ELECTRONIC THEORY FOR GRAIN BOUNDARY SEGREGATION AND EMBRITTLEMENT OF INTERMETALLIC COMPOUND Ni3Al

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ELECTRONIC THEORY FOR GRAIN BOUNDARY SEGREGATION AND EMBRITTLEMENT OF

INTERMETALLIC COMPOUND Ni3Al

K. Masuda-Jindo

To cite this version:

K. Masuda-Jindo. ELECTRONIC THEORY FOR GRAIN BOUNDARY SEGREGATION AND

EMBRITTLEMENT OF INTERMETALLIC COMPOUND Ni3Al. Journal de Physique Colloques,

1988, 49 (C5), pp.C5-557-C5-562. �10.1051/jphyscol:1988568�. �jpa-00228065�

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JOURNAL DE PHYSIQUE

Colloque C5, supplement au nD1O, Tome 49, octobre 1988

ELECTRONIC THEORY FOR GRAIN BOUNDARY SEGREGATION AND EMBRITTLEMENT OF INTERMETALLIC COMPOUND N i , A 1

K. MASUDA-JINDO

Department of Materials Science and Engineering, Tokyo Institute of Technology, Nagatsuta, Midori-ku, Yokohama 227, Japan

Abstract. The clo3va.;a s'crength of grain .boundaries in LI2 type of int?raetallic com~ounds (Ni3A1) is investigated using the tight-binding (T3) electronic theory of s, p and d-basis orbitals. The two center hopping integrals are determined by using the Harrison's universal TB theory and atomic energy levels are taken from the atomic structure calculations by Herman and Skillman. The local charge neutrality condition is used for both host metal and solute atoms. It is shown that cleavage strength of the grain boundaries depends strongly on the segregation of impurity atoms (boron) at the grain boundaries, in agree- inent with experimental results: The stoichiometry effect of the L12 compound is also discussed.

1. Introduction

It is now well established that introduction of small amount of impurity atoms can drastically improve the ductility of polycrystalline intermetallic compounds. Recently, Aoki and Izumi [I] have found that the grain boundary embrittlement of Ni3A1 (Ll2 structure) is improved significantly by an addition of boron and about 30% elongation can be attained inanarc-melted Ni3A1 sample. Similar experimental results have also been obtained using samples by thermomechanical processing [21 and rapid solidification [31. In particular, Liu and coworkers [21 have found that boron increases the ductility only in Ni-rich Ni3A1 samples;

the best ductility is achieved in samples with 24 atomic percent Al.

These authors interpreted the experimental results qualitatively on the basis of impurity segregation to grain boundaries (or the peculiar- ity of the grain boundary structure).

In contrast to the experimental works on the grain boundary

strength of intermetallic compounds, there has only been a few theoretical workson the subject [4-61. In general, theoretical works so far presented are in a primitive stage and fundamental understanding on the grain boundary stjrencjth of intermetallic compounds has not been givsn yet. It is the purpose of the present papsr to investigate the strength oP grain :>ouildaries of iiztsjr;le'cnllic ~0~20uii:s liliz I!i3nl v~it::

L I 2 structure using a microscopic electronic theory. Particular atten-

tion is focused on the effect of impurity (boron) segregation on the cleavage strength of the grain boundaries.

To investigate the cleavage strength of the grain boundaries in the Mi3A1 crystal, we perform the total energy calculations for single crystal as well as for bi-crystals including a symmetric tilt grain boundary (with and without impurity segregation): The total energies of the system are calculated as a function of the separation distance Rsep between the two crystallites. This type of cleaving test of crystals can be done quite efficiently and accurately within the present TB electronic theory. One can then calculate the cleavage force during the cleaving process. From these calculations, we can discuss the grain 5oundary strength of the Ni3A1 crystal and understand theoretically the sffect of impurity segregation on the strength of the <;-rain boundary.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988568

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C5-558 JOURNAL DE PHYSIQUE

2. Princii3le of Calculations

To calcxlate the structure and total energy (enthalpy) of the rain boundaries, we use the well established TB (or LCAO) electronic Theory for lattice defect problems [ 7 l C . The total energy of the system is assumed to he given by sum of the band structure energy Eb and the pairwise repulsive potential energy Erep between atomic sites i and j.

