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Line tension of a superdislocation
Scripta M E T A L L U R G I C A
Vol. 3, pp. 233-238, 1969 Printed in the United States
LINE TENSION
Pergamon
Press,
Inc
OF A SUPERDISLOCATION
P. Chaudhari and Ao Gangulee I B M Watson Research Center, Yorktown Heights, N e w York (Received
February
3, 1969)
The simplest case of a superdislocation in an ordered alloy consists of two dislocations
with an antiphase
region between them.
the line tension of such a superdislocation energy.
alloys,
hardening.
(3)
as for example,
1.
The lead dislocation was perturbed
amount
e
e'
L'
and
over a length
L
in understanding
the deformation
the operation of dislocation sources
The dislocation configuration assumed Fig.
n o t e we h a v e c a l c u l a t e d
for several values of the antiphase boundary
T h e v a l u e of line t e n s i o n is of i n t e r e s t
of o r d e r e d
In t h e p r e s e n t
(1, 2) a n d w o r k
in c a l c u l a t i n g the l i n e t e n s i o n is s h o w n in from its equilibrium position ABDE by a small
to a p o s i t i o n A B C D E .
were varied until a minimum
behavior
Keeping this assumed
position fixed,
in energy was obtained.
C A
~
D
~
A'
FIG. 1 Assumed
configuration of a small bow-out in a superdislocation.
233
E
234
LINE TENSION
The self energy, Table
Es,
1 were calculated
formula.
(5)
OF S U P E R D I S L O C A T I O N
and the interaction
using the expressions
cut-off parameter.
(4)
using the cut-off parameter
for dose
cut-off parameter
open structures
for more
and the line tension are larger
H
where
boundary bZ P 4~K
=
energy,
by J~ssang
e t a l . (4)
listed in based on Blin's
was included in the self energy by making
All the results
packed
of t h e s e g m e n t s
3, No. 4
structures.
presented
here were calculated
Additional calculations
using the
indicated that in this case the energy minimum
forming
structure
a superdislocation
using the expression
by 10-1 5 pct.
The
was obtained from the
(6)
_2 y
~ is the shear
one for a screw
y,
El,
than those of the close packed
H, b e t w e e n t w o d i s l o c a t i o n s
antiphase
derived
The core energy of the dislocations
use of an appropriate
spacing,
energy,
Vol.
(I)
modulus,
superdislocation
b the Burgers
vector,
and
K
a constant whose value is
a n d (1 - v } f o r a n e d g e s u p e r d i s l o c a t i o n
where
v is the
Poisson's ratio. TABLE
I.
The Self and Interaction Energy T e r m s ,
Es and ~
respectively,
of the Dislocation Configuration shown in Figure I. Self
energy
Interaction
Es(BD) energy
between segments w i t h i n l the same dislocation Interaction
energy
b e t w e e n segments o f Idifferent
dislocations
;
Es(B'D')
;
due t o
;
Es(B'C')
Ei(BD.AB)
;
Ei(B'D'.A'B')
Ei(BC.AB)
;
Ei(BC.CD)
;
Ei(BC.DE)
Ei(BICI.AIB ')
;
EI(BICI. C'D')
;
Ei(B'CI.D'E')
EI(BD.A'E')
;
Ei(B'D'.AE)
;
Ei(BD.B'D')
Ei(BC.A'B')
;
Ei(BC.B'C')
;
EI(BC.D'E')
EI(BIC'.AB)
;
Ei(B'CI.CD)
;
EI(BlCI.DE)
Es(CD)=Es(BC)
Equalities
Es(BC)
;
Es(CIDI)=Es(BIC t )
Ei(BD.DE)=Ei(BD.AB)
;Ei(BIDI.DIEI)=Ei(B
Ei(CD.AB)=Ei(BC.DE)
;
fDloAwBI)
Ei(CD.DE)=Ei(BC.AB)
IEi(CtDt.AWBI)=Ei(BICI.DIEI);Ei(CIDI.DIEI)=Ei(BICI.AIB syn~etry
of bow out
I)
;
Ei(BC.C'D')=Ei(~'C'.CD)
Ei(CD.C'D')=Ei(BC.B'C')
;
Ei(CD.b'E')=Ei(BC.A'B
Ei(C'D'.AB)=Ei(B'CI.DE)
;
EI(C'D'.DE)=Ei(B'CI.AB)
Ei(CD.A'B')=Ei(BC.D'E
')
' )
Vol.
