NASA/TM--1998-208820

An Introduction to the BFS Method

and Its Use to Model Binary NiA1 Alloys

Guillermo Bozzolo

Ohio Aersospace Institute, Cleveland, Ohio

Ronald D. Noebe

Lewis Research Center, Cleveland, Ohio

J. Ferrante

Cleveland State University, Cleveland, Ohio

C. Amador

Universidad Nacional Aut6noma de M6xico, Mexico D.F., Mexico

November 1998

https://ntrs.nasa.gov/search.jsp?R=19990009864 2018-05-11T16:58:52+00:00Z
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NASA/TM--1998-208820

An Introduction to the BFS Method

and Its Use to Model Binary NiA1 Alloys

Guillermo Bozzolo

Ohio Aersospace Institute, Cleveland, Ohio

Ronald D. Noebe

Lewis Research Center, Cleveland, Ohio

J. Ferrante

Cleveland State University, Cleveland, Ohio

C. Amador

Universidad Nacional Aut6noma de M6xico, Mexico D.F., Mexico

National Aeronautics and

Space Administration

Lewis Research Center

November 1998
Acknowledgments

Fruitful discussions with Dr. N. Bozzolo are gratefully acknowledged. C. A. acknowledges partial support from Facultad de

Quimica. Computational services by Direccion General de Servicios de Computo Academico are gratefully acknowledged. We

would like to thank F. Honecy and P. Abel for their helpful comments and suggestions in preparing this manuscript.

This work was partially funded by the HITEMP and PPM programs of NASA Lewis Research Center.

NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076

Price Code: A04

Available from

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22100

Price Code: A04
AN INTRODUCTION TO THE BFS METHOD AND ITS USE TO MODEL

BINARY NiAI ALLOYS

Guillermo Bozzolo (a), Ronald D. Noebe (b), J. Ferrante (c) and C. Amador (d)

a Ohio Aerospace Institute, 22800 Cedar Point Rd., Cleveland, OH 44142, U.S.A.

b National Aeronautics and Space Administration, Lewis Research Center,

Cleveland, OH 44135, U.S.A.

c Department of Physics, Cleveland State University, Cleveland, OH 44115, U.S.A.

d Departamento de Fisica y Quimica Teorica, Facultad de Quimica, Universidad

Nacional Autonoma de Mexico, Mexico D. F., Mexico

SUMMARY

We introduce the Bozzolo-Ferrante-Smith (BFS) method for alloys as a computationally efficient tool for aiding in the process of alloy design. An intuitive description of the BFS method is

provided, followed by a formal discussion of its implementation. The method is applied to the

study of the defect structure of NiA1 binary alloys. The groundwork is laid for a detailed progression to higher order NiAl-based alloys linking theoretical calculations and computer simulations

based on the BFS method and experimental work validating each step of the alloy design process.

Keywords: Alloys, Intermetallics, BFS Method, Defect Structure, Semiempirical Methods

NASA/TM--1998-208820 1
INTRODUCTION

A numberof industriesincludingtheelectronics,computer,aeronautics,aerospace,andother

transportationrelatedenterpriseshave long relied on the applicationof new high-performance

materialsfor their competitiveadvantages.Yet,in aeronauticsandaerospaceapplicationsin particular,thepotential lossin lives andliability that couldresultfrom anymaterialfailure dictates

thatthesesystemsmustbevery well characterizedandunderstood.Therefore,not only dothese

newmaterialshaveto pushthe envelopein termsof performance,at the sametime theyneeda

high degreeof maturity.This makesthe introductionof new materialsin theseindustriestime

consumingandcostlydueto theneedfor painstakingdevelopment,testing,andcharacterization.

At the sametime, without the largedefense-relatedexpendituresthat werepossibleduring the

cold war era,it is necessaryto find newwaysfor industryto operatethatwill providesignificant

reductionsin expenseandcycletimein the development,maturation,andcertificationprocesses

for new materials.To achievethesegoals,the experimentalapproachto materialsresearchand

designwill haveto beaugmentedwith computationalprocesses.Thegoalis to developcomputational capabilitiesfor alloy designwith crystallinematerialsto the sameor greaterdegreethan

existstodayin thepolymerandpharmaceuticalindustries.

In aseriesof articleswewill describetheintegrationof anewcomputationalmaterialsmodel,

knownasthe BFSmethod[1], into anactualmaterialsdesignprogram.The intent of this integratedexperimentalandcomputationalprogramis to developrevolutionarynew alloysbasedon

theorderedintermetalliccompoundNiA1 asareplacementfor Ni-basedsuperalloysin aero-turbine applications.NiA1 offers a numberof advantagesover Ni-basedsuperalloys,including a

highermeltingtemperature,betterenvironmentalresistance,significantlyhigherthermalconducNASA/TM--1998-208820 2
tivity, a better response to thermal barrier coatings, and a reduction in density by as much as a

third without significantly impacting creep strength [2]. However, the development of an alloy

composition with the correct balance of high temperature strength and low temperature fracture

resistance has continued to elude investigators [3], leading to continued alloy design-structureproperty investigations and the eventual integration of computational processes (BFS method) in

the search for an acceptable alloy.

Through the eventual partnership between experimental and theoretical procedures, we were

able to determine the naturally evolving role of computational methods in an alloy design program. This role is schematically illustrated in Figure 1. At first, the design approach reported in

this work was guided by past experimental work of an empirical nature. For most alloy design

programs this is essentially one continuous loop (Figure 1.a). But an integrated alloy design program that augments experimental results with theoretical modeling would follow the path

described in Figure 1.b. We started by modeling what was known about the NiA1 system. This

allowed us to establish confidence in the model, make changes and optimize the model, and to

help understand and interpret the results. The model also provided the energetic explanations for

many of the experimental observations, thus giving us greater insight into the alloy system. As

confidence in the model grew, we started filling the gaps in our experimental database with computational results. Finally, as alloy compositions became more complicated and burdensome to

evaluate, we used the computational model to direct the experimental work, and only experimentally verified those few compositions that looked promising or that we needed in order to refine

the model. The advantage of this integrated approach is that it funnels most of the alloys through a

virtual production and screening process significantly reducing the number of alloys actually produced and characterized. This saves significant time and expense since the experimental process is

NASA/TM--1998-208820 3
(

(

(

Choose an alloy system

Make series of alloys

Test/screen alloys

)

>

)

Finished

No

Figure 1.a Conventional Alloy Design Program

Choose an alloy system

Make series of alloys

Test/screen alloys

)

)

)

Model alloy_

[ (_ix/replace_

<model J

____o 1

Yes

Model compositions series of new

@ge_ Experimental

feedback for

No model

No

(Finished)

Modeled feedback for experimental design matrix

Figure 1.b Integrated Alloy Design Program with Mutual Feedback

NASA/TM--1998-208820 4
the rate and resource limiting step.

This paper represents the first in the series of articles detailing our design program on NiA1-

based alloys. In forthcoming papers, we will describe our experimental and theoretical investigation of a series of 3-, 4- and 5-element alloys based on NiA1, highlighting the connection between

the two approaches. In this first paper we will provide a detailed description of the BFS method

and use it to reproduce the basic features of the base material, with as much accuracy as possible,

since the forthcoming extension of the model to complex systems will rely only on our ability to

describe the most basic system.

Therefore we will initially concentrate on the [_-phase of the binary Ni-A1 system, which

exists over a range of stoichiometry from about 45-60 at. % Ni [4]. At the stoichiometric composition NiA1 should exist in a perfectly ordered state where the Ni atoms occupy the cube corners

and the A1 atoms occupy the cube centers of a generalized body-centered cubic lattice. Ni-rich

alloys are characterized by antisite point defects, where Ni atoms occupy sites in the A1 sublattice,

resulting in a decrease in lattice parameter and an increase in density with increasing Ni content.

A different behavior is observed on the Al-rich side of stoichiometry. There is a steep decrease in

lattice parameter as well as a greater than expected decrease in density with increasing A1 content.

The presence of vacancies in Ni sites would explain such behavior [4-8]. Recent X-ray diffraction

experiments [9] suggest a richer structure: the evidence, while strongly favoring the presence of

vacancies in Ni sites, also suggests the possibility of some vacancies in A1 sites in a 3:1 ratio.

Moreover, local ordering of vacancies may be preferred over a random distribution of individual

point defects [6].

While some of the concepts introduced in this first paper might seem too detailed for the task

at hand (i.e. describing the defect structure of [_-NiA1), they will be strictly necessary for a full

NASA/TM--1998-208820 5
understanding of the results from more complex systems that will appear in subsequent papers in

this series. In other words, we are establishing the validity of our model by eliciting a positive

response to 'Q2' in Fig. 1.b before proceeding further along the flow chart.

In most cases, theoretical techniques at the atomic level are used for validating or understanding experimental findings, a task somewhat removed from the actual process of alloy design. In

this series, we embark on a process of materials development starting with the validation of our

theoretical approach with established experimental knowledge of the basic, binary system (this

first paper), building on our theoretical predictions as the complexity of the system increases (ternary, quaternary alloys), finally allowing us to determine the structural properties of a 5-element

alloy (last paper) based on this groundwork. As a whole, this series of papers is meant to prove

another point, which goes beyond the features of the particular problems at hand: that semiempirical methods, tied to powerful numerical techniques and implemented in the framework of an economical and easy to handle operational procedure, can constitute a very useful tool in effectively

aiding in the area of materials design.

This paper introduces, in detail, the BFS method and its operational equations. After introducing the novel way of partitioning the alloy formation process proposed by the BFS method and

conceptually defining the quantities of relevance, we present details of the operational procedure.

We continue with an example illustrating the implementation of the method. As a consequence of

the tremendous advance in semiempirical techniques over the last decade, we conclude this section with a brief description of recent work done with other techniques of comparable complexity

as well as their relation to BFS. We continue by introducing the computational scheme chosen to

address the structure of NiA1, a procedure which will be used and expanded in subsequent papers

dealing with higher order alloys. We conclude by providing the BFS predictions for the defect

NASA/TM--1998-208820 6
structure and physical properties of [_-NiA1, comparing them with experimental results when possible, thus providing us with confidence in our model.

THE BFS METHOD

The BFS method is one of the latest additions to the growing family of quantum-mechanical,

approximate techniques for the calculation of materials properties. It has a strong foundation in

quantum theory, but simplifies the numerical effort usually associated with ab-initio methods.

While a great deal of detail (i.e. electronic structure) is lost when applying semiempirical methods

to specific systems, valuable insight on the general behavior of physical systems is gained, since

the simplicity of the formulation allows for a quick and sufficiently accurate estimate of general

properties.

