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Differences in deformation behavior of bicrystalline Cu
micropillars containing a twin boundary or a large-angle
grain boundary
Peter J. Imrich
a,⇑
, Christoph Kirchlechner
b,c
, Christian Motz
a,d
, Gerhard Dehm
b,c
a
Erich Schmid Institute of Materials Science, Austrian Academy of Sciences, Jahnstraße 12, 8700 Leoben, Austria
b
Department of Materials Physics, Montanuniversita
¨t Leoben, Austria
c
Max Planck Institut fu
¨r Eisenforschung, Du
¨sseldorf, Germany
d
Department of Materials Science, Saarland University, Saarbru
¨cken, Germany
Received 17 October 2013; received in revised form 9 April 2014; accepted 10 April 2014
Abstract
Micrometer-sized compression pillars containing a grain boundary are investigated to better understand under which conditions grain
boundaries have a strengthening effect. The compression experiments were performed on focused ion beam fabricated micrometer-sized
bicrystalline Cu pillars including either a large-angle grain boundary (LAGB) or a coherent twin boundary (CTB) parallel to the
compression axis and additionally on single-crystalline reference samples. Pillars containing a LAGB show increased strength, stronger
hardening and smaller load drops compared to single crystals and exhibit a bent boundary and pillar shape. Samples with a CTB show no
major difference in stress–strain data compared to the corresponding single-crystalline samples. This is due to the special orientation and
symmetry of the twin boundary and is reflected in a characteristic pillar shape after deformation. The experimental findings can be
related to the dislocation–boundary interactions at the different grain boundaries and are compared with three-dimensional discrete
dislocation dynamics simulations.
Ó2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Keywords: Micromechanics; Copper; Dislocation; Cross-slip; Compression test
1. Introduction
The mechanical behavior of samples at small dimensions
is different from that of bulk materials. The most striking
feature is a drastic increase in strength with decreasing
specimen size. This decrease in specimen size can either
be one-dimensional (1-D) as for thin films [1–3], 2-D as
for thin wires [4], or 3-D as for small compression [5], bend-
ing [6] or tensile samples [7]. In 2004 a technique was devel-
oped by Uchic et al. [5] to create and mechanically test
compression pillars that are small in all three dimensions,
using a focused ion beam system (FIB). These single-
crystalline micron- and submicron-sized compression pillars
show increasing yield strength with decreasing sample size.
This so-called mechanical size effect was confirmed by
numerous microcompression, microtension and micro-
bending experiments on samples from approximately
100 nm to 20 lm in diameter [5–13] and is now well estab-
lished and fairly well understood. The reason for the
“smaller is stronger”effect is based on the dislocation den-
sity and number of available dislocation sources as has
been shown by, for example, Bei et al. [14]. Depending
on the sample size, different processes, including source
http://dx.doi.org/10.1016/j.actamat.2014.04.022
1359-6454/Ó2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
⇑
Corresponding author. Tel.: +43 3842 804 313; fax: +43 3842 804 116.
E-mail address: peter.imrich@stud.unileoben.ac.at (P.J. Imrich).
www.elsevier.com/locate/actamat
Available online at www.sciencedirect.com
ScienceDirect
Acta Materialia 73 (2014) 240–250
statistics [15,16], dislocation starvation [17], source trunca-
tion [16] and dislocation exhaustion [18], have been pro-
posed to describe the size effect and the stochastic flow
behavior of microsamples. The critical stress to activate a
dislocation source, which can be either of Frank–Read or
single arm type, scales inversely with the source size. A
decrease in sample dimensions leads to a reduction in
source size which in turn increases the flow stress.
Efforts to quantify the size effect have mostly come up
with a power-law description of the yield strength, ry:
ry¼r0þkdnð1Þ
linking the yield strength to the diameter dof the sample
with a scaling exponent n, similar to a Hall–Petch-type
law, although the underlying mechanisms are completely
different. r0corresponds to the bulk strength of the mate-
rial and kis a constant in Eq. (1) Numbers for the scaling
exponent range from 0.2 or even 0 in ceramics [19] up to
0.6–1 in face-centered cubic (fcc) metals [20]. However,
defining an exact value for the scaling exponent has
aroused controversy, since it was shown to depend on
many factors, including dislocation and pinning point den-
sity [21]. Dunstan et al. [22] even go as far as claiming a
constant exponent of 1 for all materials, arguing that devi-
ations are due to special circumstances or ambiguous fit-
ting of logarithmized data.
Looking at real-world materials, we know that the
majority of commercially used metals are polycrystalline,
i.e. comprised of numerous grains with grain sizes reaching
from a few centimeters down to several nanometers. It was
discovered by Hall [23] and Petch [24] that the yield
strength of metals scales inversely with the square root of
the average grain size. This effect shows the importance
of grain boundaries concerning deformation behavior
and is of major importance for the strengthening of mate-
rials by grain refinement. The discovery led to many scien-
tific studies trying to understand the ongoing mechanisms
and create novel materials with advanced mechanical prop-
erties by altering the grain size [25–27].
