Materials Transactions, Vol. 55, No. 3 (2014) pp. 413 to 417
Special Issue on In Situ TEM Observation of High Energy Beam Irradiation
© 2013 The Japan Institute of Metals and Materials
Simulation of Transmission Electron Microscope Images
of Dislocations Pinned by Obstacles
Yuhki Satoh+1, Takahiro Hatano+2, Nobuyasu Nita+3, Kimihiro Nogiwa+4 and Hideki Matsui+5
Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
From a direct observation of dislocation-obstacle interaction utilizing in situ straining experiments in transmission electron microscope
(TEM), the obstacle strength factor could be evaluated from pinning angles of dislocation cusps. We simulated this process: we produced a
dislocation cusp by molecular dynamics simulation of interaction between an edge dislocation and a void or a hard precipitate in copper, and
calculated the TEM image by multislice method. In two-beam conditions, cusp images showed inside-outside contrast depending on the sign of
the diffracting vector and other variations with the specimen geometry. The pinning angles measured on TEM images ranged up to a few tens of
degrees and were between the true angles for the two partial dislocations. Characteristics and contrast mechanisms of cusp images were
discussed based on those of dislocation dipoles. [doi:10.2320/matertrans.MD201312]
(Received September 3, 2013; Accepted October 2, 2013; Published November 15, 2013)
Keywords: transmission electron microscope, image simulation, dislocation-obstacle interaction, irradiation hardening
1.
Introduction
Various crystal lattice defects induced by neutron irradiation, such as precipitates, voids, dislocation lines and loops
are responsible for degradation in mechanical properties:
increase in yield stress, loss of ductility, and increase in
ductile-brittle transition temperature. A number of researches
have been devoted for defect structural evolution and
mechanical property changes under neutron irradiation for
nuclear materials development. Understanding the mechanisms involved is necessary to construct models for
estimating the lifetime of components of nuclear power plant.
In an elementary process of dislocation-obstacle interaction, a gliding dislocation is pinned by obstacles and bows
out to form arcs between the neighboring pinning points,
which induces cusps on the dislocation at obstacles. The apex
angle of the dislocation cusp is referred to as the pinning
angle º. The dislocation breaks away by bypassing or cutting
through the obstacle when the pinning angle reaches a critical
value º c . Stronger obstacles have smaller critical angles.
The obstacle strength factor ¡ ¼ cosðºc =2Þ and the distance
between the neighboring pinning points are the key parameters that relate the defect microstructure to the change in
macroscopic mechanical properties. There have been reported a few attempts to evaluate the factor from a direct
observation of dislocation-obstacle interaction utilizing
in situ straining experiments in transmission electron microscope (TEM).1­3) The method has not been widely applied
so far, due to high technical levels required for in situ
experiments. Another difficulty comes from TEM images
with a limited resolution both in time and space for
measuring pinning angles of radiation-induced obstacles,
+1
Corresponding author, E-mail: [email protected]
Present address: Earthquake Research Institute, The University of Tokyo,
Tokyo 113-0032, Japan
+3
Present address: Central Research Institute, Mitsubishi Materials
Corporation, Naka, Ibaraki 311-0102, Japan
+4
Present address: Japan Atomic Energy Agency, Tsuruga 914-8585, Japan
+5
Present address: Institute of Advanced Energy, Kyoto University, Uji
611-0011, Japan
+2
typically less than a few nanometers in size, at a moment
of the breakaway. Alternatively, molecular dynamics (MD)
simulations have been applied extensively for the dislocationobstacle interaction.4­7)
In the present study, we performed TEM image simulation
of dislocation cusps to examine whether TEM images reveal
the cusp structure in the scale suitable for evaluating the
obstacle strength factor. For this purpose, we stopped MD
simulation of interaction between an obstacle and an edge
dislocation just before the breakaway. We then calculated
TEM images of dislocation cusps under various conditions
using the multislice method. We compared apex angles
between the cusp structure and on the calculated images,
which would support the experimental evaluation of the
obstacle strength factor.
2.
Calculation Procedure
The axes of the MD calculation cell were taken along
yMD ½110
and zMD ½1 1 1,
respectively. A typical
xMD ½112,
cell had the dimensions of 23, 23 and 15 nm, and contained
approximately 5.5 © 105 copper atoms of fcc structure
(a = 0.3615 nm). It employed the interatomic potential given
by Ackland et al.8) We introduced a spherical void (radius 1,
2 or 3 nm) or a hard precipitate (radius 1.5 nm) as an obstacle.
