Thermodynamics of the nickel-titanium system

Thermodynamics

of the nickel-t

A tight-binding-bond C. Colinet

and A. Pasture1

Anderscn

In recent ads

on

the

properties

explained cy

the phase

the

knowledge

atomic

the

For

the

numbers white

and data.

Hamiltonians meters possibtc

to

principles canonical

ref.

[I ] for

obtain

them

band structure basis

as

the

which

do

them

arc

the

of par;lit is

fitting

110~

first-

[2] or from

by Harrison

[ 31 OI

not

retical

only

tight-binding

the

Method diagram\

fol-

the

the

compound tat

phase of

system

a~-c

diagram: phase

it

ISC’C-hascd

the solid

the

wc

calculated have

phase

i\ worth

diagram

\,uluch of the energics dcterminc

but

k

verk

solution

the that

sensitilc

to

phases to

Ni’l‘i

cxpc~n~nc~l-

noting

propertic\ \ccins

that

found

boundaries

of

of the three

equilibrium

b\,

kkthe

\vetl \vith

\amc as in the

the

the

cncl-g!

the Cluster

decomposition

not

of

daci-iptioir

[ IO, I 11. The

that

\vxtern\

[O]. The

agr-ee reasonably

pcritectic

both

wc‘rc determined

using

cxpcrinicnt;itl)

Nikl‘i

toI- Thee)-

fr_aniewVrk

free-energv

(c’I3LM)

c,xhibit

also

of

mahc

generation

interest

((‘VM)

phases

Lattice

of

the

OlII

[5- SI.

that

iicw

hut

diagrams

liquid

phase

;I

great

calculations

ctctci-mind

for

c)C

method

f~~CllWd

systems

properfit

cntcring

and

have

;111op. 31.c

within

\.ariation

the

first-ordci

the

NiLTi

have

Pliasc

obtained

ctustcr

part

\\c

candidatc4

studv.

wcrc

10

and

which

cast:.

in ;I Hamiltonian

Hamiltonian.

high-p”rformancc

phenomen:i

last

the

an~ilogoti~

Ni-AI

promising

In

re\utts

studiex.

on

interaction5

details) by

previous

attention

of

[-I].

model

tnut.fin-tiil’c,rl,ital

of solid

tight-binding

calculations

proposed

directly

ix

linear

arc the sole

the choice

either

which

catcutntions.

parameters

more

pure

in

calculations.

If’. at the beginning, over

and

of

avaitabtc

tight-binding

potential

to

the sole

propertics

readily

stumbled

(see

from

first-principles

the

accura

possible

proper-tie5

of alloys

of the constituents

for

Slatcr-Kostcr required

reasonable

.lepwn

cl-band

In

can now bc

become

elemental arc

ol meth-

and

canonical

approximation.

and alloys with

stability

that

literature.

structure

thermodynamic

of

constituents

dcvclopmcnt

density

it has

particular.

examine

data.

local

and computed

In

[ I].

the

electronic

of metals

study

:I

with

years,

and accurate

based

many

sys tern

approach

I. Introduction

efficient

itanium

thl\ the

which

but bc too

the cs-

C. Colinet, A. Pusturei I Thermodynamics

othermic with respect to the other phases. It has been thought that it is due to the Bethe lattice approximation. Therefore, in this study, we decide to keep the Bethe lattice approximation only for the liquid phase and to use the coherentpotential approximation (CPA) [ 121, combined method with the generalized perturbation escribe energy-properties of (GPM) [13] to d BCC- and FCC-based solid solutions. The paper is organized as follows: in section 2 and 3, we present a brief review of the quantumand statistical-mechanical approaches used in our calculations. In section 4, we present the results of the calculations for the Ni-Ti system. A complete equilibrium phase diagram is calculated for the Ni-Ti system and compared to the experimental one.