Several methods successfully deal with single-element metallic systems, providing a wealth of

information on their physical properties and defect structures [10-15]. However, in the case of

extended defects, i.e. surfaces, some methods exhibit limitations in their predictive power [11]. In

most cases the corresponding potentials or the parameters used in the algorithm, are usually determined from bulk properties and are therefore limited in their ability to properly reproduce these

defects. However, the overall power of semiempirical techniques has enabled a better understanding of defects such as surface phenomena in spite of these shortcomings, which has lead to extensive growth in the area of computer simulations of defects.

For alloy structures, semiempirical methods have been less effective in providing results with

the same degree of reliability as those for single-element systems. This is partly due to the nature

of the formalisms. Either an impractical number of parameters is needed (modified embedded

NASA/TM--1998-208820 7
atom method [11] and tight-binding methods [12], for example) or specific potentials have to be

determined on a case-by-case basis to deal with each specific alloy composition or phase, therefore limiting their transferrability.

In spite of these limitations, there has been remarkable progress in the last few years. In combination with the development of powerful first-principles methods and an ever increasing computational power, theoretical approaches are becoming tools of great value for understanding,

modeling and designing materials suited for industrial applications.

Perhaps the main contribution of the BFS method to the field of computationally oriented

semiempirical methods, is that it is basically free of most of the restrictions that limit the application of comparable techniques [16]: there is no restriction on crystal structure or the type of

atomic species considered and the number of input parameters is reduced to a minimum. Moreover, no experimental input is necessary, as all the input parameters can be determined by simple

first-principles calculations [17].

These advantages of the BFS method make it particularly suitable for broad alloy design problems where the nature of the predictions is comparable to the information that can be obtained

from experimental analysis (in this work, for example, we attempt to study the structure of 5-element alloys, both theoretically and experimentally). It is precisely this issue that motivates the

present work: to develop a reasonably accurate and computationally simple theoretical approach

that can provide valuable input for the development of new materials, in close conjunction with

experiment.

Since its inception a few years ago [18], the BFS method has been successfully applied to a

variety of situations ranging from segregation profiles [ 16], alloy structure [19], surface alloying

of immiscible metals [20], to numerical simulations of the scanning tunneling microscopy tipNASA/TM--1998-208820 8
sample interaction [21], bulk alloy design [1,22] and alloy surface structure [23]. Two of the characteristics of the early applications of the BFS method, i.e., the need for experimentally determined input data and complete reliance on the pure constituent properties [18], initially imposed a

limitation when dealing with systems like [_-NiA1, where both constituents are fcc elements but

the alloy is bcc-based. No appropriate experimental input for bcc-Ni and A1 is available. Therefore, as shown in detail later in this work, we have reformulated the method relying only on pure

first-principles determined input, thus freeing BFS from limitations imposed by the restricted

availability of experimental data. We also avoid the potential problem of inconsistency or inaccurate data obtained from different experimental techniques [ 14]. While the method retains the same

operational algorithm used in its previous applications, the new parameterization scheme presented in this paper greatly enhances its range of applications and accuracy.

Genelvl concepts

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

AH of an arbitrary alloy (i.e. 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 = (1)

i i

where E i' is the energy of atom i in the alloy and E i its corresponding value in a pure equilibrium

monatomic crystal. In principle, the calculation of AH would simply imply computing the energy

NASA/TM--1998-208820 9
of each atom in its equilibrium pure crystal and then its energy in the alloy. In BFS, beyond

directly computing the difference e i for each atom in the alloy, we introduce a two-step approach

for such a calculation in order to identify contributions to the energy due to structural and compositional effects. Therefore, we break up the individual contributions of each atom e i to the total

energy of formation M-/of the sample into two components: a strain energy and a chemical

energy.

While there is a certain level of arbitrariness in how this separation is implemented, it is only

meaningful when a good representation of the actual process is obtained by properly linking both

contributions. In other words, a proper coupling of these two apparently independent processes

(strain and chemical effects) must be accomplished in order for the final result to approach the

result one would obtain if a straightforward calculation (i.e. ab-initio methods) was carried out.

We define the BFS strain energy as the contribution to the energy of formation from an atom

in an alloy computed as if all the surrounding atoms were of the same atomic species, while main-

(a) (b) (c)

Fig 2: (a) A pure, equilibrium crystal (reference atom denoted by the arrow), (b) a reference atom (denoted

by the arrow) in the alloy to be studied (atoms of other species denoted with other shading) and (c) the

same reference atom in a monatomic crystal, with the identical structure of the alloy to be studied, but with

all the atoms of the same atomic species as the reference atom, for the calculation of the strain energy term

for the reference atom. The strain energy is the difference in energy of the reference atom between (c) and

(a).

NAS A/TM--1998-208820 10
tainingtheoriginalstructureof thealloy.To visualizethis concept,Fig. 2.arepresentstheatomin

question(identified with an arrow) in anequilibrium position in its groundstatecrystal (arbitrarily representedby asimplecubiclattice).Fig. 2.bshowsthesameatomin thealloybeingstudied (also arbitrarily representedby a different crystallographicsymmetry).Two things canbe

differentbetweenthe referencecrystaland our alloy. First, atomsof other speciesmay occupy

neighboringsitesin the crystal and second,the crystallatticemay not be equivalentin sizeor

structureto that of thegroundstatecrystalof thereferenceatom.In Fig. 2.b,thedifferentatomic

speciesaredenotedwith different symbolsfrom thatusedfor thereferenceatom,andthe differencesin sizeand/orstructurearedenotedwith a schematicallydifferentatomicdistribution as

comparedto thegroundstatecrystalshownin Fig. 2.a.TheBFS strainenergytermaccountsfor

thechangein energydueonly to the change in the geometrical environment of the crystal lattice

(from 2.a to 2.b), ignoring the additional degree of freedom introduced by the varying atomic species in the alloy. In this context, Figure 2.c shows the environment 'seen' by the reference atom

when computing its BFS strain energy contribution. The neighboring atoms conserve the sites in

the actual alloy (fig. 2.b), but their chemical identity has changed to that of the reference atom

(fig. 2.a), thus simplifying the calculation to that of a single-element crystal. The BFS strain

energy term represents the change in energy of the reference atom in going from the configuration

denoted in Fig. 2.a to Fig. 2.c.

This choice for the BFS strain energy introduces two main advantages. First, by transforming

(for each atom) the actual alloy into a monatomic crystal, it greatly simplifies the calculation of

the energy of the reference atom in the alloy structure, making it amenable to a large number of

theoretical techniques that can efficiently deal with this simplified situation. Second, it gives partial information concerning only the structure of the alloy, which could serve later to identify fine

NASA/TM--1998-208820 11
geometrical effects on structure.

The second contribution to the BFS energy of formation is the chemical energy. Here, we are

interested in isolating the effect of compositionally different atoms occupying neighboring sites to

the reference atom. In order to do this, we leave out any structural information from the original

lattice by forcing the neighboring atoms to occupy equilibrium lattice sites corresponding to sites

in the pure cell of the reference atom, changing the composition of the atoms to match the chemical profile in the alloy lattice. Fig. 3.a shows the reference atom in the actual alloy (similar to Fig.

2.b), while Fig. 3.b indicates the atomic distribution used in computing the BFS chemical energy

(note that the lattice used in Fig. 3.b corresponds to that of the ground state crystal of the reference

atom as shown in Fig. 2.a).

The chemical energy is then based on the difference between the energy of the reference atom

in Fig. 3.b and its energy in its ground state crystal (Fig. 2.a). In order to completely free the

chemical energy from structural defects, certain provisions have to be made which will be clearly

(a) (b)

Fig. 3: (a) The reference atom (denoted

by an arrow) in the actual alloy environment and (b) the reference atom surrounded by a chemical environment

equivalent to that in (a) but with the different neighboring atoms occupying

equilibrium lattice sites corresponding to

the ground state of the reference atom.

NASA/TM--1998-208820 12
detailed later in this section.

So far, we have exploited the flexibility in partitioning the energy of formation to provide

ourselves with a simple and straightforward framework for computing both the BFS strain and the

BFS chemical energy contributions. However, a simple addition of these two quantities does not

necessarily constitute a good approximation to the actual energy of formation. After decoupling

the structural and chemical processes involved in alloy formation, we now recouple them by introducing a coupling function which, in its definition, has the necessary ingredients to ensure that

this partitioning scheme accurately reproduces a full calculation of the energy of formation when

recombined in this manner. The definition of the coupling function will be clear once we introduce the necessary tools for computing the individual (strain and chemical) contributions in the

following subsections. We therefore write the energy of formation of the alloy as

AH = __e = ___(es+gec) (2)

where e s is the strain energy, e c is the chemical energy, g is the coupling function and the sum

extends over every atom in the alloy.

Strain energy

The actual calculation of the BFS strain energy for each atom is straightforward and any technique designed to compute the energetics of pure crystals should be appropriate. However, in spite

of the apparent simplicity of this task, few techniques are actually capable of performing it successfully in general situations. The choice should be limited to only those techniques that introNAS A/TM--1998-208820 13
duce a substantial level of accuracy, regardless of the geometry of the environment, as well as

those characterized by straighforward numerical determination. While first-principles calculations

would be ideal for performing this task, they are still limited by the excessive computational effort

associated with complex geometries. It is true that because of the simplicity of the NiA1 case,

first-principles techniques could be used without major difficulties. However, this might not be

true when multiple alloying additions are considered. On the other hand, the computational

effort involved is not an issue with other semiempirical techniques comparable to BFS, but most

still fail in properly dealing with this situation in the presence of extended defects from a physical

standpoint or by relying heavily on parameters that might not be applicable or effective for the

type of alloy structure under consideration.

In all previous applications of BFS, we have used the Equivalent Crystal Theory (ECT)

[24,25], which in a straightforward manner, provides a simple algorithm for the calculation of

defect energies even in the presence of such radical defects as surfaces. The underlying concept

of ECT is simple and clear and it becomes particularly easy to understand if presented in terms of

the Universal Binding Energy Relation (UBER) [26]. Consider the ground state crystal, characterized by the equilibrium value of the Wigner-Seitz cell, rws E. An isotropic expansion or compression of such a crystal from its equilibrium shape (where all bonds are expanded or compressed

equally in all directions) yields a universal (in the sense that it has the same shape for all elements) curve for the energy vs. Wigner-Seitz radius (or atomic volume), as shown in Fig. 4.a.

