βNiTi equilibrium in Ni-Ti-Al alloys by the cluster variation method

[lIlerlllelal/ies 5 (1997) 103-109 1:: 1997 Elsevier Science Limited

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Analysis of the {3'Ni2TiAII{3NiTi equilibrium in Ni-Ti-AI alloys by the cluster variation method M. Enomoto & T. Kumeta Department of Materials Science. Ibaraki University. 4-12-1. Nakanarusawa. Hitachi 316. Japan

(Received 15 May 1996; accepted 25 June 1996)

The cluster variation method with the Lennard-Jones potential, previously used to analyze the y(Ll] Ni,AI)/-y(fcc Ni) equilibrium in Ni-AI base multicomponent alloys, was used to calculate the {3'(L2 J Ni2TiAI)/{3 (B2 NiTi) equilibrium phase boundaries in the Ni-Ti-AI system. Results are compared with the isothermal and vertical sections of the ternary phase diagram in the literature. The lattice misfit between {3' and {3 C0mpound lattices, which is considered to play an important role for the formation of a rafted structure, passed through a minimum at 50%Ni, but was considerably larger compared to y/,,; alloys. It is thus necessary to consider alloying addition for the optimization of high temperature properties of {3'/{3 alloys by controlling the size and distribution of the {3' phase and the lattice misfit. © 1997 Elsevier Science Limited. All rights reserved

Key words: A. multiphase intermetallics, B. order/disorder transformations, D. site occupancy, E. phase diagram prediction, G. aero-engine components.

with a simple phenomenological pair potential (Lennard-Jones potential) can successfully predict the "Ilyequilibrium phase boundaries in Ni-Al binary alloys. The method was extended successfully by one of the present authors to ternary and higher order experimental and commercial superalloys.6-8 Though a phenomenological potential was used, this method proved to be useful to consider the influence of alloying elements on important characteristics of the alloys in terms of a few parameters, i.e. the bond strength and atom size. Hence, the method was used to analyze the ,8'1,8 equilibrium in the ternary Ni-AI-Ti system. This study was made envisaging the potential application of the method to higher order ,8'1,8 alloys.

1 INTRODUCTION

Nickel-base heat resistant alloys utilize fine coherent precipitates of a compound phase as a major strengthener. The performance of these alloys at high temperatures is known to be improved by alloying additions. Since the number of potential alloying elements is large, it is advantageous to develop a technique of predicting theoretically the microstructure and equilibrium characteristics in muIticomponent alloys. Recently, a considerable amount of attention has been given to alloys composed of two bccbased compound phases, i.e. L2] (heusler)Ni2TiAI(,8') and B2-NiTi(,8).1-3 For instance, it has been reported that ,8'1,8 alloys have a higher specific strength than conventional "Ily superalloys in a wide temperature range. 4 Thus, phase equilibrium and mechanical properties of these alloys need to be studied in detail. One approach is to analyze the phase equilibrium theoretically and understand the influence of individual and, then, multiple alloying additions on alloy constitution. This may facilitate the design of experimental alloys. Sanchez et al. 5 demonstrated that the cluster variation method coupled

2 CALCULATION METHOD 2.1 Configurational entropy for B2 and L2, structures

Figure I shows the crystal lattice of L2] type compounds and a tetrahedron cluster used in the calculation. The configurational entropy is calculated from the number of ways of distributing 103

M. Enomoto, T. Kumeta

104

Tetrahedron Cluster

for f3' compound, where L(n=(ln(-( and k is the Boltzmann constant. t17t etc, is the probability of finding a triplet of i, j and k configuration, y17(l) and y11(2), etc, are the probabilities of finding short (1st nearest neighbor) and long (2nd nearest neighbor) pairs of i and j configuration, and x1, etc, is the probability of finding an i atom at a lattice point. 2.2 Internal energy of B2 and L2\ structures

Under the pair approximation the internal energy per lattice point of f3 and f3' compounds is given by w w E" = ~ L e(I)yAB(l) + ~ L e(~) {yAA(2) + yBB(2)} (3) "

2..

I}

w·

Ni

•

Ea, = ~ ~

Al

Ti

4 ~

"

Fig. 1. Schematic illustration of L2,-type compound lattice.

