bentley, Mat. Res. Soc. Symp. Proc. 364 (1995), 443.

17. P. Chartier, M. Balasubramanian, D. Brewe, T. Manzur, D. Pease, J. Budnick, L. Huang, C.

Law, S. Russel and C. Kimball, J. Appl. Phys. 75 (1994), 3842.

18. A. J. Duncan, M. J. Kaufman, C. T. Liu and M. K. Miller, Appl. Surf. Sci. 76/77 (1994), 155.

19. R. D. Field, D. E Lahrman, R. Darolia, Acta MetaU. Mater. 39 (1991), 2961.

20. J. D. Cotton, R. D. Noebe, M. J. Kaufman, Intermetallics 1 (1993), 1.

21. P. R. Munroe and I. Baker, J. Mater. Res. 7 (1992), 2119.

22. S. Khosla, (PhD Thesis, University of Knoxville, TN, 1993).

23. I. M. Anderson, Acta Mater. 45 (1997), 3897.

37
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October 1998 Technical Memorandum

4. TITLI=AND SUBTITLE 5. FUNDING NUMBERS

Modeling of Substitutional Site Preference in Ordered Intermetallic Alloys

6. AUTHOR(S)

Guillermo Bozzoio, Ronald D. Noebe, and Frank Honecy

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135- 3191

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National Aeronautics and Space Administration

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WU-523-22-13--00

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REPORT NUMBER

E-11387

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AGENCY REPORT NUMBER

NASA TM-- 1998-208665

11. SUPPLEMENTARY NOTES

Prepared for the "Interstitial and Substitutional Effects in Intermetallics" Fall Meeting sponsored by the Minerals, Metals,

and Materials Society, Rosemont, Illinois, October 11-15, 1998. Guillermo Bozzolo, Ohio Aerospace Institute, 22800

Cedar Point Road, Cleveland, Ohio 44142; Ronald D. Noebe and Frank Honecy, NASA Lewis Research Center. Responsible person, Guillermo Bozzolo, organization code 5140, (216) 433-5824.

12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE

Unclassified -Unlimited

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This publication is available from the NASA Center for AeroSpace Information, (301) 621--0390.

13. ABSTRACT (Maximum 200 words)

We investigate the site substitution scheme of specific alloying elements in ordered compounds and the dependence of site occupancy

on compound stoichiometry, alloy concentration. This basic knowledge, and the interactions with other alloying additions are necessary

in order to predict and understand the effect of various alloying schemes on the physical properties of a material, its response to various

temperature treatments, and the resulting mechanical properties. Many theoretical methods can provide useful but limited insight in this

area, since most techniques suffer from constraints in the type of elements and the crystallographic structures that can be modeled. With

this in mind, the Bozzolo-Ferrante-Smith (BFS) method for alloys was designed to overcome these limitations, with the intent of

providing an useful tool for the theoretical prediction of fundamental properties and structure of complex systems. After a brief

description of the BFS method, its use for the determination of site substitution schemes for individual as well as collective alloying

additions to intermetallie systems is described, including results for the concentration dependence of the lattice parameter. Focusing on

B2 NiAI, FeA1 and CoAl alloys, the energetics of Si, qfi, V, Cr, Fe, Co, Ni, Cu, Zr, Nb, Mo, Ru, Hf, Ta and W alloying additions are

surveyed. The effect of single additions as well as the result of two simultaneous additions, discussing the interaction between additions

and their influence on site preference schemes is considered. Finally, the BFS analysis is extended to ternary LI 2 (Heusler phase) alloys.

A comparison between experimental and theoretical results for the limited number of cases for which experimental data is available is

also included.

114. SUBJECT TERMS

Intermetallics; Monte Carlo; Atomistic simulations; BFS; Alloy; Semiempirical

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