Twinning in cubic superlattices

Scripta METALLURGICA

Vol. 5, pp. 949-954, 1971 Printed in the United States

Pergamon Press, Inc.

TWINNING IN CUBIC SUPERLATTICES V.S. Arunachalam and C.M. Sargent Aerospace Research L a b o r a t o r i e s W-PAFB, Dayton, Ohio 45433 Permanent Address: National Aeronautical Bangalore 17 INDIA

Laboratory

{Received July 21, 1971; Revised September 7, 1971) Laves ( I )

has o f t e n been c i t e d

[e.g.

(2,3)]

"as p r e d i c t i n g

that

t i c e s would be i n c a p a b l e of t w i n n i n g and e x p e r i m e n t a l r e s u l t s support t h i s

prediction

b e h a v i o r of two-phase n i c k e l

superalloys

deformation twinning

apparently violating

(5,6)

containing NijAI

does indeed occur i n t h i s

on the d e f o r m a t i o n (LI 2 type s u p p e r l a t t i c e ) cubic s u p e r l a t t i c e ,

Laves c r i t e r i o n .

We have re-examined the p o s s i b i l i t y superlattices

are a v a i l a b l e to

(2,4).

However, r e c e n t e l e c t r o n m i c r o s c o p i c i n v e s t i g a t i o n s show t h a t

cub'ic s u p e r l a t -

of d e f o r m a t i o n t w i n n i n g o c c u r r i n g

basing our a n a l y s i s on the d e f i n i t i o n

B i l b y and Crocker ( 7 ) :

"A d e f o r m a t i o n twin is

of t w i n n i n g

a region of a c r y s t a l l i n e

which has undergone a homogeneous shape d e f o r m a t i o n in such a way t h a t sulting

product s t r u c t u r e

is

identical

with

in cubic

adopted by body the r e -

t h a t of the p a r e n t , but o r i e n t e d

different ly". Although this

it

is

conventional

expression is

as the s t r u c t u r e constituent

of an " o r d e r e d b . c . c . and hence, we w i l l

d e r i v e d from a b . c . c .

species.

We s h a l l

r e s t o r e the ordered l a t t i c e , will

to t a l k

often misleading,

(or f.c.c.)

structure",

to a " s u p e r l a t t i c e "

phase by o r d e r i n g of the

c o n s i d e r only those t w i n n i n g modes which f u l l y

i.e.,

any s h u f f l i n g

of the atoms a f t e r

twinning

not be a l l o w e d .

Bevis and Crocker

[paper I ( 8 ) ,

and Paper I I

modes of the cubic B r a v a i s l a t t i c e s shuffles.

Table I of t h e i r

of these modes is fraction

of l a t t i c e

lattice,

listed

points

the f r a c t i o n

to c o r r e c t t w i n p o s i t i o n s

of l a t t i c e

points

from t h a t

of these f r a c t i o n s

notation:

m is the r e c i p r o c a l

having to s h u f f l e ,

i n general be d i f f e r e n t

MF are the r e c i p r o c a l s

paper, using t h e i r

m and n.

sheared d i r e c t l y

the r e m a i n i n g p o i n t s

have c a l c u l a t e d those t w i n n i n g

those t w i n n i n g modes and a s e l e c t i o n

reproduced in Table I of t h i s

For centered l a t t i c e s , positions will

(9)]

which can occur w i t h small shears and s i m p l e

paper I I

The modes are d e s c r i b e d by two i n t e g e r s , tive

(or f.c.c.) refer

n is

a labelling

shearing d i r e c t l y

of the p r i m i t i v e

of the in a p r i m i integer. to t w i n

lattice.

MI and

f o r b o d y - c e n t e r e d and f a c e - c e n t e r e d

949

950

TWINNING

lattices, fully

respectively.

IN CUBIC SUPERLATTICES

tively

Ii

we reproduce only those t w i n n i n g modes w i t h m, mI ,

and which r e s u l t in a change of o r i e n t a t i o n .

Kl ,

K2, n l ,

or

n 2 are respec-

the t w i n n i n g plane, the conjugate t w i n n i n g plane, the t w i n n i n g d i r e c t i o n

and the conjugate t w i n n i n g d i r e c t i o n . to Kl

S, No.