If the reference atom is in an equilibrium position in the ground state crystal, it sits at the bottom of this curve, with an energy -E c (cohesive energy). If this crystal is deformed anisotropically,

or if a defect is introduced in the vicinity of the reference atom (i.e., a vacancy, an interstitial

atom, etc.), it raises its energy. The curve indicates that there are two values of the Wigner-Seitz

NAS A/TM--1998-208820 14
radius for which each atom in a homogenous, ideal, crystal have the same energy as the reference

atom in the defect crystal. We will call these two crystals, equivalent crystals of the reference

atom. These two positions represent two different conditions. To the right of the minimum the

crystal has an increased lattice parameter and thus a reduced electron density. To the left of the

minimum the opposite is true.

E

(a)

Ground state

f------A}-

rWSE /r

Y

/

/

/

/

/

/

/

/

/

/

/

/

/

(b)

Surface

E

rinst rsurf r

/

/

/

/

_f

_J

J

/

/

/

/

/

/

/

(c)

Interstitial

,l{g----_

:{ g----j{

Fig. 4: Universal binding energy relation (E vs. r) for a monatomic crystal. When all atoms are in their

equilibrium lattice sites, the energy of each atom is the cohesive energy, indicated with the arrow in (a). In

the presence of a surface (b) or in the presence of an interstitial atom (c), a given atom will have a higher

energy along the binding energy curve, which also corresponds to a certain ideal perfect crystal with an

expanded (b) or compressed (c) lattice parameter. Any atom in these equivalent crystals have the same

energy as those of the reference atoms (denoted by an arrow) in (b) and (c).

NAS A/TM--1998-208820 15
For example, an atom on the surface of a metal (Fig. 4.b), after losing a large number of neighbors, thus increasing its energy substantially due to the decrease in electron density, has the same

energy in that situation that it would have in a perfect crystal but with a larger lattice parameter. In

terms of its energy, the atom would not distinguish between being on the surface or being in a

larger version of the ground state crystal. Conversely, if an interstitial atom is introduced close to

the reference atom, its energy increases to an amount equivalent to the one that it would have in a

properly homogenously compressed ideal crystal. Figure 5 reiterates these concepts, showing the

equivalent crystals of a surface atom. As a consequence, every point along the UBER of a certain

crystal is degenerate, as there is a large number of defects that would have the same energy, and

therefore, the same equivalent crystal. In other words, the reference atom can find itself in the

presence of a number of different defects that raise its energy by the same amount, therefore

assigning the same equivalent crystal to all those defects.

I_ao

:t

Atom in the surface

plane

Atom in the bulk of an expanded

(equivalent) crystal

Figure 5: Equivalent crystals of a surface atom. The surface atom is in an environment of reduced electron

density, equivalent to that found in an isotropically expanded bulk crystal.

NAS A/TM--1998-208820 16
ECT provides an efficient algorithm for finding the equivalent crystal for each atom, corresponding to a certain environment, based on quantum perturbation theory. We refer the reader to

Ref. 24 for a derivation of the ECT equations from perturbation theory, and to Ref. 25 for more

details on the implementation of the ECT operational equations. Once the reference atom is chosen, all that is needed is to know the distances between the reference atoms and a group of its

neighbors. Although no preset cut-off distance is defined in ECT, it is sufficient in most cases to

consider all neighbors up to second or sometimes third-neighbor distances. Finding the equivalent crystal for a certain atom near a defect translates into finding the equivalent lattice parameter,

i.e., the expanded or compressed lattice parameter of the equivalent crystal, such that the energy

of an atom in that crystal is the same as that of the atom in the defect crystal. This is done by solving the following transcendental equation for the lattice parameter of the equivalent crystal (see

Ref. 24 for a detailed derivation of this equation)

p ((z + S(rk))r k

= 2_.rke (3)

k

where N and M are the ideal number of nearest-neighbors (N) and next-nearest neighbors (M) in

the equivalent (ideal) crystal, R 1 and R 2 are the nearest-neighbor and next-nearest-neighbor distances, respectively, in the equivalent crystal, and where p, o_and 1 are ECT parameters that fully

describe the corresponding atomic species within the context of ECT [24]. Eq. 3 is solved for a s,

the lattice parameter of the equivalent crystal (R1 and R 2 are the nearest neighbor and next nearest

neighbor distances, respectively, in a crystal with lattice parameter as). The right hand side (r.h.s.)

of this equation, which is computed in terms of the actual distances for the known structure in the

defect crystal rk, can be interpreted as a measure of the defect (in the sense of how the actual enviNASA/TM--1998-208820 17
ronment surrounding the reference atom differs from that in its pure crystal) due to the actual

atomic environment of the reference atom where r k are the distances between these atoms and the

reference atom. S(r) is a screening function, also to be defined later. The sum runs over all neighboring atoms (at distances rk) within a sphere of radius r c, defined as the next-nearest neighbor

distance in the perfect euqilibrium lattice.

If this equation admits a solution, that means that the equivalent crystal associated with the

reference atom is properly defined and its energy can be determined by use of the binding energy

relationship (demonstrated to accurately represent the UBER) [25,26]:

_x aS

e s = -Ec(1 + as)e (4)

The scaled lattice parameter a* s is related to the solution of Eq. 3, a s , by means of the following expression

• (as - ae)

as = q 1 (5)

where a e is the equilibrium lattice parameter, E c is the cohesive energy of the ground state crystal,

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

rws E, and 1 is a scaling length to be defined later. Having computed the equivalent crystal, we now

determine the BFS strain energy contribution of the reference atom as the difference between the

energy of the atom in the defect crystal and its energy at equilibrium in its ground state crystal:

e s = E c 1-(l+as)e (6)

NAS A/TM--1998-208820 18
where the subindex S to the scaled lattice parameter denotes the fact that this equivalent crystal is

related to the strain energy component of this atom in the alloy. The parameters p, _, 1 and _ are

ECT parameters that fully describe the atomic species in question: p is 2n-2, where n is the principal quantum number, a e is the equilibrium lattice parameter, 1 is a scaling length and _ is a

screening factor to properly account for screening effects beyond first neighbors (see Ref. 24 for a

full description of the determination of these parameters).

The parameter _, which is directly related to the structure of the wave function in the overlap

region between similar atoms, has been determined in past applications of ECT by requiring that

the formalism reproduces the experimental value of the single vacancy formation energy [24,25].

However, in order to minimize the dependence on experimental or theoretical input for the determination of these parameters, we now introduce a novel way of determining the parameter _,

which provides the additional benefit of determining the range of validity of the approximations

introduced in writing the perturbation equation for the calculation of the equivalent crystal. As

mentioned before, the r.h.s, of Eq. 3 can be considered as a 'measure' of the defect (in the sense

that the departure of this quantity from its equilibrium value is representative of the difference

between the defect crystal and the perfect, equilibrium crystal). For example, the presence of

vacancies close to the reference atom reduces the value of this quantity, while other defects, like

interstitials, increase its value. While a solution exists for arbitrarily small values of this quantity,

this is not so for values much larger than the one corresponding to equilibrium (for which the

equivalent crystal is the equilibrium crystal itself). There is a maximum value for the r.h.s of Eq. 3

above which no solution to this equation can be found. We therefore determine _ by requiring that

this maximum defect corresponds to the situation when the strain energy of the reference atom

equals its cohesive energy. In doing so, _ becomes dependent on the other parameters characterizNAS A/TM--1998-208820 19
ing the atomic species (E c, a e, p, l, )_). This redefinition of the ECT parameter o_reduces the actual

necessary input to just the cohesive energy (Ec), the equilibrium lattice parameter (ae), and the

bulk modulus (Bo) since 1 is directly related to these three parameters by

E c (7)

12 __

12rCrwsEB o

where rws E denotes the equilibrium value of the Wigner-Seitz radius. Furthermore, the screening

length )v [24] is defined as

TABLE 1

COMPUTED INPUT PARAMETERS FOR NiA1 CALCULATIONS

LMTO results

Atom Lattice Cohesive Bulk Vacancy

Parameter Energy Modulus Energy

a e (_) E c (eV) B 0 (SPa) E v (eV)

Ni 2.752 5.869 249 3.0

A1 3.190 3.942 78 1.8

ECT parameters

Atom p _ 1

(_-1) (_) (_)

Ni 6 3.067 0.763 0.2717

A1 4 1.8756 1.038 0.3695

BFS parameters

ANiA1 = -0.0581 /_-1 AA1Ni = 0.0840 _-1

LMTO results and ECT parameters for bcc-based Ni and A1, and BFS

parameters for B2 NiA1.

NASA/TM--1998-208820 20
= 2.81l. (8)

Finally, the screening function S(r) is defined by S(r)=O if r<r 1, S(r) = (1/2X)(1 - cos _) if

r2<r<r 1 (where [3 = n(r-rl)/(r2-rl)) and S(r)= 1/_ if r>r 2, where r 1 and r 2 are the equilibrium

nearest- and next-nearest-neighbor distances in the equilibrium crystal [24,25].

Chemical energy

The calculation of the chemical energy involves an environment related to the ground state of

the reference atom (i.e., the reference atom, as well as its neighbors, are located in equilibrium lattice sites corresponding to the ground state crystal of the reference atom) although the actual

chemical distribution of the atoms is conserved. Given the actual chemical environment of a given

atom (those atoms within next-nearest neighbor distance from it), it is straightforward to determine an equivalent chemical environment in a perfect crystal. A set of transformations can be

defined which assigns, to each site in the perfect crystal, the appropriate chemical species. The

resulting distribution is then chemically equivalent to that found in the alloy. In essence, the same

considerations made for the strain energy apply here, with the only difference being that the

changes of electron density in the vicinity of the reference atom are now due to changes in the

atomic identity of the neighboring atoms rather than changes in the atomic locations. In this sense,

the same concepts (the existence of an equivalent crystal) apply, only now the reference atom can

have different energies than that allowed by its own UBER. As mentioned before, _ is the only

parameter within our formalism that carries all the information regarding the electron density in

the overlap region between a pair of atoms. For dissimilar atoms it is only natural then to 'perturb'

NASA/TM--1998-208820 21
such an environment by allowing the wave function in that region to be parameterized by a

slightly different parameter than that used in monatomic calculations. We therefore define the

BFS parameter AAB (and ABA), given by

(ZAB = (ZA -t- ABA (9)

when the reference atom is of species A and its neighbor of species B and

{XBA = {XB + AAB (10)

when B is the reference atom and its neighbor is an A atom. The BFS parameter AAB (or ABA)

perturbs the pure element parameter o_, indicating the mixed nature of the bond.

