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On the measurement of dislocations and dislocation substructures using EBSD and HRSD techniques

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The accumulation of the dislocations and development of dislocation structures in plastically deformed Ni201 is examined using dedicated analyses of Electron Back-Scatter Diffraction (EBSD) acquired orientation maps, and High-Resolution Synchrotron Diffraction (HRSD) acquired patterns. The results show that the minimum detectable microstructure-averaged (bulk) total dislocation density (r T) measured via HRSD is approximately 1E13 m À2 , while the minimum GND density (r G) measured via EBSD is approximately 2E12 m À2 e the EBSD technique being more sensitive at low plastic strain. This highlights complementarity of the two techniques when attempting to quantify amount of plastic deformation (damage) in a material via a measurement of present dislocations and their structures. Furthermore, a relationship between EBSD-measured r G and the size of HRSD-measured Coherently Scattering Domains (CSDs) has been mathematically derived e this allows for an estimation of the size of CSDs from EBSD-acquired orientation maps, and conversely an estimation of r G from HRSD-measured size of CSDs. The measured evolution of r T , and r G is compared with plasticity theory models e the current results suggest that Ashby's single-slip model underestimates the amount of GNDs (r G), while Taylor's model is correctly predicting the total amount of dislocation (r T) present in the material as a function of imparted plastic strain.
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On the measurement of dislocations and dislocation substructures
using EBSD and HRSD techniques
O. Mur
ansky
a
,
b
,
*
, L. Balogh
c
,M.Tran
d
,
a
, C.J. Hamelin
e
,
a
, J.-S. Park
f
, M.R. Daymond
c
a
Australian Nuclear Science and Technology Organisation, Lucas Heights, NSW, Australia
b
School of Mechanical and Manufacturing Engineering, UNSW Sydney, Sydney, Australia
c
Queen's University, Mechanical and Materials Engineering, Kingston, ON, Canada
d
University of California, Mechanical and Aerospace Engineering, Davis, CA, USA
e
EDF Energy, Barnwood, Gloucestershire, UK
f
Advanced Photon Source, Argonne National Laboratory, Lemont, IL, USA
article info
Article history:
Received 26 March 2019
Received in revised form
14 May 2 019
Accepted 17 May 2019
Available online 7 June 2019
Keywords:
Dislocation density
Metal plasticity
Electron back-scatter diffraction (EBSD)
High-resolution synchrotron diffraction
(HRSD)
Peak broadening
abstract
The accumulation of the dislocations and development of dislocation structures in plastically deformed
Ni201 is examined using dedicated analyses of Electron Back-Scatter Diffraction (EBSD) acquired
orientation maps, and High-Resolution Synchrotron Diffraction (HRSD) acquired patterns. The results
show that the minimum detectable microstructure-averaged (bulk) total dislocation density (
r
T)
measured via HRSD is approximately 1E13 m
2
, while the minimum GND density (
r
G) measured via
EBSD is approximately 2E12 m
2
ethe EBSD technique being more sensitive at low plastic strain. This
highlights complementarity of the two techniques when attempting to quantify amount of plastic
deformation (damage) in a material via a measurement of present dislocations and their structures.
Furthermore, a relationship between EBSD-measured
r
Gand the size of HRSD-measured Coherently
Scattering Domains (CSDs) has been mathematically derived ethis allows for an estimation of the size of
CSDs from EBSD-acquired orientation maps, and conversely an estimation of
r
Gfrom HRSD-measured
size of CSDs. The measured evolution of
r
T, and
r
Gis compared with plasticity theory models ethe
current results suggest that Ashby's single-slip model underestimates the amount of GNDs (
r
G), while
Taylor's model is correctly predicting the total amount of dislocation (
r
T) present in the material as a
function of imparted plastic strain.
©2019 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Transmission electron microscopy (TEM) is commonly used to
investigate dislocations and their structures on a nanometre scale.
However, discrete TEM measurements cannot be readily used to
elucidate macroscopic material properties [1]ehence it is difcult
to use TEM measurement for the purposes of whole-of-life tness-
for-service (FFS) monitoring required for most industrial plant.
Promising research in the eld of dislocation measurement on a
mesoscopic
1
scale is underway, led by two approaches that build on
the foundation of well-established techniques: (i) high-resolution
neutron/synchrotron diffraction (HRND/HRSD); and (ii) electron
backscatter diffraction (EBSD). Diffraction line prole analyses
(DLPA) of HRND/HRSD-measured diffraction patterns provide
insight into the dislocations and their structures from the shape
and width of diffraction peaks (lines). HRND/HRSD-measured
patterns are by default averaged over many grains and thus
directly convey microstructure-averaged (bulk) information. On
the other hand, analyses of EBSD orientation maps provide infor-
mation on dislocations at a mesoscopic scale by averaging many
discrete measurements. Information on dislocations (plastic dam-
age) on a submicron scale is obtained from the variation in the
EBSD-measured crystal orientation across the microstructure and
these are then averaged to obtain microstructure-averaged (bulk)
information.
In the present work, we employ EBSD and HRSD techniques to
investigate the accumulation of dislocations and the consequent
*Corresponding author. Australian Nuclear Science and Technology Organisation,
Lucas Heights, NSW, Australia.
E-mail address: ondrej.muransky@ansto.gov.au (O. Mur
ansky).
1
Mesoscopic (continuum length-scale) stretches across the microstructure, thus
representing the behaviour across variously oriented grains within a polycrystalline
aggregate.
Contents lists available at ScienceDirect
Acta Materialia
journal homepage: www.elsevier.com/locate/actamat
https://doi.org/10.1016/j.actamat.2019.05.036
1359-6454/©2019 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Acta Materialia 175 (2019) 297e313
formation of dislocation structures within a plastically deformed
polycrystalline material. As part of this work, a number of solution-
annealed Ni201 specimens were deformed to varying levels of
macroscopic plastic strain through uniaxial tensile loading. The
development of a stored dislocation density as a function of
imparted plastic strain is then determined by the analysis of (i)
EBSD-measured orientation maps and (ii) HRSD-measured
diffraction patterns. The results obtained from each technique are
compared against each other and also against the values predicted
by classical plasticity models that link the presence of dislocations
with macroscopic material behaviour.
2. Classifying dislocations
In a general sense, the plastic deformation of a polycrystalline
aggregate proceeds by the heterogeneous formation, movement,
annihilation and/or storage of dislocations [2]. Heterogeneous
dislocation storage can be observed across different length scales,
and the underlying phenomenology for each response depends on
the length scale in question. When talking about pure metals and
alloys with a high stacking fault energy (SFE), at the microscopic
length scale, a cellular dislocation structure is typically formed
[1,3,4] whereby relatively dislocation-free regions are surrounded
by dislocation walls [1]. These dislocation walls are formed as
accumulated/stored dislocations arrange themselves into energet-
ically favourable congurations [4]. At the mesoscopic length scale,
a polycrystalline aggregate comprising variously oriented grains
will contain intergranular and intragranular microstrain gradients,
caused by the plastic anisotropy inherent to crystallites (grains)
[2,5]. Because dislocation accumulation is approximately propor-
tional to the level of imparted plastic strain [2,5] within the strain
ranges of interest, this anisotropy will naturally lead to intergran-
ular variations in the density and distribution of dislocations across
the microstructure. The mesoscopic dislocation behaviour is
further complicated when the concept of strain compatibility is
addressed. Fig. 1 schematically illustrates how the intergranular
variations in the plastic response of differently oriented grains are
accommodated in a polycrystalline aggregate according to the
single-slip model put forth by Ashby [2]. Intragranular deformation
must occur to ensure continuity across the microstructure, thus
preventing the formation of voids and unrealistic overlaps between
grains.
The maintenance of strain compatibility across the microstruc-
ture occurs by storing a portion of the dislocations formed [6].
These dislocations have a net non-zero Burgers vector, which gives
rise to a curvature of the crystal lattice. Because the storage of these
dislocations is deemed necessary for maintaining the strain
compatibility across variously orientated grains, they are referred
to as geometrically-necessary dislocations (GNDs) [7,8]. At the
microscopic level, internal energy minimisation causes the GNDs to
arrange themselves into energetically favourable congurations
(dislocation walls), thereby dividing a grain into subgrains [9]as
schematically shown in Fig. 1. Since these newly-formed dislocation
walls are comprised of GNDs, they are referred to as geometrically-
necessary boundaries (GNBs)
2
[9,10]. At the mesoscopic level, var-
iations in the GND density and the size of the subgrains will be
dependent on local (intergranular and intragranular) variations in
imparted plastic strain.