The hard precipitate was modeled by a group of immobile
atoms that were coherent with the matrix crystal.7) We
induced also a perfect edge dislocation with Burgers vector
b ¼ a=2½110,
and it dissociated into two Shockley partial
dislocations b1 ¼ a=6½211
(leading partial) and b2 ¼
a=6½121 (trailing partial). The glide plane bounded by the
two partials involved a stacking fault with the displacement
vector R = ¹a/3[111]. The geometry of the cell was
schematically shown in Fig. 1(a). Periodic boundary condition was applied in xMD and yMD directions. In MD
calculation at 300 K, the dislocations were driven toward
¹yMD direction at a strain rate of 8 © 106 s¹1; the lower
(¹zMD) surface was moved along ¹yMD direction while the
upper surface was fixed. The model supposed the two partial
dislocations of infinite length in xMD direction to glide along
414
Y. Satoh, T. Hatano, N. Nita, K. Nogiwa and H. Matsui
(a)
electron beam
(b)
d
perfect
cell
obstacle
t
defect
cell
φ
e
id
gl
perfect
cell
z MD[111]
z TEM[011]
y MD[110]
y TEM[100]
x MD [112]
x TEM [011]
(d)
40
20
b2
b
zTEM [011] (b)
y TEM[100] (b)
(c)
40
20
b1
0
0
0
20
0
40
x TEM [011] (b)
20
40
y TEM [100] (b)
Fig. 1 Schematic illustration of the system for MD simulation (a) and for TEM image simulation (b). The shaded plane shows the glide
plane of the dislocations. (c) (d) The geometry of the void (1 nm radius) and dislocations in the defect cell. The defect structure was
visualized by showing the atoms which do not have 12 nearest neighbors. Note that the Burgers vectors b, b1 and b2 do not lie on zTEM
plane.
¹yMD direction under shear strain, and to interact with
periodically arranged obstacles in a thin film. See Refs. 5)
and 6) for the method and results of the MD simulation; we
do not go into further details.
The MD calculation was stopped just before the leading
partial broke away from the obstacle, and each atom was
relaxed to the equilibrium position by a static method. Then
the defect cell that contained the obstacle and the dislocation
cusp was cut from the MD cell. The cell was sandwiched
between two perfect lattices in the TEM image simulation
system, as is schematically shown in Fig. 1(b). The size of
the defect cell was 14.3 nm (56b), 14.5 nm (40a), 14.1 nm
yTEM ½100 and zTEM ½01 1
axes,
(55b), along xTEM ½011,
respectively, where b denotes the atomic distance
(0.2556 nm). Figures 1(c) and 1(d) show the cusp structure
projected along ¹zTEM and xTEM directions. The glide plane
was inclined to zTEM plane by about 35 degree. The total
system was divided into slices (thickness b) normal to zTEM
direction, and TEM images of the system were simulated
based on the multislice method with assuming a pseudo
supercell structure in xTEM and yTEM directions. Because the
defect cell did not satisfy the periodic boundary condition,
TEM images involved an artifact at the cell boundaries. The
variables in the system geometry were the total thickness t
of the system and the depth d of the defect cell. The latter
was defined as the thickness of the perfect cell above the
defect cell.
In the simulation of 200 kV TEM images, we employed
the electron scattering potential for copper given by Radi.9)
The beams up to 066 and 800 were taken into the image
calculation. The beam incident direction was along ¹zTEM
with a slight offset (about 7 degree) in order to excite 200
systematic reflections. The bright-field and dark-field images
were formed by 000 and 200 reflections respectively,
including also the pseudo superlattice spots which passed
through an objective aperture. The aperture radius was
22.5% of the distance between 000 and 200 reflections. The
deviation from the Bragg condition n was positive and varied
from 1.4 to 6.5 in the conventional notation g = 200(ng).
For example, g = 200(5g) means that 5g fulfills the Bragg
condition, and the Kikuchi band is between 2g and 3g,
centered at 2.5g. Another procedural details and the parameters used in the image calculation were provided in the
reports.10,11)
3.
Results and Discussion
3.1 Two-beam bright-field images
In bright-field (BF) images under a small deviation from
the Bragg condition (i.e., two-beam condition), a dislocation
produces a dark and broad image at the position with long
distances (up to a few nm) from the dislocation core. The
condition has been widely used for observing dislocations,
including the in situ straining experiments for evaluating the
Simulation of Transmission Electron Microscope Images of Dislocations Pinned by Obstacles
(a) structure
(b) BF (n =1.8)
g=200
415
(c) WBDF (n =6.5)
g=200
g=200
precipitate 1.5nm
void 3nm
void 2nm
void 1nm
g=200
5 nm
0.00
0.02
10-4
0.04
10-2
100
image intensity
Fig. 2 Comparison between defect structure (a) and typical simulated images for g = 200 and g ¼ 200.