Recently it hasbecome possible to justlfy this type of division of the

.a. _,cL.:l.*

-

snergy from first principle electronic theory [ a ] . The TB hopping 2araneters are taken from Harrison's universal theory [91 and atomic Energy levels are from the atomic structure calculations by Herman and Skillman [ 10

I.

The band structure energy E are calculated for s , p a n d d-subbands by using the recursion method [I?, 121. For the sake of numerical calculations of Eb, we use the following expression for the local density of states ( D O S )

X X

pi(E) =: (l/a){blb2

- - .

bn/D2n(~)} 'J4b,

-

^{( E}- am12 /2b,, (1) with

X 2n-1 +

...

D2n(E) = A: E~~

+

A 2 E X

+ A 2 n + ~ 1 (2)

where all the recu.rsion coefficients (a,,b. ) after n-level are assuined to be equal to their infinite values a, an8 b,. A specifies the nine.

valence orbitals s, p In equation 2, tfie expansion coefficients

,

^p

,

pzr xy, yz, zx, x2-y2 and 3 2 r2.

AA

for D2 ( E )

K -

are simple functions of the recursion coefficients ( a l l

...,

^{an, a}^,^? and ( b l ,

...

,

bn, b,). F'or instance, within the approx~mation of n=3 (6-th moment approximation) A 4 are given as

The similar expressions are also given for A4, X A' A? and A?. The asymp- totic recursion coefficients a, and bm are re?:tea to the bandwidth and center of the band, and can be determined by the method proposed by Beer and Pettifor [131. Using equations (1) and (21, the local DOS functions are now given by the form

~ h r e E denotes the j-th (complex3 root of the equation D $ ~ ( E ) = 0

,

and C j ?he weight of the partial fractions 1 /(E

-

Ej). Then, the inte-

gration of the DOS can be evaluated analytically in terms of elementary integrals [121, and results for Eb are much more accurate compared to those obtained by using a direct numerical integration method.

For the repulsive potential between atomic sites i and j, we take three kinds of LJovler law dependence:

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where

aAA , aAB

and

aBB

denote the repulsive potential for transition-metal atom pair, transition-metal sp-valence metal atom pair and sp-valence atom pair (Al-A1 pair in the present calculation), respectively. Ro is the nearest-neighbour interatomic distance of the perfect crystal. We have chosen this type of power law dependence, since within the TB formalism a repulsive contribution falls off as the square of the TB hopping integrals [14]. It is known that such repulsive potentials are auite successful for the studies on the structural trends of p-d bonded -AB-com ounds [ 15 3.

In view of tl?e success in Refs. 1 4 and 1 5, the spatial dependence of the repulsive potentials are chosen such that ~ A A = 8, ~ A B = 6 and n a ~ = 4. CAAl CBB and CAB are determined so as to satisfy the equilib- rium condition and to reproduce the experimental values of elastic constants (bulk modulus or usual elastic constants C..kl), using the condition of OAB

= ./- .

We have also consulted tie experimental data on the change in the lattice constant of the intermetallic compounds due to deviation of stoichiometry.

The repulsive potential parameter between an impurity (boron) and host atoms is determined by taking into account the atomic volumes of the impurity and host atoms: For this determination, we use the formula of volume change AV due to the solute atom [I61

!?here 4 Zi and

2

Leeote ;he gosition an2 force vectors for host ato.nic site i around tke solute atom, respectively, and K the bulk modulus of the crystal. In the lack of the reliable experimental results on the AV for boron in Ni3Al, we simply take the atomic volume of boron as

I 7 , ( B ) : (0.833)~ . v ~ ( A ~ ) , VO(A1) being the atomic volume of A1 (for de-

t-ils see Ref.17).

