Gottstein G., Shvindlerman L.S. Grain Boundary Migration in Metals: Thermodynamics, Kinetics, Applications
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466 5 Computer Simulation of Grain Boundary Motion
5.3 Migration of [001] Twist Grain Boundaries
5.3.1 Procedure
Almost all results on GB migration and GB self-diffusion of planar [001] twist
GBs that will be presented were obtained by utilizing the 3D LJ potential and
applying a 3D periodic boundary concept. For the GB migration simulations
due to an OCDF different boundary concepts and various atomistic potentials
were applied. This is especially true for the simulations of the Σ5 [001] twist
GB, where simulations with the Cu EAM potential of Doyama and Kogure
also were performed.
After the simulation cell has been set up, the grain boundary structure has
to be relaxed prior to its motion. The GB properties and atomic-level struc-
tures of fully relaxed GB systems are obtained by performing lattice-statics
(LS) simulations according to the steepest gradient method. These zero Kelvin
relaxation simulations are necessary for most GBs due to their strictly geo-
metric CSL generation scheme. The must to perform such simulations results
from the fact that most GBs differ substantially in their equilibrium structure
from the geometrical CSL GB construct. An important feature of equilibrated
GBs is the volume expansion ΔV
GB
which was found to be proportional to
the GB energy and, therefore, a key parameter of any GB [503]. The volume
expansion manifests the tendency of a GB to expand or shrink parallel to the
GB normal. This is the case since for all performed simulations the in-plane
dimensions of the CSL GBs are held fixed. If MD simulations were performed
on unrelaxed high energy GBs with fixed finite time steps, the time integra-
tion scheme would fail to yield the true atomistic trajectories of each atom
and cause a crash of the MD simulation by a destruction of the crystalline GB
system. Thus, an initial relaxation precodure is essential for the simulation of
GB migration. Furthermore, other important information on GBs like the GB
energy and the volume expansion can be obtained from these computations.
In the case of [001] twist GBs a sequence of LS simulations is sufficient to
yield fully relaxed GB properties, whereas for tilt GBs a combined LS/MD
strategy has to be used to finally yield relaxed GBs.
5.3.2 Atomistic Structure of the Relaxed [001] Twist GBs
Since the atomistics of twist GBs are difficult to visualize the arrangement
within (002) planes facing each other at the GB is used to gain insight into
the atomic-level situation and will be referred to as in-plane data plots. It is
accomplished by showing the closest A-type (002) planes of either of the two
grains in one plot. This illustrates to what extent atomic-level relaxation had
occurred in the vicinity of the GBs, e.g. by the coherency of the CSL points
of the relaxed GB structures. The exact atomistic arrangement of the Σ29
© 2010 by Taylor and Francis Group, LLC
5.3 Migration of [001] Twist Grain Boundaries 467
TABLE 5.1
Studied [001] Twist GBs
No. of
GB plane θ Σ E
GB
[
J
m
2
] CSL cells ΔV
GB
[a
o
]
(001) 43.60
◦
29 0.716 1;4 0.12204
(001) 36.87
◦
5 0.715 9 0.12080
(001) 28.07
◦
17 0.700 4 0.11623
(001) 22.62
◦
13 0.677 4 0.11144
(001) 16.26
◦
25 0.625 2 0.10350
(001) 12.68
◦
41 0.582 1 0.09801
(001) 8.80
◦
85 0.490 1 0.08517
(001) 6.03
◦
181 0.404 1 0.07324
Note: The table gives the misorientation angle, θ, the GB energy,
E
GB
, the number of CSL cells within each (002) plane, and the volume
expansion of the GBs, ΔV
GB
.
and Σ5 twist GBs is given by Figs. 5.5 and 5.6. For [001] twist GBs in fcc
crystals one finds the following relation for the volume of the primitive CSL
cell, V
CSL
,namely
Σ · Ω=V
CSL
= d
(001)
· A
CSL
=
1
2
a
o
· A
CSL
=Σ·
a
3
o
4
⇔ A
CSL
=Σ·
a
2
o
2
In general this means that the in-plane CSL area, A
CSL
,isgivenbyA
CSL
=
a
CSL
·b
CSL
where a
CSL
and b
CSL
are the in-plane CSL vectors. Assuming a
square for the in-plane CSL cell, the in-plane area A
CSL
is related in a simple
manner to the magnitudes of the in-plane CSL vectors, a
CSL
and b
CSL
,
A
CSL
=
Σ
2
· a
o
·
Σ
2
· a
o
⇔ a
CSL
= b
CSL
=
Σ
2
· a
o
The relaxed low-angle twist grain boundaries reveal the structural screw dis-
location in terms of non-ideal fcc atoms. The Σ25 twist GB is on the border
between low-angle and high-angle twist GB but also reveals discrete structural
dislocations.
