Grain boundary structure and energetics
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[26]
T h e case when rn < 0 can be t r e a t e d by t r a n s f o r m i n g E(k) for a n e g a t i v e m o d u l u s . T h u s , the total e n e r g y is
ET = Ee + e s : bEo[ln ( R c / R ) -
1/2 (1 + m)]
+ 4~cR(1 + rn)E(k),
m =0.33
\
-,
\',3k \
\ Kn 1.0
.
Aa~_\
"'2-. 15
30
45
60
75
90
a, POLAR ANGLE IN ~-PLANE
Fig. 5--Normalized strain and surface energies for an elliptical core for different eccentricities with no constraints. Note as r n ~ 1, the difference at c~ = 0, becomes larger. However, it is e a s i e r to c a r r y out the o p e r a t i o n s i n d i cated in Eqs. [28] and [29] on ET and obtain two s i m u l t a n e o u s equations for R and rn, n a m e l y , [31]
and [32]
K(k) is the c o m p l e t e elliptic i n t e g r a l of the f i r s t kind with modulus given by Eq. [26]. Two l i n e a r l y i n d e p e n d e n t s o l u t i o n s for R and m a r e obtained f r o m Eqs. [31] and [32], m = +1,
[33]
and
R --
[28]
[34]
8y c 9
T h e s e s o l u t i o n s c o r r e s p o n d to a finite s l i t e i t h e r on the x - a x i s (rn = 1) o r along the y - a x i s (m = - 1). F o r m = 1, the t o t a l e n e r g y is
ET(rn=l)=bEo[lnRc-lnbE~
and
aCT - O. ~R
....
[27]
f r o m which the e x t r e m a of ET can be i n v e s t i g a t e d . Conditions for e x t r e m a to e x i s t for ET in the ( R - m ) space are, aCT - 0
m = 0.46
( bEo ~ Ir 1 R = \ 2 ~ Y c / ~ (1 + m ) E ( k ) ;
where E(k) is the complete elliptic i n t e g r a l of the second kind with modulus, he =
~
\\
2.5
2.0
m =0.00
l
( bEo ~ ~ m R = \ 2 ~ c 1 ~ (1 + m)E(k) - (1 - m ) K ( k ) '
The surface or chemical energy for an elliptically shaped core is just Es : 4ycR(1 + m) E(k),
',
a.o
\ Kn
[29]
The type of e x t r e m a r e s u l t i n g f r o m E q s . [28] and [29] u s u a l l y r e q u i r e i n v e s t i g a t i o n of the sign of A, where
c
[35]
and f o r m = - 1 , er(m=--l)=bEo
[ ln Rc - ln sbE~ s c- 1 + bEo.
[36]
We note h e r e that A=
( 02s ~amaR]
a2ET O2ET
am 2 aR 2 "
[301
e T ( r n = 1) < ET(m = - - 1 )
[37]
a s s u m i n g that R c is the s a m e in each c a s e . 1378 VOLUME 8A, SEPTEMBER 1977
METALLURGICALTRANSACTIONSA
We now c o m p a r e the c a s e of a c i r c u l a r d i s l o c a t i o n core (m = 0) with the s l i t - l i k e c o r e s (m = +1). F o r the c i r c u l a r core, we s e t m = 0 in Eq. [29] and R = to, which y i e l d s e T ( m = O) : bEo [ l n R c - In ro - 1/2] + 2~Tcro.
[38]
The e x t r e m u m o c c u r s when OET Oro - O,
which y i e l d s the s o l u t i o n bEo ro - 27rye,
[39]
in a g r e e m e n t with Eq. [13]. The m i n i m u m e n e r g y is found by s u b s t i t u t i n g Eq. [39] into Eq. [38], n a m e l y E T ( m = O) = bEo
InR c-indic
+ In
-1/2
+ bEo;
[40] thus, c o m p a r i s o n of v a r i o u s e l l i p t i c a l core s h a p e s p r o d u c e s the following r a n k i n g of total e n e r g i e s : s
rn
bEo ~ ( bEo = 1, R = 8 Y c / < ~T m = O , R = 2~]/c]
bEo) .
< E T rn = - l , R = ~ c c
[41]
tive g e o m e t r
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