2. Model 2.1.

The tight-binding-bond

model

The quantum mechanical model is based on a tight-binding (TB) approximation for the Hamiltonian of the alloy which includes both diagonal and off-diagonal disorder. The oneelectron TB Hamiltonian is described using a basis of five d orbitals per atom: H=

C E,~l&L)(&I ‘P

+ C P,,.,.Iir.L)(jvl ‘F.,U

.

- E”J~!,qL”

+ K&,“$q

in which the bandwidth is 254, d is the bond length and s is the Wigner-Seitz radius. The first term C - E,, corresponds to E in eq. (1) and is taken to be the fixed-energy zero and center of the d-band. Therefore, the volume and structure dependence enters only in the variation of the Slater-Koster parameters. Paxton et al. [15] have shown that it is a good approximation. In this study of the Ni-Ti system, we use Andersen and Jepsen’s most localized Slater-Koster and potential parameters for each element at their equilibrium atomic volume [ 141. Two difficulties are arising from alloying effects: (i) the Slater-Koster parameters for mixed pairs are assumed to be given by the geometric mean of the corresponding parameters for the elemental metals [7]. (ii) The shift between the on-site energies of the two metals is determined self-consistently by imposing local charge neutrality, a reasonable approximation for a transition metal alloy, where charge transfers are known to be negligible. Imposing the condition of local charge neutrality, the main contribution from the d-electrons to the energy of formation, AE, is given by (4)

(1)

In this expression, lip) is the ket for the orbital p on site i, the on-site and hopping energies E,, and pIfiL.,,, give respectively the effectively atomic energy of the p orbital on site i and its coupling to the orbital v on site j. It can be shown [4] that the TB Hamiltonian is directly analogous to the first-order LMTO Hamiltonian H ip.,r, = K,

239

of the Ni-Ti system

where the alloy (U~~~‘,‘Y) and (UL,,nd) bond energies are given

pure by

elements

E,.

u ~~~~~= c In

( (E - E,,)n,,(E) -x

dE ,

’ EF

U’h<~“d =c

f (E -

E:h)&(E)dE

T

(2)

where C,, - EYIp and & are standard potential parameters and Sllr,,” is the matrix of LMTO structure constants [4,14]. Matrix elements in the second term, in the usual tight-binding model, take the Slater-Koster form [14]:

where n(E) and n:(E) are the electronic density of states for the alloy and pure elements i, respectively. E,, (Ej:) are respectively the onside energy for species i and orbital (Y in the alloy (pure element).

The assumption that each atom is assu~~uxi to remain charge neutral by varying the on-site hamiltonian matrix in such ;I way that the cncrgh splittings between the different orbitals on the same atom arc prcscrved ensures that the tightbinding bond model is consistent with the force theorem [ 161.

The quantity which is of central interest in OUIcalculations (see eqs. (5)-(h)) is the electronic density of states. According to the state which characterizes the alloy at ;I given composition, i.e. liquid state, solid solution or compound, three different steps are required to calculate the electronic density of states of the alloy. 2.2. I. Stoichiometric~ cornpounds To calculate the electronic density of states of compounds, we have used the recursion method 1171,which constructs ~1 new orthogonal hasi { I
where ([I,,} and {/I,,) are the matrix elements. The coefficients N,, and hi are calculated up to ;I given step tz,, and the continued fraction is then terminated in the usual way: (I,, ,i,, = (I, and h2,I ,,,,= bf these calculations fol [ 171. W,t r epcat AI the nonequivalent sites of the compound studied and the electronic density of states of the compound is given by the sum of these local densities of statcs. _._.-. ’ 3 3 Solid solutions For solid solutions, the configuration-averaged Green function at arbitrary composition is required. The best approach is the zeroth-order mean field approximation. i.e. the coherent

potential approximation average Green function using the identit!

