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Effect of elemental combination on friction stress and Hall-Petch
relationship in face-centered cubic high / medium entropy alloys
Shuhei Yoshida
a
,
*
, Takuto Ikeuchi
a
, Tilak Bhattacharjee
a
,
b
, Yu Bai
a
,
b
,
Akinobu Shibata
a
,
b
, Nobuhiro Tsuji
a
,
b
a
Department of Materials Science and Engineering, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, 606-8501, Japan
b
Elements Strategy Initiative for Structural Materials (ESISM), Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto, 606-8501, Japan
article info
Article history:
Received 25 November 2018
Received in revised form
15 March 2019
Accepted 8 April 2019
Available online 11 April 2019
Keywords:
Strengthening mechanism
Dislocation
Grain refinement
Ultrafine-grained materials
Element-element interaction
abstract
In this study, we report the effect of elemental combinations on the friction stress and Hall-Petch
relationship in medium entropy alloys (MEAs) and high entropy alloys (HEAs) which are defined as
the alloys composed of four or less and five or more principal elements, respectively, with (near-) equi-
atomic concentrations. The MEAs (CoCrFeNi, CoCrNi, etc.), which are subsystems of equi-atomic CoCr-
FeMnNi HEA, were highly deformed by high-pressure torsion (HPT) and subsequently annealed at
different temperatures. The specimens with fully-recrystallized microstructures of FCC single-phase with
various mean grain sizes down to sub-micrometer scale were obtained. Subsequently, tensile tests were
performed at room temperature to obtain precise Hall-Petch relationships and friction stresses of the
materials. Co
20
(CrNi)
80
was successfully predicted as the alloy showing the highest strength among the
MEAs by the modified Labusch model (so-called mean field Labusch model) for solution hardening.
Experimental values of the friction stresses were found to fit with the model very well, indicating that
the strength of the alloys was closely related to entirely distorted crystal lattice acting as high-density
obstacles for dislocation motion in the alloys. At the same time, values of the average lattice distortion
in the alloys were found comparable to those in some dilute alloys, although “severe”lattice distortion
had been believed as a reason for the higher strength than dilute systems. Finally, a strengthening
mechanism by element-element interaction was proposed as an additional mechanism to enhance the
strength in FCC HEAs and MEAs.
©2019 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
1. Background and purpose
The emergence of high entropy alloys (HEAs), proposed by Yeh
et al. [1] and Cantor et al. [2] independently in 2004, has vastly
expanded the possibility of new multicomponent metallic alloys
[3e5]. HEAs are defined as the alloys composed of five or more
principal elements with concentrations of 5e35 at. %, and it has
been believed the following four kinds of “core effects”characterize
the nature of HEAs [3]: (1) high entropy effect, (2) cocktail effect, (3)
sluggish diffusion kinetics and (4) severe lattice distortion effect. In
the high entropy effect (1), it has been said that single-phase crystal
structure such as simple FCC and BCC is stabilized owing to the
large configuration entropy of HEAs, and single-phase solid solu-
tion could be easily obtained by increasing the number of alloying
elements. However, Senkov et al. [6] reported that complex
multiphase structures including intermetallics could be more likely
to form with increasing the number of alloying elements based on
their calculation results of a large-scale thermodynamical
screening with CALPHAD method. Their results suggested that the
configuration entropy was not only an important factor but the
mixing enthalpy also contributed to the phase stability in multi-
component alloys significantly, which agrees with the fact that
every equi-atomic alloys of 5 or more elements do not necessarily
show single-phase solid solution independent of combinations of
elements [7]. In the cocktail effect (2), the alloys are believed to
exhibit characteristics of each composing pure metal. For example,
to realize light-weighted HEAs, light elements such as Al have been
added to FCC HEAs [8e11 ]. Li et al. [12e15] achieved a good
*Corresponding author.
E-mail addresses: yoshida.shuhei.26x@st.kyoto-u.ac.jp (S. Yoshida), ikeuchi.
takuto.56m@st.kyoto-u.ac.jp (T. Ikeuchi), bhattacharjee.tilak.6w@kyoto-u.ac.jp
(T. Bhattacharjee), bai.yu.6m@kyoto-u.ac.jp (Y. Bai), shibata.akinobu.5x@kyoto-u.
ac.jp (A. Shibata), nobuhiro-tsuji@mtl.kyoto-u.ac.jp (N. Tsuji).
Contents lists available at ScienceDirect
Acta Materialia
journal homepage: www.elsevier.com/locate/actamat
https://doi.org/10.1016/j.actamat.2019.04.017
1359-6454/©2019 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
Acta Materialia 171 (2019) 201e215
strength-ductility balance in a Co-Cr-Fe-Mn-Ni-C HEA by adjusting
the amount of Mn, C, and so on, which are elements to lower the
stacking fault energy of the corresponding alloys. The sluggish
diffusion kinetics of HEAs (3) was firstly reported by Tsai et al. in a
CoCrFeMnNi equi-atomic HEA [16], comparing its activation energy
for diffusion with those of pure metals and austenitic stainless
steels. Da˛browa et al. [17] recently confirmed sluggish diffusion
kinetics in Al-Co-Cr-Fe-Ni HEAs. Further details are necessary to
clarify the mechanisms of sluggish diffusion in future. Severe lattice
distortion (4) has been said to be the main reason for the high
strength of HEAs [18,19]. Particularly, Cr was reported to act as a
large-sized atom in Ni-containing FCC equi-atomic alloys such as
CoCrFeMnNi HEA, and the concentrations of Cr had a huge impact
on the strength of such alloy systems [20e22]. However, a quan-
titative understanding of the severe lattice distortion effect is
lacking. One of the reasons for making it difficult for us to deal with
the effect of lattice distortion is that there has been no theoretical
model effectively explaining the strength of HEAs.