3. Numerical Results and Discussions

The present numerical calculations on the grain boundaries in the intermetallic compound Ni3A1 are performed using the two conter hoppinq int~grals proposed by Harrison [14]. The atomic energy levels E,, Ep

for s, p and d-basis orbitals are taken from atomic structure by Herman and Skillman [to]; s, p and d-basis orbitals are used for the transition metal atoms and s and p-basis orbitals are use?

Eor the sp-valence rnetal atoms (A1 and R in the present calculation).

Charge neutrality conditions are imposed for each atom.

In order to assess the reliability of the present (sp3d5)-basis

'1'3 scheme for the intermetallic compounds lilce Ni3A1, we have calcu-

lated the shear elastic constants C A 4 and C' ( = (Cll

-

C12)/2 ) both for Ni and Ni3A1 and presented the results in Table 1. One can see in Table 1 that calculated values of C and C ' both for Ni and Ni3A1 are in good agreement with the correspoAding experimental values. ~t is interesting to note that the calculated elastic anisotropy ratio A for Xi331 is r~~l.clo. I - a r ~ s r t h a n that of ?Ti: riecent experimental results on

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the elastic constants for Ni-A1 system clearly show this trend, i

.

^e

. ,

the anisotropy ratio A increases quite sig- nif icantly with in- creasing the content of A1 up to 25 atomic percent of A1 [I 8

1.

Furthermore

,

^we

have checked that anti-phase boundary

Table 1. Calculated shear elastic constants C44 and C' for Ni and Ni3A1, in units of 10' Pa.

cal. 11.43 5.74 2.00 Ni

exp. 12.47 4.96 2.52 cal. 12.60 4.14 3.05 Ni3A1 ;

exp. 1 1 .80 3.57 3.31 (APB) energy of (1 1 1 )

plane is about 150 Experimental values for Ni are taken from J. P.

m ~ / m ~ and about 1 - 4 Hirth and J. Lothe, "Dislocation ~heory", (John times larqer than Wiley & Sons, 1982).

that of (i00) plane

(stability of LI2 structure relative to structure). We also note that the present TB scheme gives a reasonable description for the one- electron properties of the Ni3A1 compound. In Fig.:, we present the local DOS for Ni3Al crystal calculated by the sp d basis recursion method [71. These DOS curves

are similar to those obtained

by using much more elaborate 1.5.

calculational schemes such as ASW or LMTO method [ 19 1.

The ca:Lculation of 'the cleavage strength of grain

boundary is performed as fol- 1.0-

lows: The two halves of the ordered alloy lattice with surface index (l,m,n) are sep- arated in the <l,m,n) direc- tion with no surface relaxa-

tion and no reconstruction.

t

0.5-

cn This is based on the fact that ⁰FI for transition metal svstems ^el the energy reduction d;e to

the surface relaxation or re- constructiori is generally small (less than a few percent) compared to the surface energy [201, and not of pri- mary importance for the present

problem. Actual numerical calculations

the symmetric are performed tilt to01 l for grain 0O - 8 - 4 5 k

boundaries with X =5 (210) and ^E- X =I3 (320) structures in the

LI2 Ni3A1 crystals. The stable Fiq.1 Local DOS for ~ i ~ A 1 crystal, atomic struc:ture of the grain (a) Ni-site and (b) Al-site. Recur- boundary is calculated by sion coefficients are calculated up using a direct total energy to n=ll levels and energies, relative minimization procedure [ 21 I, to the Fermi energy, are given in from unrelaxed initial atomic units of eV.

configurations proposed by

Takasugi and Izumi [22] and Farkas and Ran [23].

