dislocation model for migration
TRANSCRIPT
Dislocation model for interface migrationKedarnath Kolluri and M. J. Demkowicz
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
Acknowledgments: B. Uberuaga, X.-Y. Liu, A. Caro, and A. Misra
• 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
b1
!1
Set 2
Set 1
L
a2 a1 Set 1
Set 2
a1
a2
L
Lb1
!1
b1
!1
Set 1
Set 2
3L
• Thermal kink pairs nucleating at adjacent MDI mediate the migration
• Migration barriers 1/3rd that of migration barriers in bulk
KJ1
KJ3´KJ4
Cu
〈112〉
〈110〉Cu
KJ2´
KJ4
KJ3
KJ2KJ1
Cu
〈112〉
〈110〉Cu
a bIVacancy
Step 1
! (reaction coordinate)
t
ca
I
t t
b
" E
(eV
)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t
I
t
b
t
"Ea-b = 0.06 - 0.12 eV"Ea-I = 0.25 - 0.35 eV"Ea-t = 0.35 - 0.45 eV
VacancyInterstitial
Isolated point defects in CuNb migrate from one MDI to another
b1
!1
Set 2
Set 1
L
a2 a1 Set 1
Set 2
a1
a2
L
Lb1
!1
b1
!1
Set 1
Set 2
3L
KJ1
KJ3´KJ4
Cu
〈112〉
〈110〉Cu
KJ2´
KJ4
KJ3
KJ2KJ1
Cu
〈112〉
〈110〉Cu
a bIVacancy
Step 1
! (reaction coordinate)
t
ca
I
t t
b
" E
(eV
)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t
I
t
b
t
"Ea-b = 0.06 - 0.12 eV"Ea-I = 0.25 - 0.35 eV"Ea-t = 0.35 - 0.45 eV
VacancyInterstitial
Isolated point defects in CuNb migrate from one MDI to another
• Thermal kink pairs nucleating at adjacent MDI mediate the migration
• Migration barriers 1/3rd that of migration barriers in bulk
Set 2
b1
!1
a1
a2
Set 1
L
L
b1!1
Set 1
Set 2
b1
!1
Set 1
Set 2
3L
KJ1
KJ3´KJ4
KJ2´
Cu
〈112〉
〈110〉Cu
cb IVacancy
Step 2
! (reaction coordinate)
t
ca
I
t t
b
" E
(eV
)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t
I
t
b
t
"Ea-b = 0.06 - 0.12 eV"Ea-I = 0.25 - 0.35 eV"Ea-t = 0.35 - 0.45 eV
VacancyInterstitial
Thermal kink pairs aid the migration process
• Thermal kink pairs nucleating at adjacent MDI mediate the migration
• Migration barriers 1/3rd that of migration barriers in bulk
Restrictions/Simplifications for dislocation model
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
Point defect is a dislocation mechanism
b1
!1
Set 2
Set 1
a
La2
a1
I1
b1
!1
Set 1
Set 2
a1
a2
L
Lb1
!1
Set 1
Set 2
b
3L
(a) (b) (c)
Figure 5: (color online) A simplified model capturing the key steps in the point defect
migration process. The model corresponding to the kink pair configuration shown in Fig. 1
is marked by red box.
interaction energy due to a parallel shift by distance a of a dislocation segment of length L
that is part of a long straight dislocation and is given by Eqn.(2). The second term is the
total negative elastic interaction energy between all pairs of parallel dislocation segments
created due to the nucleation of a kink-jog pair, each of length ai, and separated by L⇥i (a
function of L and ai) and is given by Eqn.(3), and the third term is the self energy of the
dislocation segments that form the kink-jog pair. In Eqns. (2) and (3), µ = 42 GPa is the
shear modulus of bulk copper and b = aCu⇤2
is the magnitude of the Burgers vector of all the
dislocation segments; aCu is the 0 K lattice constant of copper. From simulations, we obtain
a1 = aCu⇤3, a2 = aCu⇤
2, and L = 3aCu⇤
2. Energy expressions for all the states in our simulations
are readily obtained as a combination of Eqns. (2) and (3) with appropriate values for the
variables L, L⇥i, and ai.