In addition to the storage of GNDs, a deformed polycrystalline
aggregate also stores dislocations via statistical (random), mutual
trapping. These dislocations are referred to as statistically-stored
dislocations (SSDs) [2]. Unlike GNDs, SSDs have a net-zero Burgers
vector at the microscopic length scale and thus have no geometrical
consequence (i.e. they do not promote lattice curvature at this
length scale). Like GNDs however, SSDs can also form dislocation
substructure [11] as a result of energy minimisation within the
system [9]. The relatively high-density cell walls formed by SSDs
are referred to as incidental dislocation boundaries (IDBs) as shown
Fig. 1. To maintain compatible deformation across variously oriented grains in a polycrystalline aggregate, the voids and overlaps between the individual grains, which would
otherwise appear due to the orientation-dependent anisotropy, are corrected by storing a portion of dislocations in the form of geometrically-necessary dislocations (GNDs).
Plastically deformed material also stores statistically-stored dislocations (SSDs), which are stored by mutual random trapping. Both GNDs and SSDs arrange themselves into
energetically favourable congurations, forming geometrically-necessary boundaries (GNBs) and incidental dislocation boundaries (IDBs), respectively.
2
In general, grain boundaries and twin boundaries are also considered
geometrically necessary boundaries [9].
O. Mur
ansky et al. / Acta Materialia 175 (2019) 297e313298
in Fig. 1. The equiaxed, cell-type structures formed by IDBs are
considerably smaller than those formed by GNBs [10].
As plastic deformation proceeds, the polycrystalline aggregate
stores an increasing number of both GNDs and SSDs [2,5]. Hence, in
pure metals overall material work-hardening behaviour is
controlled by the presence of all stored dislocations(GNDs, SSDs) via
their short- and long-range interactions with moving dislocations
[2]. However, while the amount of GNDs and SSDs increase with the
level of imparted plastic strain, the underlying mechanism of their
accumulation differs. In general, the density of stored GNDs is
controlled by microstructural characteristics (grain size, particle size
and their distribution, texture, etc.), while the density of stored SSDs
is a material characteristic independent of the characteristics of the
microstructure [2]. Note that in alloys an interplay of various
hardening mechanics (e.g. dislocation hardening, solid-solution
hardening, precipitation hardening, dispersion hardening, etc.)
take place and the overall hardening behaviour is more complicated.
3. Analysis of EBSD-measured orientation maps
EBSD is used to measure the crystal orientation eoften repre-
sented by Euler angle notation (
f
1
,
F
,
f
2
)[12]eof crystalline
material within a diffracting volume. Implementing a progressive
point-by-point scan over the specimen surface produces a crys-
tallographic orientation map, where each point (pixel) of the map
contains information on crystallite orientation. This surface scan-
ning process is shown schematically in Fig. 2. The presence of
dislocations in the crystal lattice of a crystallite in such orientation
maps is manifested in two ways:
(i) The collective effect of GNDs and SSDs [13 ] leads to a
degradation in quality of the Kikuchi diffraction pattern used
to determine crystal orientation; this degradation can be
readily quantied [14e16]. However, the quality of Kikuchi
diffraction patterns is also affected by other factors ee.g.
phase diffraction intensity, grain orientation, sample prepa-
ration and beam conditions [14,17]esuch that a determi-
nation of the dislocation density based on the quality of
Kikuchi patterns is inherently ambiguous.
(ii) The presence of GNDs leads directly to local variations in the
crystal orientation between neighbouring measurements
(pixels) in the EBSD orientation map ethis is schematically
shown in Fig. 2. Small variations in crystal orientation
(misorientation,
3
Dq
) between neighbouring measurements
are caused by the lattice curvature, which is in turn caused by
the presence of GNDs. Hence, one can quantify the density of
GNDs [18e23] by calculating the amount of GNDs needed to
produce the observed lattice curvature. Note: (a) the pres-
ence of SSDs do not lead to the observable lattice curvature
due to their net-zero Burgers vector at the length scale of
interest, as schematically shown in Fig. 2; and (b) the EBSD-
measured lattice curvature is insensitive to the dislocation
arrangements, as it is schematically shown in Fig. 3.
3.1. Quantifying lattice curvature
Fig. 4 schematically shows how EBSD-measured crystal lattice
orientations (
f
1
,
F
,
f
2
)[12] can be used to calculate components of
the lattice curvature tensor (
k
ij
) between measurement points
(pixels):
Fig. 2. The presence of GNDs is manifested in EBSD-acquired orientation maps as a variation in crystal orientation (
f
1
,
F
,
f
2
) caused by lattice curvature, which is then used the
quantify the density of GNDs. Note that the lattice curvature (as quantied by the misorientation,
Dq
) is unaffected by the presence of SSDs because they demonstrate a net-zero
Burgers vector at the length scale of interest. It is further important to note that the lattice curvature and thus measured density of GNDs is strongly dependent on the user-specied
pixel spacing.
3
The misorientation angle is an angle by which one crystal coordinate system is
rotated relative to the reference crystal coordinate system, about a common axis of
rotation ½uvw.
O. Mur
ansky et al. / Acta Materialia 175 (2019) 297e313 299
k
ij
¼2
6
6
4
k
11
k
12
k
13
k
21
k
22
k
23
k
31
k
32
k
33
3
7
7
5
(1)
In a conventional 2D EBSD orientation map, the out-of-plane
curvatures (j¼3) are not accessible, thus only six out of nine lat-
tice curvature tensor components can be resolved. The lattice cur-
vatures are calculated using the misorientation angle (
Dq
), which
denes the difference in the crystal orientation between two
measurement pixels, separated by distance
D
x. The lattice curva-
ture across the pixel separation distance (
D
x) is then expressed as
follows [18]:
k
ij
¼v
q
i
vx
j
y
Dq
i
D
x
j
(2)
where
Dq
i
are direction cosines determined by the unit vector ½uvw
dening the common axis of rotation:
Dq
1
¼j
Dq
ju
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
2
þv
2
þw
2
p;
Dq
2
¼j
Dq
jv
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
2
þv
2
þw
2
p;
Dq
3
¼j
Dq
jw
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
2
þv
2
þw
2
p
(3)
The six accessible lattice curvatures are then explicitly written
as:
k
11
y
Dq
1
D
x
1
;
k
21
y
Dq
2
D
x
1
;
k
31
y
Dq
3
D
x
1
k
12
y
Dq
1
D
x
2
;
k
22
y
Dq
2
D
x
2
;
k
32
y
Dq
3
D
x
2
(4)
Note that the spatial resolution of conventional 2D EBSD
orientation maps is typically the same in both in-plane di-
rections, thus
D
x¼
D
x
1
¼
D
x
2
.Inmostcases,theEBSDscanning
step size used during data acquisition (h) is used as the pixel
separation distance (
D
x) for lattice curvature calculation. How-
ever, one can attempt to minimise the measurement error when
calculating the local lattice curvature by increasing
D
x.Fig. 5 il-
lustrates how the separation distance can be increased by using
measurement points located further from the reference pixel
(p
(0)
). Note that care must be taken when increasing
D
xto ensure
the nominal cell-block size is not exceeded. As pointed out by
Wilkinson et al. [20], increasing
D
xincreases the probability of
crossing multiple GNBs ethis results in an inaccurate measure-
ment of lattice curvature, and thus the corresponding GND
density. Fig. 2 illustrates how the EBSD-based misorientation
(lattice curvature) analyses can miss the presence of GNDs when
passing over GNBs with opposing net Burgers vectors. In this
instance, the observed misorientation (
Dq
) between p
(0)
and p
(2)
would equal zero, incorrectly implying a lack of GNDs between
p
(0)
and p
(2)
.
3.2. Quantifying GND density
The incomplete lattice curvature tensor obtained from Eq. (4)
can be used to derive a lower-bound GND density (
r
G
). A number
of authors [18e20,24,25] have employed Ney's dislocation tensor
formulation (
a
ij
) when resolving
r
G
from the lattice curvature
tensor. Neglecting the long-range elastic stress eld [7,8], Ney's
Fig. 3. While the EBSD-measured lattice curvature is sensitive to the GND density, it is
insensitive to their arrangement in the crystal.
Fig. 4. A schematic representation of lattice curvature components, calculated between two neighbouring crystals misoriented by a rotation (
Dq
) about a common crystallographic
axis [100]
c
([uvw]
c
) and separated by pixel separation distance (
D
x2). Note that in this example:
k
12z
Dq
1=
D
x2; and
k
22;
k
32 ¼0.
O. Mur
ansky et al. / Acta Materialia 175 (2019) 297e313300
total dislocation density tensor relates all possible dislocation
types
4
(t) as follows:
a
ij
¼X
N
t¼1
r
t
G
b
!
t
i
l
!
t
j
(5)
where b
!
t
is the Burgers vector and l
!
t
is the line vector for a
dislocation conguration of type t. Pantleon [18] has shown that
from a conventional 2D EBSD measurement the six accessible lat-
tice curvature tensor components (Eq. (4)) can be used to derive
ve components of Ney's dislocation tensor:
a
12
¼
k
21
;
a
13
¼
k
31
;
a
21
¼
k
12
;
a
23
¼
k
32
;
a
33
¼
k
11
k
22
(6)
and one difference between two of the dislocation tensor compo-
nents can be calculated as:
a
11
a
22
¼
k
11
k
22
(7)
Four components of the full dislocation density tensor are
missing:
a
31
¼
k
13
;
a
32
¼
k
23
;
a
11
¼
k
22
k
32
;
a
22
¼
k
11
k
33
(8)
These missing components require out-of-plane curvature in-
formation ð
k
i3
Þthat is not accessible in conventional 2D EBSD.