Since we are concerned here only w i t h those modes which

restore the l a t t i c e ,

mF = l ,

Vol.

S is the plane of shear, p e r p e n d i c u l a r

and K2 and g i s the magnitude of the shear s t r a i n .

represents f o u r d i s t i n c t

modes, i . e .

Each mode i n Table I

Kl K2 n l n 2 i m p l i e s the existence of

K2 Kl n 2 n l (the conjugate mode), n l n 2 Kl K2 and n 2 n l K2 KI . I t is poini:ed out in Bevis and Crocker's paper I I t h a t the signs of the old or new K2 and ~l must be reversed to preserve the c o r r e c t r e l a t i v e signs of the f o u r elements in the l a s t two modes.

In general i n t e r c h a n g i n g Kl , n l and K2, n 2 (as i n the

l a s t two modes) would r e q u i r e a change from d i r e c t to r e c i p r o c a l l a t t i c e This is not necessary i n p r i m i t i v e Cubic l a t t i c e s .

parameters.

However, since the face-

centered and body centered cubic l a t t i c e s are r e c i p r o c a l to each o t h e r , i t

is

necessary to interchange mI and mF f o r these modes. K1

K2

nI

n2

S

1.2

I00

120

010

210

001

will

If

1

2

2

I00

III

011

2ii

011

2

2

1

a+ll

a-ll

a-ll

a+ii

0il

4

2

2

1.6

I00

524

012

2~2

021

5

2

2

1.7

Ob+l

Ob-I

O~b+

Olb-

I00

5

2

2

2.2

III

111

211

211

011

2

4

1

2.8

II0

i74

ili

311

i12

6

1

4

TWINNINGMODES IN CUBIC LATTICES restore the cubic Bravais

a c r y s t a l s t r u c t u r e has more than one atom associated w i t h each

p o i n t , then only those t w i n n i n g modes which preserve

be a l l o w a b l e .

only i f

mF

1.4

TABLE I

lattice

mI

1.3

Table I contains those t w i n n i n g modes which f u l l y lattices.

2 2 m g

m. n

this

A l i n e a r array of atoms at each l a t t i c e

t h i s arrangement p o i n t is preserved

l i n e is contained in Kl or K2, the u n d i s t o r t e d planes.

array of atoms,

it

For a planar

is necessary t h a t the array be p a r a l l e l to Kl or K2.

We have

examined three cubic s u p e r l a t t i c e s : I)

L2 o s u p e r l a t t i c e e.g.

CuZn

This is the s u p e r l a t t i c e discussed by Laves ( 1 ) .

I t has a cubic P

Bravais l a t t i c e w i t h two atoms associated w i t h each l a t t i c e

point, e.g.,

Cu at

Vol.

5, No.

ii

TWINNING

( 0 , 0 , 0 ) , Zn at (½,½,½).

IN CUBIC S U P E R L A T T I C E S

951

In order to preserve t h i s s t r u c t u r e a f t e r t w i n n i n g , a

mode must be chosen from Table I w i t h Kl or K2 c o n t a i n i n g [ I l l ] . mode 1.3 f i t s

this

c r i t e r i o n with

K1 = ( 2 1 1 ) , A sketch

of

this

A v a r i a n t of

mode i s

K2 = ( 0 1 1 ) , shown i n

n I = [III]

Fig.

n2 = [I00],

g :

2.

1

of

l)

I

rlO0] ~

Fig. 2) DO3 S u p e r l a t t i c e e 9.

[0111

•

Bravois lattice point

[]

Bravais

"

"

•

Body center position

G

Body

"

"

in plane below in plane below

1 - Twinning i n L2 o S u p e r l a t t i c e Fe3Be

This has a cubic F Bravais l a t t i c e w i t h f o u r atoms associated w i t h each lattice

point

Again,

~II]

in Table

Be a t

(0,0,0),

s h o u l d be c o n t a i n e d

I which

and, h e n c e , produce

eg.,

completely

twinning

twins

in the

Fe at

(k,k,k),

i n K1 ( o r

restore

the

K2).

f.c.c,

by a homogeneous s h e a r w i l l DO3 s u p e r l a t t i c e

(½,½,½),

(3/4,

3/4,

None o f the t w i n n i n g lattice

fulfills

this

n o t be p o s s i b l e .

have been u n s u c c e s s f u l

(2,4).