Recently, micromechanical testing was applied to speci-
mens containing two or more grains [28–48] in order to
gain information on the influence of extrinsic and intrinsic
feature size on the stress–strain behavior as well as on the
underlying deformation mechanisms. In a polycrystalline
macrosample the behavior of individual grain boundaries
is not clearly identifiable due to simultaneous plasticity in
many grains. By studying bicrystalline micropillars, specific
gain boundary types can be tested in uniaxial compression,
which avoids the complication of an unknown stress state
in polycrystalline samples. Thus, pillar microcompression
offers the possibility to apply different stress states to dislo-
cations transmitting the boundary. Up to now, there have
been experiments on multiphase nanolayered composites
[28–31], nanocrystalline [31–35] and nanotwinned materials
[36,37]. The studies most relevant to this work, however,
are on fcc compression samples that include one single
grain boundary. Ng and Ngan [38] first conducted
experiments on bicrystalline Al pillars approximately
6lm in diameter with a large-angle grain boundary
(LAGB) running through the pillar along the compression
direction. They found that the bicrystals show higher
strength, stronger hardening and smaller strain bursts with
a higher frequency. The less jerky deformation and the
hardening behavior are attributed to the high density of
stored dislocations that accumulate due to the boundary
and lead to more frequent dislocation–dislocation interac-
tions. Forest hardening, dipole interaction, junction forma-
tion and dislocation pile-up impede easy propagation of
dislocations which leads to less serrated materials flow. A
similar stress–strain response was found by Fan et al. [39]
using 3-D discrete dislocation dynamics (DDD) to simulate
Al micropillars with a side length of 500 nm. The simula-
tions include a 10°tilt boundary that allows dislocation
penetration and dislocation emission depending on a
threshold stress and energetic considerations. Again, higher
strength and less serrated flow is found for the bicrystals
and explained by the impeded movement and accumula-
tion of dislocations.
Experiments by Kunz et al. [40] on smaller Al bicrystals
400 nm–2 lm in diameter stand in stark contrast to these
results. They report lower hardening and larger strain
bursts which is explained by low dislocation storage in
the pillars leading to a larger stochasticity of deformation.
Transmission electron microscopy (TEM) analyses reveal a
low dislocation density near the boundary indicating that
the boundary might act as a dislocation sink. Another
explanation given is a sudden localized breakdown of the
boundary that triggers large bursts. An interesting molecu-
lar dynamics (MD) study by Tucker et al. [41] shows that
the deformation behavior of 30 nm bicrystalline nanowires
can be either smoother or jerkier compared to single-
crystalline samples. This is said to be determined by the
dislocation–dislocation interactions inside the material. A
smooth behavior was found when dislocations intersect
each other and form defects, lowering the mean free path,
while jerky deformation was predominant when the dislo-
cations easily escaped the crystal leading to dislocation
starvation. Kheradmand et al. [43–46] claimed that there
is a change in the hardening mechanism when decreasing
the pillar size. While larger Ni pillars (3–5 lm) are domi-
nated by hardening due to dislocation–dislocation interac-
tions and the flow curve is controlled mainly by the
strongest component crystal, the influence of dislocation–
boundary interactions increases in smaller pillars (61–
2lm) and leads to strengthening compared to the single
crystals.
From the current understanding an impenetrable
boundary leads to an increase of dislocation density
through dislocation accumulation while a boundary acting
as a sink will lower the dislocation density. Depending on
the sample size and dislocation density, this has different
effects on the mechanical behavior: (i) small samples tend
to harden through dislocation starvation so an increase
in dislocation density leads to easier dislocation
P.J. Imrich et al. / Acta Materialia 73 (2014) 240–250 241
multiplication, thus easier deformation; (ii) large samples
tend to harden through dislocation–dislocation interac-
tions so an increase in dislocation density leads to more
restrained dislocation motion, thus hardening. At this
point it is hard to say how big the influence of the continu-
ity constraints at the boundary and the activation of differ-
ent slip systems on the deformation behavior are. The fact
that grain boundaries can also act as sources should not be
omitted, but it is clear that multiple factors have to be con-
sidered to describe the full impact of grain boundaries on
the deformation behavior.
In the present study the authors try to advance the
understanding of bicrystalline samples by investigating the
deformation behavior of Cu microcompression pillars con-
taining a LAGB or a coherent twin boundary (CTB). For
comparison, single-crystalline pillars of the same size were
tested for all four crystal orientations forming the bicrys-
tals. The sample shape as well as the stress–strain behavior
will be discussed and related to dislocation interactions and
finally compared to 3-D DDD simulations of samples with
an impenetrable and a semipermeable boundary.
2. Experimental details and simulation setup
In the following part the sample preparation route, sam-
ple dimensions and testing procedure as well as the setup
for the DDD simulations will be described. All samples
were prepared with a LEO 1540XB dual-beam FIB work-
station operating with Ga
+
ions at an accelerating voltage
of 30 kV. The currents used range from 5 nA down to
100 pA for coarse to fine milling, respectively. At least
two pillars were tested for each pillar type, i.e. for each sin-
gle-crystal orientation and each boundary type but only
one representative set of pillars is shown for clarity. The
exact micropillar preparation route differed due to the dif-
ferent geometries and availability of the bulk material and
is described in detail below.