Equations (5) and (6) can be combined as follows:
k
ij
¼X
N
t¼1
b
t
j
l
t
i
1
2
d
ij
b
t
m
l
t
m
r
t
G
(9)
where
d
ij
is the Kronecker delta.
Considering 12 independent f111g110 dislocation modes found
in a face-centred cubic (fcc) crystal structure and assuming that all
stored GNDs are either pure screw ( b
!kl
!) or pure edge ( b
!l
!)
dislocations, one needs to account for 6 pure screw and 12 pure
edge dislocations. The resultant 18 dislocation types are listed in
Table 1. However, one also need to be mindful of dislocations with
the same Burgers vector and an opposite line vector. Therefore, the
total number of dislocation types that must be considered in the
analysis (Nin Eq. (9))is36[18]. With a set of 6 linear equations (Eq.
(10)) and 36 unknowns (
r
1
;
r
2
;/;
r
36
), a large number of possible
solutions exists whereby a unique solution cannot be obtained. We
can thus only obtain a lower-bound GND density of dislocations
necessary to create observed lattice curvature, based on a subset
comprising those 6 dislocation types with the highest density (
r
1
;
r
2
;/;
r
6
). Equation (9) can thus be explicitly written as:
2
6
6
6
6
6
6
6
6
4
k
11
k
21
k
31
k
12
k
22
k
32
3
7
7
7
7
7
7
7
7
5
¼
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
1
2b
1
1
l
1
1
1
2b
2
1
l
2
1
/1
2b
6
1
l
6
1
b
1
1
l
1
2
b
2
1
l
2
2
/b
6
1
l
6
2
b
1
1
l
1
3
b
2
1
l
2
3
/b
6
1
l
6
3
b
1
2
l
1
1
b
2
2
l
2
1
/b
6
2
l
6
1
1
2b
1
2
l
1
2
1
2b
2
2
l
2
2
/1
2b
6
2
l
6
2
b
1
2
l
1
3
b
2
2
l
2
3
/b
6
2
l
6
3
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
2
6
6
6
6
6
6
6
6
4
r
1
G
r
2
r
3
«
«
r
6
3
7
7
7
7
7
7
7
7
5
(10)
This solution is an approximation that is dened by imposing
the following physically-based constraints [18,20]:
(i) One can assume that plasticity proceeds in an energetically
favourable way, which requires a minimum amount of in-
ternal work; although in practice we know that some non-
minimum-energy dislocation arrangements will exist
locally. Nonetheless, it is a reasonable assumption that the 6
prevalent dislocation congurations would be such as to
produce a minimum total dislocation density:
r
G
¼X
36
t¼1
r
t
G
zmin X
6
t¼1
r
t
G
(11)
(ii) One can further constrain the number of solutions by ac-
counting for the fact that not all of the 36 possible dislocation
types are equally energetically favourable [26]. The energy of
screw dislocations is lower than that of edge dislocations by
an amount proportional to Poisson's ratio,
y
:
E
screw
¼ð1
y
ÞE
edge
(12)
Pantleon [18] suggested using the total line energy of the
dislocation conguration, which considers the energy ratio given in
Eq. (12), as the most physically appropriate weight function (w
t
)to
constrain Eq. (11):
X
6
t¼1
w
t
r
t
G
¼min (13)
where the total line energy of a dislocation conguration is w
t
¼
b
!
t
l
!
t
.
A direct minimisation method to nd the solution(s) to Eq. (13)
checks all the possible permutations of 6 dislocation types out of 36
to determine the most favourable combination(s) of dislocation
types that satisfy Eq. (13). Although the direct minimisation
Fig. 5. The accessible components of the lattice curvature tensor were calculated be-
tween the measurement pixels separated spatially by h;2hand 3h. Only the mea-
surements within the grain boundaries dened by a misorientation angle less than 7
were used in the calculations. For instances where the diffraction pattern could not be
measured, the lattice curvature and associated dislocation density was not calculated.
4
Dislocation types are dened by their slip system, as well as their Burgers
vector orientation with respect to their line vector.
O. Mur
ansky et al. / Acta Materialia 175 (2019) 297e313 301
method is robust and simple to implement, it requires a large
number of calculations (1,947,762) to be performed, and is thus
computationally expensive and ultimately impractical. Fortunately,
underdetermined systems of linear equations can be readily solved
by a number of numerical optimisation schemes. One approach that
has been utilised with success is the classical simplex method
[23,27]. In the present work, a Matlab implementation of the
simplex method has been employed wherein the weight function
w
t
is used as the objective function in the optimisation algorithm.
We have veried that this method predicts an identical solution to
the direct minimisation method, while being orders of magnitude
more efcient. On the other hand, unlike the direct minimisation
method the classical simplex method nds only one possible so-
lution (i.e. the rst minimised solution) satisfying Eq. (13), where a
number of equivalent solutions might exist.
4. Analysis of HRSD-measured diffraction patterns
The kinematical theory of X-ray/neutron scattering shows that
diffraction patterns collected on materials comprising large, defect-
free crystallites contain sharp reections with vanishingly small
widths [28,29]. In practice, diffraction peaks measured by X-ray/
neutron diffractometers will always have well-dened shapes and
non-zero widths. This is due to: (i) instrumental broadening caused
by the diffraction apparatus; and (ii) physical broadening caused by
the microstructure and defects present in the specimen. The most
common microstructural features in plastically deformed metals
that contribute to peak broadening are small crystallites (i.e. sub-
grains), dislocations (SSDs, GNDs) and planar faults (twin/stacking
faults) [30e35]. An understanding of these peak broadening phe-
nomena [30e35] has led to the development of diffraction line
prole analysis (DLPA) methods [30,36,37]. DLPA allows the analyst
to remove instrumental broadening from measurements through
their comparison against a standard (dislocation free) sample. The
observed physical broadening can be subsequently decoupled into
size broadening caused by the presence of small crystallites/sub-
grains (Section 4.1) and strain broadening caused by the presence
inhomogeneous strain elds ein particular, those due to disloca-
tions (Section 4.2)[38,39]. Hence, DLPA can derive the micro-
structural characteristics of the specimen such as the size of
crystallites, the density of dislocations, their arrangement and type,
and the type and density of planar faults (twin/stacking faults)
[38,39]. Further details of how each broadening mechanism is
measured via DPLA are described below.
4.1. Size broadening
In general, the width of a diffraction peak increases as the size of
the measured crystallites decrease [28]. Diffraction is sensitive to
very small misorientations (as discussed in more detail in section
6.2), thus for plastically deformed metals one needs to consider not
only the high-angle grain boundaries but also any domains having
low-angle relative misorientations [32]. In the case of pure metals
and alloys with high SFE which tend to form well-pronounced
subgrain boundaries the average size of these will be represented
by size broadening. In the case of metals and alloys not showing
well-pronounced sub-grains, the characteristic size of dislocation
structures developing due to the spatial non-randomness found in
the dislocation population will correspond to domains having low-
angle relative misorientations [32]. Subgrains, low-angle grain
boundaries (LAGBs) and dislocation cells are regions delimited by
GNBs (Fig. 1), which create domains that have a continuous spec-
trum of misorientation with their neighbours [40]. Due to this
shared boundary denition and the fact that diffraction is sensitive
to very low-angle misorientations (as discussed in more detail in
section 6.2), the average crystallite size determined from DPLA
cannot be readily decoupled into those of grains, but represent the
average size of these sub-structures [41]. Hence, when talking
about the crystallite size broadening in polycrystalline materials we
refer to the size of these structures, which in general are referred to
as coherently scattering domains (CSDs).
In order to register CSDs as separate regions by X-ray/neutron
DLPA they need to be misoriented by about 1
for regions with a
size of about 30 nm, or by about 0.1
for regions with a size of about
300 nm [42]. Comparison between X-ray DLPA and TEM has shown
that the size of CSDs obtained through DPLA correlates well with
the size of subgrains observed directly by TEM [41]. It has also been
shown [43] that even dipolar dislocation walls, which cause only a
shift and not a misorientation in the lattice, break down the co-
herency of scattering and are thus registered as a boundary when
analysed from diffraction patterns.