(b) Two-beam bright-field images
at n = 1.8, d = 254b and t = 414b. (c) Weak-beam dark-field images at n = 6.5, d = 219b and t = 417b.
obstacle strength factor.1­3) We examined bright-field images
of cusps at the deviation parameter n between 1.4 and 2.7.
Figures 2(a) and 2(b) compare the cusp structure and
typical calculated images at n = 1.8. The extra contrast
observed at the cell boundaries is the artifact to be ignored.
The cusp images for 1 nm void are less clear because of small
size and low contrast. The other cusps formed by the strong
obstacles are identified by their dark images. The two partial
dislocations are not distinguished from each other, which is
consistent to a general image of dissociated dislocations in
fcc metals under two-beam conditions.12,13) The cusp images
depend on the sign of the diffracting vector; the image
appears inside and outside the leading partials for g = 200
and g ¼ 200,
respectively. This contrast variation was
common for all the obstacles examined, and would be
similar to the “inside­outside” contrast known for dislocation
loops of interstitial- and vacancy-types.
Defect images depend also on the specimen geometry. The
cusp images showed periodic variations with the specimen
thickness t and the depth d of the defect cell; the period of
the oscillation was the effective extinction distance ð² eff
g Þ.
Accordingly, Fig. 3, the images with various t and d for one
effective extinction distance, represents all variations.
The pinning angle º was defined as the apex angle
between the two tangent lines of the dislocation at the
obstacle surface. We measured the apex angle on zTEM plane,
which is referred to as the projected pinning angle º0 , by
using either dark or bright image which had larger contrast in
each condition in Fig. 3. The angle was not determined when
neither of the images was clear. Several examples of the
tangent lines are shown with broken lines in Fig. 3. Figure 4
compares the angle º0 measured on TEM images with the
true angles obtained from the configuration of the leading
and trailing partials shown in Fig. 2(a). The distribution
ranges about 30 degrees, and is between the true angles for
the two partials. The angles are slightly larger in the outside
contrast condition ðg ¼ 200Þ.
The pinning angle of the hard
precipitate was zero, which was identified on TEM images
both in the inside and outside conditions.
3.2 Weak-beam dark-field images
In the MD simulation the leading and trailing partials
have different interaction with an obstacle and have different
critical angles.5) The weak-beam dark-field (WBDF) technique might reveal the fine scale behavior of the two partials
in a cusp, because dislocation images with narrow width and
high contrast lies close to the projection of the dislocation
core. We examined the cusp images on weak-beam dark-field
images at the deviation parameter n between 4.5 and 6.5.
Typical weak-beam images of dislocation cusps are shown
in Fig. 2(c) using a logarithmic grey scale. Because the scale
displays the intensity over a wide range without saturation,
detailed contrast that is darker than the backgound may not
be observed on practical weak-beam images. The configurations of the two partials are observed at g = 200. On
reversing the diffracting vector ðg ¼ 200Þ,
the stacking fault
image has larger intensity while dislocation images are not
clear. The two partials might be detected as borders of the
416
Y. Satoh, T. Hatano, N. Nita, K. Nogiwa and H. Matsui
(b) g=200
390
5 nm
image
intensity
414
0.06
0.04
438
specimen thickness, t (b)
366
(a) g=200
0.02
0.00
222
246
270
294
222
246
270
294
depth of defect cell, d (b)
Fig. 3 Variation of the cusp images for 2 nm void with the sign of the diffracting vector, the specimen thickness, and the depth of the
defect cell, at n = 1.8 ð² eff
g 96bÞ.
(a) void 2 nm, g=200
number of counts
10
leading
partial
trailing
partial
0
10
(b) void 2 nm, g=200
0
0°
30°
60°
90°
120°
N.D.
120°
N.D.
(c) void 3 nm, g=200
number of counts
10
leading
partial
trailing
partial
0
(d) void 3 nm, g=200
10
0
0°
30°
60°
90°
projected pinning angle φ’
Fig. 4 Distribution of the projected pinning angles measured on TEM
images at n = 1.8. The arrows indicate the true angles obtained from the
configuration of the core of the leading and trailing partials.
stacking fault band. It has been known that stacking fault
image shows systematic asymmetry, depending on the sign of
g · R; the bright band corresponds to the strong condition in
the asymmetry.12) These image variations were common for
the obstacles excepting for 1 nm void. We note that irregular
dark bands appear around the obstacle in weak-beam images.