2.1. Large-angle grain boundary
For the microcompression tests on the LAGB, thin slices
were cut off from a bulk bicrystalline copper sample with an
arbitrary LAGB. These slices were then electrochemically
etched to produce a wedge with a sharp edge for faster
manipulation by the ion beam to form micropillars, follow-
ing the approach of Moser et al. [49]. On each of the wedges
that were prepared with this method, two pillars from each
grain and one incorporating the LAGB were cut with a
nominal size of 7 721 lm
3
leading to an aspect ratio
of 1:3. All pillars were milled with the ion beam perpendic-
ular to the compression axis to ensure parallel sidewalls, i.e.
the cross-section is constant throughout the whole pillar
length. This avoids strain gradients that are caused by a
tapered sample geometry. To achieve a top surface that is
perpendicular to the compression axis the sample was over-
tilted by 1–2°. The grain boundary intersects the diagonal of
the rectangular pillar and lies parallel to the compression
axis. Micro-Laue measurements show a difference in grain
orientation of 22.8°around ½1
34and compression direc-
tions close to ½40
1and ½310for grain 1 and grain 2,
respectively. A rough estimation of the grain boundary
plane using scanning electron microscopy (SEM) images
and micro-Laue data leads to 3
112
in respect to grain 1.
The pillars were loaded in 2–4 steps with a strain rate of
10
3
s
1
to a total plastic strain of 8–18% using the dis-
placement-controlled straining device described in Ref.
[50]. Force displacement data was recorded during com-
pression and high-resolution SEM images were taken
before and after deformation.
2.2. Coherent twin boundary
The sample that was used for testing the mechanical
properties of a twin boundary was grown from oxygen-free
high-conductivity copper of 99.999% purity by the Bridg-
man method and contained a CTB along the growth direc-
tion. A small rod with square cross-section was cut off,
electropolished, glued to a sample holder and then pro-
cessed with the FIB to form micropillars with a nominal
size of 4.3 4.3 13 lm
3
resulting in an aspect ratio of
1:3. All pillars were shaped near the sample edge to be able
to observe the deformation process in situ in the scanning
electron microscope. The side surfaces were cut perpendic-
ular to the compression axis and the front and back sur-
faces parallel to the compression axis. To ensure a
constant cross-section and avoid tapering of the sample
throughout the entire gage length, the front and back sur-
faces were cut with an overtilt of 1–2°. This method led to a
pillar with a constant, but slightly trapezoidal cross-section
(2°of FIB-taper on the sidewalls from front to back) for
this type of sample. One compression sample in each grain
and one sample incorporating the CTB was produced. The
bicrystalline sample contains a CTB that runs through the
middle of the pillar parallel to its sidewalls straight from
top to bottom and extends through the whole bulk sample.
Both crystals are oriented such that the compression axis
is close to a h0
11idirection. After mechanical testing, the
same strip of material was ground and electropolished
and another set of pillars was made.
Mechanical testing was carried out in situ in a LEO 982
scanning electron microscope using an ASMEC UNAT
microindenter. The displacement-controlled compression
experiments were performed in two consecutive steps to a
plastic strain of 15–19% with a displacement rate of 10 nm s
1
leading to a strain rate of 810
4
s
1
. During the experi-
ment force–displacement data as well as secondary electron
images of the deforming pillar were collected (for a time-lapse
of the compression experiment see Supplementary Movie S1).
2.3. Setup of the 3-D discrete dislocation dynamics
simulation
To assist the interpretation of the bicrystalline micro-
compression experiments 3-D DDD simulations were
242 P.J. Imrich et al. / Acta Materialia 73 (2014) 240–250
performed. Only a short overview on the simulation
method will be given here; for further details the reader is
referred to Refs. [51–54].
The boundary value problem is solved according to the
superposition principle of Ref. [53], where a finite-element
(FE) framework is utilized to apply the boundary condi-
tions, image forces, etc. From the superimposed stress
fields of the FE framework and the dislocation structure
the Peach–Koehler force acting on each dislocation seg-
ment is calculated and the equation of motion is solved
using standard time integration schemes. Subsequently,
the dislocation structure is updated and a new time step
is calculated.