I}

I}

I}

tetrahedron atom clusters of various atom configurations. The atom configuration is specified by the tetrahedron cluster probabilities z1711. Here, the subscripts, i, j, k and I denote the component atom species (1 = Ni, 2 = Ti and 3 = AI) and the superscripts A, Band C denote the Ni, Ti and Al sublattices, respectively. In the B2 type compound, the Ti and Al sublattices are disordered. The tetrahedron cluster probabilities in this compound are thus designated as Z17~1 where A and B denote the Ni and (Ti,AI) sublattices, respectively. The entropy per lattice point is given by 9,10

= -k[ ~I 6L(z17~1) 3

+ ~ 2-{L(Y1J<2») + lj

~{6L(t17~) + 6L(t171))

L(yf7(2))) + L. 4L(y17(l») lj

-~~{L(x1) + L(x1)}]

(1)

for f3 and S{3' =-{

~I 6L(z171~ - ~{6L(t171) + 3L(t171) 3

+ 3L(t1;1)) +~2-{L~11(2») + L(y1;(2»)} + ~ {2L(y17(l») lj

lj

(Y1;(l»))-~ f{2L(x1) + L(x1) + L(X7)} ]

(2)

eO) {yAB(l) + yAC(l)} I}

I}

I}

I}

+ W2 _ 4

+

..

lj

and

o

S{3

4

I}

lj

~ e(~) ~ I}

{yAA(2) + yBC(2)} I}

(4)

I]

I}

respectively. wI(=8) is the first and w2(=6), the 2nd and nearest neighbor coordination numbers e(J> denote the interaction potential for short (1st neighbor) and long (2nd neighbor) atom pairs. As previously, a phenomenological Lennard-lones potential

e(p

is used in the calculation, where eij and rij are the parameters to be determined for specific atom pairs and r is the atom distance. m = 8 and n = 4 can adequately describe the cohesive energy of metals. 5,6 Since we now have expressions for the entropy and the energy, we are able to calculate the most probable values of basic cluster probabilities by minimizing the grand potential

n =E -

TS + PV - LiJ-iXi

(6)

in each phase. Here, P is the pressure, V the volume and iJ-i the chemical potential of atom i. Equilibrium phase boundaries are obtained from the cluster probabilities which ensure the same value of the grand potential at common iJ-i values for the two phases. An equilibrium atom distance (i.e. lattice parameter) can be obtained by incorporating another procedure of minimizing the

(3'Ni]TiAII{3NiTi equilibrium

105

Table 1. Potential parameters used in the calculation. N is Avogadro's number i-j

Ne/,kJ/mol

eijolell

Ni-Ni Ti-Ti AI-AI Ni-Ti Ni-AI Ti-AI

62·63 70·12 44·02 78·53 7H7 70·12

1·000 1·119 0·703 1·254 1·173 1·120

°

rij,nm

rjrll

0·2515 0·2933 0·2890 0·2668 0·2544 0·2803

1·000 1·166 1·149 1·061 1·012 1·115

Ti concentration, 51

49

48

47

4

°

45

15 co ....

-

.....'GfHii~:ht-"8--fl· ....

3

c:

--0--

Q)

(,)

c:

2 .................

o(,)

AI in Ni/800'C AI in Ti/800 • C AI in Ni/1100 AI in Til1100

49

51

50

Ni concentration,

A Ne= -Ii B2EI

46

5r-~'-~~~~~~~~~~~

:;:;

grand potential with respect to r. Since the phenomenological potential mimics the interaction only in the vicinity of equilibrium distance, the potential parameters, eS and rij' were re-evaluated from the data on bcc metals and compounds. Rewriting eqn (3) for a pure metal, and optimizing the energy with respect to the atom distance, the following relationships are obtained:

50

at%

·c ·c

52

53

at%

(7)

Fig. 2. Site occupancy of Al atoms in (:lNiTi phase calculated at 800 and 1100°C.

(8)

(10)

and

where N is Avogadro's number, El <0) is the cohesive energy and a i is the lattice parameter of an element i.

where W2

and

r.

8

A= -NeO(~) I 4 /I a B2

4

Wz

O( -rii) B= -Ne 4 Ii I

and B = ~ + ~2 (~})4 222

The cohesive energy of the hypothetical bcc Ni and Al was evaluated from the lattice stability parameter reported by Kaufman and Nesor.ll For most metallic elements, the atomic volume of fcc and bcc differ by less than a few percent in the temperature range of interest. 12 Here, the lattice parameter of a hypothetical bcc allotrope was calculated assuming that the atomic volume increases by 1% from the fcc form. The Ni-Ti and Ni-AI interaction parameters were calculated from the relationships NeS

2 (Bi + B + WI

and

Ai + ~ +

EB2

etc. EB2 «0) and a B2 are the cohesive energy and the lattice parameter of the B2 type compounds, i.e. NiAI or NiTi. The above relationships are obtained by optimizing an equation of energy for a B2 compound such that the cohesive energy passes through a minimum at an observed interatomic distance. 5 The Ti-AI interaction parameters were calculated from Miedema's formula for cohesive energyl3 and the volume size factor compiled by King. 12 The potential parameters thus determined are shown in Table 1. They significantly differ from those determined for the calculation of y/y equilibrium,? though no systematic difference is observed.