In Fiq.2, we present the calculated atomic configurations in the core of the tilt grain boundaries with X=5 (21 0) (a) and X=13 (320)

( b ) structures in Ni3A1 crystal. Symbols 0 and O represent Ni and A1 atoms, respectively. One can notice in Fig.2 that the distances be- t~reen (210) atomic planes or between (320) ato7Fc 213nes ar= changed

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strongly near the grain boundaries. 8 = S I Z I ~ J

These characteristic grain bound- _a

ary structures are not so much ₀⁰ _q

changed from those reported in 0 0 0

Ref .4. ⁰ ₀⁰ 0 ⁰

In Fig.3ar we present the cal- 0 0

culated cleavage strength f, as a ⁰ 0 ⁰ 0 ⁰ ⁰

function of the separation Rseg 0 0

0 q 0 0 0

between a pair of the crystallltes, 0 o 0 0

for C=5 tilt grain boundary in --- o --- 0 .--... o- ...-- Q--.

Ni3A1 crystal. The corresponding

result of the cleavage strength ⁰ ⁰

o n

⁰

for C=13 tilt grain boundary is O o 0 O o 0

shown in Fig.3b. In Figs.3a and n o O 0 3b, solid curves represent the O 0 o

cleavage strength for the (single ⁰ 0 0

q 0

and bi-) crystals, without impu- O o 0 rity segregation, while symbols A ⁰ represent the cleavage strength

for the boron segregated grain b

.-boundaries. The cleavage strength o 0 0 for the Ni enriched grain bound- c3 D O

ary is shown by symbols

,

ⁱⁿ ^{0 (3} ⁰ ^O 0 O 0 both Fig.3a and 3b:' In the boron q 0 0 0 O n segregated or Ni enriched grain 0 O

. O 0 0

boundaries, all of the Al-sites o O O O o o o at the boundary Ni/A1 layer are -0 ---- 0 ---..+ .--. q Q . . . ~ .

substituted by boron or Ni atoms: ₀ ₀ ₀ ₀

Cleavage plane is chosen to the 0

boundary plane (dashed line in o O O O o o 0 Q O Fig.2). We can discuss, from o O 0

these cleavage force calcula- 0 0

tions, the strength of the grain 0 O Q O 0

boundaries in the L I Z NijAl ⁰ 0 crystal (more directly than the

calculations of the grain bound- _1=13(320) ary energy). The importance of

the cleavage strength calcula-

tion has been discussed by Tyson Fiq.2 Calculated atomic configura- [ 2 4 1 and Kelly et al. [ 2 5 1 , in tions of the tilt grain boundaries relation to the fracture tough- in Ni3Al crystals, with (a) C = 5 ness of crystalline materials. (210) and ( b ) C = l 3 ( 3 2 0 ) structures.

Fiq. 3 Cleavage force f (1

o1

Opa), as a functioil of RseD for the Z = 5 (210) (1) axld C = 13 (320) tilt grain boundaries in Ni3Al crystal.

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One can see in Figs.3a ana 3b that the cleavage strength of the tilt grain boundaries in Ni3A1 reduces significantly (about 80%) compared to that for the perfect single crystal. This result can be understood qualitatively from the local DOS on the atoms near the grain boundaries. The local DOS on the Ni atom (at the first Ni layer) around C = 5 (210) tilt grain boundary in Ni3A1 crystal is shown in Fig.4. One can see that local bandwidth decreases substantially and this gives rise to the reduction of grain boundary strength.

On the other hand, cleavage forces in Fig.3 give us other important

information: One can see in Fig.3

,.-

that segregated boron atoms con-

-

⁸ ^-4 ^o ⁴

siderably increase the cleavage E- strength of the grain boundary. It

is most interesting that the ef- Fig.4 Local DOS on the Ni atom fect of Ni enrichment on the grain (first Ni layer) around X=5 (210) boundary strength is fairly im- grain boundary in Ni3A1 crystal.

portant and much stronger than

that of boron. This indicates that the increase of d-electron bonding strongly enhances the grain boundary cleavage strength of Ni3A1 crystal.

These theoretical findings (we havz obtained the similar results for interstitial boron) are in good agreement with the experimental results of Refs.1 and 2 (stoichiometry effect) and also with Ref.26

(grain boundary strenghening due to solute atoms with unfilled d-shell)

.

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