1. Dislocation model for interface migration
Kedarnath Kolluri and M. J. Demkowicz
Acknowledgments:
B. Uberuaga, X.-Y. Liu, A. Caro, and A. Misra
Financial Support:
Center for Materials at Irradiation and Mechanical Extremes (CMIME) at LANL,
an Energy Frontier Research Center (EFRC) funded by
U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences
2. •
This deck contains only the dislocation model for migration of isolated
vacancies and intersitials at CuNb KS interface
•
The atomistic results are available at
•
•
http://bit.ly/cunb-defect-migrate
The link to papers published with these and other results are
•
http://bit.ly/cunb-migrate-paper
•
http://bit.ly/cunb-pointdefects-paper
3. 0.45
0.4
Vacancy
I
KJ1t
t
Se
KJ3
!1
b1
•
t
KJ3´
a2
0.15
0.1
a1 L
!1
0.05
b1
b
t1
L
t1
L
0.2
t
t
I
I
0 a
•
KJ4
〈110〉
Cu
KJ4
0.25
Set 2
〈110〉
Cu
Se
t1
Step 1
t
Se
" E (eV)
a2 a1
KJ1
KJ2´
0.35
0.3
b
KJ2
〈112〉
Cu
a
〈112〉
Cu
Isolated point defects in CuNb migrate from
one MDI to another
Set 2
3L
Set 2
!1
b1
b
Interstitial
Vacancy
"Ea-b = 0.06 - 0.12 eV
"Ea-I = 0.25 - 0.35 eV
"Ea-t = 0.35 - 0.45 eV
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Thermal kink pairs nucleating at adjacent MDI mediate the migration 0.
Migration barriers
1/3rd
! (reaction coordinate)
that of migration barriers in bulk
4. 0.45
0.4
Vacancy
I
KJ1t
t
Se
KJ3
!1
b1
•
t
KJ3´
a2
0.15
0.1
a1 L
!1
0.05
b1
b
t1
L
t1
L
0.2
t
t
I
I
0 a
•
KJ4
〈110〉
Cu
KJ4
0.25
Set 2
〈110〉
Cu
Se
t1
Step 1
t
Se
" E (eV)
a2 a1
KJ1
KJ2´
0.35
0.3
b
KJ2
〈112〉
Cu
a
〈112〉
Cu
Isolated point defects in CuNb migrate from
one MDI to another
Set 2
3L
Set 2
!1
b1
b
Interstitial
Vacancy
"Ea-b = 0.06 - 0.12 eV
"Ea-I = 0.25 - 0.35 eV
"Ea-t = 0.35 - 0.45 eV
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Thermal kink pairs nucleating at adjacent MDI mediate the migration 0.
Migration barriers
1/3rd
! (reaction coordinate)
that of migration barriers in bulk
5. Thermal kink pairs aid the migration process
b
0.4
Vacancy
I
KJ1
KJ2´
KJ4
I
3L
0.15
t1
a2
Se
L
t1
0.2
t
t
KJ3´
0.25
a1 L
!1
b1
t
〈110〉
Cu
I
2
Set 0.1
•
t
t
b
Se
" E (eV)
Se
t1
Step 2
•
c
0.35
0.3
!1
t
〈112〉
Cu
0.45
0.05
b1
0 a
Set 2
Set 2
!1
b1
b
Interstitial
Vacancy
"Ea-b = 0.06 - 0.12 eV
"Ea-I = 0.25 - 0.35 eV
"Ea-t = 0.35 - 0.45 eV
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Thermal kink pairs nucleating at adjacent MDI mediate the migration 0.
Migration barriers
1/3rd
! (reaction coordinate)
that of migration barriers in bulk
6. Point defect is a dislocation mechanism
I1
a2
Set 2
b1
t1
Set 2
Se
t
!1
Se
t1
a1
L
1
a2
L
b
Se
a
!1
a1 L
b1
3L
Set 2
!1
b1
(a)
(b)
Restrictions/Simplifications for dislocation model
(c)
1. Isotropic linear elastic solutions for dislocation interactions
2. Interactions are considered between kinks/jogs and set 1 dislocation only
3. Interactions neglected between kinks/jogs and the dislocation network
7. created due to the nucleation of a kink-jog pair, each of length ai , and separated by L⇥i (a
Dislocation )model forEqn.(3), anddefectterm is the self energy of the
function of L and a and is given by point the third migration
i
dislocation segments that form the kink-jog pair. In Eqns. (2) and (3), µ = 42 GPa is the
I1
b
a
shear modulus of bulk copper and b =
L
aCu
⇤
2
is the magnitude of the Burgers vector of all the
Se
Se
t1
t1
L
dislocation segments; aCu is the 0 K lattice constant of copper. From simulations, we obtain
a2
Thermal
⇤
⇤
a1 = aCu , a2 = aCu , and L a13aCu . 2Energy expressions for all the states in our simulations kink pair
= ⇤2
3
2
3L configuration
Set
a2
are readily obtained as a combination of Eqns. (2) and (3) with appropriate values for the
Set 2
!1
t1
and ai .