((‘PA). ‘fhe proJected of xi i atom is calculated

(i),(E)

CJ)[

L

7

(;,,,,():

I

1 + t,(;,;,,(b-

tr)]

where C; ’ (-_ fr) i4 the Circcn operator corresponds to the C’PA Hamiltonian. X CT,,1;~) (ipl ‘I’ and

(8)

,I

that

+ X CJ,,,, ,<, /i/l. i Xit11 /,, /I

is the Green function of the pure metal R (tfetincd hy EL = 0 and W,, 01. The potential CT is dctermincd self-conby the condition on the scitttcring

(; '(z)

refercncc ,!3,,,,,,,, )_

sistently operators

whet-c

( IO)

(OhlC

C;,;,,(z) = J c

(z)lOh)

(III

\ ‘l‘o arialyre the ordering effects in solid solutionx, it i\ nectzssa~-1’ to use the gc:neralked pcrturhation method (GPM) [ l.31. the ordering king cxpresscd in terms of pair and energy many-body interactions using ;I perturbation expansion of the random CPA energy in concentration fuctuations. Now ’I

where 7‘,, is the diagonal part of the scattering matrix and 11 is the order of the cluster cxpansion. It has heal shown that for binary transition metal alloys [ 11, the contributions of the triplet and larger cluster interactions to the ordering energy are small compared with that of the pairs. so we shall retain only the first-nearest-neighbor

C. Coliner, A. Pasture1 I Thermodynamics

interactions for FCC-based phases whereas and second-nearest-neighbor interactions been considered for BCC-based phases.

firsthave

2.2.3. Liquid alloys To calculate the density of states of the liquid phase, we have used the scalar version of the alloy Cluster Bethe Lattice Method (CBLM) [ 111 which is a very appropriate method for describing liquid systems. The Green’s function of atom I is given by G,(Z) = c %G,(z).
(13)

G,(Z) is scalar inside the invariant subspaces (Y(I) defined by the types of orbitals chosen (d in our case) [ll]; it is given by the following set of equations:

(144

of the Ni-Ti system

computed self-consistently for each composition and degree of SRO. The bond energy is also calculated as a function of SRO and the effective pair interactions of the liquid alloys are obtained from its dependence [7,19].

3. Phase diagram

calculations

As discussed previously, the internal energies can be considered as the starting point of the phenomenological treatment of alloy phase equilibria by means of statistical-mechanics techniques like the cluster variation method (CVM). In order to compute the phase diagram, we need to know the total free energy of the binary alloy in a given phase, then its energy and also its entropy of formation. For the alloy in its phase (Y, the total free energy may be written as F”=xF*,+(l-x)F”,+E~-TS;

Hou is the hamiltonian without hybridization and 4fi is the part of the mean second moment of the DOS on a state (Y due to the coupling with states p of the neighbors:

(16)

where x is the concentration of the A element, FP the free energy of pure element I in phase (Y, Ep and ST respectively the enthalpy and entropy of alloy formation. The first simplifying assumption regarding the evaluation of this total free energy is to assume that the free energy for the pure elements can be written as F; = E; - TS;

where n, is the degeneracy of the subspace LX and g,,(R) the pair correlation function between atoms of type I and J, provided by the reference system. To describe the liquid state in the metallic alloys, it is also possible to take a Bethe lattice with a coordination number roughly equal to 12, a value which is consistent with the local coordination found in the liquid metallic alloys [18]. In this case, to describe SRO the pair correlation functions are replaced by the pair probabilities. It has been shown that for thermodynamic data, i.e. AL? and hS, there two approaches give similar results [19]. The density of states is

241

(17)

in which the structural cohesive energy EP and entropy SF are both temperature independent. The second assumption assumes that Sy is a purely configurational term, which means that the remaining entropy depends only on the concentration via the first two terms on the right hand side of eq. (16). For the calculations of the electronic density of states and energies of formation, we have seen in the previous section that three families of phases have to be distinguished. In the present approach, we use also different configurational entropy approximations depending on the nature of the phase being considered.

this

In taken

cast.

equal

the

configurational

entropy

Liquid

i4

alloys

shown

been

to zero.

to take their

it into

For

the

entropy

solid

solutions,

is dcscribcd

maximum hedron

cluster

used

containing

the BCC

lattice:

entropy

configurational

of the CVM.

in our

first

study

is the

and second

in this

of a BCC

the

by means

approximation,

disordered

in

the molar

system

is given

of

hard

diamctcr

but

interact

tetra-

neighbors

[Xl

by

account

ill

through

display

SK0

I].

it i4 csscntial

and

the

data.