3/4). modes condition,

Attempts

to

952

TWINNING

IN CUBIC SUPERLATTICES

Vol.

S, No. ii

3) 1.12. eg. Ni3Al This has a cubic P Bravais l a t t i c e with four atoms associated with each l a t t i c e point, eg., A1 at ( 0 , 0 , 0 ) , Ni at (½,0,½), (½,½,0), (0,½,½) forming a tetrahedron. I t is convenient to describe the r e l a t i v e positions of the atoms in a d i f f e r e n t but c r y s t a l l o g r a p h i c a l l y equivalent way: Al at (0,0,0) Ni a t (½,0,½), (½,½,0), and (0,-½,½), i e . , the t h i r d nickel atom is translated by [OTO] Thus, a planar array lying on ( T l l ) is associated with each l a t t i c e point. System 1.3 (Table I) has K2 = ( I l l ) , and i s , therefore, a possible twinning mode for the Ll 2 s u p e r l a t t i c e . The conjugate mode (K l = ( I l l ) , K2 = (lO0), nl=[2TT ] , n 2 = [Oil] and g = 2) is sketched in Fig. 2. This is the system observed by Guimier and Strudel (5). A pole mechanism f o r twinning this s u p e r l a t t i c e by this system has been suggested by Kear et al. (lO).

-q, --r 21ij

= ['OI I ]

\\

[IOO1 ~ [ 0 1 1 1

Fig.

X

Troceof K~=[III]

•

Brovois lattice point

O X

Brovois . . . . i n p l o n e / [ O I ] : ] below Face centering positions (depthindicoted by fraction of [01i.1 distance

2 - Twinning in L12 Superlattice

Vol.

S, No.

II

TWINNING

IN CUBIC SUPERLATTICES

The above a n a l y s i s shows t h a t t w i n n i n g i s superlattices which do t w i n ,

examined.

It

do not e x h i b i t

from which they were formed. crystal

structures

953

p o s s i b l e in two of the t h r e e

is not s u r p r i s i n g

cubic

t h a t those cubic s u p e r l a t t i c e s

the same modes as those of the d i s o r d e r e d l a t t i c e s The s u p e r l a t t i c e s

do, in f a c t ,

which must obey s e p a r a t e c r i t e r i a .

It

r e p r e s e n t new

is t h i s

distinction

which has r e s u l t e d in c o n f u s i o n i n the p a s t . We would l i k e

to thank P r o f e s s o r s

vious v e r s i o n of t h i s

R.W. Cahn and A.G. Crocker f o r

reading a p r e -

manuscript.

REFERENCES I.

F. Laves, Naturwissenschaften, 39, 546 (1952).

2. R.W. Cahn and J.A. Coil, Acta Met. 9, 138 (1961). 3. G.F. Bolling and R.H. Richman, Acta Met., 13, 709 (1965). 4. M.J. Marcinkowski

and R.M. Fisher, J. Appl. Phys. 34, 2135 (1963).

5. A. Guimier and J.L. Strudel, Proceedings of the Second I n t e r n a t i o n a l Conference on the Strength of Metals and Alloys, American Society for Metals, p. I145 (1970). 6. B.H. Kear, J.M. Oblak and A.F. Giamei, Proceedings of the Second I n t e r n a t i o n a l Conference on The S t r e n g t h of Metals and A l l o y s , p. 1155 (1970). 7. B.A.

Bilby

and A.G. Crocker, Proc.

Roy. Soc A 288, 240 (1965).

8. M. Bevis and A.G. Crocker, Proc.

Roy. Soc. A 304, 123 (1968).

9. M. Bevis and A.G.

Roy. Soc, A 212, 509 (1969).

I0.

B.H. Kear, A.F.

Crocker, Proc.

Giamei, G.R. L e v e r a n t , J.M. Oblak, S c r i p t a M e t . , _3, 123 (1969).