Following the guidelines for writing Eq. 3 and taking into account that the neighboring atoms

are by definition located in equilibrium lattice sites of the reference atom, the trascendental equation to be solved for the (chemical) equivalent crystal is simply

N,,PA %R_ pA (% + _A)R2

K 1 e +MR 2 e

_'_ t)A ((_A + AkA)rI, A t)A ( + AkA +I_Ajr2, ^1 )\(_AA

= 2_rl, ae +Zrz, ae (11)

k k

where rlA denotes the nearest neighbor distance in an ideal crystal A and r2A the corresponding

next-nearest neighbor distance. All other terms have their previous meanings. The calculation of

the chemical energy proceeds from solving Eq. 11 for a c, the lattice parameter of the equivalent

crystal for which R 1 and R 2 are the nearest- and next-nearest neighbor distances, respectively,, and

then determining the scaled lattice parameter a* c given by

NASA/TM--1998-208820 22
* (a C - aA)

a c = q 1A (12)

so that the chemical energy is

e c = YEc 1-(l+ac_e )

(13)

where y= 1 if a* c > 0 andy= -1 if a* c <0 [18].

Finally, a safeguard should be introduced in the definition of the chemical energy contribution

to account for those situations where the reference atom does not have full coordination (i.e., the

number of nearest-neighbors is less than that found in the perfect, equilibrium crystal). Consider

the reference atom occupying a surface site of an alloy. In computing the chemical energy, information about the existence of the surface is introduced by the fact that there are not enough atoms

in the vicinity of the reference atom to correspond to the assumption in Eq. 11. The chemical

energy obtained in this way would carry information not only on the chemical effect but also on

the structural effect due to the absence of some neighbors. In order to completely free the chemical energy from this structural information, we reference the previously defined chemical energy

to a similar structural state but where all the atoms surrounding the reference atom are of the same

identity as the reference atom. A reference chemical energy ec0 is thus computed in this manner,

so that the total chemical energy contribution from the reference atom is

e c = e c- eco. (14)

where eCo is computed using Eq. 11 but setting the BFS parameters to zero. The energy is then

computed using Eqs. 12 and 13.

NASA/TM--1998-208820 23
Coupling the strain and chemical energy

As defined, the chemical energy does not depend on the actual geometry of the alloy which is,

obviously, unrealistic. Therefore the BFS strain and chemical energy terms need to be recoupled

by properly accounting for the influence of structural effects on the chemical contribution. In

order to properly describe the asymptotic behavior of such a quantity ( ec should vanish at large

separations and should drastically increase at small separations), we introduce a coupling function

g linking both terms. The coupling function introduces this asymptotic dependence by means of

a* s, the strain scaled lattice parameter (Eq. 5), which can be understood as a measure of the strain

effect. If a's= 0 it means that the reference atom finds itself in an environment that resembles

equilibrium; a positive value of a* s results from average expansions with respect to equilibrium

and negative values of a* s relate to average compressions. We therefore define the coupling function g as:

g = e (15)

so that defects that involve expansions reduce the effect of the chemical energy on the total energy

of formation and viceversa. Fig. 6 displays the effect of the coupling function g on the chemical

energy, which otherwise would amount to a constant contribution to AH, independent of the size

of the cell. Summarizing these concepts, the BFS contribution e from an atom i to the energy of

formation AH is given by

e = e s+g(e c-eco) = e s+gec (16)

A graphical representation of the calculation of e is shown in Fig. 7.

NASA/TM--1998-208820 24
gec

1

/

/

/

/

/

/

Fig. 6: Schematic respresentation of the strain and chemical

energy contributions of an atom

A in an alloy as a function of

the (strain) equivalent lattice

parameter a s . The dashed line

indicates the constant value of

the chemical energy, which

amounts to the total contribution

to the energy of formation only

when the lattice parameter of the

alloy coincides with that of the

equilibrium crystal A. The total

chemical energy contribution,

gec, vanishes at large interatomic separations. For example, if the equivalent (strain)

lattice parameter is al, we show

the corresponding strain (es) and

chemical (gee) energy contributions. If the equivalent (strain)

lattice parameter is ae, then e s =

0 and e C corresponds to its constant value ( g = 1).

Applying BFS to the calculation of the energy of formation of a bcc binary alloy

To demonstrate the ideas presented thus far, we apply the BFS formalism to the calculation of

the energy of formation of a binary A-B alloy in the B2 structure, as a function of the alloy lattice

parameter a o . Fig. 8 represents the alloy formation process where two pure bcc crystals A and B,

each with equilibrium lattice parameters aA and aB are reassembled to form a B2 alloy with lattice

parameter a o. In the abscence of defects, this is a particularly simple example due to its high symmetry and uniformity. All A atoms are equivalent, and so are the B atoms. It is only necessary to

NASA/TM--1998-208820 25
compute the energy contribution of one generic A atom and one B atom. In what follows, we will

compute eA only, since eg would be computed in the same manner. The different contributions to

eA are shown in Fig. 9.

In writing the equation for the calculation of the BFS strain energy (see Eq. 3), we note that in

the actual alloy an A atom has 8 nearest neighbor B atoms at a distance r 1 and 6 next-nearest

neighbor A atoms at a distance a o. Following the method for evaluating the strain energy environment, we must remember to 'flip' the identity of these neighbors to that of the reference atom. The

BFS strain energy equation for atom A is then

e = es + g ( ec eco )

>-_2_ _

m

m +g

m m

-- { _.....__

Fig. 7: Schematic representation of the BFS contributions to the total energy of formation. The left hand

side represents the reference atom (denoted by an arrow) in an alloy. The different terms on the right hand

side indicate the strain energy (atoms in their actual positions but of the same atomic species as the reference atom), the chemical energy term (atoms in ideal lattice sites) and the reference chemical energy

(same as before, but with the atoms retaining the original identity of the reference atoms).

NASA/TM--1998-208820 26
_nPA _ARI ,_nPA k. A (_ PA _Arl PA )_A

_5/_ 1 e + o/_ 2 e = _sr I e + 6a o e (17)

This equation is trivially solved, with the lattice parameter of the equivalent crystal being just

A

as ao . The corresponding BFS strain energy contribution is

e s = E l+(l+a s )e ) (18)

where the scaled lattice parameter of the (strain) equivalent crystal is given by

A* (ao-aA)

a s = q IA (19)

A similar calculation is carried out for atom B, replacing PA, 1A, aA and )_A with PB, 1B, aB and

)_B in Eqs. 17-19. From the calculation of the strain energy contributions for atoms A and B one

A (aA) B (aB)

A-B (a0)

• qw qr

© ©

Fig. 8: Crystals A and B (with lattice parameter aA and aB) are

d •

reassembled to form a B2 A-B alloy with lattice parameter ao.

NASA/TM--1998-208820 27
O a0 0 a0

O O

0 aA

I

o

aA

o

I

o

Fig.9: Schematic representation of the contribution of an A atom (center of the cube) to the energy of

formation of B2 A-B compound with lattice parameter ao. The left hand side represents the actual B2

structure. The first term in the left hand side indicates the strain energy environment (all atoms are of

type A, in the lattice of the alloy and the second and third term (between brackets) indicate the chemical

energy environment and the reference chemical energy environment, respectively.

would also derive the corresponding coupling functions gA and gB, according to Eq. 15.

Fig. 10 shows the environment assumed in the calculation of the chemical energy. Since in the

example given, both pure crystals are of the same crystallographic structure as the alloy, the calculation is particularly simple, and it only requires noticing that the lattice parameter for each case

is the equilibrium lattice parameter of the pure crystals (a A and as).

The BFS equation for the calculation of the chemical energy contribution for atom A is

,o ,e

o"

I I

O •

A (aA) B (aB)

Fig. 10: Environment seen by atoms A and B, respectively, for the calculation of the BFS chemical energy

contribution. Note that the actual atomic composition of the alloy is preserved, but the lattice parameters

used for each case corresponds to that of the equilibrium pure crystals A and B, respectively.

NASA/TM--1998-208820 28
((XA + _A)R2 ((Xa + _A)aa

_ nPA _ARI .- nPA \ A _ PA ((_A + ABA)rA PA A

_/_1 e + oK 2 e = _51"A e + 6a A e

(20)

where rA is the equilibrium nearest neighbor distance in a crystal A. Note that atom A interacts

with 8 nearest neighbors of species B (thus the factor 0_A+ABA in the first term of the r.h.s, of Eq.

20) and that all 6 of its next-nearest neighbors are of the same atomic species A. This equation is

solved for the equivalent lattice parameter a c and the chemical energy contribution is

e c = 7A E 1-(1 +a c )e ) (21)

where the scaled lattice parameter of the (chemical) equivalent crystal is given by

A* ( aA -- aA)

a c = q IA

(22)

and where YA = 1 or -1, depending on whether the scaled lattice parameter is positive or negative.

A similar calculation is performed for atom B, replacing the appropriate parameters. In particular,

note that the interaction parameter in the exponential in the first term of the r.h.s, of Eq. 20 is now

0_B+AAB, as all the nearest neighbors of the reference B atom are of species A (Fig. 10).

In this particularly simple example, it is not necessary to compute the reference chemical

energy (ec0) for either type of atom, as the corresponding environments correspond to perfect

equilibrium A and B crystals already. Therefore, the total chemical energy contributions are

A A B B

Ec ec and Ec ec .

NASA/TM--1998-208820 29
We finally write the following expression for the energy of formation M-/of the B2 structure

A-B:

1 A A B B

AH = _(e s + gAEC + e S + gBEC) (23)

While this expression applies to this stoichiometric binary system characterized by one atom

of each atomic species, in more complex situations the calculation might involve considering

more non-equivalent atoms. In this example the Ni atoms locate themselves in one simple cubic

sublattice and the A1 atoms in the other. As such, all Ni or A1 atoms are respectively identical and

contribute the same amount to the energy of formation. Therefore, the BFS calculations have to

be performed for just one atom of each species. The simplicity of this calculation allows for a

straighforward determination of the BFS parameters in terms of the energy of formation and equilibrium lattice parameter of the binary base alloy (NiA1). In the following subsection, we will

refer to the scheme used in this work for parameterizing the binary system at hand.

Parameterization of the BFS Method

The previous example indicates that beyond the approximate scheme developed to simulate

the process of alloy formation (i.e. separating strain and chemical contributions), the power and

accuracy of the method are heavily dependent on the parameters used both for the individual elements (p, l, a e, E c, )_) and those used for each type of binary combination that might appear in the

system (AAt 3 and ABA). Also, this example highlights the extreme simplicity of the method when

the crystallographic structure of the alloy is the same as that of the constituents (in the example,

NASA/TM--1998-208820 30
both A and B were bcc elements and the alloy formed is also bcc-based). While in general it is not

known what is the phase structure of an alloy for a certain composition, some basic knowledge

can be used a priori to simplify the calculations.