Theoretical widths and shapes of reections broadened by the
size effect of CSDs can be calculated theoretically based on physical
models of the microstructure. The diffraction peak model
describing the size broadening applied in the present work as-
sumes: (i) the average shape of the CSDs is equiaxed; and (ii) the
size of CSDs follows a log-normal distribution [41,44]:
I
size
hkl
ðgÞ¼ð
0
m
sin
2
ð
pm
gÞ
2ð
p
gÞ
2
erfc2
6
4
ln
m
m
ffiffiffi
2
p
s
3
7
5d
m
(14a)
where mand
s
are the median and variance of the log-normal CSD
size distribution, gis the length of the diffraction vector and the
variable of integration
m
represents the CSD size. These variables
can be used to estimate the area-weighted average CSD size
[44,45]:
Table 1
Table of considered pure dislocation congurations: (1e6) six screw dislocation
congurations with Burgers vector parallel to the line vector b
!kl
!, and (7e18 )
twelve edge dislocation congurations with Burgers vector perpendicular b
!l
!to
the line vector. Note, that the same 18 dislocation congurations with the opposite
line vector need to be also considered in the dislocation density calculation (see
section 2), leading to the total of 36 dislocation congurations to be considered in
the GNDs calculations.
Type Burgers Vector Line Vector
1b
!kl
!b
!
1
¼a=2½011l
!
1
¼b
!
1
¼½011
2b
!kl
!b
!
2
¼a=2½101l
!
2
¼b
!
2
¼½101
3b
!kl
!b
!
3
¼a=2½011l
!
3
¼b
!
3
¼½011
4b
!kl
!b
!
4
¼a=2½101l
!
4
¼b
!
4
¼½101
5b
!kl
!b
!
5
¼a=2½110l
!
5
¼b
!
5
¼½110
6b
!kl
!b
!
6
¼a=2½110l
!
6
¼b
!
6
¼½110
7b
!l
!b
!
1
¼a=2½011l
!
7
¼b
!
1
n
!
1
¼½211
8b
!l
!b
!
1
¼a=2½011l
!
8
¼b
!
1
n
!
2
¼½211
9b
!l
!b
!
2
¼a=2½101l
!
9
¼b
!
2
n
!
3
¼½121
10 b
!l
!b
!
2
¼a=2½101l
!
10
¼b
!
2
n
!
4
¼½121
11 b
!l
!b
!
3
¼a=2½011l
!
11
¼b
!
3
n
!
5
¼½211
12 b
!l
!b
!
3
¼a=2½011l
!
12
¼b
!
3
n
!
6
¼½211
13 b
!l
!b
!
4
¼a=2½101l
!
13
¼b
!
4
n
!
7
¼½121
14 b
!l
!b
!
4
¼a=2½101l
!
14
¼b
!
4
n
!
8
¼½121
15 b
!l
!b
!
5
¼a=2½110l
!
15
¼b
!
5
n
!
9
¼½112
16 b
!l
!b
!
5
¼a=2½110l
!
16
¼b
!
5
n
!
10
¼½112
17 b
!l
!b
!
6
¼a=2½110l
!
17
¼b
!
6
n
!
11
¼½112
18 b
!l
!b
!
6
¼a=2½110l
!
18
¼b
!
6
n
!
12
¼½112
O. Mur
ansky et al. / Acta Materialia 175 (2019) 297e313302
hX
A
me
2:5
s
2
(14b)
4.2. Strain broadening
Strain broadening is caused by microstrains [46] that vary the
atomic distances throughout the microstructure. These micro-
strains are caused by the presence of intergranular and intra-
granular stress elds, which either balance out over the grain (i.e. as
Type II stresses) or on the scale of crystallite (CSD) size (i.e. as Type
III stresses) [47]. Microstrains from Type II stresses are typically
caused by elastic and/or thermal incompatibilities between
neighbouring grains, while microstrains from Type III stresses are
caused by various crystal defects present in the crystal lattice. Even
though microstrains on both length scales contribute to diffraction
peak broadening, it has been shown that dislocation strain elds
dominate the diffraction peak broadening observed in the cold-
worked material of the present study [48].
Theoretical widths and shapes of diffraction peaks broadened by
dislocations can be calculated using the method of Wilkens [31,49].
The Wilkens model describes a theoretical dislocation structure
with a restricted random distribution (RRD) that is characterized by
a density value, an arrangement parameter and a given combina-
tion of possible dislocation types [31]. In terms of Fourier co-
efcients the shape of a strain-broadened diffraction peak can be
then written as follows:
FI
str
hkl
ðLÞ¼exp"
r
T
p
b
2
2
g
2
hkl
C
hkl
L
2
fð
h
Þ#(15)
where hkl are the Miller indices of the reections; C
hkl
, is the
average dislocation contrast factor;
r
T
is the total dislocation den-
sity; bis the Burgers vector; g
hk:l
is the diffraction vector length; Lis
the Fourier variable; fð
h
Þdenotes the Wilkes function, which in-
corporates the mean-square microstrain hε
2
hkl
ias a function of L;
and
h
¼L=R
e
,where R
e
ethe outer cut-off radius eis a charac-
teristic length corresponding to the RRD whereby a domain size of
R
e
will comprise randomly-distributed dislocations with a net
Burgers vector of zero [31]. The dimensionless arrangement
parameter (M) of the dislocation structure is dened through R
e
as
M¼R
e
ffiffiffiffiffi
r
T
p[31]. A large Mvalue means that the dislocation
structure has a large congurational energy, which corresponds to
a highly randomized, uncorrelated arrangement, for example [50].
On the other hand, a low Mvalue means a low congurational
energy, which corresponds to a highly correlated arrangement
[31,51]. The average dislocation contrast factor C
hkl
is related to the
types of dislocations. In the case of cubic crystals the average
contrast factor has the following form [51 ]:
C
hkl
¼C
h00
1qH
2
;where (16)
H
2
¼h
2
k
2
þh
2
l
2
þk
2
l
2
h
2
þk
2
þl
2
2
(17)
C
h00
is the average contrast factor of h00-type reections, and qis a
parameter depending on the proportion of edge and screw dislo-
cations present in the material.
4.3. Combined size and strain broadening
A set of experimental diffraction peaks collected on a poly-
crystalline specimen with small CSDs and a high dislocation density
(as illustrated in Fig. 1) will exhibit the combined effect of size and
strain broadening. If the measured diffraction pattern has a suf-
cient number of diffraction peaks etypically >5 for cubic and >8 for
hexagonal symmetry materials ethe effects of size and strain
broadening can be separated [32]. Fig. 6a shows an example of a
theoretically calculated diffraction pattern for pure nickel,
assuming X
A
¼60 nm, and containing a dislocation structure
with a total density
r
T
¼21E14 m
2
, a dislocation arrangement
ratio M¼1.6, and q¼2, which corresponds to ~75% of the total
dislocations having screw character. The wavelength of hard X-ray
(synchrotron) was set to
l
¼0.018972 nm, in order to match the
experiments presented later in the manuscript. Fig. 6a clearly il-
lustrates the different behaviour of broadening caused by CSD size
and by dislocation density, as a function of hkl reections (diffrac-
tion peaks). A Williamson-Hall plot in Fig. 6b then shows that the
size broadening component to the overall diffraction peak width
(measured as full width at half maximum, FWHM) remains con-
stant for all the diffraction peaks unlike the strain broadening
component. The strain broadening varies with hkl ethis variation is
related to the anisotropy of dislocation stress elds as a function of
crystallographic orientation [52].
The theoretical calculations shown in Fig. 6 demonstrate the
fundamentally different effects of size broadening and strain
broadening on the width of diffraction peaks. In summary, diffrac-
tion line prole analysis (DLPA) is able to use the measured
Fig. 6. Diffraction peak broadening due to the size of the coherently scattering domains (CSDs) is the same for all {hkl} diffractionpeak s, while thebroadening component due to the
microstrain caused by the presence of dislocations varies with {hkl}. This latter variation arises due to orientation-dependent plastic anisotropy.
O. Mur
ansky et al. / Acta Materialia 175 (2019) 297e313 303
diffraction pattern and the instrument calibration data (i.e. instru-
mental broadening) to derive the physical broadening and peak
shapes in terms of size and strain broadening. DLPA [30,36,37]can
thus be used to infer the characteristics of dislocation substructure
and average crystalline (CSD) size [39,53]. This inference is per-
formed by modelling the distinctive broadening and shape of
diffraction peaks [33] of diffraction patterns acquired using either X-
ray or neutron diffraction instruments. In the present work, DLPA is
performed using the Convolutional Multiple Whole Prole (CMWP)
software package [39,54]. CMWP constructs a theoretical diffraction
pattern based on the physical models of size and strain broadening
(Eqs. (14) and (15)), and instrumental broadening, which is matched
to the measured diffraction pattern by iterative renement of the
variables dening both size and strain broadening.
5. Experimental
5.1. Material
The material used in the present study was solution-annealed
Ni201 in rod form, with a chemical composition listed in Table 2.
Fig. 7a presents an EBSD orientation map of the as-received (so-
lution-annealed) material, revealing an average equiaxed grain size
of 25
m
m and a random crystallographic texture. A number of
ASTM-standard tensile specimens were machined from this rod
and subjected to interrupted-tensile testing. The specimens were
deformed to varying levels of plastic strain ethe nal imparted
plastic strain (ε
p
) was then measured after unloading of specimens.
Seven specimens were tested to the following levels of ε
p
: (1) 1.1%;
(2) 3.2%; (3) 6.0%; (4) 7.8%; (5) 9.6%; (6) 11.5%; and (7) 13.9%. Fig. 7b
presents the average response observed during the tensile tests.