Because the position of the band depended on the diffraction
condition and the specimen geometry (i.e., the specimen
thickness t and the defect depth d), the band will not represent
a certain defect structure but will be similar to a bend contour,
suggesting a high strain around the obstacle.
3.3 Discussion
Characteristics of the calculated cusp images could be
interpreted from images of a dislocation dipole, i.e., a pair
of single straight dislocations of opposite Burgers vectors.
Figures 5(a) and 5(b) schematically show that an edge
dislocation forms a screw dipole near the obstacle; and
vice versa. The approximation will be better for stronger
obstacle, because the cusp is elongated to form a dipole as the
case of the hard precipitate shown in Fig. 2(a). Dipole images
are known to show the inside­outside contrast.12,13) Because
the two dislocations have opposite Burgers vectors, one image
will lie on one side of the core and the other on the opposite
side. The order reverses on reversing diffracting vector.
Figures 5(c) and 5(d) compare image intensity profiles
among single and dipole perfect dislocations of edge and
screw types, lying parallel to the specimen surface. The
calculation was based on the multibeam dynamical equations
with the column approximation.12) We assumed a strain
field of a perfect dislocation in an isotropic medium for the
single dislocation and their simple superposition for the
dipole. The image intensity was shown as normalized by the
background intensity, IBG. The dipole with a small separation,
corresponding to a small cusp, has very weak contrast both in
inside and outside conditions. This is because the two strain
fields with opposite Burgers vectors almost cancel to each
other. With increasing separation, the contrast increases for
the inside condition because of an additive effect of the two
strain fields, while the outside contrast is weaker than that
of single dislocations due to a partial cancellation. The
characteristics of dislocation cusp images observed in the
multislice simulation were interpreted qualitatively by the
conventional dynamical theory applied for dislocation dipole.
A quantitative difference in the dipole images between edge
and screw types would reflect the narrower image for a single
screw dislocation.
Finally, we note very briefly another method useful to
determine the apex angle:3) the total configuration of bow out
dislocation was extrapolated to the cusp with assuming the
dislocation configuration that minimized the total elastic
energy in the standard line tension model. Understanding
Simulation of Transmission Electron Microscope Images of Dislocations Pinned by Obstacles
(a)
(b)
egde dislocation
screw dislocation
dipole
glide
obstacle
y
p
x
(c) edge dislocation, n =2.5
(d) screw dislocation, n =2.5
single
single
1
normalized image intensity (I BG)
g.b=+1
g.b=-1
g.b=+1
g.b=-1
0
dipole (inside contrast)
dipole (inside contrast)
1
417
two-beam bright field images, large cusps formed by strong
obstacles were identified by dark images. The images showed
the inside­outside contrast on reversing the diffracting vector
and other variations depending on the specimen geometry.
Local configurations of dislocation cusps were not always
revealed in two-beam conditions. Then the distribution of
pinning angles measured on TEM images ranged up to a few
tens of degrees and were between the true angles for the two
partial dislocations. In some weak-beam dark-field images,
the two partial dislocations and the stacking fault were
distinguished from one another. Characteristics and contrast
mechanisms of cusp images could be interpreted from those
of dislocation dipoles. The knowledge of the cusp images
is expected to reduce errors in estimating obstacle strength
factor using in situ straining experiments.
p =1 nm
Acknowledgments
4 nm
10 nm
A part of the numerical simulation was performed using
the Center for Computational Materials Science, Institute for
Materials Research, Tohoku University.
0
p =1 nm
1
4 nm
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10 nm
dipole (outside contrast)
dipole (outside contrast)
0
-20
0
distance, x /nm
20
-20
0
20
distance, x /nm
Fig. 5 (a) (b) Schematic illustration of an edge dislocation before and after
the pinning by an obstacle. Image intensity profiles of single and dipole
of (c) pure edge and (d) pure screw dislocation ðb ¼ a=2½110Þ. The
separation p of the dipole was 1, 4 or 10 nm. Bright field images,
g ¼ 200 and n = 2.5. The specimen thickness was 12² eff
g , and the depth
eff
of the dislocations was 6:5²eff
g ð² g 14:5 nmÞ.
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presented here as well as the application of the line tension
model to the dislocation configuration are expected to reduce
errors in measuring the pinning angle on TEM images.
4.
Conclusion
We simulated TEM image of the cusp on an edge
dislocation that was pinned by an obstacle in copper. In
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