3 RESULTS AND DISCUSSION 3.1 Substitution behavior in {3' and {3 compounds

EB2)2

j = - - ------'----

a B2

(9)

To begin with, the substitution site of Al atoms in {3NiTi and the atom distribution in the Ti and Al

M. Enomoto, T. Kumeta

106

a)

-7.165 , - - - - - - - - - - - - - - - - ,

100. 0

~

..

AI in Ti-site AI in AI-site

800'C -7.166

80.0

"#. .... til

r::

-7.167

'+=i til

.

60.0

0

.......c:

•....

a

-7.168 40.0

....•....

Q)

u

.'

....

••••••••••(3 •• ,

•....

c: 0

u

20.0


J.i.Ni=O.6 J.i.AI=-O.2

-7.17 0.0 20.0

l'

40.0

30.0

50.0

-7.171 t-~--'--~--r--~--'-~--r-~---1 0.550 0.552 0.554 0.556 0.558 0.560

NiTi

Ni2TiAI

Bulk Ti concentration, at%

Chemical potential, J.i.Ti Fig. 4. Grand potential versus chemical potential plot for determination of equilibrium composition.

b)

1 00.

r----;;;:::-=====-----t---tIIr-----~r::lI

o 80.0

Liquid

1600 --- -... - - - - -

\ ___!t: _____-_-: ~ ~

60.0 }.)

~ Cii (j) c. E Q)

40.0

1200

"

, /II

/,

:::J

20.0

•

I

,

'I

800 /

'

\1

'. I

•

h /

\

"

I

/3

\

\ I

-_-_-_~-_

.. \

f-

Ti in Ti-site Ti in AI-site

0.0 J=i-t=:::l..L.....,--~--~--.__-~--_1 20.0 30.0 40.0 50.0

t

Ni2 TiAl

I __, ____ -_______

'/'

(3'

NiTi

Bulk Ti concentration, at% Fig. 3. Site occupancy of Ti and Al atoms in {3'Ni2TiAI phase calculated at 800°e.

sublattices of non-stoichiometric {3' phase were calculated. Figure 2 shows the variation of the Al concentration in the Ni and Ti sublattices of the {3 compound with the bulk Ni or Ti concentration calculated at 800 and 1100°C. The added Al concentration is 2 at%. It is seen that Al atoms enter the Ti sublattice as long as the sum of the Al and Ti concentrations are less than 50%. On the other hand, as the sum exceeds 50%, a significant proportion of Al atoms enter the Ni sublattice, the proportion being increased with decreasing Ni concentration from stoichiometry. This can be interpreted along the same lines as the substitution of Cr, Mn and Fe in the L12 Y phase. 6,8 When the Ti concentration in the bulk is,

400

o

(3'

+ (3

L-~_L_~~~~_~-L~_~~~

20.0

30.0

40.0

50.0

Titanium, at%

Fig. 5. Calculated {3'/{3 equilibrium phase boundaries in quasibinary Ni2TiAI-NiTi section. Solid curves assume constant e/1ell and r;/r", long-dashed curves incorporate the temperature dependence of rii and short-dashed curves, the temperature dependence of rii and e/. Thin dashed curves are from Ref. 1 and thin dot-and-dash curves are from Ref. 14.

say 49·5% (arrowed), the Al concentration in the Ni sub lattice is close to 3%, which implies that three quarters of added Al atoms are squeezed out from the Ti sublattice. Thus, although Al basically enters the Ti sublattice, a larger proportion of Al can enter the Ni sublattice, finally ending up with an entire occupation in the Ni sublattice. It is noted that the structural vacancies which are likely to be present in the Ni-Iean side of {3 compounds are not considered in this discussion.

f3'NiJiAllf3NiTi equilihrium 20.00 19.00

0

E -,

~

18.00 17.00 16.00 15.00

0

Qf

14.00 13.00

a)

1 2.00

t----,.--,~-.,.~__,_~__.__~_._-,.-_r_~._...__l

0

200

400

600

800

1000 1200 1400 1600

Temperature, °C

0.310 , - - - - - - - - - - - - - - - - , 0.300 0.290

E

C

0.280

..:.,=

0.270

Ti

0.260

Ni

0.250 L--I!r_-A---/.~-lr---0.240

b)

I-....--,~-.,.~___r_~_r_~_._-,.-_r_~._...__l

o

200

400

600

800

1000 1200 1400 1600

Temperature, °C

Fig. 6. Temperature dependence of potential parameters: (a) e~ and (b) r ii .