The specimen size was chosen with wtl
=0.5 0.5 1.5 lm
3
and 1.0 1.0 3.0 lm
3
to keep
computational time low. The material properties of Al were
used as standard material, namely shear modulus
l= 27 GPa, Poisson’s ratio m= 0.347 and lattice constant
a= 404 pm. The authors know that there is a difference
in cross-slip probability between Al and Cu. However,
the full anisotropic behavior of Cu cannot yet be simulated
by DDD and only the principle response is of interest here
which can be well captured with the material parameters
used. For the simulations displacement-controlled bound-
ary conditions are utilized, where the top surface is moved
downwards in the [0 0 1] direction (compression) and the
bottom surface is fixed in that direction. The strain rate
was chosen in the region of 10
3
s
1
, so that strain rate
effects due to inertia are avoided. As an initial dislocation
structure, randomly distributed Frank–Read sources with
a size of L= 500awere used. This value was chosen since
it correlates with the mean dislocation distance of 224 nm
for the chosen dislocation density of 2 10
13
m
2
and well
describes the experimental observations. A single crystal
with [00 1] orientation aligned with the compression axis
was simulated. At the center plane at w= 0.5 lm (similar
to the experiments) three different types of boundaries were
introduced: (i) no boundary, all dislocations can freely
move within the crystal; (ii) an impenetrable boundary
where all dislocations are blocked; and (iii) a semiperme-
able boundary where only dislocations on a specific glide
system (glide plane normal n
3
=(1 –1 1), Burgers vector
b
2
= [01 1]) can pass. The choice of orientation was deliber-
ately chosen to deviate from the experimental orientation.
To approximate the CTB boundary to the best possibility
(the introduction of a R3 boundary is not supported by
the DDD code) a R1 boundary was introduced where only
dislocations of one glide system can be transmitted. To
ensure that cross-slip is the dominant deformation mecha-
nism at the boundary, no dislocation source was placed on
the permeable system. To keep the Schmid factor high and
constant before and after the cross-slip event, as is the case
in the CTB bicrystal, an orientation with symmetric slip
systems had to be chosen. Since a [1 1 0] compression direc-
tion would lead to cross-slip into a slip system parallel to
the compression direction with a Schmid factor of zero,
[10 0] was used as compression direction. This orientation
was also used for the single crystal and R1 crystal with
impenetrable boundary to allow best comparability
between the simulation results. The DDD results will be
presented in Section 4.3 to interpret and discuss the exper-
imental results.
3. Microcompression results
In this section the SEM micrographs of the micropillars
and the corresponding stress–strain curves are presented;
only one representative set of LAGB and CTB samples is
depicted for clarity.
3.1. LAGB
The single- and bicrystalline pillars after compression
are shown in Fig. 1. All single crystals have very distinct
slip steps as is characteristic for this material and pillar size
[55]. On every single-crystalline pillar two f111gh
110i
glide systems can be identified that account for the defor-
mation. The bicrystal, on the other hand, has many small
but few large slip steps. These steps extend solely through
one grain and stop at the grain boundary. The only larger
surface steps that are visible near the sample edges fade
away when getting closer to the boundary (Fig. 1b). On
both side surfaces of the pillar a small step of varying
height was formed where the boundary intersects the free
surface.
The stress–strain curve depicted in Fig. 2 confirms the
expected behavior. While the single crystals show the typi-
cal serrated flow with very little hardening, the bicrystal
exhibits a higher yield strength, much stronger hardening
and less pronounced load drops. The yield strengths of
the pillars are 52, 44 and 62 MPa for grain 1, grain 2 and
the bicrystal, respectively. This leads to a 19% and 41%
strength increase for the bicrystal compared to grain 1
and grain 2, respectively. All yield strengths are measured
at 1% strain or at the peak yield stress for smooth or abrupt
elastic–plastic transitions, respectively. It should be noted
that the slope of the unloading curves deviates from the
expected behavior. This is due to a faster unloading speed
of the strongly damped indenter but does not affect the
loading part of the experiment and the stress measurement.
3.2. CTB
The bicrystalline pillar containing a CTB is shown in
Fig. 3 together with two single-crystalline compression
samples after deformation. As for other multiple slip ori-
ented single-crystalline pillars, these deform in a character-
istic manner. The distinct glide steps that were formed at
the surface can be clearly distinguished. The bicrystalline
pillar shows similar behavior when separately looking at
each component crystal, i.e. the two f
1
11gh
110iglide sys-
tems activated for each twin orientation match those of the
corresponding single crystal. Contrary to the LAGB
bicrystal, each of the two slip systems of grain 1 is
P.J. Imrich et al. / Acta Materialia 73 (2014) 240–250 243
connected at the twin boundary with the corresponding slip
systems in grain 2, as can be clearly seen in Fig. 3b. The
planes of the individual systems always meet perfectly at
the CTB and all slip steps extend through both grains,
which leaves the boundary unbent and produces large slip
steps which can easily be identified in the side view in
Fig. 4. The deformation of the pillar occurs through collec-
tive slip of both grains into the same direction along a set
of two
1
11 planes that enclose an angle of 141°.
While one could think that the introduction of the CTB
would lead to a strengthening of the compression sample
like for the LAGB, Fig. 5 proves the opposite. The
stress–strain curves show similar elastic slopes, yield points
and hardening behavior for both kinds of pillars. Both sin-
gle crystals show a smooth elastic–plastic transition at yield
stresses of 120 and 135 MPa, while the bicrystal exhibits a
peak yield stress of 131 MPa and shows slight softening
before the stress increases and reaches a plateau at
145 MPa.
4. Discussion
The opposing results of the LAGB and the CTB is the
central aspect of the following discussion which is comple-
mented by 3-D DDD simulations. The results for the sin-
gle-crystalline Cu samples that were tested in this paper
are comparable regarding deformation morphology,
stress–strain behavior, reproducibility and results docu-
mented in literature [4–13,55–59], and thus are not further
discussed.