;I refcrencc

mixture

same

The

I?()- 7

thermodynamic

calculations. ;I

also

ma>

rcccntlh

diffcrcnt

[ 32.231 ix ;I good Garting method.

If WC arc only

dynamic

data.

ximilar

point

in

the

the

disordered

if the

of the

FCC‘

within

the

Bcthe

ticluid

tree but

trantctra-

cnergb with

now

interactions

lattice

for

conligura

by the

CVM

structure

first-nearest-neighbor

potential

in the therms-

is appr~~ximatcd

cncrgy

the

which

and

arc found

tional

approximation

I,!

have

foi- ;I var.iationaJ

intercstcd

allo\;\

fret

all

Coulomb

sition-metal-h~lsc~i hcdron

\uch

charactcrired

charges

results

()I

perform

which

;I hcrccncd

has

dctcrmination To

system

sphcra

24

calculated

a~~l~r~~ximation

[Xl.

( IS)

whcrc

it’,,/,,. l,,i.

rcspcctivcly

1

and

yj;’

the probability

triangles.

second-neighbor

pairs

and

points

their

subscripts

alloy).

R

For

in (i

and

pairs.

the

x,

A

or

In

cienotc

grand

first-neighbor B

given in

;I

structure

potential

WC have [S]

6 2: yi,') In j.j,' ' + 5 x X) In .t-, ) / // cluster

following

probabilities

consistency

arc

related

the

equilibrium

convenient

f1 given

phase

to minimi;lc

the

b!

by

whcrc

p

is

( 19)

by

the

relationships:

the

the present potential

The

to dctcrminc it is more

binary

gas constant. FCC

order

diagram.

tctrahcdra.

configuration

equals

is the perfect

the disordered

y{l’

of finding

cffectivc

work,

chcmicnt

is carried

Out with

independent

configurational

cast

the

tctrahcdron

configurational

variables

of

tetrahedron peraturc taking

7‘ and

the

respect

to 2 set

variables:

in

approximation.

chemical

01 the

these to

bc

w#,~, at constant

effective

account

In

of the grand

a~-e chosen

probabilities

into

potential.

the minimisation

the tcn-

potential

normalization

I-(.

constraint

(7’) iI,,

z 7

u’,,h/

(2Oa)

-

This (1) .‘,,

=

;

M’,,hi

’

minimisation

Iteration

(Xb)

(Nl)

[?a]. The !,A(?I = T

Wiii,

.

lz’,,i,

(Xc)

( Xd

equations

have been presented bc repcatcd The

Y/ = x ill

NI

is

)

done

method

using

the

Natural

developed

by

Kikuchi

used in the present elsewhere

[X]

and

model will

not

hcrc.

equilibrium

phases

I

and

II

scheme

as proposed

phase is

diagram

computed by Kikuchi

bctwcen

using

the

two same

and De Fontaine

C. Co&et,

A. Pasture1 I Thermodynarrks

[26]. For the same initial value of the effective chemical potential p, the grand potential of phases I (0,) and II (n,,) are calculated using the procedure described above. If 0, = fl,,, the equilibrium conditions are realised. but if not, the value of p is modified until 0, = .12,,.