Take, for example, the system to be studied in this work: [_-NiA1. This binary alloy is known

to have the B2 structure, in spite of the fact that each of its constituents are fcc elements. Assuming the existence of a bcc phase of Ni as well as for A1 would reduce the complexity of the calculation to a level similar to the example just discussed. If the BFS method relied only on

experimental input for the determination of the necessary parameters listed above, it would make

this approach impossible to implement under most circumstances. However, first-principles

methods can be used in a straightforward manner to determine the physical properties of these

elements ( bcc-Ni and bcc-A1) even if they do not exist naturally. In doing so, not only do we

eliminate the dependence of the BFS method on sometimes uncertain or non-existent experimental input, but we also generate a consistent approach. This approach simplifies the numerical

complexity of the problem and also introduces a systematic procedure for the generation of the

necessary parameterization, since they would all be computed by the same method, with the same

level of approximation.

To determine these parameters, we used the Linear-Muffin-Tin-Orbital (LMTO) [27] method

in the Atomic Sphere Approximation (ASA). This scheme was used to calculate the equilibrium

properties ( cohesive energy E c, equilibrium lattice parameter a e, and bulk modulus Bo) of the elemental solids in the same crystal symmetry as that of the compound to be studied (bcc). This set

of parameters is accurately described by the Local Density Approximation [28]. In this work, we

generated parameters for Ni, A1, Ti, Cr and Cu in the bcc symmetry, as well as the B2 structures

of each of the pairs formed (only B2 NiA1 and NiTi exist in nature) for the determination of the

NASA/TM--1998-208820 31
BFS parameters AAB and ABA. The parameters for Ti, Cr and Cu will be used and validated in

subsequent papers in this series.

The LMTO method uses a minimal basis set. For this work, we only used s, p and d orbitals.

All calculations were done with equivalent sampling of the Brillouin zone using, for the bcc lattice, 120 k-points in the irreducible wedge. The basis set and sampling of the Brillouin zone used

for the calculation of the B2 compounds was equivalent to that used for the pure elements. We

refer the reader to Refs. 27-28 for a detailed discussion of the LMTO method. The pure element

LMTO-generated parameters p, l, E c, a e and the resulting value of o_ are displayed in Table 1 for

Ni and A1.

The simplicity of the BFS formalism allows for a straighforward method for determining the

BFS parameters AAB and ABA by an analytical procedure that not only eliminates numerical

errors but also sheds light on the range of validity of the resulting parameters. Leaving the details

of the determination of the BFS parameters to Appendix 1, we list in Table 2 the resulting values

for ANiA1 and AA1Ni.

Other theoretical methods applied to intermetallics

As mentioned in the Introduction, there has been substantial progress in the last two decades

in the area of theoretical techniques and their relevance in dealing with systems of practical

importance. Current techniques range from those that impose an empirical parameterization of the

constants in Ising Hamiltonians [29-33] or those that parameterize experimental databases [34],

families of semiempirical methods based on the determination of interatomic potentials [11-15],

to all-electron first principles calculations [35-44]. Recent first-principles calculations include the

NASA/TM--1998-208820 32
full-potential linearized augmented Slater-type orbital method (LASTO) used by Alatalo et al., in

their analysis of the complex Zr-A1 phase diagram [40], the first-principles discrete variational

method study of disordered Fe-V alloys by Krause et al. [41], a detailed study of composition and

phase diagrams of Cu-Au, Ag-Au, Cu-Ag and Ni-Au by Ozolins et al. [42], and the determination

of formation energies for F%A1 by Mayer et al. [43]. These, and other similar studies, concentrate mostly on monatomic [12,44] or binary systems.

In terms of semiempirical techniques, current work on NiA1 alloys include the extensive analysis of this system by Farkas et al., in a series of recent articles where the embedded atom method

is modified for its use in this system [45-46]. Other techniques currently used include Finnis-Sinclair potentials specifically determined for NiA1 [47], tight-binding methods [12,44] (still mostly

restricted to single element systems), and potentials determined from first-principles calculations.

This last group includes the work of Zhang et al. in Fe-A1 and Ni-A1 alloys [48], and the study of

transition metal aluminides by Moriarty and Widom [49].

Other relevant work in this area includes the extension of Miedema's semiempirical model for

point defects in B2 compounds by Bakker et al. [34] and the development of semiempirical Nbody non-central potentials for FeA1 alloys by Besson and Morillo [50].

In terms of ternary NiAI+X systems, very little progress has been made relative to that made

for binary alloys, although it is expected that the development of accurate potentials and efficient

computational methods, will generate growth in this area in the near future. Most notably, recent

theoretical work of the influence of ternary additions to NiA1 includes the Monte Carlo analysis of

NiAI+Ti alloys by Sumin et al. [51], and the statistical theory of ordering combined with the electronic theory of alloys in the pseudopotential approximation by Mekhrasov et al. [52].

As we increase the complexity of the system at hand, as will clearly be the case in the subseNASA/TM--1998-208820 33
quent papers in this series, we will have little if any previous theoretical work for comparison.

Once we move beyond ternary systems, all our results will become purely of a predictive nature,

relying on experimental verification.

RESULTS AND DISCUSSION

The BFS calculations for B2 NiA1 were performed on a 72-atom cell, allowing for atomic

relaxation induced by the presence of vacancies and substitutional atoms. However, relaxations

were allowed only at a global level in the sense that the lattice parameter of the cell as a whole

was varied until the energy was minimized. A more detailed study, which we consider unnecessary for the goals of this work, would include the possibility of local atomic relaxations in the

vicinity of defects. This issue, and its consequences on the phase structure, will be dealt with in

forthcoming papers.

In a cell of this size there is a large number of possible distributions of Ni and A1 atoms for

any given alloy composition. Knowing the behavior of each and every one of them would be

unnecessary (in this case, considering all the possible distributions of Ni and A1 atoms as well as

vacancies, in a 72-site cell), as most of the information needed for determining the defect structure can be obtained from a few selected situations where the essential defect structures are considered. Therefore, instead of searching for an absolute energy minimum for a given number of Ni

and A1 atoms located in every available lattice site with any specific short and long order scheme,

we chose to construct a selected, but sufficiently large number of "candidate' distributions to

obtain information on the energetics of the system close to the ground state. These configurations

are still, to a certain extent, arbitrary, and may not necessarily include the ground state for a given

NASA/TM--1998-208820 34
composition.However,it is clearthatwhencreatingacatalogueof selectedconfigurationswhich

considerthe mostlikely atomicdistributions,onewould obtainnot only thenecessaryinformation (i.e.,the lowestenergystatefor eachconcentration)but alsoadditionalinformationongeneral trendsin defectstructures.While mostof the configurationsincludedin the catalogueare

experimentallyunaccessible,theyservethe purposeof determiningtrendsin thebulk properties

of the actualalloys.Moreover,if the setof configurationssampledis sufficiently largeandthe

structuresareproperlychosen,respectingthesymmetriesthat characterizethe system,onewould

expectto findthegroundstateor statessufficientlycloseto it ateachcomposition.

As aconsequence,anadditionaladvantageof this approachis thatit enablesusto determine

an 'energyspectrum',showinghow alternativeorderingschemescompareenergeticallywith the

lowestenergystatefor eachcomposition.Themagnitudeof the energygapsbetweenalternative

distributionsandthegroundstateshouldbeproportionalto theprobabilityof findingtheseconfigurationsin theactualalloy.Figure 11includesrepresentativesamplesof thecatalogueof configurationsusedin this work, first showingthebasic72-atomcell (Fig. 11.a)followedby a fewcases

thatincludesubstitutionaldefects,antistructureatomsandvacancies(Fig. 11.b-d).

BasedonstoichiometricNiA1,Appendix2introducesa simplewayof characterizingeachone

of theconfigurationsincludedin thecatalogueof structuresweexamined.Due to thelargenumber of configurationsincluded,only a small but representativenumberof them aregraphically

shownin Fig. 11.Therestaredescribedin anotationalshorthanddiscussedin Appendix2.

The analyticalprocedureis thenstraightforward.The BFSmethodis appliedto eachoneof

theconfigurationsincludedin thecataloguein orderto determineits energyof formationaswell

as the equilibrium lattice parameter(for which the total energyof formation is minimized).

Appendix2 alsoincludestheseBFSresultsfor eachconfiguration.

NASA/TM--1998-208820 35
(a)

I

I

I

[ll NiA1

I , I I I bi_

I .. I ..:::_ • I . ::..... _::.

::: - ¢-1:T-k .'_ _ _....... _ "_ ..... _iiii_ I

LI ;I i'Ll ,I ,.,_ I, I ,.L j

II ' I : At" I , I ,_ _ I , I ,

I I ' ' iF ' ' I _1 ,_1"1 , ..,L.d"], I ,__ ,

__iii_._-_-

_I _

_iii_.._

I -- I

-_ : @

-;-o

_ A14_2,43 [4] Niso,s 2

Fig. 11: (a) The computational cell used to generate the catalogue shown in Appendix 2. (b) Sample of a NiA1 cell showing two site exchanges (see [4] in the catalogue).

NASA/TM--1998-208820 36
(c)

®

(d)

[25] A13241,46 "'-"_ N i_

..@ =@ .@

--@, --_,---'®--x:-._@---':---@ @-" _ _,,--- T-- _ f--

I I I

_i I'II _,._i_ , I --_I

.... r-" - .:;iijJJiJ

I I I I

"+:+" I

I I

@ O • @ • O

[38] A! 42,_11_- V + Ni so ,_II_A! + Ni31,_i_- ¥-

Fig. 11 (continued): (c) Sample of a Ni52.78A147.l 2 alloy where A1 atoms at lattice sites 38, 41 and

46 are replaced by Ni atoms (see [25] in the catalogue in Appendix 2). (d) Sample of a

Ni47.83A152.17 alloy, where the A1 atom at lattice site 42 is replaced by a vacancy, the Ni atom at

lattice site 30 is replaced by an A1 atom, and the Ni atom at lattice site 31 is replaced by a vacancy

(see [38] in the catalogue).

NASA/TM--1998-208820 37
Fig. 12 displays some of the results for representative configurations in the catalogue. The

results are shown in the form of an energy spectrum, which shows a clear comparison of the energetics of the various atomic arrangements. Each column in this spectrum corresponds to specific

compositions of Ni50+xA150_ x alloys, denoted by the Ni atomic concentration. Each energy level is

identified by the catalogue number used in Appendix 2. The lowest energy configuration (lowest

level in the first column in Fig. 12) corresponds to the perfect B2 structure, where all Ni and all A1

atoms populate their own interpenetrating sublattices. Higher energy configurations represent a

wide variety of defects. Moving up in the spectrum, we first find a group of levels very close in

energy which correspond to the presence of antistructure atoms, where a pair of Ni and A1 atoms

switch places. The small differences in energy between these levels relates to the relative position

of the two antistructure atoms (being smallest when they are nearest-neighbors). After another

comparable gap in energy, we found states with two antistructure pairs and so on. Similar situations are found at higher Ni concentrations. The lowest energy configuration for a given concentration in Ni-rich alloys always corresponds to a pure substitutional state and higher energy states

include additional swaps of atoms between Ni and A1 sites. For the highest Ni concentration

shown (55.55 at. % Ni), the purely substitutional case distinguishes between the configuration

where the substitutional atoms are as far away from each other as possible, to configurations with

slightly higher energy where the substitutional atoms display a clustering tendency.