4.1. LAGB
Two very distinctive differences are visible by analyzing
the LAGB pillar by SEM compared to the adjacent single-
crystalline pillars. The glide steps at the surface are much
weaker and more glide steps are observed, indicating less
localized plasticity. Additionally, the different flow behav-
ior of the individual grains leads to a large distortion of
the pillar, with a deflection of the grain boundary in the
center of the sample of 17°compared to the pristine ori-
entation. This distortion is most likely caused by the differ-
ent mechanical properties of the two crystal orientations in
the LAGB pillar combined with the constraints imposed by
the testing setup. The Young’s moduli in compression
direction are fairly similar: 75 and 79 GPa for grain 1
and grain 2, respectively, as calculated from the elastic con-
stants [60]. Elastic buckling can be excluded due to the
aspect ratio of length to thickness of 3; however, elasto-
plastic buckling as discussed in Ref. [61] could cause the
observed bending.
Fig. 1. SEM micrographs of micropillars after compression. (a) Compression sample of grain 1. (b) Bicrystalline pillar containing a LAGB that is
composed of grain 1 and grain 2. (c) Compression sample of grain 2.
Fig. 2. Engineering stress–strain curve of the single-crystalline Cu pillars
of grain 1 (gray), grain 2 (black) and the bicrystalline pillar containing a
LAGB (red). While the single crystals show similar behavior, the bicrystal
exhibits increased yield strength and shows stronger hardening. (For
interpretation of the references to color in this figure legend, the reader is
referred to the web version of this article.)
244 P.J. Imrich et al. / Acta Materialia 73 (2014) 240–250
The stress–strain curve (Fig. 2) of the LAGB pillar is
dominated by a higher yield strength, stronger hardening
and smaller load drops compared to the corresponding sin-
gle-crystalline pillars. The most apparent reason for the
strength increase of 19–41% at 1% strain is the reduction
in source size in the component grains through truncation
hardening. It should be noted that the evaluation of yield
strength in micromechanical samples is influenced by the
strain at which it is analyzed. Since an evaluation at 0.2%
strain as for macroscopic samples is not sensible due to
the lack of accuracy in micromechanical tests, we per-
formed our measurements at 1% strain. As discussed in
the Introduction, the use of a constant size effect exponent
for the strengthening power law is controversial; however,
to obtain an estimate for the increase in yield strength, the
phenomenological scaling law by Dou and Derby [20] and
the source activation stress formula by Rao et al. [18] were
used. Rao et al. calculated the critical resolved shear stress
of double-ended and single-ended dislocation sources of
30°mixed type with varying lengths from 233 to 933 Bur-
gers vectors and fitted the simulation data with the follow-
ing equation:
sðLÞ¼kG lnðL=bÞ
ðL=bÞð2Þ
Here Lis the source length, Gthe shear modulus, bthe
Burgers vector and ka constant with values of 0.06–0.18
for single-ended to double-ended sources. Since the
Fig. 3. SEM micrographs of micropillars after compression; the CTB is nearly parallel to the viewing direction going from top to bottom of the pillar. (a)
Compression sample of grain 1. (b) Bicrystalline pillar containing a CTB that is comprised of grain 1 and grain 2. It can be clearly seen that the slip steps of
both grains meet perfectly along a straight line, the CTB. (c) Compression sample of grain 2.
Fig. 4. Side view of the deformed bicrystalline pillar from Fig. 3b. The
CTB runs from left to right through the top surface of the pillar and is
nearly parallel to the viewing plane. Both grains are displaced simulta-
neously into the same directions parallel to the image plane.
Fig. 5. Engineering stress–strain curve of the single-crystalline compres-
sion samples of grain 1 (gray), grain 2 (black) and the bicrystalline sample
containing a CTB (green). (For interpretation of the references to color in
this figure legend, the reader is referred to the web version of this article.)
P.J. Imrich et al. / Acta Materialia 73 (2014) 240–250 245
bicrystal is slightly smaller in size compared to the single-
crystalline samples, it must first be calculated how much
this size difference contributes to the strengthening of the
pillar. This only amounts to 9% or 13% using the concepts
of Dou and Derby or Rao et al., respectively. Assuming
that the size of the dislocation sources is limited by the free
surfaces and the grain boundary, the smallest distance to
the surface or boundary defines the strength of the disloca-
tion source for continuous operation. The largest possible
source size is the radius of the largest circle that can be
inscribed into the triangular base area of each grain. Using
this length, a strength increase of 55% or 81% can be calcu-
lated for Dou and Derby or Rao, respectively. Both calcu-
lations overestimate the experimental values, so one could
question whether source truncation is the sole mechanism
that defines strengthening of the bicrystal similar to obser-
vations by Ng and Ngan [38]. However, the largest suitable
single-ended sources have an activation shear stress of only
4–7 MPa, calculated by the Rao model. Thus it is easily
reasoned that the expected sources in the micropillar are
much smaller (155–535 nm when using the shear stresses
from the experiment). This could be the cause for the
less-pronounced source truncation effect for these large
samples.