4. Results for the Ni-Ti

system

The Ni-Ti phase diagram displayed in fig. 1 is characterized by a liquid phase, a FCC (A,) phase in the Ni-rich part, a BCC (AI) phase in the Ti-rich part for high temperatures; two stoichiometric compounds NiTi, and Ni,Ti and one intermediate phase with variable range of solubility NiTi [27]. 4.1.

Energies

of formation

of compounds

As a first step, we present the values of the energies of formation of the three intermediate phases observed in the equilibrium phase diagram. We have also calculated energies of formation of metastable phases, which are essentially superstructures of BCC or FCC lattices, to study if our TB Hamiltonian is able to predict the correct ground states. In table 1, we compare our values obtained by coupling TB-LMTO Hamiltonian and the recursion method with values obtained by performing self-consistent-

0.10

0.33

0.50

0.70

0.90

x NI Fig.

1. Experimental

phase

diagram

of the Ni-Ti

system

243

of the Ni-Ti system

Table I Calculated

energies

Structure

of formation

in the Ni-Ti

system.

AH, (kJ/mol) TBB

LMTO

Ref. 1291

NiTi,-EY,

-3s

-35

-29

NiTi-L IO NiTi-B2 NiTi-B32

-36.8 -46 -36.5

-30.6 -40.9 -38.5

-34

Ni,Ti&Ll~ Ni,Ti-DO, Ni,Ti-DO

-45.8 -44.4 -46.4

-49.4 -46.X -49.7

21

-43

LMTO total-energy calculations. This comparison is done to check our alloying model, i.e. TB Hamiltonian (with only d orbitals. Shiba approximation, . . .) and bond energy contribution to the total energy. Neglecting the repulsive part of the total energy is acceptable since all the studied compounds or solid solutions are based on closepacked structures and thus have similar local environment [28]. Retaining the d contribution to the TB-LMTO Hamiltonian is sufficient to explain the strong negative values of the alloying formation energies in the Ni-Ti system. The comparison between both sets of data shows that the different approximations used in our TB-bond (TBB) model are acceptable. More particularly, for x,, = 0.5 we have found that the B2 structure is more stable than the Ll,, or B32 structures. For xNI = 0.75 we have checked that the DO,, hexagonal structure is more stable than the Ll, or DO, cubic structures. For both compositions, self-consistent LMTO total-energy calculations and TBB energy calculations point out strong competitions between the different BCC or FCC superstructures. More particularly, for xN, = 0.75 it seems to be possible to obtain the metastable Ll, phase from the Al phase if rapid cooling is used for instance. Both theoretical approaches give values of formation energies which are roughly 20% larger than the experimental ones [29]. It is in fact necessary to perform full-potential calculations to obtain a better agreement [30,31].

The previous attempt to calculate the phase diagram of the Ni-Ti system is based on the CBLM approximation for BCC- and FCC-based solutions. In fig. 7 arc compared the conccntration dependence of the calculated energy of mixing E,~,,,,,(x) for th e completely random KY’ and BCC solid solutions using both the TBCBLM and TB-CPA approaches. We can set that the agreement between the two sets of calculations is better for the FCC-based structure than for the BCC-based structure. This results from the fact that the BCC structure is more sensitive to the Bethc lattice approximation. However, this difference will be sufficient to obtain the phase boundaries of the peritectic decomposition of the NiTi, compound in agrccment with the experimental phase diagram. as shown in the next section.

It is also essential to have a good description of the thermodynamic data of the liquid phase since a great part of the phase diagram is from equilibrium between the liquid

and solid phases. The scalar \,ersion of the alloy (‘BLM coupled with the TU-LMTO Hamiltonian gives calculated values of mixing cnergicx which are displayed in tig. 3. The agreement with the experimental data [.12-331 is very good. capccinlly for the Ni-rich part. Let us mention that the evolution as a function of the composition is \‘er!’ similar to those of B<‘(‘- and FCC-based solid soiutions (see fig. 2).