3, No.
4
LINE
TENSION
The change in energy of the system in Fig.
OF S U P E R D I S L O C A T I O N
235
d u e t o t h e b o w - o u t of t h e s u p e r d i s l o c a t i o n
shown
1 is given by:
z~E = Z E s ( B C )
+ ZEi(BC.AB)
+ 2Ei(B'C'.A'B') + 2Ei(BC.A'B')
+ ZEi(BC. D E ) + Ei(BC. C D ) - E s ( B D )
- ZEi(BD. A B ) + ZEs(B'C')
+ 2Ei(B'C'. D'E') + Ei(B'C'. C'D') - Es(B'D') - 2Ei(B'D'.A'B') + ZEi(BC.B'C')
+ ZEi(BC.D'E'}
+ 2Ei(B'C'. D E ) + Ei(BD. B'D') - Ei(BD.A'E')
+ ZEi(B'C'.AB)
- Ei(B'D'.AE)
+ Z(B'C'.CD)
+ y(areaBCD
- areaB'C'D')
(z) Typical variations of
Z~E with
8' and L' at fixed values of 8, L, and
Fig. 2a and Fig. 2h for an edge and a s c r e w superdislocation, of the B u r g e r s length
AE
vector, b, w a s taken as Z. 943 A . U . ~
as 2 ram.
The minimum
Z~E , increased with increasing increasing
1 .~5
II H
value of
e, L
and
y.
y
respectively.
T h e magnitude
the Poisson's ratio as 0. 33, and the
A~. as a function of 8' and L', designated F o r small
8, A~.
increased linearly with
e.
I /
I I
EDGESUPERDISLOCATION .
\ \\\\\
eo,-,L=,OOOA.U.
\ \\\\\
~--oo~(,~,..)
l/a\\\\\\
i 7.5
SCREW SUPERDISLOCATION 8=P,L=IOOOA.LI J
/'~
\\
\ \
I
\ \\ \ \
MINIMUM ENERGY,Z~E~
" 5
0
are s h o w n in
L
IO00
2000
i
0
3000
i
i
I
I000
2000 C (AU.)
t~ (A.u.)
(a)
(b) FIG.
Typical
I0 15 25
variations
with changes
2
of the minimum
energy
in units of ~bg/4=
i n L ' o r e ' a t f i x e d v a l u e s o f y, L , a n d e.
3000
236
LINE TENSION OF SUPERDISLOCATION
Vol. 3, No. 4
The line tension, T, of a dislocatlon is conventionally defined as: T
.tim :
(~LE)
(3)
AT.--O
In the case of a superdistocation consisting of two dislocations, the corresponding definition would be:
=
T
t im
s
AT.
~
(~_~) (4)
0
s
where
AT.
s
= AL + AT.,, AT. = B C D - B D ,
and
AT.'
= B'C'D'-
the change in length corresponding to A E . While T
B'D',
and
AT., r e f e r s
to
conforms to the usual definition of s
line tension, it suffers from the drawback that L
and L' are not necessarily equal.
Calculation of the change in energy, knowing the change in length of one of the dislocations, is therefore
not feasible.
Typical values of
V, L , L ' ,
the ratio of the two areas,
A'/A,
generated
by the bowed out segments, and T are listed in Table 2 for both edge and screw s o superdislocations at 0 = 1 As V or L increases, the values of L and L' approach
each other and the ratio of the areas
approaches
unity, i.e.
t~/e t w o d i s l o c a t i o n s t e n d t o b e
parallel. The results
of the present
dislocation is larger dislocation increases a disordered
material
calculations
show that the line tension of a screw super-
than that of an edge superdislocation. monotonically with except when
found to be true for all values of L
¥
V or
L.
approaches
It i s l a r g e r zero,
substantially
smaller.
than that of a dislocation in
where the two are equal.
estimates
with a pure screw component compare
(2, 3} b u t t h a t o f t h e p u r e e d g e c o m p o n e n t i s
The ratios of the line tension of a superdislocation
dislocation in a disordered edge and screw components.
material
This is
investigated.
The values of line tension of a superdislocation favorably with some of the earlier
The line tension of a super-
are within a factor of approximately
In a r r i v i n g
at t h i s f i g u r e ,
to that of a single 1 to 2.5 for both
the values of line tension of a single
d i s l o c a t i o n (4} c a l c u l a t e d b y a t e c h n i q u e s i m i l a r t o t h e o n e u s e d h e r e w e r e e m p l o y e d .