This spectrum emphasizes both the completeness of the catalogue selected (i.e. the lowest

energy state is properly identified for every concentration) as well as its shortcomings: some high

energy states (2 for 50 at. % Ni; 19 for 51.39 at. % Ni), characterized by clustering of antistructure

atoms) are missing at higher concentrations. On the other hand, the high energy state 27 for 55.55

at. % Ni is comparable to 8 for 50 at. % Ni, both including specific alignment of the antistructure

NASA/TM--1998-208820 38
o

>

o

o

0.00 _

-0.30_

-0.60_

15

12

14

4

3

5

9

11

10

2

18

17

19

1 16

27

22 28-32

20 _ 23 ;24 _ 26

21 25

50.00 51.39 52.78 54.17 55.55

at. % Ni

Fig. 12: 'Energy spectrum' for substitutional defect structures in Ni-rich NiA1 alloys. Each column

represents a specific binary alloy concentration. The levels represent the energy of formation of configurations specified in the catalogue (the numbers by each level identify the configuration in Appendix 2). All the configurations included in this Figure correspond to substitutional alloys exclusively

(no vacant sites). In all cases, the lowest energy configuration corresponds to Ni atoms occupying

available A1 sites. Higher energy states include various antistructure defects.

NASA/TM--1998-208820 39
atomsin specificplanesanddirections(seeAppendix2). The smallerenergygapbetweenthis

typeof configurationandthelowestenergystatefor higherNi concentrationshintsatthepossibility thatthis typeof defectmightoccurin theactualalloy.

Fig. 13 showssimilar resultsfor Al-rich alloys, indicatingnot only that the lowestenergy

statesarisefrom thepresenceof Ni vacanciesbut alsothat selectedantistructure-vacancycombinationscouldalsobepresent.Experimentalevidenceconfirmstheseresults[4-9].

Thedeterminationof thelatticeparameterfor eachconfigurationis doneby minimizing the

energyof the cell with respectto thelattice spacingof the cell asa wholebut not allowing for

individual anisotropicrelaxations.While performingdetailedcalculationsincluding individual

atomicrelaxationsis well within thecomputationalefficiencyof themethod,suchlengthycalculations would becomeimpracticalfor higher order alloys. We thereforecompromisebetween

accuracyand simplicity assumingthat whole-cell isotropic relaxationswill be sufficient for

describingtheenergeticsof thesealloys.

The wealth of information embeddedin Fig. 12 canbe usedto gain further insight in the

dependenceof thelatticeparametera(x) of Ni50+xA150_ x alloys. While it is natural to determine

a(x) from the values of a corresponding to the lowest energy states for each concentration, as

depicted in Fig. 12, it is interesting to first visualize the lattice parameter of alternative, higher

energy, structures by choosing characteristic values for each 'band' (group of energy levels close

in energy) in Fig. 12.

Fig. 14 displays the lattice parameter as a function of Ni concentration for a portion of the set

of "candidate' configurations, regardless of the energy of the configuration. Each point corresponds to a distinct arrangement of Ni and A1 atoms as well as vacant Ni or A1 sites in the calculational cell. We focus our attention on three particular regimes that can be seen in Fig. 14: an

NASA/TM--1998-208820 40
invertedV-shapedsetof filled symbolswith its apexat the stoichiometriccomposition,a line of

circlesindicating structureswith increasinglattice parameterwith increasingA1content,and a

largenumberof configurationsenclosedby theboundaryline consistingof filled symbols.The

configurationsdenotedby filled symbolsrepresenthelowestenergyconfigurationsat eachcomposition. The ascendingline of circles correspondsto configurationswhereno vacanciesare

presentandincreasingnumbersof A1atomsoccupyNi sites,atypical substitutionaldefectstruc-

-0.15_

©

-0.25_

(D

©

-0.35--

©

(D

-0.45--

m 35

33

34

m36

39

37

38

42

_41

m 40

45

44

43

47

46

49.29 48.57 47.83 47.06 46.27

at. % Ni

Fig. 13 : Energy spectrum for Al-rich NiA1 alloys containing vacancies. For XNi = 49.29, the lowest energy state corresponds to an A1 atom in a Ni site and a nearest-neighbor vacant A1 site (34),

a vacant Ni site (33) and an A1 atom in a Ni site and a distant vacant A1 site (35), in order of

increasing energy. For XNi = 48.57, the lowest (two-vacancy) state corresponds to an A1 antistructure atom and a nearest-neighbor pair of vacancies in Ni and A1 sites (38) with almost the

same energy of formation as a configuration with two vacant Ni sites (37). For higher A1 concentrations, the lowest energy state is always one with vacant Ni sites (40, 43, 46).

NAS A/TM--1998-208820 41
2.90

o< 2.85

(D

.4..a

(D

(D

¢9

.4..a

.4..a

.a 2.80

2.75

X

©

©

X

X

×

NN N

I I I I I I I

@

Types of Detect Structures

_-NiA1

Al-antisite

[--] Ni-antisite

Ni and A1

antisite

/_ Ni vacancies

A1 vacancies

Ni and A1

vacancies

Ni-antisite and Ni vacancies

t Al-antisite and

A1 vacancies

48 50 52 54

at. % Ni

o

Fig. 14: Lattice parameter (in A) as a function of Ni concentration for various candidate defect

structures. Different symbols indicate alloys with different types of general defect structures. Similar symbols for the same composition correspond to different distributions of the same defect

within the computational cell. The shaded symbols correspond to those configurations with the

lowest energy for that composition.

NASA/TM--1998-208820 42
ture. While substitutional states have the lowest energy for each concentration in the Ni-rich side,

the opposite is true for the 45-50 at. % Ni concentration range. The cluster of points enclosed by

the inverted V-shaped boundary correspond to configurations characterized by random distributions of vacancies in A1 or Ni sites as well as random exchanges of Ni and A1 atoms from their

equilibrium stoichiometric locations. Stoichiometric [_- NiA1 is noted by an asterisk.

A comparison of the modeled results with experimental data for the lattice parameter of various NiA1 alloys is displayed in Fig. 15. The theoretical results, shown as a continuous line, consist

of the calculated lattice parameter and density for the lowest energy configuration observed at

each composition. The lattice parameter results from the lowest energy configurations on the Nirich side of stoichiometry, show excellent agreement with experimental data (Fig. 14). The linear

regime in the density (Fig. 15.a) and lattice parameter (Fig. 15.b) vs. Ni concentration for the

range 50-60 at.% Ni is almost exactly reproduced by our calculations. A recent review paper by

Noebe et al. [53] provides a convenient linear description of the available experimental results,

both for the lattice parameter a and density 9 vs. Ni concentration. To highlight the agreement

between theory and experiment, we therefore compare those expressions with ones corresponding

to the BFS results. For comparison purposes, both quantities were normalized to the stoichiometric NiA1 values (a o and 9o, for the lattice parameter and density, respectively), as there is a slight

o o

difference of 1.2 % between the experimental (a o = 2.887 A) and LMTO values (aTheo r = 2.85 A)

for the lattice parameter of the stoichiometric B2 NiA1 structure. The expressions for experimental and theoretical results on the Ni-rich side of stoichiometry are

(if0/ = 0.5339+0.009322xNi (24)

Exp

and

NASA/TM--1998-208820 43
<

ZQ._

Q._

<

1.10

1.00

0.90 I I I

48 52 54

at. % Ni

F 1

r.:y

1.000

0.996

0.992

0.988

48 52 54 56

Fig. 15 : (a) Density and (b)

lattice parameter of binary

alloys as a function of Ni concentration, normalized to

their stoichiometric values.

The solid symbols denote

experimental results from different investigators (see Ref.

53). The lines denote the BFS

predicted properties (see

text), corresponding to the

structure with the lowest

energy state for each composition.

at. % Ni

NASA/TM--1998-208820 44
(fro) = 0"5324 + 0"009352XNi

Theor

for the density (relative to the stoichiometric NiA1 density) and

(25)

(a_) = 1.0384 _ 0.0007914XN; (26)

Exp

and

(_-o) = 1.0396-0.0007689XNi (27)

Theor

for the lattice parameter (relative to the stoichiometric NiA1 value) for Ni concentrations, xNi,

between 50 and 60 at. %.

On the Al-rich side of stoichiometry a more complicated defect structure is observed. The

lowest energy states on the Al-rich side correspond to arrangements of vacancies with no change

in site occupancy for the Ni and A1 atoms with respect to their original sublattice in NiA1. In the

lowest energy configurations, the Ni vacant sites are at next-nearest neighbor distance from each

other. In other words, clustering of the defects results in the lowest energies. Once again, the ensuing values for the lattice parameter and the density as a function of Ni concentration (closed triangles in Fig. 14) can be described by linear relationships and compared to the experimental values

[19]:

(L) = O'02XNi

Exp

and

(28)

NASA/TM--1998-208820 45
_-0.19914+001 02xNi

Theor

for the density (relative to the stoichiometric NiA1 value) and

(a_] = 0.924148+0.001517XNi (30)

Exp

and

(a_) = 0.840261+0.003184XNi (31)

Theor

for the lattice parameter (relative to the stoichiometric NiA1 equilibrium value) for XNi between 45

and 50 at. %.

While not directly apparent from Fig. 14, this work also showed that locally ordered arrangements of defects were energetically preferred to random distributions of point defects. For example, in Fig. 14 at 47.82 at. % Ni there are an open and filled triangle of very similar lattice

parameter. The difference in these points is that the input structure of the lowest energy configuration (solid triangle) was composed of a greater degree of ordering of Ni-vacancies. While overall

structures composed of Ni-vacancies had the lowest energy at a given composition, a similar trend

of lower energy with increasing ordering of point defects was observed within the other types of

defect arrangements as well. For example, within the family of configurations composed of A1-

vacancies plus Ni-vacancies, the configurations (hexagons) with lattice parameters closest to

those of the ground state consisted of locally ordered arrangements of vacancies (i.e. an A1

vacancy shared by two Ni-vacancies at the nearest neighbor distance) while those configurations

with higher energy consisted of more random distributions of point defects.