After testing, the specimens were sectioned by electrical discharge
machining (EDM) into 10 81 mm samples. These samples were
extracted from the central region of the specimen gauge length; as
schematically shown in Fig. 7b, the extracted samples were ori-
ented such that the scan area was aligned parallel to the loading
axis. The specimens were mechanically polished in multiple stages
to an EBSD standard, using diamond paste (3-
m
m solution, followed
by 1-
m
m solution) and colloidal silica before being electro-polished
using a solution of methanol, 2-butanol and perchloric acid at 60 V
for 15 s. It is important to note the same samples were used for both
EBSD and HRSD measurements, thereby permitting direct com-
parisons of measurement data.
5.2. EBSD measurement and analysis
EBSD-acquired orientation maps were collected using a Zeiss
®
UltraPlusscanning electron microscope (SEM) equipped with an
Oxford Instruments
®
AZtecEBSD system and a Nordlys-SEBSD
detector. The following SEM and EBSD detector settings were used
during the EBSD data acquisition: 70
sample tilt; 20 keV acceler-
ating voltage; 150 magnication; 14.5 mm working distance;
60
m
m aperture size; 200 nm EBSD step size (h); and 4 4 data
binning. These orientation maps were then analysed using in-
house-developed Matlab analysis software (attached with an
example as supplementary material to this paper) utilizing the
theory outlined in Section 3above and in greater detail by Pantleon
[18]. Analyses were performed using the MTEX toolbox [55]. First,
post-processing (ltering) of the raw EBSD orientation data is
performed. During this process, all measurement points (pixels)
Table 2
Chemical composition (wt.%) of Ni201.
Ni C Si P Fe Mn Cr Mo Cu V Nb Ti Al
bal. <0.01 0.07 <0.01 0.03 <0.01 <0.01 <0.01 0.01 <0.01 <0.01 0.07 <0.01
Fig. 7. (a) EBSD-acquired, grain-averaged orientation map revealing the equiaxed microstructure of the undeformed Ni201 benchmark; and (b) seven ASTM-standard tensile
specimens were individually loaded to a nominally different strain in order to observe microstructure evolution with εp.
O. Mur
ansky et al. / Acta Materialia 175 (2019) 297e313304
with a mean angular deviation
5
(MAD) above 0.8
or a band
contrast
6
(BC) below 0.1 were not considered in subsequent anal-
ysis. Note, that values for both MAD and BC are calculated for each
point by AZtec during data acquisition.
Once data pre-processing is complete, the six components of the
lattice curvature tensor were calculated for each pixel in the
manner outlined in Section 3.1 (see Fig. 4). To examine the effect of
the pixel separation distance (
D
x, see Eq. (2)), on the calculated
GND density (
r
G
), different values of
D
x¼h;2hand 3h(h
¼0.2
m
m) are considered (Fig. 5). A bracketed superscript (1, 2 or 3)
is used to indicate the pixel separation distance employed. If the
magnitude of the misorientation angle ðj
Dq
between the two
measurement points exceeds 7
(chosen experimentally, note
choice of 10
or 15
does not inuence the results shown), it is
assumed that the points lie within different grains, such that they
are separated by a high-angle grain-boundary (HAGB). Hence, the
EBSD-measured density of GNDs does not consider those disloca-
tions which form HAGBs. The six lattice curvature components
determined at each pixel are used to determine
r
G
as per Section
3.2 by employing the simplex method [27] in Matlab. Recall that
should a non-unique solution exist where multiple congurations
satisfy Eq. (13), the only conguration considered via the simplex
method is that which is rst obtained.
5.3. HRSD measurement and analysis
HRSD measurements were performed at the 1-ID beamline
High-Energy Beamline (HEB) at the Advanced Photon Source,
Argonne National Laboratory. HEB was set up for high-resolution
measurement as schematically shown in Fig. 8. The HRSD mea-
surements were obtained in transmission mode, using a
200 200
m
m monochromatic beam with a wavelength of
0.018972 nm, corresponding to an energy of 65.351 keV. Diffraction
patterns were collected using three area at panel detectors (GE-
41RT), each having 2048 2048 pixels of 200 200
m
m size [56].
The 1-mm thickness of the extracted samples (Fig. 7b) was chosen
to minimize instrumental peak broadening while sampling a large
number of grains. The use of thin specimens also ensures low
attenuation of the transmitted X-rays. A set of 10 measurements
were performed at different positions along the loading axis of each
sample as schematically depicted in Fig. 8. Each individual mea-
surement is comprised of 200 diffraction snapshotstaken over
0.3-s intervals, to acquire sufcient counting statistics without
saturating the individual detector images. The diffraction patterns
were collected with high angular resolution by placing the detector
panels 1.873 m away from the sample. The exact position and tilt of
the detectors was calibrated using the diffraction pattern of a CeO
2
standard powder. The instrumental broadening was determined
using an undeformed reference sample, which possessed a dislo-
cation density that was undetectable via DLPA.
The collected 2D diffraction patterns were converted into 1D
diffraction (2
q
-intensity) patterns by integrating 10
sections
(cakes) of the measured Debye-Scherrer rings by employing FIT2D
software [57]ean example of this output is shown in Fig. 9. The
diffraction peak broadening in these patterns was then analysed
using the CMWP DLPA software package [38,39]. The following
microstructure-model parameters were rened in the iterative
DLPA solution: (i) the area-weighted CSDs (subgrain) size (X
A
);
(ii) the total dislocation density (
r
T
); (iii) the outer cut-off radius
(R
e
); and (iv) the parameter describing the screw/edge character of
the dislocation structure (q). Fig. 9b presents an example of such a
tting (red continuous line), for the sample with 13.9% of imparted
ε
p
(the experimental data is represented by black circles). Fig. 9b
also presents the experimental diffraction pattern measured on the
undeformed sample so that one can directly observe the effect of ε
p
on diffraction peak broadening. Note that the observed change in
diffraction peak intensity is related to the development of crystal-
lographic texture during plastic deformation. The microstructure
results for each sample were obtained as follows: a 10
wide cake
between the azimuth angles of 150
and 140
, the location of
which is represented schematically in Fig. 8, was integrated for
every measurement and evaluated using CMWP. Thus, for each
sample 10 sets of microstructural results were obtained corre-
sponding to the 10 measurements performed at different positions
on each specimen. The microstructure values and their un-
certainties for each sample was calculated as the mean and stan-
dard deviation of the CMWP results obtained for the 10 different
measurements.
6. Results and discussion
6.1. EBSD-measured development of GND density as a function of
plastic strain
Fig. 10 presents the lower-bound GND density (
r
G
) maps as
calculated from EBSD-measured orientation maps employing the
methodology outlined in Section 3using a pixel separation distance
D
x¼h¼200 nm for the undeformed sample, and for samples with
ε
p
¼7.8% and 13.9%. As noted above,
r
G
is not calculated across
Fig. 8. High-resolution synchrotron diffraction (HRSD) set-up at the 1-ID high-energy
beamline (Advanced Photon Source, Argonne National Laboratory).
5
Mean angular deviation (MAD), given in degrees, species the averaged angular
mist between measured and theoretical (simulated) Kikuchi bands [17 ].
6
Band contrast (BC) is an EBSD map quality factor derived from the Hough
transform. It describes the average intensity of the Kikuchi bands with respect to
normalized background intensity. BC is affected by phase diffraction intensity,
crystal lattice defects (e.g. dislocations, vacancies) and crystal orientation [17 ].
O. Mur
ansky et al. / Acta Materialia 175 (2019) 297e313 305
HAGBs (
Dq
>7
). In Fig. 11 we further plot
r
G
in terms of average
dislocation spacing (d
G
):
d
G
¼1
ffiffiffiffiffi
r
G
p(18)
It becomes clear from Figs. 10a and 11a that in the undeformed
specimen, GNDs are fairly evenly distributed across the micro-
structure eno distinctive dislocation substructure is observed e
and overall
r
G
is relatively low (<5E10
14
m
2
).
With increasing levels of ε
p
the material stores a growing den-
sity of GNDs, in order to maintain strain compatibility across vari-
ously oriented grains and subgrains [2]. This trend can be seen from
Fig. 10b and c and 11b,c. One can further see that GNDs are non-
uniformly distributed across the microstructure of plastically
deformed samples. This illustrates both intergranular and intra-
granular effects of orientation-dependent work-hardening behav-
iour in the polycrystalline aggregate. As ε
p
increases, the GNDs
arrange themselves into energetically-favourable congurations
[4,9] forming GNBs (namely, LAGBs), which split grains into sub-
grains [9,10]. Recall that these newly-formed LAGBs are like any
other boundaries present in the microstructure in that they are
formed by arrangement/pile-up of GNDs. The GND density (
r
G
)
across these GNBs is in the order of 10
15
m
2
, which corresponds to
a dislocation spacing (d
G
) below 100 nm. In contrast,
r
G
within the
interior of these subgrains is in the order of 10
13
-10
14
m
2
, which
corresponds to a value of d
G
above 150 nm.