Vol.
3, No. 4
LINE T E N S I O N TABLE
OF S U P E R D I S L O C A T I O N
237
2.
Typical Values of Line Tension of an Edge and Screw Superdislocation as Functions of the Antiphase Energy, N, and Perturbed Length
Edge Superdislocation Y
L
L'
A'/A
L.
Screw Superdislocation T
L'
A'/A
S
T S
(~b2 ) (A. U.) 0.001
4000
13320
0.665
2.7
15200
0.578
12.5
0.001
10000
20420
0.876
3.9
30800
0.830
16.2
0.001
20000
27820
0.929
4.7
37940
0.882
20.4
0.005
2000
3920
0.883
3.0
5600
0.823
12.9
0.005
4000
5310
0.934
3.8
7106
0.884
16.7
0.005
i0000
10740
0.980
4.4
11938
0.952
19.8
0.005
20000
20460
0.994
4.9
21188
0.982
21.2
0.010
i000
1790
0.897
2.6
2730
0.814
11.4
0.010
2000
3620
0.944
3.4
3480
0.893
15.2
0.010
4000
4400
0.974
3.8
5040
0.937
17.7
0.010
i0000
I0200
0.994
4.6
10640
0.998
20.0
0.050
400
490
0.945
2.4
638
0.890
11.4
0.050
i000
1050
0.981
3.1
1146
0.959
14.3
0.050
2000
2030
0.994
3.7
2092
0.985
16.2
0.050
4000
4010
0.998
4.3
4052
0.995
18.5
0. I00
200
240
0.950
2.0
304
0.901
9.8
0. i00
400
430
0.982
2.6
476
0.949
12.1
0. I00
I000
i010
0.995
3.3
1040
0.984
14.7
0. I00
2000
2004
0.999
4.0
2026
0.995
17.1
238
LINE TENSION OF SUPERDISLOCATION
Vol. 3, No. 4
REFERENCES Io
P. Chaudhari, Acta Met., 14, 69 (1966).
2.
M. F. Ashby, Acta Met., I_~4, 679 (1966).
3.
G. Schoeck, Scrlpta Met., --2, 283 (1968).
4.
T. J~bssang, J. L o t h e , and K. Skylstad, A c t a M e t . ,
5.
J. Blin, A c t a M e t . , _3, 199 (1955).
6.
M. J. Marcinkowski, Electron Microscopy and Strength of Crystals, p. 333, John Wiley & Sons, N e w York (1963).
I.~3, 271 (1965).
Vol. 3, pp. 233-238, 1969 Printed in the United States
LINE TENSION
Pergamon
Press,
Inc
OF A SUPERDISLOCATION
P. Chaudhari and Ao Gangulee I B M Watson Research Center, Yorktown Heights, N e w York (Received
February
3, 1969)
The simplest case of a superdislocation in an ordered alloy consists of two dislocations
with an antiphase
region between them.
the line tension of such a superdislocation energy.
alloys,
hardening.
(3)
as for example,
1.
The lead dislocation was perturbed
amount
e
e'
L'
and
over a length
L
in understanding
the deformation
the operation of dislocation sources
The dislocation configuration assumed Fig.
n o t e we h a v e c a l c u l a t e d
for several values of the antiphase boundary
T h e v a l u e of line t e n s i o n is of i n t e r e s t
of o r d e r e d
In t h e p r e s e n t
(1, 2) a n d w o r k
in c a l c u l a t i n g the l i n e t e n s i o n is s h o w n in from its equilibrium position ABDE by a small
to a p o s i t i o n A B C D E .
were varied until a minimum
behavior
Keeping this assumed
position fixed,
in energy was obtained.
C A
~
D
~
A'
FIG. 1 Assumed
configuration of a small bow-out in a superdislocation.
233
E
234
LINE TENSION
The self energy, Table
Es,
1 were calculated
formula.
(5)
OF S U P E R D I S L O C A T I O N
and the interaction
using the expressions
cut-off parameter.
(4)
using the cut-off parameter
for dose
cut-off parameter
open structures
for more
and the line tension are larger
H
where
boundary bZ P 4~K
=
energy,
by J~ssang
e t a l . (4)
listed in based on Blin's
was included in the self energy by making
All the results
packed
of t h e s e g m e n t s
3, No. 4
structures.
presented
here were calculated
Additional calculations
using the
indicated that in this case the energy minimum
forming
structure
a superdislocation
using the expression
by 10-1 5 pct.