All studied GBs were simulated in three-dimensional atomistic systems.
Hence, the GBs were truly two-dimensional defects. For the LJ potential,
simulations with 3D periodic boundary conditions (BC) as well as with 2D
periodic border conditions within the GB plane and stabilized free (002) sur-
faces were performed. Apart from the LJ potential, simulations with the EAM
potential by Doyama and Adams and the Finnis-Sinclair potential according
to Sutton and Chen were conducted as well. An overview of the performed
GB migration simulations is given in Table 5.2.
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468 5 Computer Simulation of Grain Boundary Motion
-2 -1 0 1 2
-2
-1
0
1
2
Relaxed
Σ
29 [001] Twist GB
y
B
y
A
x
B
x
A
θ
= 46.40°
y [a
o
]
x [a
o
]
FIGURE 5.5
Atomistics of the Σ29 [001] 43.60
◦
twist GB. Here the closest A-type (002)
planes facing each other at GB
2
are plotted. Atoms within plane A of grain
A are given by filled circles while the atoms within plane A
of grain B are
given by squares.
TABLE 5.2
Overview of conducted GB migration simulations on the studied [001]
twist GBs
Σ θ Elastic DF Orientation-Correlated DF
29 43.60
◦
LJ 3D periodic BC —
Doyama 2D periodic BC and
free (002) surfaces
5 36.87
◦
LJ 3D periodic BC LJ 2D periodic BC and
stab. free (002) surfaces
LJ 3D periodic BC
17 28.07
◦
LJ 3D periodic BC [530] LJ 2D periodic BC and
stab. free (002) surfaces
LJ 3D periodic
13 22.62
◦
LJ 3D periodic BC [530] LJ 2D periodic BC and
stab. free (002) surfaces
25 16.26
◦
LJ 3D periodic BC [530] —
41 12.68
◦
LJ 3D periodic BC —
85 8.80
◦
LJ 3D periodic BC LJ 3D periodic BC
181 6.03
◦
— LJ 3D periodic BC
© 2010 by Taylor and Francis Group, LLC
5.3 Migration of [001] Twist Grain Boundaries 469
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
primitive CSL
Relaxed
Σ
5 [001] Twist GB
y
B
y
A
x
B
x
A
θ
= 36.87°
y [a
o
]
x [a
o
]
FIGURE 5.6
Atomistics of the Σ5 [001] 36.87
◦
twist GB. Here the closest A-type (002)
planes facing each other at GB
2
are plotted. Atoms within plane A of grain
A are given by filled circles while the atoms within plane A
of grain B are
given by squares.
© 2010 by Taylor and Francis Group, LLC
470 5 Computer Simulation of Grain Boundary Motion
(001) Plane Stacking Sequence :
Grain A = ABAB and Grain B = A'B'A'B'
A
B
A
B
A
B
A
B
A'
B'
A'
B'
A'
B'
A'
B'
GB
1
r
z
|| [ 0 0 1]
x || [ 14 6 0 ]
GB
2
Grain A
Grain B
-1.04
-1.02
-1.00
-0.98
-0.96
-0.94
-0.92
-0.90
-0.88
-20
-15 -10 -5 0 5 10 15 20
E
pot
[eV/atom]
r
z
[a
o
]
T = 0K
ε
=−
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
0015 0030 0
0030 0015 0
000
..
..
Energy profile for strained case
Energy profile for unstrained case
GB
1
GB
2
(b)
Σ
29 (001) 43.60° Twist GB
(a)
FIGURE 5.7
Σ29 [001] 43.60
◦
twist GB, γ = 0.708
J
m
2
: (a) shows a schematic of the GB
system showing the stacking sequence of (002) planes in the GB system. GB
1
and GB
2
identify the GB positions. (b) presents the verification of an elas-
tic driving force for ε =0.03at0Kfora3DLJsystem.Herethebulk
regions of either grain for the unstrained case can be identified by having
a potential energy equal to the cohesive energy of the ideal crystal of E
coh
= −1.037824 eV/atom. Applying the strain homogeneously to the MD box
leads to a shift of the unstrained energy profile by the amount of stored elas-
tic energy within each grain. Thus this allows one to obtain the magnitude of
stored elastic energy by substracting the energy profiles of the strained and
unstrained system. Here the two GBs, GB
1
and GB
2
, are associated with the
energy peaks in the profiles of (b).