This also hints at the possibility that the actual defect arrangements on the Al-rich side of stoichiometry may be more complicated than normally assumed. A complete treatment of temperaNASA/TM--1998-208820 46
ture and local relaxation effects would be necessary to absolutely rule out the existence of these

more complicated structures consisting of ordered arrangements of Ni and A1 vacancies in Al-rich

alloys. But even that would not rule out the possibility that these structures may actually be

observed as metastable states since their energy is not that different from defect structures composed solely of Ni-vacancies.

CONCLUSIONS

We introduced a formalism that combines the type of results achieved by first-principles

methods with the convenience of semiempirical methods for a simple and straightforward analysis of situations that cannot be treated in a similar fashion by either individual technique. An intuitive as well as a formal description of the BFS method was presented, with the goal of

familiarizing the reader with the concepts to be used in this and subsequent papers in this series on

NiAl-based alloy design. Beyond the operational equations, we introduced other elements that

will be heavily used when dealing with higher order alloys. These include the definition of a catalogue of predetermined configurations (as opposed to lengthy computational algorithms for the

search of minimum energy structures), the use of energy spectra to identify trends and to gain

insight on metastable structures (as opposed to just dealing with lowest-energy states), and the

description of the observed behavior in terms of BFS strain and chemical energies.

As our first test of the BFS method we analyzed the defect structure of binary NiA1 alloys.

Our theoretical modeling of the defect structure of [_-NiA1 alloys is fully consistent with the conclusions drawn from experimental evidence indicating that substitutional defects dominate in Nirich alloys and that the presence of vacancies is responsible for the sharp decrease in lattice

NASA/TM--1998-208820 47
parameter with increasing A1 content in Al-rich alloys. Moreover, the results hint to the possibility that a more complex defect structure exists: the presence of vacancies both in the Ni and A1

sublattice in Al-rich alloys and a preference for clustering or local ordering instead of a random

distribution of point defects with any deviation in stoichiometry. In future papers we will build on

these results by modeling ternary and higher order alloys.

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2. Darolia, R., in Structural Intermetallics, ed. by R. Darolia et al., The Minerals, Metals and

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The Minerals, Metals and Materials Society, Warrendale, PA, 1997, pp. 573.

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9. Kogachi, M., Minamigawa, S. and Nakahigashi, K., Acta Metall. 40 (1992) 1113.

10. Faulkner, J. S., Moghadam, N. Y., Wang, Y. and Stocks, G. M., Phys. Rev. B 57 (1998) 7653.

11. Baskes, M. I., Materials Chemistry and Physics 50 (1997) 152.

NASA/TM--1998-208820 48
12.Yang,S.H., Mehl.,M. J.andPapaconstantonopolous, D. A., Phys.Rev.B 57(1998)R2013.

13.Norskov,J.K., Phys.Rev.B 20(1979)446.

14.Foiles,S.M. andDaw,M. S.,J.Mater.Res.2 (1987)5.

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16.Good,B., Bozzolo,G.andFerrante,J.,Phys.Rev.B 48(1993)18284.

17.Bozzolo,G.,Amador,C.,Ferrante,J.andNoebe,R. D.,ScriptaMetall. 33(1995)1907.

18.Bozzolo,G.,Ferrante,J.andSmith,J.R.,Phys.Rev.B 45 (1992)493.

19.Bozzolo,G. andFerrante,J., Phys.Rev.B 45 (1992) 12191;Bozzolo,G., andFerrante,J.,

ScriptaMetall.26 (1992)127.

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21.Bozzolo,G. andFerrante,J.,Ultramicroscopy42/44(1992)55.

22.Bozzolo,G. Noebe,R. D., Garg,A., Ferrante,J.andAmador,C.,Mat.Res.Soc.Syrup.Proc.

460(1997)443.

23.Kobistek,R.,Bozzolo,G.,Ferrante,J.andSchlosser,H., Surf.Sci. 307-309(1994)390.

24. Smith, J. R., Perry,T., Banerjea,A., Ferrante,J. andBozzolo,G., Phys.Rev.B 44 (1991)

6444.

25. Bozzolo,G., Ferrante,J. andRodriguez,A. M., J.Computer-AidedMater.Design1 (1993)

285.

26.Rose,J. H., Smith,J.R. andFerrante,J.,Phys.Rev.B 28 (1983)1835;Rose,J.H., Smith,J.

R.,Guinea,F.andFerrante,J.,Phys.Rev.B 29(1984)2963.

27.Andersen,O.K., Postnikov,A. V.andSavrasov,S.Y.,Mat.Res.Soc.Syrup.Proc.253(1992)

37.

NASA/TM--1998-208820 49
28. Hohenberg, R, and Kohn, W., Phys. Rev. 136 (1964) B864; Kohn, W. and Sham, L. J., Phys.

Rev. 140 (1965) A1133.

29. Shockley, W., J. Chem. Phys. 6 (1938) 130.

30. Cowley, J. M., Phys. Rev. 77 (1950) 669.

31. van Baal, C. M., Physica (Amsterdam) 64 (1973) 571.

32. Golosov, N. S., Popov, L. E. and Pudan, L. Y., J. Phys. Chem. Solids 34 (1973) 1149.

33. Kikuchi, R., J. Chem. Phys. 60 (1974) 1071.

34. Bakker, H., Modder, I. W., and Kuin, M. J., Intermetallics 5 (1997) 535, and references

therein.

35. Terakura, K., Oguchi, T., Mohri, T. and Watanabe, K., Phys. Rev. B 35 (1987) 2169.

36. Sanchez, J. M., Stark, J. R and Moruzzi, V. L., Phys. Rev. B 44 (1991) 5411.

37. Amador, C. and Bozzolo, G., Phys. Rev. B 49 (1994) 956.

38. Wolverton, C. and Zunger, A., Comput. Mater. Sci. 8 (1997) 107.

39. Wolverton, C., Ozolins, V. and Zunger, A., Phys. Rev. B 57 (1998) 4332.

40. Alatalo, M., Weinert, M. and Watson, R. E., Phys. Rev. B 57 (1998) R2009.

41. Krause, J. C., Padvani, C., Schaff, J. and da Costa, M. I., Phys. Rev. B 57 (1998) 587.

42. Ozolins, V., Wolverton, C. and Zunger, A., Phys. Rev. B 57 (1998) 6427.

43. Mayer, J., Meyer, B., Oehrens, J. S., Bester, G., Barnsen, N. and Fahnle, M., Intermetallics 5

(1997) 597.

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45. Farkas, D., Zhou, S. J., Vailhe, C., Mutasa, B. and Panova, J., J. Mater. Res. 12 (1997) 93, and

references therein.

46. Mishin, Y. and Farkas, D., Philos. Mag. A 75 (1997) 169.

NASA/TM--1998-208820 50
47. Gu, Q., Wang, Y. X., Wang, B. Y., Xu, G. Y. and Wang, T. M., Science in China Series E -

Technological Sciences 39 (1996) 551.

48. Zhang, W. Q., Xie, Q., Ge, X. J. and Chen, N. X., J. Appl. Phys. 82 (1997) 578.

49. Moriarty, J. A. and Widom, M., Phys. Rev. B 56 (1997) 7905.

50. Besson, R. and Morillo, J., Phys. Rev. B 55 (1997) 193.

51. Sumin, V. V., Povarova, K. B., Fadeeva, N. V., Kazanskaya, N. K. and Bespaldova, M. D.,

Russian Metallurgy 1 (1997) 1998.

52. Mekhrasov, A. O., Akdeniz, M. V. and Arer, M. M., Acta Mater. 45 (1997) 1077.

53. Noebe, R. D., Bowman, R. R. and Nathal, M. V., Int. Maters. Rev. 38 (1993)193.

NASA/TM--1998-208820 51
Appendix A - BFS Parameterization

An advantage of the BFS method is that, due to its simple formulation, it allows for a straightforward (analytical) determination of the BFS AAB and ABA interaction parameters, therefore

avoiding numerical uncertainties inherent to any numerical fitting procedure. Moreover, the input

data used, whether it is obtained experimentally or from other theoretical calculations, 'localizes'

the accuracy of the ensuing BFS predictions for those alloys in the vicinity of the phase diagram

of the ordered structure used as input.

For simplicity, we reduce the following derivation to the case where the ordered structure corresponds to a cubic lattice characterized by a single lattice parameter a o (simple bcc or fcc alloys

with no tetragonal distortion).

Consider an alloy A-B, where due to the symmetry of the structure, there are N x non-equivalent atoms of species X (X = A, B) and nxi denotes the multiplicity of the ith non-equivalent atom

of species X, so that

Nx

_ nx, = N c (A1)

Xi 1

where N c is the total number of atoms in the cell. In this context, two atoms are deemed to be

equivalent if they have the same environment within a sphere of radius R, the nearest-neighbor

distance in the ground state crystal. In this case, the two conditions used to determine the BFS

parameters consist of exactly reproducing the heat of formation AH 0 of the ordered structure and

the corresponding lattice parameter, which are determined via LMTO calculations for a given

structure [27,28]. In this work, we used the B2 NiA1 base alloy as the basis for the LMTO calculation.

NASA/TM--1998-208820 52
The two conditions are

E _----, IIX i

z__ _--_ex_ = AHo

X i 1 C

(A2)

and

_x_nx, 3ex,

ix"1N--cc0-a ]ao = 0 (A3)

where % represents the BFS contribution to the energy of formation, given by

S C Co

exi = exi + gx_(ex_- ex_ ) (A4)

Co Co

In Eq. A4, J/ is the BFS strain energy and Exc/ _xc % is the BFS chemical energy, where % is

the reference energy.