The
was obtained from the
(6)
_2 y
~ is the shear
one for a screw
y,
El,
than those of the close packed
H, b e t w e e n t w o d i s l o c a t i o n s
antiphase
derived
The core energy of the dislocations
use of an appropriate
spacing,
energy,
Vol.
(I)
modulus,
superdislocation
b the Burgers
vector,
and
K
a constant whose value is
a n d (1 - v } f o r a n e d g e s u p e r d i s l o c a t i o n
where
v is the
Poisson's ratio. TABLE
I.
The Self and Interaction Energy T e r m s ,
Es and ~
respectively,
of the Dislocation Configuration shown in Figure I. Self
energy
Interaction
Es(BD) energy
between segments w i t h i n l the same dislocation Interaction
energy
b e t w e e n segments o f Idifferent
dislocations
;
Es(B'D')
;
due t o
;
Es(B'C')
Ei(BD.AB)
;
Ei(B'D'.A'B')
Ei(BC.AB)
;
Ei(BC.CD)
;
Ei(BC.DE)
Ei(BICI.AIB ')
;
EI(BICI. C'D')
;
Ei(B'CI.D'E')
EI(BD.A'E')
;
Ei(B'D'.AE)
;
Ei(BD.B'D')
Ei(BC.A'B')
;
Ei(BC.B'C')
;
EI(BC.D'E')
EI(BIC'.AB)
;
Ei(B'CI.CD)
;
EI(BlCI.DE)
Es(CD)=Es(BC)
Equalities
Es(BC)
;
Es(CIDI)=Es(BIC t )
Ei(BD.DE)=Ei(BD.AB)
;Ei(BIDI.DIEI)=Ei(B
Ei(CD.AB)=Ei(BC.DE)
;
fDloAwBI)
Ei(CD.DE)=Ei(BC.AB)
IEi(CtDt.AWBI)=Ei(BICI.DIEI);Ei(CIDI.DIEI)=Ei(BICI.AIB syn~etry
of bow out
I)
;
Ei(BC.C'D')=Ei(~'C'.CD)
Ei(CD.C'D')=Ei(BC.B'C')
;
Ei(CD.b'E')=Ei(BC.A'B
Ei(C'D'.AB)=Ei(B'CI.DE)
;
EI(C'D'.DE)=Ei(B'CI.AB)
Ei(CD.A'B')=Ei(BC.D'E
')
' )
Vol.
3, No.
4
LINE
TENSION
The change in energy of the system in Fig.
OF S U P E R D I S L O C A T I O N
235
d u e t o t h e b o w - o u t of t h e s u p e r d i s l o c a t i o n
shown
1 is given by:
z~E = Z E s ( B C )
+ ZEi(BC.AB)
+ 2Ei(B'C'.A'B') + 2Ei(BC.A'B')
+ ZEi(BC. D E ) + Ei(BC. C D ) - E s ( B D )
- ZEi(BD. A B ) + ZEs(B'C')
+ 2Ei(B'C'. D'E') + Ei(B'C'. C'D') - Es(B'D') - 2Ei(B'D'.A'B') + ZEi(BC.B'C')
+ ZEi(BC.D'E'}
+ 2Ei(B'C'. D E ) + Ei(BD. B'D') - Ei(BD.A'E')
+ ZEi(B'C'.AB)
- Ei(B'D'.AE)
+ Z(B'C'.CD)
+ y(areaBCD
- areaB'C'D')
(z) Typical variations of
Z~E with
8' and L' at fixed values of 8, L, and
Fig. 2a and Fig. 2h for an edge and a s c r e w superdislocation, of the B u r g e r s length
AE
vector, b, w a s taken as Z. 943 A . U . ~
as 2 ram.
The minimum
Z~E , increased with increasing increasing
1 .~5
II H
value of
e, L
and
y.
y
respectively.
T h e magnitude
the Poisson's ratio as 0. 33, and the
A~. as a function of 8' and L', designated F o r small
8, A~.
increased linearly with
e.
I /
I I
EDGESUPERDISLOCATION .
\ \\\\\
eo,-,L=,OOOA.U.
\ \\\\\
~--oo~(,~,..)
l/a\\\\\\
i 7.5
SCREW SUPERDISLOCATION 8=P,L=IOOOA.LI J
/'~
\\
\ \
I
\ \\ \ \
MINIMUM ENERGY,Z~E~
" 5
0
are s h o w n in
L
IO00
2000
i
0
3000
i
i
I
I000
2000 C (AU.)
t~ (A.u.)