Se
variables L,
L⇥i ,
!1
b1
W (L, a, {L⇥i }, {ai })
(a)
= 2
a1 L
!1
b1
dis
Winter (L, ⇥, a)
(b)
Set 2
+
⌥
b1
jog
Winter (L⇥i , ai )
⌥
+ (c) 2
µb2 ai
ai ⇥
ln
4⇧(1 ⌅)
b
⌅⌃
i
i
⇧
⇤
µb2 ⇧ 2
2L
dis
Winter (L, ⇥, a) =
L + a2 L a + L ln ⇧
4⇧
L 2 + a2 + L
⇧
⇤
⌅⌃
⇧
µb2
L
jog
Winter (L, a) =
2L 2 L2 + a2 2a ln ⇧
4⇧(1 ⌅)
L 2 + a2 + a
M
WA EP (L, a, {L⇥i }, {ai }, s) = WA B (L, a, {L⇥i }, {ai }, s) + A⇥ GSF (s)
B
J. P. Hirth and J. Lothe, Theory of Dislocations, (Wiley, New York, 1982)
•
The parameter
is related to the dislocation (in this case, jogs and kinks) core radius and
α can not be determined with in linear elastic theory of dislocations
can not be estimated within the linear elastic theory of dislocations. We obtain
= 0.448
by = 0.458 is
E = 0.27 eV, of the the expression for formation the
• αfitting the energy,obtained by fitting kink pair configuration corresponding toenergy
configuration in Fig. 1[schematic of the kink pair is marked in Fig. 5(b) and] from our
thermal kink pair from simulations (ΔE = 0.27 eV)
of a
8. are readily obtained as a combination of Eqns. (2) and (3) with appropriate valu
variables L, L⇥i , and ai .
Augmenting with
Peierls-Nabbaro framework
a
I
L
+
⌥
jog
Winter (L⇥i , ai )
1
a2
=
L 2
dis
Winter (L, ⇥, a)
1
W (L, a, {L⇥i }, {ai })
b
+
⌥
µb
2
4⇧(1
⌅⌃
Se
t
1
Se
t
Se
t
i
i
⇧
⇤
2
3L
µba2 ⇧ 2
2L
Set 2
dis
2
⇧
Winter (L, ⇥, a) =
L +a
L a + L ln Area swept by
4⇧
L 2 + a2 + L
Set 2
Set 2
incipient kink pair
⇧
⇤
2
⇧
a1 µb
L
!
jog
2 + a2
L
!
Wb (L, a) =! b
2L b 2 L
2a ln ⇧
inter
4⇧(1 ⌅)
L 2 + a2 +
M
WA EP (L, a, {L⇥i }, {ai }, s) = WA B (L, a, {L⇥i }, {ai }, s) + A⇥ GSF (s)
B
a1
1
1
1
1
1
1
fractional Burgers is related to
Generalized Stacking fault function
The parameter vector contentthe dislocation (in this case, jogs and kinks) core r
s ϵ [0,1]
= 0.175 sin2(πs) J/m2
can not be estimated within the linear elastic theory of dislocations. We obtain
Solutions are expected to be greater than energies from the simulations
by fitting the energy, E = 0.27 eV, of the kink pair configuration correspond
At the interface
configuration in Fig. 1[schematic of the kink pair is marked in Fig. 5(b) and]
•
•
The shear modulus is thought to be lower than in bulk
7
The unstable stacking fault energies are thought to be lower than in bulk
9. Entire migration path can be predicted
0.5
0.5
0.45
0.45
0.4
0.4
0.35
0.35
0.3
0.3
0.25
0.25
I
0.2
0.15
0.2
I
0.15
0.1
0.1
0.05
0.05
0
a
0
Dislocation model
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
KJ1
1
KJ1
0 KJ2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
s
Key inputs to the dislocation model
b
Atomistics
0
〈112〉
Cu
Δ E (eV)
0.55
KJ2´
〈110〉
Cu
KJ3
KJ4
•
Interface misfit dislocation distribution
•
KJ4
1
s
KJ3´
K. Kolluri and M. J. Demkowicz,
Phys Rev B, 82, 193404 (2010)