5.3.3 GB Migration Due to an Elastic Driving Force
According to Fig. 5.7(b) the elastic energy stored within each grain can be
determined by subtracting the energy profiles of the strained and unstrained
system. The obtained energy profile thus represents the elastic energy varia-
tion along the [001] direction within the GB system which can be compared
with the analytical linear elastic model as discussed in Sec. 5.2.2. Fig. 5.8
shows such a plane-by-plane elastic energy profile for the Σ29 twist GB at a
strain of ε = 0.001 and 0 K. In addition to the computed simulation data the
analytical linear elastic model data are given as well by the broken lines which
demonstrate that for a strain in the range of the classical elastic range the
computed and model data agree very well. Since the simulation covers only
© 2010 by Taylor and Francis Group, LLC
5.3 Migration of [001] Twist Grain Boundaries 471
0 1020304050607080
0,0
0,5
1,0
1,5
2,0
2,5
Lin. elast.
model : +
θ / 2
Lin. elast.
model : -
θ
/ 2
Σ
29 [001] Twist GB at 0 K and
ε
=0.001
computed prole
E
ela
[10
-5
eV/atom]
Plane number [1]
FIGURE 5.8
Plane-by-plane elastic energy profile of the Σ29 twist GB at 0 K for ε = 0.001.
The zick-zack energy profile in the bulk of the two grains is due to the fact
that in this example a not fully relaxed Σ29 GB system was used.
a time interval of maximum ns, the GB velocity must be at least in the range
of m/s in order to be detected within these simulations. Hence, the applied
strain ranged from 0.01 up to 0.035, where effectively a linear elastic behavior
was observed, since no other defects were present in the system.
5.3.4 Data Analysis with Respect to GB Migration
To determine and track the position of the GBs throughout the MD simu-
lations the so-called common-neighbor structural analysis (CNA) [516, 517]
method was applied. A few simple criteria were imposed on the atomistic
data to distinguish true GB atoms from the rest. First, according to the CNA
method true ideal fcc atoms were determinedinthesystemwhereeachtrue
ideal fcc atom could be recognized by its sharing of (421) triplets with the
twelve nearest neighbors. Any atom that was not identified a true ideal fcc
atom was then considered a potential GB atom. These non-ideal fcc atoms
sometimes were bulk vacancies, and further data analysis was needed to rule
them out as true GB atoms. Therefore, the system was sliced into slabs of
e.g. discrete (002) planes, and a histogram of all non-fcc atoms was obtained.
From these histograms, the centers of mass of the GBs were calculated and
their temporal evolution was tracked. This data analysis was not carried out
during the MD simulation runs but rather on stored atomistic configurations.
The storing intervals used were one box after every 1000 time steps. This data
analysis yielded the GB position vs. time, the slope of which corresponded to
the GB velocity.
© 2010 by Taylor and Francis Group, LLC
472 5 Computer Simulation of Grain Boundary Motion
0246810
0
1
2
3
4
5
6
Acc. to Linear Fit :
1000K : slope = 4.14 eV/
Ω
900K : slope = 4.66 eV/
Ω
800K : slope = 5.26 eV/
Ω
700K : slope = 5.85 eV/
Ω
Linear Best Data Fits
Σ
5 [001] Twist GB : Elastic DF acc. to DQS
1000K
900K
800K
700K
p
ela
[0,001 eV/
Ω
(0K)]
ε
2
[0,0001]
FIGURE 5.9
Σ5 [001] twist GB. Elastic DF according to the DQS scheme for temperatures
ranging from 700 K up to 1000 K and different strains.
5.3.5 Mobility of [001] Twist GBs
MD simulations were conducted for a Σ5 GB in a temperature range from
700 K up to 1100 K. Utilizing the DQS scheme it was possible to calculate
the elastic DF in situ during the MD GB migration simulations. The DF is
proportional to the square of the strain, ε
2
, at a fixed temperature (Fig. 5.9).
The slope equals c
A
· sin(2 · θ). It only differs by about 10 percent from the
theoretical value calculated from linear elastic theory due to non-linear effects.
Fig. 5.10 shows the temporal evolution of the GB position vs. time at 800 K
for different elastic DFs. Due to the 3D periodic bicrystalline box geometry
and the direction of the imposed DF, the upper grain usually shrank finally
leaving behind a single grain with the orientation of the lower grain. The pre-
sented data manifest a linear dependency, the slope of which corresponds to
the GB velocity.
The GB mobility for a certain temperature was then determined as the
slope of a best linear data fit of GB velocity vs. elastic DF plots (Fig. 5.11).
It is important to realize that many GB migration simulations for a fixed
temperature at different DF levels are needed to yield reliable data of GB
mobility at a given temperature.
An Arrhenius plot of the temperature dependence of the GB mobility for
all studied [001] twist GBs in this study is given in Fig. 5.12. The slope of a
best fit of each of the GB data yielded the activation parameters, namely the
activation enthalpy, Q
GBM
, and the GB mobility pre-exponential factor, m
o
,
as listed in Table 5.3. The results demonstrate that [001] twist GBs in an fcc
material move continuously in the presence of an elastic DF. Strains of 0.01 to
© 2010 by Taylor and Francis Group, LLC
5.3 Migration of [001] Twist Grain Boundaries 473
0 1000 2000 3000 4000 5000
-100
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
10
20
30
40
50
ε
xy
= 0.0
ε
xy
= 0.0
Σ 5 [001] Twist GB : T=800 K
ε
xy
= 0.030
ε
xy
= 0.025
ε
xy
= 0.020
ε
xy
= 0.015
GB Position [10
-10
m]
t [ps]
FIGURE 5.10
Σ5 [001] twist GB. GB position vs. time at 800 K for different elastic DFs.