It is also clear from these orientation maps that GNDs do not
simply pile-up at the HAGBs and triple junctions as one would
expect based on the simplied model put forward by Ashby in
Ref. [2] and schematically shown in Fig. 1. There is an observed
increase in
r
G
at the HAGBs as the level of accumulated plasticity
increases; however, the increase is not signicantly higher than the
increase within grains. Recall the hardening mechanics and thus
the accumulation of GNDs will be different between the relatively
pure Ni studied here and most common engineering alloys, where
dislocation pile-up near grain boundaries and triple junctions may
occur. In addition to this distinction, Hughes et al. [10] explain that
the assumptions made by Ashby are valid when single slip is
considered - multiple slip systems are required to accommodate
strain compatibility across the grain, which is supported by the
present ndings. Furthermore,
r
G
is dependent not only on the
macroscopic average of ε
p
but also on microscopic characteristics
that include grain size, second-phase distribution (precipitates)
Fig. 9. (a) 2D synchrotron diffraction pattern showing partial Debye-Scherrer diffraction rings and a schematic representation of the sectioning (caking) processes, which involves
integration of the 10section into a standard 1D HRSD pattern, shown in (b); and (b) comparison of 1D HRSD patterns from the undeformed (εp¼0%, blue symbols) specimen with
a heavily-deformed (εp¼13.9%, black symbols) specimen and the corresponding DLPA model (red line) for the for the heavily-deformed specimen. (For interpretation of the
references to colour in this gure legend, the reader is referred to the Web version of this article.)
Fig. 10. GND density (
r
ð1Þ
G) maps calculated from EBSD-acquired orientation maps: (a) undeformed benchmark (εp¼0%); (b) εp¼7.8%; and (c) εp¼13.9%. Bracketed superscript (1)
infers a pixel separation distance
D
x¼h¼200 nm was used for these calculations .
O. Mur
ansky et al. / Acta Materialia 175 (2019) 297e313306
and crystallographic texture. While grain size is accounted for in
Ashby's model, further microstructural considerations are required.
Signicant variations in plastic anisotropy can exist within these
randomly-textured specimens, leading to an increase in
r
G
required
to satisfy strain compatibility as ε
p
increases.
While the spatial variation in GNDs across the microstructure is
of fundamental interest, the ability to discern bulk
r
G
trends is of
technological importance in any tness-for-service (FFS) assess-
ment of engineering components. The most straightforward way to
determine the microstructure-averaged (bulk)
r
G
from discrete
measurements is by nding the arithmetic mean of all available
measurements across the microstructure. However, Kamaya et al.
have shown [58e60] that the intragranular misorientation distri-
bution eand hence, the intragranular
r
G
distribution (see Fig. 12)e
across a microstructure follows a lognormal distribution. Githinji
et al. [61] have recently argued that when the amount of accu-
mulated plasticity is high, misorientations follow a gamma distri-
bution. Therefore, we have considered both distribution models
(lognormal, gamma) to describe the measured distribution of
r
G
across the microstructure. It has been found, based on a Bayesian
criterion, that in the present case a lognormal distribution de-
scribes the measured distribution of
r
G
slightly better than the
gamma distribution for all levels of ε
p
. Therefore, we have
calculated bulk
r
G
assuming a lognormal distribution of the
measured discrete values of
r
G
across the microstructure, as follows
[62]:
fð
r
G
j
m
;
s
Þ¼ 1
x
s
ffiffiffiffiffiffi
2
p
pexp ðlnð
r
G
Þ
m
Þ
2
2
s
2
!(19)
where
m
and
s
are the location and scale parameter of lognormal
distribution, respectively. Under this denition, the mean ðmÞand
variance ðvÞof the lognormal distribution are dened as:
mð
r
G
Þ¼ exp
m
þ
s
2
2(20a)
vð
r
G
Þ¼ exp2
m
þ
s
2
exp
s
2
1(20b)
In this analysis, mð
r
G
Þis the microstructure-averaged (bulk)
value of
r
G
, while vð
r
G
Þreects the heterogeneity of the GND dis-
tribution within the microstructure.
Fig. 12a presents distributions of the discrete GNDs shown in
Fig. 10. It is clear from this gure that both mð
r
G
Þand vð
r
G
Þincrease
with ε
p
increasing; however, steps must be taken to ensure these
Fig. 11. GND spacing (dð1Þ
G) maps calculated from the same orientation map data used in Fig. 10:(a) undeformed benchmark (εp¼0%); (b) εp¼7.8%; and (c) εp¼13.9%. Bracketed
superscript (1) infers a pixel separation distance
D
x¼h¼200 nm was used for these calculations .
Fig. 12. (a) Distribution of all discrete GNDs density (
r
ð1Þ
G) measurements shown in Fig. 10 for samples with 0%, 7.8% and 13.9% of imparted plastic strain (εp). (b) Corresponding
evolution of the mean mð
r
ð1Þ
GÞof lognormal distribution of the discrete
r
ð1Þ
Gmeasurements as a function of number of grains included in the distribution until all the grains (i.e.
number of discrete
r
ð1Þ
Gmeasurements) in the
r
ð1Þ
Gmaps shown in Fig. 10 are included.
O. Mur
ansky et al. / Acta Materialia 175 (2019) 297e313 307
values are truly representative of the bulk response of the material
by validating the statistical signicance of the results. Fig. 12b
shows the evolution of mð
r
G
Þas a function of the number of grains
included in the lognormal distribution. This study is based on the
data shown in Fig. 10, starting with the grain in the left bottom
corner and expanding outwards to include additional grains. The
number of grains required for solution convergence appears to
increase with ε
p
, to the point that convergence for the sample with
13.9% plastic strain (Fig. 10c) requires approximately 125 grains in
the analysis. Such results can be expected since the level of het-
erogeneity in the GND distribution increases with ε
p
(clear from
Figs. 10 and 11 and in Fig. 12a). By ensuring the number of grains
sampled is well in excess of these minimums, we ensure statistical
signicance in these results. Furthermore, at least two EBSD scans
were acquired for each sample, thereby considering the potential
for any macroscale variations in
r
G
.
Using the aforementioned analysis method, the evolution of
GNDs as a function of accumulated plastic strain is shown in Fig. 13.
To examine the measurement sensitivity to the pixel separation
distance (
D
x, see Fig. 5), the results of
r
ð1Þ
G
,
r
ð2Þ
G
and
r
ð3Þ
G
calculations
are presented, which respectively use the pixel separation dis-
tances of h,2hand 3h. Before discussing the results we need to
address the uncertainty in the calculation of
r
G
, which is based on
the uncertainty in the EBSD-measured crystal orientations. Wil-
kinson and Randman [20] have estimated the expected noise on the
GND calculations by
Dr
G
¼
d
=b
D
x, where
d
is the uncertainly in the
crystal orientation measurement (~0.1
, upper-bound), where band
D
xare Burgers vector (0.2546 nm) and the pixel separation distance
(h,2hand 3h,h¼20 0 nm), respectively. This leads to uncertainties
in
r
G
calculations of ~1.7E13, 8.6E12, and 5.7E12 for
D
xof h,2hand
3h, respectively. By this reasoning, a higher
D
xresults in a reduction
of the noise in
r
G
calculations erecalling that care must be taken
when increasing
D
xto ensure the nominal subgrain/cell-block size
is not exceeded [20].
Turning now to the results presented in Fig. 13, the following
observations can be made:
i. Increasing the pixel separation distance (
D
x) from hto 3h(Fig. 5)
results in a systematic reduction of calculated
r
G
. It is likely that
as
D
xincreases a higher portion of GNDs with opposing non-
zero Burgers vectors exist across the separation distance
(Fig. 2). This leads to a reduction in the measured lattice cur-
vature and thus in
r
G
. Reducing
D
xwill minimise this under-
estimation, which is as signicant (2 - 3E12 m
2
) as the
measurement uncertainty. Therefore, a compromise must be
made between noise reduction via increasing
D
x, and using a
minimum value of
D
xthat ensures an appropriate spatial reso-
lution is used for
r
G
calculations.
ii. The increase of
r
G
with ε
p
follows a linear trend (Fig. 13a) when
ε
p
>1%. This qualitative feature can be seen independent of
D
x.