Structure of the accommodated point defects
Analysis of the interface structure may help predict quantitatively
point-defect behavior at other semicoherent interfaces
11. It should be easier for this to happen!
0.12
a1
L
0.06
Se
t1
•
b1
!1
a1 0 1L
b1
3L
from dislocation model
0.04
0.02
!1
b
0.08
a2
Set 2
I
Se
t1
L
ΔW (eV)
a2
0.1
Se
t1
a
Set 2
1.5
Set 2
!1
2
2.5
3
b
1
L/Lo
Migration of a jog, one neighbor at a time, should occur readily according
to linear elastic theory of dislocations
•
This mechanism, however, is not observed in atomic-scale simulations
13. Position of the jog (x-axis for previous plot)
4
0
2
1
6
5
3
first jog (stationary) is here
S=6
14. 0.6
66
0.8
0.8
!E
0.6
0.6
!Ejog core
0.4
!E
!E!E
!V/"o
!E!E
!Ejog core
!E!Ejog core
jog
!E core
0.4
0.4 0.4
0.4
0.3
0.3
0.3
0.2
0.3
!E 0.3 0.30.3
jog
!Ejog core core !E
!V/"!W
o
jog core
!V/"
!V/"oo
!V/"
!V/"o o
!W
!W
0.2
0.2
0.2
0.1
0.5
0.4 4
2
3
0.3
1
0
6
5
!V/"o
!E (eV)
!E (eV)
88
0.5
0.4
0.5 0.5
0.5
!V/"
1
!E
!V/"o o
!V/"
!V/"o o
1.2 1
11
0.6
!V/"o
1.4
!V/"o
22
!V/"o
1.4
The self energies of the jog change with0.6
position
0.4
44
0.6 0.6
0.5
0.5 0.6
Not accounted for in the dislocation model
1.2
0.20.2
0.2
0.1
0.1
0.4
0.2
first jog (stationary) is here
!V/"o 0.2 0.1 0.1 0
2
!W
2
!W
0.1
!V/"o
!W
0
0
1 0
2 2
3
45
5 0 6
0.1
1
3
4
0.2
!W 6
0
0
S=6
00 1 1
2 2
3 3S
4
5
4
5 0.1 6 0 0
6
S
0 !W 1
2
3
4
5
6
1 barrier for SS
2
3
4 is much greater than that from0
5
6
atomistics
dislocation model!
• The
0
S
0
1 S
2
3
4
5
6
4
E4 core
jog
0.4
•
The barrier in this path can S thought of as the difference in the formation
be 0
3
4
5
6
energies of the jog at the MDI and on set 1 misfit dislocation
S
•
In this interface, the barrier is much larger than that we observed (but there
may be other interfaces where such a mechanism could occur)
15. The energy differences and volumes are comparable
to those for jogs in bulk screw dislocation in Cu
!(#%
!+!,-./
!(
!V/"o range of a
〈112〉jog on a
!E range of a〈112〉
screw dislocation
in Cu
jog on a screw
dislocation in Cu
!"#*
!"#%
!"#'
!"#)
!"#&
!"#%
!"#$
!+3435-6
!"#$
!+789:
!"
!"#(
!.0"1
!"
!(
!$
!)
!.0"1
!(#$
!"#&
!%
!*
!&
!"
2
The volume and energies of the “jog core” at S={3,4} are comparable to those of a
<112> jog in a screw dislocation in bulk copper
16. The self energies of the jog are different at different
positions
4
0
2
1
6
5
3
S=6
At S = {3,4}, the migrating jog resides on set 1 misfit dislocation and away
from MDI
The core energy for s={3,4} is much more than that at s={1,2,5,6}
17. Summary
•
A dislocation model for point defect migration was developed
•
•
It is predictive!
Dislocation model applied in its originally derived form suggests that
there are alternate, lower-barrier paths
•
However, atomistic calculations do not support that
•
Reason: The moving jog has different self energies along the path
•
This too can also be incorporated into the dislocation model