‘1‘0 obtain the phase diagram given by the competition between different structures (HC’P. BC‘C. FCC or liquid) it is essential to know the thermodynamic propertics of pure Ni and Ti OI-. in other words. the difference between the free energies of the liquid phase and the different crystalline phases. Two ways can be usetl to obtain such data. The first one i\ to use thermodynamic compilations [ 3.51. The second one airsumes that the structural energies between FCC‘. BCC and HCP structures for Ni and Ti elements arc fixed to the values obtained from the selfconsistent LMTO total-energy calculations. We obtain A,!?,,,., , (( = 2.9 (0.7) kJ/atom and for the Ni (Ti) AE I!< I’ , <‘C= 3.3 (-3.5) kJ/atom parameters. i.c. clement. The remaining A(;, ,.( ,.(Ni) and AC;, ,(( ( (Ti), are obtained from the experimental thermochemical data. melting temperatures and latent heats of melting. while AS H( ( ,,( ,,(Ti) is obtained from the experimental HCP-B(‘<’ transformation tcmperature. In fig. 4. we compare both approaches for the Ti elcmcnt. of the Ni-l‘i \ystcm. The phase diagram

C. Colinet.

A. Pasture1

I Thermodynamics

+Ref35 a LMTO fee-hcp

600

2000

1200 T(I<)

Fig. 4. Gibbs

energy

of phases

of the Ti element.

calculated by combining both FCC- and BCCbased free-energy curves with the liquid freeenergy and the free-energies of the three compounds is shown in fig. 5. Although the cubic B2 phase is experimentally reported to be stable over a small concentration range, it has been described as a stoichiometric compound. At high temperatures, the Ni-Ti phase diagram can be viewed as resulting from a BCC-FCC competition; FCC dominates on the Ni side with the FCC solid solution and the metastable Ll, (Ni,Ti) FCC superstructure. On the Ti side, BCC dominates with a BCC Ti terminal phase above 710 K. The solid A2 solution displays a range in good agreement with the experimental one, contrary to the previously reported calculations based on the CBLM [S]. The eutectic I ”

20001

0

Fig. 5. The calculated

1”

0.25

Ni-Ti

I,

0.50 XNi phase

”

r

0.75

diagram

“,

,

1

of the Ni-Ti

system

245

equilibrium is found now between the NiTi, compound and the A2 solid solution, the eutectic temperature being 60 K higher than the experimental one. The congruently melting temperatures of Ni,Ti and NiTi compounds are correctly predicted. The stability of the B2 phase can be understood using simple arguments similar to those used to explain the BCC-FCC competition for pure transition metals. Mixing Ni and Ti corresponds to an average number of d electrons more favourable for the BCC structure than for the FCC structure. Correspondingly, the experimental phase diagram does not exhibit a B2 ground state, although the B2 phase appears above about 900 K [27]. Experimental observations are not conclusive on which path leads to the most stable state. It is either a martensitic transformation to a monoclinic phase not studied here [36], or a decomposition at about 900 K into Ni,Ti-DO,, and NiTi,-E9, [37]. Such discussion is beyond the scope of our TBB analysis but preliminary results based on the self-consistent LMTO calculations indicate that the decomposition between DO,, and E9, structures is the most likely way [38].

5. Conclusion The tight-binding calculation of the Ni-Ti system coupled with CPM-CPA for solid solutions and scalar-CBLM for liquid alloys give results in good agreement with experimental informations. (i) The large interactions between the d states of Ni and the d states of Ti are the main factors governing the highly negative formation energies of the different compounds occurring in this system. We have found for xNI = 0.5 and 0.75 that the compounds crystallize in the B2 and DOzJ structures respectively. (ii) The formation energy of the A2 terminal solution on the Ti-rich side obtained by CPA calculations displays less exothermic values than those given by alloy-CBLM calculations. This leads to a eutectic equilibrium between the NiTi, compound and the A2 solution which is now in agreement with experimental data.

1IS/

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