For bulk ordered alloys, such as B2 NiA1, the BFS strain energy is the same for all atoms of

the same species and it is uniquely determined by the input value of the lattice parameter a0 ,

ex_ = E 1-(l +axi)gx_

where

(A5)

(A6)

S* X

axi = _x(ao - a e )

x x

where Ec and a e are the cohesive energy and equilibrium lattice parameter, respectively, for

q where q is a structure constant ( q3 3 for bcc elements) and 1x

atoms of species X and [_x ix, 8_

is a scaling length for species X [24,25]. The coupling function included in Eq. A4 is given by

S*

ax i

= e (A7)

gx i

The BFS energy depends on the lattice parameter of the alloy structure only via the BFS

strain energy and glue, therefore, Eqs. A2-A3 can be written as

NASA/TM--1998-208820 53
E nx, 2.. -_-_ex, f tao) , + gx,(ao)ex, c} = AHo

Xi 1 c[

and

Nx } X i 1 c[da ao "-d-a ao

(A8)

= 0. (A9)

Eq. A8 and A9 can then be written as

Nx flXi S , ,

= AHo- E E -_-ex, tao)

Xi 1 c

and

Nx Hv v S*

Xi 1 c

Nx flXi C

X i 1 c

(AIO)

(All)

If we concentrate now only on binary alloys that form fcc or bcc ordered structures characterized by a single lattice parameter (LI2, LI0, B2, B32, etc.), then fiA + fib N c . Simple expressions

can then be obtained for the BFS chemical energies EAc and Ec"

C _B_I -- 52

EA rlAg(AO)(_B__A)

and

(A12)

C 52 -- _ASI (A13)

EB = nBg(O)(_B__A)

where

' S

51 = (n A + ne)AH o - (nAesA(ao) + neee(ao)) (A14)

and

x (o)

52 = __nx_xEcgx (ao)

X

(A15)

with g(x°) gx(ao). The BFS chemical energies can then be determined with Eqs. A12 and A13, so

NASA/TM--1998-208820 54
that we can then search for the set of parameters (AAB , ABA ) that simultaneously satisfy these conditions. This is done by starting with the use of the following expression for the BFS chemical

energy in terms of the equivalent chemical) lattice parameter

ex = YxE I i - ( l + a x )e

(A16)

C •

where v 1 if ax > o and v 1 otherwise. The scaled lattice parameter for the chemical energy

ax , given by

C* C X

ax = x(ax- ae) (A17)

is related to the BFS parameters (AAB , ABA ) by means of the BFS equation for the chemical energy

(C_x+_l)R2

Nr.Px _xR_ Px z,x )

1_ 1 e +MR 2 e

.c-, ,• px (C_x+ Akx)rxl ( c_x+ Akx+ _l A,xj)rx 2

: 2_jVxkrx,e + ZMxkrPx_e (118)

k k

where N(M) is the number of nearest-neighbors (next-nearest-neighbors) in the equivalent crystal

x x (e "/3 for bcc). If we define of speciesX, R1 cac R2 ac rI ca e and rx2 ae , ,

{

Px e _x Cax _ ]C C* _ Px (_x + rx 1

OX =N rx +--_x +M a_+-_x _ e

X _ • PX (XXrXI

ql = IVXXFXI e

(A19)

(A20)

and

x Px (ox+ )rx2

q2 = Mxxrx2 e

(A21)

and considering the typical magnitude of the exponent in Eq.A21, it is reasonable to make the

approximation rx2 = rx, only in that term, then the 1.h.s. of the BFS equation (Eq. A18) reads

NASA/TM--1998-208820 55
(_1 l"

X X Dx ((_x+Ayx)FxI DX _(_x+AyX+_x) X2

Qx- ql - q2 = Nxyrx_e + Mxyrx2 e

(A22)

Let

(_x+_l_rx

X _ r PX (_xrxl Px _'x) 2

qd = lVxyrxle + Mxyrx2 e

(A23)

so that

X X X AyxrxI

Qx - ql - q: = q_e (A24)

then the BFS parameter ayx is given by

X X

1 rjQx_ ql - q2

Ayx - ln[ x

(A25)

rx_ qd

This result is exact if second-neighbor contributions are not taken into account, or if for any

particular structure Mxy 0 for all X and Y, as is the case in this work. The B2 structure is such

that any given atom has an atom of its own species as a next-nearest-neighbor, i.e. Mxx 6 and

Myx 0 . With the exception of the numerical solution of Eq. (A16), the procedure for the determination of ayx using Eq. (A25) is straightforward and simple. Moreover, it can be easily shown

that in most cases a quadratic approximation to the Rydberg function (1 + z)e z suffices to guarantee accuracy up to 10 % of the exact results, with the added advantage of a completely analytical

determination of the BFS parameters AAB and A_A. The parameters used in this work were

obtained by following the procedure described in this Appendix, including second neighbor interactions and a numerical solution of the the transcendental equations involved.

NASA/TM--1998-208820 56
APPENDIX 2

In this appendix, we list the catalogue of configurations used in this work for different

atomic distributions in a 72-atom cell, shown in Fig. 11. The B2 NiA1 alloy corresponds to the

atomic distribution shown in Fig. 11.a, where Ni atoms are denoted by black disks (labeled 1, 2,

3 .... ) and the A1 atoms are denoted with open circles (labeled 13, 14, 15 .... ). A configuration is

defined by changing the occupancy of these sites by exchanging atomic positions, substituting

atoms of one species by atoms of the other species, or by introducing vacancies. The following

table lists most of the configurations used in this work (only those used to construct the energy

spectrum shown in Figs. 12 and 13). For example, a Ni atom in site n substituting for an A1 atom

in site m is denoted with Ni n -> Alm; V n denotes a vacancy in site n and A1n <-> Ni m denotes an

exchange of A1 and Ni atoms between sites n and m. The configurations are listed according to

their Ni concentration. The table also includes the values of the energy of formation of the configuration as well as its equilibrium lattice parameter (obtained by minimizing the energy of the

cell with respect to a).

NASA/TM--1998-208820 57
XNi

50.00

51.39

52.78

54.17

Configuration

1. B2 NiA1

2. Ni 31 <-> A142.

3. Ni30,32 <->A141,43

4. Ni30,32 <-> A142,43

5. Ni30,31 <-> A142,43

6. Nil_ 9 <-> Al13_24

7. Nil,3,5,7,9,11 <-> Al13,15,17,19,21,23

8. A165_68 <-> Ni53_56

9. Ni30,31 <->Al18,42

10. Ni31 <-> A142

11. Ni30 <-> A146

12. Ni30,31,34,35,54,55,58,59 <-> Al14,17,19,22,62,65,67,70

13. Ni30,31,34,35,54,55,58,49 <->A113,15,21,23,61,63,69,71

14. Ni50,54,58 -> A1 + A162,66,7o->Ni

15. Ni3o->A1 + Ni2,5,7,10,50,53,55,58 <>Al13,14,17,18,37,38,41,42

16. A142 -> Ni

17. A142 -> Ni + [12]

18. A142 -> Ni + [13]

19. Al18,3O,42 -> Ni

20. A142,43 -> Ni

21. A142,44 -> Ni

22. A138,42,46 -> Ni

23. A142_44 -> Ni

24. A142,46,47 -> Ni

25. A138,41,46 -> Ni

AH

(eV/atom)

-0.60310

-0.50816

-0.41377

-0.41261

-0.43544

-0.07384

-0.16910

-0.31271

-0.44184

-0.50814

-0.49795

-0.11404

-0.06528

-0.09057

-0.06414

-0.59223

-0.12280

-0.06828

-0.50607

-0.57990

-0.58163

-0.56609

-0.56781

-0.56759

-0.53671

a

(A)

2.8500

2.8588

2.8678

2.8679

2.8656

2.9006

2.8914

2.8773

2.8650

2.8588

2.8597

2.8964

2.9011

2.9050

2.9020

2.8457

2.8902

28955

2.8537

2.8418

2.8416

2.8380

2.8378

2.8379

2.8259

NASA/TM--1998-208820 58
XNi

55.55

XNi

Configuration

26. A138,40,61,63 -> Ni

27. A165_68 -> Ni + A161_64 <-> Ni53_56

28. Al18,19,42,43 -> Ni

29. A138,39,41,42 -> Ni

30. Al19,23,42,46 ->

31. Al18,22,23,46 -> Ni

32. A165_68 -> Ni

Al-rich Ni(5o_x)Al(5o+x)

Configuration

49.29 33. Ni31 -> V

34. Ni 31 -> A1 + A142 ->V

35. Ni31 -> A1 + A146 -> V

48.57 36. Ni3o,32 -> V

37. Ni3o,31 -> V

38. Ni3o -> A1 + Ni31 -> V + A142 -> V

39. Ni3o,31 -> A1 + Al18,42 -> V

47.83 40. Ni26,3o,34 -> V

41. Ni3o,31,32 -> V

42. Ni3o,34,35 -> V

47.06 43. Ni3o,31,54,55 -> V

44. Ni26,27,29,30 -> V

45. A165_68 -> V + Ni53_56 -> A1

46.27 46. Ni5,29,3o,31,32,53 -> V

47. Ni5,7,29,31,53,55 -> V

AH

(eV/atom)

-0.56044

-0.33123

-0.55365

-0.55547

-0.55702

-0.55523

-0.55418

AH

(eV/atom)

-0.43876

-0.45149

-0.42323

-0.27055

-0.35110

-0.35407

-0.34075

-0.34334

-0.26174

-0.24887

-0.18313

-0.14426

-0.10201

-0.09769

-0.06919

a

(A)

2.8335

2.8541

2.8342

2.8340

2.8338

2.8340

2.8341

a

(A)

2.8426

2.8474

2.8518

2.8338

2.8349

2.8383

2.8442

2.8276

2.8263

2.8281

2.8218

2.8205

2.8430

2.8070

2.7960

NASA/TM--1998-208820 59
Form Approved REPORT DOCUMENTATION PAGE

OMB No. 0704-0188

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources,

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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

November 1998 Technical Memorandum

4. TITLE AND SUBTITLE

An Introduction to the BFS Method and Its Use to Model Binary NiAL Alloys

6. AUTHOR(S)

Guillermo Bozzolo, Ronald D. Noebe, J. Ferrante, and C. Amador

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

National Aeronautics and Space Administration

Lewis Research Center

Cleveland, Ohio 44135- 3191

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546- 0001

5. FUNDING NUMBERS

WU-523-22-13-00

8. PERFORMING ORGANIZATION

REPORTNUMBER

E-11426

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA TM--1998-208820

11. SUPPLEMENTARY NOTES

Guillermo Bozzolo, Ohio Aersospace Institute, 22800 Cedar Point Road, Cleveland Ohio 44142; Ronald D. Noebe

NASA Lewis Research Center; J. Ferrante, Cleveland State University, Cleveland, Ohio 44115; C. Amador

Universidad Nacional Aut6noma de M6xico, Mexico D.F., Mexico. Responsible person, Ronald D. Noebe, organization

code 5120, (216) 433-2093.

12a. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified - Unlimited

Subject Category: 26 Distribution: Nonstandard

This publication is available from the NASA Center for AeroSpace Information, (301) 6214)390.

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

We introduce the Bozzolo-Ferrante-Smith (BFS) method for alloys as a computationally efficient tool for aiding in the

process of alloy design. An intuitive description of the BFS method is provided, followed by a formal discussion of its

implementation. The method is applied to the study of the defect structure of NiA1 binary alloys. The groundwork is laid

for a detailed progression to higher order NiAl-based alloys linking theoretical calculations and computer simulations

based on the BFS method and experimental work validating each step of the alloy design process.

14. SUBJECT TERMS

Intermetallics; Atomistic simulations; BFS method; Alloys

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15. NUMBER OF PAGES

65

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A04

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