The numerous small slip steps at the sample surface, the
stronger hardening and the less serrated flow can be
explained by exhaustion hardening as defined by Rao
et al. [21] that is enhanced due to the grain boundary. Ng
and Ngan [38] and Fan et al. [39] state that a LAGB can
act as a barrier to dislocations, leading to a dislocation
pile-up and increase in dislocation density. These accumu-
lated dislocations act as forest dislocations and lower the
mean free path for dislocation-slip. Therefore sources will
exhaust faster due to dislocation–dislocation and disloca-
tion–boundary interactions and the developing backstress-
es. Further deformation needs higher stresses to reactivate
these sources or activate other less favorable ones [21],
explaining the strong hardening and the numerous slip
steps. The continuous increase in dislocation density results
in steady hardening. Since the measurement of strain-hard-
ening rate is prone to instrumental boundary conditions,
e.g. the lateral compliance of the indenter system [62],no
quantitative values will be given here. However, the curves
clearly show that the strain hardening in the bicrystal is
considerably higher compared to the single crystals. Note
that the same indenter and the same macrosample were
used, ensuring the same unknown lateral compliance.
Regarding the serration amplitude we will consider a
single dislocation with a Schmid factor of 0.5 that traverses
a pillar from one side to the other. The generated displace-
ment of 1.8 A
˚in compression direction leads to a load drop
of 0.65 MPa in a 21 lm long sample when using a Young’s
modulus of 75 GPa. When a new dislocation source gets
activated and produces a dislocation burst in a single crys-
tal, where dislocations can easily escape to the surface, a
pronounced load drop will be the consequence. In the
bicrystal the lower mean free path leads to smaller but
more frequent dislocation steps, resulting in a smaller ser-
ration amplitude.
These observations agree well with the results from Ng
and Ngan [38]. Interestingly, their TEM investigations
show no evidence of pile-up at the grain boundary but a
high density of stored dislocations in the whole pillar. This
can be caused by a spreading of dislocations onto different
parallel slip planes by double cross-slip when approaching
a developing pile-up, a reorganization of the dislocation
structure after the removal of the compressive load as
expressed by the authors or the dissolution of the pile-up
to the surface during TEM lamella preparation [63]. Con-
trary to their experiments, no barreling and no pronounced
sliding of the boundary is observed in our study. There is a
step <100 nm in size where the grain boundary intersects
the surface; however, it is much smaller than the 1lm
wide step seen by Ng and Ngan [38]. Further agreement
can be found with the simulations of Fan et al. [39] even
though their simulated samples allow dislocation transmis-
sion through the boundary at a threshold stress.
Other experimental results, however, are highly contrary
to the findings presented here. It seems as though the prop-
erties of the grain boundary are significantly different to the
LAGB that we tested. Bicrystalline Al compression pillars
in the size range of 400 nm–2 lm tested by Kunz et al. [40]
exhibit higher strain bursts and lower hardening rates,
completely opposing the findings presented above. As men-
tioned in the Introduction, this behavior is said to be due to
the absorption of dislocations at the boundary that is
claimed to act as a sink for dislocations, based on post-
mortem TEM studies revealing no dislocation pile-ups at
the boundary. If the boundary truly acts as a dislocation
sink, it could enhance the dislocation starvation process,
explaining the large strain bursts. The lower hardening
rate, however, would need a stress-independent dislocation
generation mechanism for this theory to be coherent, e.g.
the grain boundary acting as a source of dislocations as
well. Kheradmand et al. [43–46] claim that there is a change
in hardening mechanism when decreasing the pillar size. If
this boundary was also acting as a dislocation sink, the
increased influence of the boundary in smaller samples
would make sense due to enhanced dislocation starvation.
For larger samples that harden due to dislocation–disloca-
tion interactions the hardening and softening mechanisms
would have to cancel each other out, leading to a similar
behavior to single crystals. The discrepancy in strain bursts
and hardening behavior found in MD simulations by
Tucker et al. [41] matches this argumentation, showing
more continuous flow for low mean free paths with
enhanced dislocation–dislocation interactions and larger
serrations for unimpeded dislocation movement leading
to starvation.
4.2. CTB
Contrary to the sample containing a LAGB the CTB
pillar behaves similarly to the single-crystalline samples,
246 P.J. Imrich et al. / Acta Materialia 73 (2014) 240–250
i.e. having similar yield strength and hardening behavior. It
deforms in a very interesting way, showing glide on conju-
gated slip systems. Since the direction of deformation at the
slip steps is identical for both grains, it is easy to argue that
the Burgers vector in both slip systems must be the same.