Figures 3(a) and (b) show the site occupancy of Al and Ti atoms in the {3' compound in the 50% Ni section calculated at 800°C. In the Ti-rich side, the excess Ti atoms are seen to enter the Al sublattice and as the Ti concentration exceeds ~40%, the two sublattices are no longer distinguishable. This is the transformation of {3' to {3.

107

Boettinger et al. I and Nash and Liang l4 in the quasibinary section (solid curves). It can be seen that the agreement is quite satisfactory. The use of constant parameter values in Table I (eijole ll and fij/fll) implies that the temperature dependence of cohesive energy and the thermal expansion are all equal to those of Ni. For pure metals the temperature dependencies of E; and a; are available in the literature. 15 .16 A I 'j";) expansion of the specific atom volume was assumed for a; of Ni and Al of a hypothetical bcc form. The lattice stability parameter I I was used to calculate the temperature dependence of E; of a bcc form of these elements. The e;;o and f;; thus calculated are plotted as a function of temperature in Fig. 6. The temperature dependence of the parameters of unlike atoms (eijo and f;) was not considered because of the lack of data. The results obtained incorporating only the temperature variation of f;; and incorporating the variation of both r ii and ei ? are shown by long- and short-dashed curves, respectively, in Fig. 5. The influence of temperature dependence on calculated {3'I(3 boundaries are appreciable at high temperatures. Figures 7(a)-(c) show calculated (3'I{3 boundaries in the isothermal section at 800, 900 and 1000°C, respectively, together with experimental data. In Fig. 7(b) the slope of calculated tie-lines appears to be opposite to those reported by Nash and Liang,14 whereas some of the SEM-EDX results in a Ni-43 at% Ti-7 at% Al alloyl5 show the slope of the same sign as calculated. Since {3' precipitates are very fine, an APFIM analysis is being made to determine the composition of precipitates. 16

3.3 Lattice misfit Figure 8 shows the variation of the lattice misfit with the Ni concentration calculated at a fixed temperature. The lattice misfit is here defined as 8=

(11 )

3.2 {3'I{3 equilibrium phase boundaries

The difference in the free energy between the {3' and f3 phases is expected to be small because it is only due to ordering in one sublattice. As shown in Fig. 4, the n versus f..LTi curves for each phase do cross each other and thus, equilibrium composition can be determined. In Fig. 5 the calculated equilibrium phase boundaries are compared with those proposed by

whereas a~ and af3 are the lattice parameter of the {3' and {3 phases, respectively. It is seen that the misfit passes through a minimum at the Ni concentration of 50%. This is because the compositions of the {3' and {3 phases are closest to each other at this composition. y/y alloys are known to exhibit optimum properties when the lattice misfit is minimal. Accordingly, we are interested III alloys containing ~50% Ni subsequently.

M. Enomoto, T. Kumeta

108

b)

a)

25.0

25.0

900°C

Ti. at%

AI. at%

AI. at%

10.0

10.0

5.0 55.0 45.0

5.0 55.0

0 50.0

55.0

60.0

65.0

70.0

75.0

45.0

55.0

60.0

Ni. at%

65.0

70.0

75.0

Ni. at%

A 6. SEM-EDX(17)

EPMA(14)

(a)

(b) c)

25.0

lOOO°C

(3' Ti. at%

AI. at% 10.0

5.0 55.0

~

45.0

____~__~~____~__~~____~__~O 50.0

55.0

60.0

65.0

70.0

75.0

Ni. at%

A . 6.

SEM-EDX
(c)

Fig. 7. Calculated {3'I{3 equilibrium phase boundaries at (a) 800, (b) 900 and (c) IOOO°C compared with experiment. 17•18

Figure 9 shows the variation of the lattice misfit of a 50% Ni alloy with temperature. Ost is the misfit between the stoichiometric {3 and {3' compounds. Thus, the proximity of calculated results at room temperature to Ost indicates that the lattice parameter of {3' compound was reproduced very well from those of NiAI and NiTi by the present method.