The linear proportionality of
r
G
on ε
p
shows that although
r
G
develops due to microstructural characteristics that inuence
strain compatibility (i.e. it is a mesoscale phenomenon), the
storage of
r
G
by the deformed material is strongly dependent on
ε
p
(i.e. a macroscale phenomenon). For comparison, we plot in
Fig. 13a the estimated
r
G
based on Ashby's model [2]:
r
G
y
ε
p
4bD (21)
where bis the Burgers vector (0.2546 nm) and Dis the grain size
(30
m
m). It is clear that Eq. (21) underestimates
r
G
even when
compared to lower-bound
r
G
calculations. As discussed above, this
can be attributed to the simplication of the single-slip model [2],
which presumes impenetrable grain boundaries (HAGBs) where
dislocations simply pile up. It is also possible that the electro-
chemical polishing did not remove deformation introduced in the
mechanical polishing stage of the sample preparation (Section 5.1);
however, one would not expect such an error to be systematic
across each specimen measured and the degree of linearity
observed in Fig. 13a belies such a claim.
iii. Calculated
r
G
in the undeformed reference sample lies be-
tween 2E12 and 5E12 m
2
and it does not follow the linear
dependency on ε
p
observed at strains above 1.0% [2]. Based
on an assumed linear trend,
r
G
in the undeformed sample
should be <1E9 m
2
; this value is well below the expected
noise in the measurement (5.7E12 m
2
when
D
x¼3h). The
results in Fig. 13a for samples with ε
p
¼0% and 1% therefore
suggest we can estimate the minimum detectable
r
G
to be
approximately 2E12 m
2
.
iv. The choice of EBSD scan location also appears to produce
measurement uncertainty, particularly for samples with low
ε
p
. The analysis of multiple orientation maps is therefore
recommended to ensure condence in obtaining accurate
bulk
r
G
values.
v. The consistent increase in the variance of
r
G
as a function of
ε
p
(Fig. 13b) suggests increasingly non-uniform distributions
of GNDs across the microstructure. This can be linked to ar-
rangements of GNDs into low-energy congurations (GNBs)
Fig. 13. (a) Mean and (b) variance of the microstructure-averaged (bulk) GND density (
r
ð1Þ
G;
r
ð2Þ
G;
r
ð3Þ
G) as a function of plastic strain, assuming a lognormal distribution. The bracketed
superscripts (1, 2, and 3) refer to the pixel separation distance (
D
x, see Eq. (2) and Fig. 5) used for data processing.
O. Mur
ansky et al. / Acta Materialia 175 (2019) 297e313308
and the formation of subgrains. These subgrains comprise
interior regions of relatively low GND density, delimited by
relatively high-density walls (Figs. 10 and 11).
It is further useful to investigate the dislocation types present in
the obtained
r
G
solutions. Recall that in
r
G
calculation (Eq. (11))we
are looking for six individual
r
t
G
, where t¼1 . 36, whose sum rep-
resents the minimum
r
G
ðmin P
6
t¼1
r
t
G
Þ. Recall also that
r
t
G
are
weighted by the total line energy (w
t
, Eq. (13)) of dislocation
conguration, which favours screw dislocations over the edge
dislocations due to their lower energy (Eq. (12)). Fig. 14 shows the
ratio of screw dislocation types to the total number of 6 dislocations
types allowed in each
r
G
solution (Eqs. (11) and (13)) for
r
G
maps
shown in Fig. 10c. It is clear that portion of screw dislocations in the
solutions varies across the microstructure but seems to stay be-
tween 0.33 and 0.66, which corresponds to 2 and 4 screw dislo-
cation types out of 6, respectively. Hence, it is clear that active GND
types are inuenced by both the crystallographic orientation and
intragranular variations in ε
p
.Fig. 14b also presents the mean of
ratio of screw dislocations to the total number of 6 dislocation types
allowed in each
r
G
solution for all ε
p
. It is clear from Fig. 14b that the
average ratio of screw dislocations is 0.33, which corresponds to a
r
G
solution comprising 2 screw and 4 edge dislocation types. This
was not expected since the line energy (w
t
) of a pure screw dislo-
cation is lower than that of a pure edge dislocation, thus warranting
preferential selection in the minimum solution (Eq. (13)). However,
one needs to keep in mind that the solution found using the sim-
plex method represents the minimum
r
G
but it is not necessarily
unique and other solutions may exist with a higher proportion of
screw dislocations. Similar results were obtained when analysing
dislocation types in
r
ð2Þ
G
and
r
ð3Þ
G
solutions.
6.2. HRSD-measured development of total dislocation density and
the size of coherently scattering domains (CSDs) as a function of
plastic strain
As mentioned above, HRSD patterns were evaluated using the
CMWP DLPA method to obtain quantitative characteristics of dis-
locations and dislocation structures. Because of a relatively large
grain size in the studied material (Fig. 7a) and the relatively low
dislocation density at early stages of deformation, tting the
dislocation arrangement parameter M¼Re ffiffiffiffiffi
r
T
pproved unreliable.
Difculties arise since the peak signal is weak when the dislocation
density is small. Unsurprisingly then, the uncertainty of the
arrangement parameter was the lowest for data collected on the
sample with the highest amount of plastic deformation (13.9%). For
this case the arrangement was determined to be M¼2.9 ±0.3. For
all other samples, microstructure renement was conducted under
the assumption that the arrangement parameter remains un-
changed, i.e. by constraining Mto 2.9. The results obtained by DLPA
are present in Fig. 15;Fig. 15a shows development of
r
T
while
Fig. 15b shows the development of X
A
as a function of ε
p
.
As it is schematically shown in Fig. 8, ten measurements were
made along the sample loading axis ewe present the individual
measurements and the mean of these measurements together with
the standard deviation in Fig. 15. Note that because the undeformed
reference sample was used to measure instrument broadening and
are thus a baseline measurement, the strength of the peaks
measured in the deformed samples must be stronger than this
lower-bound intensity for effective DLPA. As a result, a reliable
measure of the peak broadening in the sample with ε
p
¼1% could
not be performed, even though EBSD measurements revealed an
increase in GNDs relative to the undeformed sample (Fig. 13).
Looking at the sample with ε
p
¼3.2% (Fig. 15), the measurement of
r
T
and X
A
is realistic; however, one still needs to be mindful of the
large scatter in measured values. The results thus indicate the
minimum detectable density of dislocations is lower for the EBSD
technique (2E12 m
2
,Fig. 13a) than for the HRSD technique, which
is about an order of magnitude higher (1E13 m
2
,Fig. 15a) -
remember however that the EBSD technique is measuring GNDs
while HRSD is measuring total dislocation density. The value of q,
the parameter describing the dislocation edge/screw character,
scattered around values between 2 and 2.4, indicating that the
dislocation structure has a strong screw character. As discussed in
the previous section, this is in agreement with the expectations
based on dislocation line energy considerations, even though the
EBSD solutions indicate a lower screw dislocation character for
majority of measurements across the microstructure (Fig. 14).
Since the studied material can be considered a pure metal
(Table 2) without any signicant presence of precipitates or second
phase particles that would contribute to its work-hardening
behaviour, one can consider the hardening behaviour of the stud-
ied material to be controlled by dislocation hardening eand the
combined effect of SSDs and GNDs. Hence, we can estimate
r
T
from
Fig. 14. (a) orientation dependence of the GND ratio, which denes the portion of screw dislocation types to the total number of the six prevalent dislocation types in a lower-
bound
r
ð1Þ
Gsolution (map of sample with εp¼13.9% shown); and (b) mean ratio of screw-type GNDs as a function of εp.
O. Mur
ansky et al. / Acta Materialia 175 (2019) 297e313 309
the measured stress-strain curve shown in Fig. 7b by employing the
modied Taylor equation [63]:
s
T
¼
s
y
þ
a
MGb ffiffiffiffiffi
r
T
p(22)
r
T
¼
s
T
s
y
a
MGb
2
(23)
where
s
T
is the applied stress;
s
y
is the yield stress (65 MPa); Gis
the shear modulus (72 GPa); bis the Burgers vector (0.2546 nm); M
is the Taylor factor(3.06 for a randomly-textured polycrystal), and
a
is the Taylor constant ranging from 0.2 to 0.4 [63,64]. Fig. 15a
compares HRSD-measured and Taylor-estimated values of
r
T
when
using 0.2 and 0.4 as extreme values of
a
. As seen in Fig. 15a, an
excellent agreement is achieved when
a
¼0.2 is used in the strain
range where condence in HRSD measurements exists (i.e. ε
p
>
3.2%), which provides condence in the HRSD-measured
r
T
.
Fig. 16a presents the comparison of HRSD-measured total
dislocation density (
r
T
) and EBSD-measured GNDs density (
r
G
)
calculated with the separation distance
D
x¼h. Note that the error
bars represent the standard deviation of individual measurements.
It is clear that the material stores a higher amount of SSDs relative
to GNDs. Hence, the work-hardening of this material is governed
predominantly via SSDs ethis is perhaps expected for a relatively
coarse-grained material (Fig. 7a).