To understand the geometry of the CTB pillar it is best
to imagine a Thompson tetrahedron that is mirrored across
the (1 1 1) plane (Fig. 6). This common mirror plane repre-
sents the twin boundary, leaving three other
1
1 1 planes on
each tetrahedron. As for every fcc metal, there are 12 pos-
sible slip systems in each grain (four slip planes with three
Burgers vectors each). Only three Burgers vectors are
shared by both grains, i.e. lie on the common (1 1 1) mirror
plane. Since the CTB plane in this experiment is parallel to
the compression direction, there is no shear stress acting on
it. However, dislocations with one of the three shared Bur-
gers vectors can glide on one slip plane in each grain. These
dislocations will always have the highest Schmid factor
when the twin boundary is parallel to the compression
direction as shown by Li et al. [48]. The two Burgers vec-
tors with Schmid factor m–0 in this experiment that
can lead to collective sliding of both sides are depicted by
arrows in Fig. 6. Activation of such combined systems
leads to the observed geometry of the deformed pillars.
Different explanations can be given to explain the lack
of source truncation strengthening due to the separate
grains. Analogous to the LAGB, the size of sources can
be calculated from the resolved shear stresses by the Rao
method [18]. The calculated size of the single-arm sources
ranges from 79 to 180 nm for the CTB bicrystal, thus being
significantly smaller than half the pillar size. As for the
LAGB, this could lead to a diminished influence of the
source truncation process. Irrespective of that, there are
some scenarios that can explain the unchanged
strength (note that the source morphologies in Fig. 8
should be seen as simplified illustrations to support the
basic understanding):
(1) The CTB does not pose a barrier at all for the dislo-
cations and they can easily transfer from one grain to
the other while experiencing a bend at the boundary
(Fig. 8a).
(2) A single-arm dislocation source that has a pinning
point in the grain boundary operates (Fig. 8b).
(3) A dislocation source that has one pinning point in
each grain and spans over the twin boundary oper-
ates (Fig. 8c).
(4) A source (single arm or Frank–Read type) in one
grain is activated and produces dislocations that
approach the boundary and cross-slip into the second
grain after piling up at the boundary (Fig. 8d).
The points above are discussed in detail below. In all
cases it is assumed that the dislocations have a Burgers vec-
tor parallel to the twin plane, as depicted in Fig. 6. In case 1
the sources can operate like in a single crystal since the dis-
locations are not blocked by the twin boundary and imme-
diately transfer to the other grain, when meeting the
boundary. Case 2 illustrates a special case of example 1
where the pinning point is exactly at the grain boundary,
e.g. formed during the growth process. The source operates
and two cross-slip events are needed for each full rotation.
Another possibility (case 3) is a dislocation created during
the growth process that spans from one grain to the other
and is then pinned on both sides (Fig. 8c). With easy cross-
slip, the sources would split up into two single-arm sources
and operate similar to case 1, while reconnecting when the
dislocation segments meet again in between the pinning
points. With hard cross-slip an additional revolution of
one side would lead to a screw dislocation laid down at
the boundary that would annihilate upon operation of
the second source.
If cross-slip is not easily achieved, but sufficient stress is
needed to transfer dislocations through the boundary, case
1 would transform to case 4 (Fig. 8d). While in situ TEM
experiments on twinned Cu films by Dehm et al. [64] show
a single dislocation crossing a twin boundary at shear stres-
ses in the order of 25 MPa calculated by fitting the disloca-
tion radius, other publications show the need for much
higher stresses. MD simulations by Jin et al. [65] demon-
strate that global shear stresses of 465–510 MPa are needed
for a screw dislocation to cross the twin boundary. Chassa-
gne et al. [66] calculate a critical reaction stress of close to
400 MPa for transmitting a screw dislocation into a nano-
twin, and in situ TEM studies by Lee et al. [67] on austen-
itic steel show a considerable dislocation pile-up of suitably
oriented screw dislocations at a CTB. In situ experiments
by Chassagne et al. [66] also show that a pile-up of eight
dislocations is needed for dislocations to be transmitted
through an annealing twin. This indicates that unimpeded
cross-slip might not always be the case in Cu and the pos-
sibility of a higher cross-slip resistance should be discussed.
When the CTB acts as a barrier with a certain threshold
stress, dislocations would approach the boundary and pile
up along it (Fig. 8d). A pile-up can produce very high
Fig. 6. Thompson tetrahedron ABCD that is mirrored across the (1 1 1)
plane (ABC) creating the geometry of a twin boundary. The h
110iBurgers
vectors BA,CB and CA are shared by both grains. Green arrows depict
the acting Burgers vectors. The compression direction (close to ½0
11in
the experiments) is from top to bottom of the image along the dotted line.
In this setup the screw dislocation CB can cross-slip from the glide plane
BCD in the red grain into BCD’ in the blue grain. (For interpretation of
the references to color in this figure legend, the reader is referred to the
web version of this article.)