The decrease in the misfit with increasing temperature is primarily due to the narrowing of the two phase field at higher temperatures (Fig. 5). The misfit value was reported to be -0·01·6% in Ref. 2 which is consistent with calculation. On the whole, the misfit in {3'1{3 alloys is large compared to yly alloys. Hence, as a next step, the possibility

{3'Ni]TiA1I{3NiTi equilibrium - 0.024 . - - - - - - - - - - - - - - - - - - - ,

o -0.022

--

GOot

..... Qo....

800t

., ... ,6'1'.u

900l:

-0.020

\09

metals and bee-based compounds. The results agree well with those reported in the literature . Coupled with a minimum amount of experiment, this method may be useful in the development of a new class of nickel-base heat resistant alloys which are composed of bee-based compound phases.

CJ')

~ -0.Q18

ACKNOWLEDGEMENTS

-0.016

50.0

52.0

54.0

56.0

Bulk nickel concentration, at%

The authors acknowledge the financial support from the Atom Arrangement and Design Control Project of Japan Research and Development Corporation, Tokyo. Thanks are due to Mr Goukon (formerly a student of Ibaraki University, now at Tokyo Institute of Technology) for his assistance in calculation.

Fig. 8. Calculated variation of lattice misfit between {3' and {3 phases at equilibrium.

REFERENCES -0.030

,----,---,----r--,.-----,--.,---;--,

-0.020

-0.010

o

400

800

1200

1600

Temperature, "C

Fig. 9. Calculated variation of lattice misfit with temperature. For solid and dashed curves, see Fig. 5.

of the reduction of misfit by quaternary alloying addition may be explored. 4 SUMMARY The equilibrium phase boundaries and the lattice misfit in f3'Ni2 TiAI/f3 NiTi alloys were calculated by the cluster variation method with the phenomenological Lennard-Jones pair potential. The potential parameters were evaluated from the thermochemical and lattice parameter data on bee

I Boettinger, W. J., Bendersky, L. A., Biancaniello, F. S. and Cahn, J. W., Mater. Sci. Eng., 1988, 98, 273. 2. Bendersky, L. A., Voorhees, P. W., Boettinger, W. J. and Johnson, W. C, Scripta Metall., 1988,22, 1029. 3. Field, R. D., Darolia, R. and Lahrman, D. F., Scripta Metall., 1989,23, 1469. 4. Koizumi, Y., Ro, H., Nakazawa, S. and Harada, H., Report of the 123rd Committee on Heat-Resistant Metals and Alloys, Vol. 35. Japan Society for the Promotion of Science, 1994, p. 195. 5. Sanchez, J. M., Barefoot, J. R., Jarrett, R. N. and Tien, J. K., Acta Metall., 1984,32,1519. 6. Enomoto, M. and Harada, H., Metall. Trans., 1989, 20A, 649. 7. Enomoto, M., Harada, H. and Yamazaki, M., CALPHAD, 1991, 15, 143. 8. Enomoto, M., Harada, H. and Murakami, H., Tetsu-toHagane, 1944,80,487. 9. Enomoto, M. and Harada, M., in Computer Aided Innovation of New Materials II, eds M. Doyama, J. Kihara, M. Tanaka and R. Yamamoto. North Holland, Amsterdam, 1993, p. 725. 10. Sigli, C and Sanchez, J. M., CA LPHA D, 1984,8,220. 11. Kaufman, L. and Nesor, H., CALPHAD, 1977, 1,27. 12. King, H. W., J. Mater. Sci., 1966, 1, 79. 13. Miedema, A. R. de Boer, F. R. and Room, R., CALPHAD, 1977, 1, 341. 14. Nash, P. and Liang, W. W., Metall. Trans., 1985, 16A, 319. 15. Pearson, W. B., A Handbook of Lattice Spacing and Structure of Metals and Alloys. Pergamon Press, London, 1967. 16. Hultgren, R., Deseai, P. D., Hawkins, D. T., Gleiser M. and Kelly, K. K., Selected Values of the Thermodynamic Properties of the Elements. ASM, 1973. 17. Kumeta, T. and Enomoto, M. Unpublished research at Ibaraki University, 1994. 18. Warren, P., Murakami, H. and Harada, H., Unpublished research at National Research Institute for Metals, 1994.