01234567
-2
0
2
4
6
8
10
12
1100 K
950 K
Σ
5 [001] Twist GB : elastic DF LJ 3D
1000 K
900 K
800 K
700 K
700 K
800 K
900 K
950 K
1000 K
1100 K
< v
GB
> [m/s]
p
ela
[0,001 eV/
Ω
(0K)]
FIGURE 5.11
Σ5 [001] twist GB. GB velocity vs. elastic DF from 700 K up to 1100 K.
© 2010 by Taylor and Francis Group, LLC
474 5 Computer Simulation of Grain Boundary Motion
10 11 12 13 14 15 16 17 18 19 20
1
10
100
600 K
Σ
29
1100 K
Σ
85
Σ
41
Σ
5
700 K
800 K900 K1000 K
Σ
25
Σ
13
Σ
17
m
GB
(T) [10
-8
m
4
/Js]
1/k
B
T[1/eV]
FIGURE 5.12
Arrhenius plot of GB mobility for the [001] twist GBs studied. The determined
activation parameters are listed in Table 5.3.
TABLE 5.3
GB Migration Activation Parameters of the [001] Twist GBs Studied
GB plane θ Σ Q
GBM
[eV] m
o
[10
−6
m
4
Js
]
(001) 43.60
◦
29 0.438
1
73.06
3
43.60
◦
29 0.226
2
± 0.008 2.57
4
(001) 36.87
◦
5 0.319 ± 0.001 10.21
(001) 28.07
◦
17 0.267 ± 0.040 4.32
(001) 22.62
◦
13 0.284 ± 0.030 9.39
(001) 16.26
◦
25 0.346 ± 0.020 35.94
(001) 12.68
◦
41 0.109 ± 0.030 2.19
(001) 8.80
◦
85 0.121 ± 0.010 4.69
Note: The results were obtained for the LJ potential with 3D periodic BC
and an elastic DF.
1
according to low temperature regime
2
according to high temperature regime
3
according to low temperature regime
4
according to high temperature regime
© 2010 by Taylor and Francis Group, LLC
5.3 Migration of [001] Twist Grain Boundaries 475
0 5 10 15 20 25 30 35 40 45
0,0
0,1
0,2
0,3
0,4
0,5
0,6
and this work
Winning - [001] Twists in Al
[001] Twist GBs
Σ
29 GBM
at high
T
Σ
29 GBM
at low T
Σ
85
Σ
41
Σ
25
Σ13
Σ
17
Σ
5
Σ29
LJ with 3D per . BC
and elastic DF
Q
GBM
[eV]
θ
[°]
FIGURE 5.13
Misorientation dependence of the activation enthalpy of GB migration of the
studied [001] twist GBs. Results of LJ 3D simulations with an elastic DF and
experimental data of planar [001] twist GBs in Al according to Winning [521].
0.03 provide sufficient DF to move these GBs continuously over distances of
a few ten nanometers in a few nanoseconds. The misorientation dependence
of the elastic DF expressed by a sin(2·θ) term (Eq. (5.24)) was confirmed by
the simulation results. This substantiates that the elastic DF becomes smaller
when changing from high-angle GBs to low-angle GBs at a fixed strain.
For [001] twist GBs, Table 5.3 summarizes the obtained results with respect
to the computed activation parameters. In Fig. 5.13 the activation enthalpy
of GB migration, Q
GBM
, is plotted vs. the misorientation angle, θ.Forcom-
parison, experimental data of planar [001] non-CSL twist GBs in Al according
to Winning [521] are included.
For all twist GBs except Σ29 the activation parameters were constant over
the whole temperature regime. In the case of the Σ29 GB a low tempera-
ture and a high temperature regime were found. For the other boundaries the
activation parameters could be associated with a low temperature behavior.
This usually meant that only the 700 K to 900 K GB mobility data could
be considered for the Arrhenius analysis. The GB mobility data at 1000 K
suggested that some form of GB structural transition occurred since typically
the 1000 K mobility did not fit into the general trend of the low temperature
dependence of the GB mobility. An example is the Σ5 twist GB where the
deviation from the expected behavior occurred again at about 900 K. Since
the simulations did not allow to extend the temperature range to tempera-
tures much higher than 1100 K, it was difficult to assess how the GB mobility
profile would have continued for much higher temperatures.
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