W (L, a, {L⇥i}, {ai}) = 2�W disinter(L,⇥, a) +
⌥
i
W joginter(L
⇥i, ai) +
⌥
i
2µb2ai
4⇧(1� ⌅)ln
� ai
�b
⇥
�W disinter(L,⇥, a) =
µb2
4⇧
⇧⇧L2 + a2 � L� a + L ln
⇤2L⇧
L2 + a2 + L
⌅⌃
W joginter(L, a) = � µb2
4⇧(1� ⌅)
⇧2L� 2
⇧L2 + a2 � 2a ln
⇤L⇧
L2 + a2 + a
⌅⌃
�WMEPA�B (L, a, {L⇥i}, {ai}, s) = �WA�B(L, a, {L⇥i}, {ai}, s) + A⇥GSF (s)
The parameter � is related to the dislocation (in this case, jogs and kinks) core radius and
can not be estimated within the linear elastic theory of dislocations. We obtain � = 0.448
by fitting the energy, �E = 0.27 eV, of the kink pair configuration corresponding to the
configuration in Fig. 1[schematic of the kink pair is marked in Fig. 5(b) and] from our
7
• α can not be determined with in linear elastic theory of dislocations
• α = 0.458 is obtained by fitting the expression for formation energy of a
thermal kink pair from simulations (ΔE = 0.27 eV)
Thermal kink pair configuration
Dislocation model for point defect migration
b1
!1
Set 2S
et 1
a
La2
a1
I1
b1
!1
Set 1
Set 2
a1
a2
L
Lb1
!1
Set 1
Set 2
b
3L
(a) (b) (c)
J. P. Hirth and J. Lothe, Theory of Dislocations, (Wiley, New York, 1982)
Solutions are expected to be greater than energies from the simulations
At the interface
• The shear modulus is thought to be lower than in bulk
• The unstable stacking fault energies are thought to be lower than in bulk
Figure 5: (color online) A simplified model capturing the key steps in the point defect
migration process. The model corresponding to the kink pair configuration shown in Fig. 1
is marked by red box.
interaction energy due to a parallel shift by distance a of a dislocation segment of length L
that is part of a long straight dislocation and is given by Eqn.(2). The second term is the
total negative elastic interaction energy between all pairs of parallel dislocation segments
created due to the nucleation of a kink-jog pair, each of length ai, and separated by L⇥i (a
function of L and ai) and is given by Eqn.(3), and the third term is the self energy of the
dislocation segments that form the kink-jog pair. In Eqns. (2) and (3), µ = 42 GPa is the
shear modulus of bulk copper and b = aCu⇤2
is the magnitude of the Burgers vector of all the
dislocation segments; aCu is the 0 K lattice constant of copper. From simulations, we obtain
a1 = aCu⇤3, a2 = aCu⇤
2, and L = 3aCu⇤
2. Energy expressions for all the states in our simulations
are readily obtained as a combination of Eqns. (2) and (3) with appropriate values for the
variables L, L⇥i, and ai.
W (L, a, {L⇥i}, {ai}) = 2�W disinter(L,⇥, a) +
⌥
i
W joginter(L
⇥i, ai) +
⌥
i
2µb2ai
4⇧(1� ⌅)ln
� ai
�b
⇥
�W disinter(L,⇥, a) =
µb2
4⇧
⇧⇧L2 + a2 � L� a + L ln
⇤2L⇧
L2 + a2 + L
⌅⌃
W joginter(L, a) = � µb2
4⇧(1� ⌅)
⇧2L� 2
⇧L2 + a2 � 2a ln
⇤L⇧
L2 + a2 + a
⌅⌃
�WMEPA�B (L, a, {L⇥i}, {ai}, s) = �WA�B(L, a, {L⇥i}, {ai}, s) + A⇥GSF (s)
The parameter � is related to the dislocation (in this case, jogs and kinks) core radius and
can not be estimated within the linear elastic theory of dislocations. We obtain � = 0.448
by fitting the energy, �E = 0.27 eV, of the kink pair configuration corresponding to the
configuration in Fig. 1[schematic of the kink pair is marked in Fig. 5(b) and] from our
7
Generalized Stacking fault function= 0.175 sin2(πs) J/m2
Area swept by incipient kink pair
fractional Burgers vector contents ϵ [0,1]
b1
!1
Set 2S
et 1
La2
a1
a
b1
!1
Set 1
Set 2
b
3L
I
b1
!1
Set 1
Set 2
a1
a2
L
L
Augmenting with Peierls-Nabbaro framework
Entire migration path can be predicted
Key inputs to the dislocation model
• Interface misfit dislocation distribution
• Structure of the accommodated point defects
Analysis of the interface structure may help predict quantitatively
point-defect behavior at other semicoherent interfaces
Δ E
(eV
)
s s
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
I
a 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
b
IDislocation model
Atomistics
K. Kolluri and M. J. Demkowicz, Phys Rev B, 82, 193404 (2010)
KJ1
KJ3´KJ4
Cu
〈112〉
〈110〉Cu
KJ2´
KJ4
KJ3
KJ2KJ1
Another possible mechanism
http://bit.ly/cunb-alternate
This movie is available at http://bit.ly/cunb-alternate
This mechanism was manually constructed - not observed in simulations
It should be easier for this to happen!