(a)
(b) FIG.
Typical
I0 15 25
variations
with changes
2
of the minimum
energy
in units of ~bg/4=
i n L ' o r e ' a t f i x e d v a l u e s o f y, L , a n d e.
3000
236
LINE TENSION OF SUPERDISLOCATION
Vol. 3, No. 4
The line tension, T, of a dislocatlon is conventionally defined as: T
.tim :
(~LE)
(3)
AT.--O
In the case of a superdistocation consisting of two dislocations, the corresponding definition would be:
=
T
t im
s
AT.
~
(~_~) (4)
0
s
where
AT.
s
= AL + AT.,, AT. = B C D - B D ,
and
AT.'
= B'C'D'-
the change in length corresponding to A E . While T
B'D',
and
AT., r e f e r s
to
conforms to the usual definition of s
line tension, it suffers from the drawback that L
and L' are not necessarily equal.
Calculation of the change in energy, knowing the change in length of one of the dislocations, is therefore
not feasible.
Typical values of
V, L , L ' ,
the ratio of the two areas,
A'/A,
generated
by the bowed out segments, and T are listed in Table 2 for both edge and screw s o superdislocations at 0 = 1 As V or L increases, the values of L and L' approach
each other and the ratio of the areas
approaches
unity, i.e.
t~/e t w o d i s l o c a t i o n s t e n d t o b e
parallel. The results
of the present
dislocation is larger dislocation increases a disordered
material
calculations
show that the line tension of a screw super-
than that of an edge superdislocation. monotonically with except when
found to be true for all values of L
¥
V or
L.
approaches
It i s l a r g e r zero,
substantially
smaller.
than that of a dislocation in
where the two are equal.
estimates
with a pure screw component compare
(2, 3} b u t t h a t o f t h e p u r e e d g e c o m p o n e n t i s
The ratios of the line tension of a superdislocation
dislocation in a disordered edge and screw components.
material
This is
investigated.
The values of line tension of a superdislocation favorably with some of the earlier
The line tension of a super-
are within a factor of approximately
In a r r i v i n g
at t h i s f i g u r e ,
to that of a single 1 to 2.5 for both
the values of line tension of a single
d i s l o c a t i o n (4} c a l c u l a t e d b y a t e c h n i q u e s i m i l a r t o t h e o n e u s e d h e r e w e r e e m p l o y e d .
Vol.
3, No. 4
LINE T E N S I O N TABLE
OF S U P E R D I S L O C A T I O N
237
2.
Typical Values of Line Tension of an Edge and Screw Superdislocation as Functions of the Antiphase Energy, N, and Perturbed Length
Edge Superdislocation Y
L
L'
A'/A
L.
Screw Superdislocation T
L'
A'/A
S
T S
(~b2 ) (A. U.) 0.001
4000
13320
0.665
2.7
15200
0.578
12.5
0.001
10000
20420
0.876
3.9
30800
0.830
16.2
0.001
20000
27820
0.929
4.7
37940
0.882
20.4
0.005
2000
3920
0.883
3.0
5600
0.823
12.9
0.005
4000
5310
0.934
3.8
7106
0.884
16.7
0.005
i0000
10740
0.980
4.4
11938
0.952
19.8
0.005
20000
20460
0.994
4.9
21188
0.982
21.2
0.010
i000
1790
0.897
2.6
2730
0.814
11.4
0.010
2000
3620
0.944
3.4
3480
0.893
15.2
0.010
4000
4400
0.974
3.8
5040
0.937
17.7
0.010
i0000
I0200
0.994
4.6
10640
0.998
20.0
0.050
400
490
0.945
2.4
638
0.890
11.4
0.050
i000
1050
0.981
3.1
1146
0.959
14.3
0.050
2000
2030
0.994
3.7
2092
0.985
16.2
0.050
4000
4010
0.998
4.3
4052
0.995
18.5
0. I00
200
240
0.950
2.0
304
0.901
9.8
0. i00
400
430
0.982
2.6
476
0.949
12.1
0. I00
I000
i010
0.995
3.3
1040
0.984
14.7
0. I00
2000
2004
0.999
4.0
2026
0.995
17.1
238
LINE TENSION OF SUPERDISLOCATION
Vol. 3, No. 4
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