6.3. Interconnection between EBSD-measured density of GNDs and
HRSD-measured size of coherently scattering domains (CSDs)
It is important at this point to emphasise that the EBSD-
measured density of GNDs (
r
G
) and the HRSD-measured size of
CSDs (X
A
) are not independent of each other, since the boundaries
of CSDs are dislocation walls formed by GNDs (Fig. 1)ethis fact is
discussed in detail by Zilahi et al. [65]. Fig. 16b compares the EBSD-
derived d
G
with the HRSD-derived X
A
. Both values decrease
monotonically but with different rates as a function of plastic strain,
with the values of X
A
being larger when ε
p
<6% and smaller when
ε
p
>6%. No experimental X
A
values are shown for ε
p
¼0% and ε
p
¼1% because for both of these samples, strain and size broadening
were undetectably small. As a result, the average size of CSDs (X
A
)
cannot be measured in the samples, and it can only be stated that
their value is larger than ~1
m
m(Fig. 16b). The qualitatively-similar
trend of X
A
and d
G
values as a function of plastic strain conrms
there is a correlation between these features. This correlation can
Fig. 15. (a) Total dislocation density (
r
T) and (b) size of the coherently scattering domains (CSDs) obtained by DLPA of the HRSD pattern shown in Fig. 9, as a function of εp. Open
symbols represent individual measurements along the sample loading axis (schematically shown in Fig. 8); solid symbols represent the mean value of these measurements.
Fig. 16. (a) Comparison of the HRSD-based total dislocation density (
r
T) and the EBSD-based GND density (
r
G) with theoretical dislocation densities calculated using the modied
Taylor model (Eq. (23)) and the single-slip Ashby model (Eq. (21)); (b) comparison of the HRSD-determined CSD size (red circles) with EBSD-determined GND spacing (blue squares)
and the estimated minimum size of CSDs (green triangles) based on
r
G(Eq. (29)). (For interpretation of the references to colour in this gure legend, the reader is referred to the
Web version of this article.)
O. Mur
ansky et al. / Acta Materialia 175 (2019) 297e313310
be quantied through a simple model, which takes into account the
pixel separation distance (
D
x) used in EBSD measurements and the
minimum misorientation required to detect two grain regions as
individual CSDs.
As outlined in Section 3, EBSD measures the orientation of
crystallites at discrete locations across the sample surface with a
pixel separation distance of
D
x(h;2h;3h). The difference in orien-
tation between two neighbouring locations is caused by the local
lattice curvature proportional to the array of GNDs present. A
simple scenario is depicted in Fig. 2, where arrays of GNDs shown in
location p
(1)
and p
(2)
are the cause of the measured local misori-
entation, quantied by misorientation angle
Dq
. Within the bounds
of this simple representation we interpret the measured local
misorientation between any two neighbouring pixels as being
caused by the presence of a GND array. The multitude of such GND
arrays over the whole misorientation map constitutes the GND
structure of the sample. Such a GND structure can be represented
by two characteristic distances:
D
xand. The dislocation density
corresponding to such a hypothetical GND structure is [65]:
r
G
¼1
D
xL (24)
This relationship shows that
r
G
is inherently dependent on the
value of
D
xchosen by the user when carrying out an EBSD scan (See
Section 6.1 and Fig. 13). The misorientation (
u
G
Þcorresponding to
the presence of GNDs has a reciprocal dependence to L(Fig. 2):
u
G
yb
L(25)
where bis the length of Burgers vector. By combining Eq. (24) with
Eq. (25), and by assuming a fcc material with a lattice constant a
(and thus where b¼a=ffiffiffi
2
p), a relationship between
r
G
,
u
G
and
D
x
can be dened:
r
G
y
u
G
D
xb ¼ffiffiffi
2
p
u
G
D
xa (26)
The HRSD-measured hX
A
irepresents the average size of CSDs,
which have sufcient relative misorientations against each other so
that they scatter the X-rays incoherently. A criterion for two do-
mains to scatter incoherently is that none of their broadened re-
ections in reciprocal space should overlap [66]. Unless an extreme
amount of strain broadening is present, which is not the case here,
this criterion can be simplied such that the two domainsrecip-
rocal lattice points with the shortest diffraction vectors should not
overlap. For Ni these are the {111}-type reciprocal lattice points. If
these lattice points have a broadening of B, which can be imagined
as the intensity distribution around the lattice point having a
spherical symmetry and a FWHM of B, the minimum
u
G
required
for two neighbouring domains to scatter incoherently is:
u
u
min
yB
g(27)
where g¼ffiffiffi
3
p=ais the distance of the {111}-type reciprocal lattice
points from the origin. The width of the size-broadened reciprocal
lattice points is inversely related to the domain size By1=hX
A
i.
Hence, the relation between HRSD-measured hX
A
iand the mini-
mum relative misorientation (
u
G
) between CSDs is dened as:
u
u
min
ya
ffiffiffi
3
phX
A
i(28)
By combining Eq. (26) and Eq. (28), a relation between the
minimum size of CSDs (X
A;min
measured by HRSD and the density
of GNDs (
r
G
) measured by EBSD using a pixel separation distance of
D
xcan be established:
hX
A
iX
A;min
¼ffiffiffi
2
3
r1
D
x
r
G
¼ffiffiffi
2
3
rd
2
G
D
x(29)
This shows that it is possible to make a qualitative link between
the HRSD-measured X
A
and the EBSD-measured
r
G
. Equation (29)
applies to fcc materials but can be generalized for materials with
other crystal structures.
Fig. 16b presents the comparison of HRSD-measured X
A
with
estimated X
A;min
values that are determined using EBSD-
measured d
G
and Eq. (29). It is clear from this comparison that
the two trends are in good agreement. Note that CSDs larger than
~1
m
m do not cause a measurable size-broadening effect on
diffraction peaks; the calculated X
A;min
values indicate that the
expected size of CSDs for samples with ε
p
1% would be larger than
the detection limit of the instrument. This analysis shows that one
can estimate the minimum size of CSDs from the EBSD measure-
ment of
r
G
. Inversely, one can estimate the minimum
r
G
expected
in an EBSD map from the HRSD-measured size of CSDs:
r
G
ffiffiffi
2
3
r1
D
xX
A
(30)
This assessment is based on the assumption that all boundaries
that dene CSDs are dislocation walls, which is true for most
plastically deformed materials where the dominant plasticity
mechanism is dislocation slip. This methodology cannot be applied
for materials which have signicant proportion of other micro-
structural features that act as domain boundaries, such as stacking
faults, nano-twin boundaries and martensitic structures.
7. Conclusions
The present work examines the accumulation of the dislocations
and development of dislocation structures in Ni201 when plasti-
cally deformed under uniaxial tension, employing the analysis of
EBSD-acquired orientation maps and HRSD-acquired diffraction
patterns. The following conclusions are drawn:
- The minimum detectable total dislocation density (
r
T
)
measured via HRSD is approximately 1E13 m
2
, which occurs
when ε
p
>3%, while the minimum GND density (
r
G
) measured
via EBSD is approximately 2E12 m
2
, which occurs when ε
p
>
1%. These results suggest EBSD techniques are more sensitive to
the early stages of plastic ow, which is expected given CSDs
have not sufciently developed at this stage.
- The accuracy of HRSD measurements increases with increasing
amount of imparted plastic deformation within the material,
while the accuracy of EBSD will start decrease when plastic
deformation will prevent indexing of Kikuchi diffraction pat-
terns. Such behaviour highlights the complementarity of the
two techniques when attempting to quantify dislocations in a
material that has experienced an unknown and potentially
broad range of accumulated plasticity. Selection of the most
appropriate technique for quantifying plasticity should there-
fore be based on the expected material condition.
- A relationship between
r
G
estimates using EBSD techniques and
the size of CSDs estimated using HRSD techniques has been
derived. Strong agreement between the two features has been
demonstrated, which serves to validate the underlying meth-
odology used in each technique and lend condence to the ac-
curacy of results presented herein. This approach also permits
the estimation of CSD size based on EBSD data alone.
O. Mur
ansky et al. / Acta Materialia 175 (2019) 297e313 311
- A comparison of EBSD-inferred
r
G
with the theoretical GND
density model of Ashby has been performed. This comparison
quanties how signicantly Ashby's single-slip model can un-
derestimate the number of dislocations present in a poly-
crystalline material.
- A comparison of HRSD-inferred
r
T
with the total dislocation
density model of Taylor shows excellent agreement when ε
p
>
3% (i.e. above the detection limit for HRSD). These results lend
condence in the approach taken to quantify
r
T
via the HRSD-
based DLPA.
- The combined analyses of dislocation types via EBSD and HRSD
for ε
p
>3% illustrate that the SSD density is greater than the GND
density at all times; however, the linear proportionality of these
two densities with plastic strain is not the same, and suggests
GNDs may be the dominant dislocation type at low levels of
plasticity.
Acknowledgement
The authors would like to acknowledge work of Mr Tim Palmer
(ANSTO) and fruitful discussions with Dr Roman Voskoboynikov
(ANSTO) as well as support from Professors Michael R. Hill (UC
Davis) and Lyndon Edwards (ANSTO). This research relied on data
acquired using the Advanced Photon Source, a U.S. Department of
Energy (DOE) Ofce of Science User Facility operated for the DOE
Ofce of Science by Argonne National Laboratory under Contract
No. DE-AC02-06CH11357. LB and MRD were supported by an NSERC
Discovery Grant.
Appendix A. Supplementary data
Supplementary data to this article can be found online at
https://doi.org/10.1016/j.actamat.2019.05.036.
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