P.J. Imrich et al. / Acta Materialia 73 (2014) 240–250 247
stresses locally since the shear stress at the first dislocation
s
1
can be described by s1¼sn, with sbeing the global
shear stress and nthe number of dislocations in the pile-
up [68]. The number of dislocations as a function of shear
stress s, shear modulus G, Burgers vector band pile-up
length Lfor screw dislocations can be calculated as follows:
n¼Lsp
Gb :ð3Þ
For the resolved shear stress of 53 MPa that was acting on
the glide planes during deformation of the twin pillar, a
pile-up of 10 dislocations can be calculated assuming a
pile-up length of as little as 0.7 lm (approximately a third
of the grain size). This pile-up leads to a stress at the dislo-
cation front of 530 MPa, which is comparable with the
values calculated by MD simulations. Therefore, even a
moderate threshold stress for dislocation transmission
can lead to similar results. Of course there is the possibility
of more complicated processes; however, no deformation
along the twin boundary or other features that would indi-
cate the formation of dislocations with different Burgers
vectors could be observed. To really resolve the barrier
strength of the CTB and confirm the transfer process, fur-
ther research using micro-Laue diffraction and TEM are
necessary.
4.3. DDD simulations
The DDD simulations were carried out to validate the
completely different mechanical behavior of the bicrystal-
line samples. The simulations of both the bicrystals with
impenetrable boundary as well as with semipermeable
boundary resemble the behavior of the LAGB and CTB
samples, respectively (Fig. 7). All simulated pillars with
an impenetrable boundary are characterized by tremen-
dous hardening coupled with a strong rise in dislocation
density due to dislocation pile-up/accumulation near the
boundary. The smaller pillar with a semipermeable bound-
ary (0.5 0.5 1.5 lm
3
) reaches a plateau at the same
stress level as the single crystal, while the larger pillar with
semipermeable boundary (1 13lm
3
) exhibits a yield
strength close to the one with an impenetrable boundary
and quickly levels off to a flow stress close to the single
crystal. Since the pillars with a semipermeable boundary
contain no dislocation source on the slip system that is able
to cross the boundary, all passing dislocations have to
undergo cross-slip first. It is interesting to see that these pil-
lars behave similarly to the single crystals, even though
they are less likely to generate cross-slipping dislocations
compared to the experimental orientation. With a h100i
direction parallel to the compression axis, eight equivalent
slip systems with high Schmid factors can be activated in
the DDD simulation, of which one is allowed to cross
the boundary. The CTB pillar of the experiment has four
equivalent slip systems for its h
110iorientation, while
two of them can cross the boundary by cross-slip. This
leads to the conclusion that a single easy deformation path-
way determines the deformation behavior as long as the
size and dislocation density of the sample provide enough
sources on the required slip system.
5. Summary and conclusion
Compression experiments on single-crystalline and
bicrystalline micrometer-sized Cu pillars were conducted.
The pillars were deformed in displacement-controlled
mode to strains between 8% and 19% and show strongly
varying behavior. Bicrystals containing the LAGB exhibit
a bent pillar shape with a strong deflection of the grain
boundary. Numerous small slip steps at the sample surface
indicate the activation of many sources and no slip traces
could be found that cross the interface. The stress–strain
data shows increased yield stress and stronger hardening
compared to the single-crystalline pillars. This is attributed
mostly to source truncation hardening in the smaller indi-
vidual grains and the need for activation of less favorable
sources due to exhaustion hardening. The examined
boundary proves to be an effective barrier to dislocations,
Fig. 7. DDD simulations of (a) 0.5 0.5 1.5 lm
3
and (b) 1 13lm
3
compression pillars. Bold lines: engineering stress; thin lines: dislocation
density; black: single crystals; red: bicrystals with impenetrable boundary;
green: bicrystals with a semipermeable boundary (only dislocations of one
slip system can transfer to the other grain). (For interpretation of the
references to color in this figure legend, the reader is referred to the web
version of this article.)
248 P.J. Imrich et al. / Acta Materialia 73 (2014) 240–250
making deformation more difficult and leading to a notice-
able hardening.
The CTB pillar, on the other hand, shows no strengthen-
ing compared to the single crystals. This is explained by the
special CTB orientation where screw dislocations can
cross-slip from one grain into the other without impedi-
ment of the CTB at the applied stress. This proves that
the strengthening effect of twin boundaries strongly
depends on their orientation. While twin boundaries may
strengthen bulk materials and even micropillars if the twin
density is high enough, there are scenarios where
twin boundaries do not significantly change the material’s
behavior.
Additional 3D DDD simulations show considerable
hardening for an impenetrable boundary, while a semiper-
meable boundary exhibits a behavior that is very close to
that of the single crystals. In conclusion, the influence of
boundaries can differ significantly, depending on the
boundary type, orientation, sample size and dislocation
density.
Acknowledgements
The authors thank L.L. Li, X.H. An and Z.F. Zhang of
the Shenyang National Laboratory for Materials Science in
China for providing the bulk Cu bicrystals. J. Kreith of the
Montanuniversita
¨t Leoben in Austria is acknowledged for
the help with crystal orientations and coordinate
transformations.
Appendix A. Supplementary data
Supplementary data associated with this article can be
found, in the online version, at http://dx.doi.org/10.1016/
j.actamat.2014.04.022.
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