L/Lo
ΔW
(eV
) 0
0.02
0.04
0.06
0.08
0.1
0.12
1 1.5 2 2.5 3
from dislocation model
• 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
b1
!1
Set 2
Set 1
La2
a1
a
b1
!1
Set 1
Set 2
b
3L
I
b1
!1
Set 1
Set 2
a1
a2
L
L
Energy and activation volume in alternate scenario from atomistic calculations
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 0
0.1
0.2
0.3
0.4
0.5
0.6
!E
(eV
)
!V
/"o
S
!E
!Ejog core
!V/"o
!W
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 0
0.1
0.2
0.3
0.4
0.5
0.6
!E
(e
V)
!V
/"o
S
!E
!Ejog core
!V/"o
!W
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 0
0.1
0.2
0.3
0.4
0.5
0.6
!E
(e
V)
!V
/"o
S
!E
!Ejog core
!V/"o
!W
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 0
0.1
0.2
0.3
0.4
0.5
0.6
!E
(e
V)
!V
/"o
S
!E
!Ejog core
!V/"o
!W 0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 0
0.1
0.2
0.3
0.4
0.5
0.6!
E (
eV
)
!V
/"o
S
!E
!Ejog core
!V/"o
!W
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 0
0.1
0.2
0.3
0.4
0.5
0.6
!E
(eV
)
!V
/"o
S
!E
!Ejog core
!V/"o
!W
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 0
0.1
0.2
0.3
0.4
0.5
0.6
!E
(eV
)
!V
/"o
S
!E
!Ejog core
!V/"o
!W
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 0
0.1
0.2
0.3
0.4
0.5
0.6
!E
(e
V)
!V
/"o
S
!E
!Ejog core
!V/"o
!W
dislocation model Energy differences from manually constructed atomic configurations
Energy differences for few atoms surrounding the moving jog
activation volume of the moving jog
Position of the jog (x-axis for previous plot)
S = 6
5
4
3
2
1
6
0
first jog (stationary) is here
• The barrier for atomistics is much greater than that from dislocation model!
• The barrier in this path can be thought of as the difference in the formation
energies of the jog at the MDI and on set 1 misfit dislocation
• In this interface, the barrier is much larger than that we observed (but there
may be other interfaces where such a mechanism could occur)
The self energies of the jog change with positionNot accounted for in the dislocation model
S = 6
5
4
3
2
1
6
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 0
0.1
0.2
0.3
0.4
0.5
0.6
!E
(e
V)
!V
/"o
S
!E
!Ejog core
!V/"o
!W
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 0
0.1
0.2
0.3
0.4
0.5
0.6!
E (
eV
)
!V
/"o
S
!E
!Ejog core
!V/"o
!W
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 0
0.1
0.2
0.3
0.4
0.5
0.6
!E
(eV
)
!V
/"o
S
!E
!Ejog core
!V/"o
!W
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 0
0.1
0.2
0.3
0.4
0.5
0.6
!E
(e
V)
!V
/"o
S
!E
!Ejog core
!V/"o
!W 0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 0
0.1
0.2
0.3
0.4
0.5
0.6
!E
(e
V)
!V
/"o
S
!E
!Ejog core
!V/"o
!W
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 0
0.1
0.2
0.3
0.4
0.5
0.6
!E
(e
V)
!V
/"o
S
!E
!Ejog core
!V/"o
!W
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 0
0.1
0.2
0.3
0.4
0.5
0.6
!E
(eV
)
!V
/"o
S
!E
!Ejog core
!V/"o
!W
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 1 2 3 4 5 6 0
0.1
0.2
0.3
0.4
0.5
0.6
!E
(eV
)
!V
/"o
S
!E
!Ejog core
!V/"o
!W first jog (stationary) is here
!"
!"#$
!"#%
!"#&
!"#'
!(
!(#$
!(#%
!" !( !$ !) !% !* !&!"
!"#(
!"#$
!"#)
!"#%
!"#*
!"#&!+!,-./
!.0"1
2
!+3435-6
!+789:
!.0"1
!E range of a〈112〉jog on a screw dislocation in Cu
!V/"o range of a〈112〉jog on a
screw dislocation in Cu
S = 6
5
4
3
2
1
6
0
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
The energy differences and volumes are comparable to those for jogs in bulk screw dislocation in Cu
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}
The self energies of the jog are different at different positions
S = 6
5
4
3
2
1
6
0
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