For conventional binary alloys, Fleisher [23,24] and Labusch [25]
theoretically formulated the solid solution strengthening effect.
Especially, Labusch concerned relatively concentrated systems
having solute concentrations up to 20 at. % based on the statistical
theory that had been used to understand superconductivity of
materials. Their models have been validated by countless experi-
mental data, and have sometimes been modified to apply to dilute
multicomponent alloys [26].
Recently, Wu et al. [27] tried to apply the Labusch model to
medium entropy alloys (MEAs), which are defined as the alloys
composed of four or less principal elements with equimolar or
near-equimolar concentration. They arbitrary distinguished solutes
and solvent in the MEAs that were subsystems of CoCrFeMnNi HEA,
and assumed that the strength of the alloys could be expressed by a
linear combination of Labusch model. Finally, they compared the
calculated strength with their experimental data and found that the
strength could not be perfectly explained by the Labusch model.
Several other attempts have been made to establish a new
model that can predict the strength of high-alloy systems like HEAs
and MEAs. Varvenne et al. [28] utilized the effective matrix model
that had been employed to solve many-body problems in quantum
mechanics to predict the strength of HEAs. By comparing obtained
results with the experimental data of noble metal HEAs, they
confirmed that the model worked well [29]. Okamoto et al. [20]
found a correlation between the mean square atomic displacement
(MSAD) (lattice distortion in other words) and the strength of
subsystems of the CoCrFeMnNi HEA, by using first-principle
calculation. Chen et al. [19] also obtained a similar correlation in
BCC HEAs and MEAs. Although some of the previous studies suc-
cessfully predicted the strength of HEAs and MEAs, underlying
physics including the contribution of lattice distortion had not been
fully understood yet.
In the present study, the detailed strengthening mechanism and
essential nature of high-alloy systems such as HEAs and MEAs are
studied by experimentally investigating the effect of elemental
combinations on the friction stress and Hall-Petch relationship for
various MEAs. We fabricate fully-recrystallized specimens of
various kinds of MEAs having FCC single-phase with wide ranges of
different mean grain sizes by the use of high-pressure torsion (HPT)
and subsequent annealing processes. Tensile tests are performed at
room temperature, and precise Hall-Petch relationships are sys-
tematically obtained for the alloys. As was described in our previ-
ous paper, the friction stresses of the alloys are evaluated from the
obtained Hall-Petch relationship [22]. Afterward, the Labusch
model for solution hardening is modified to establish an effective
model for predicting the strength of high-alloy systems, so-called
the mean field Labusch model, and our experimental data are
compared with the model. Finally, strengthening mechanisms
which are responsible for the high strength of high-alloy systems
such as HEAs and MEAs are discussed by comparing with the data
of dilute systems.
2. Materials and experimental methods
2.1. Starting materials
We selected five equi-atomic subsystems of the CoCrFeMnNi
HEA, i.e., CoCrFeNi, CoFeMnNi, CoFeNi, CoMnNi, FeMnNi, as well as
two non-equi-atomic subsystems (CoNi)
(100-x)
Cr
x
(x ¼5.0, 20 at. %)
as the starting materials for the experiments carried out in the
present study. Wu et al. [7] have already confirmed the phase sta-
bility of some of these alloys. Additionally, we also selected
Co
20
(CrNi)
80
that was predicted as the alloy showing the highest
strength among the subsystems of CoCrFeMnNi HEA, as will be
discussed later. Ingots of the alloys were produced by arc-melting of
constituent elements with a purity higher than 99.9 wt % in a
water-cooled copper mold under a protective Ar atmosphere. Since
Mn was oxidized easily, the pieces of pure Mn were cleaned in a
solution of 25 vol % nitric acids and 75 vol % C
2
H
5
OH at room
temperature just before melting. Additionally, because of the high
vapor pressure of Mn, pure Mn was sandwiched between the pieces
of other constituent metals to minimize evaporation. Even after
such procedures, there was a loss of Mn during melting, so that an
extra amount Mn (þ5.0 wt %) was added to compensate for the
weight loss in melting and casting of each Mn-containing alloy. The
arc-melted buttons were flipped and re-melted at least five times to
improve compositional homogeneity.
The arc-melted buttons were also subsequently cold-rolled by
80e90% reduction in thickness and homogenized (fully-recrystal-
lized) at 1200
C for 24 h under a vacuum environment
(~8.0 10
4
Pa) using a vacuum tube furnace to remove macro-
segregations in the alloys. By the use of energy dispersive X-ray
(EDX) spectroscopy, it was confirmed that the chemical composi-
tions of the constituent elements in the alloys were very closeto the
designed compositions and the standard deviations of the con-
centrations were small enough to neglect the segregation of the
elements. (See the supplementary material S1.)
2.2. Deformation and annealing
Disc-shaped specimens with a diameter of 10 mm and a thick-
ness of 0.80 mm for HPT were prepared by cutting the homoge-
nized materials. The HPT process was performed at room
temperature under a pressure of 6.0 GPa at a speed of 0.20 rpm
(rotation per minute). The total rotation angle applied was 1800
(five rotations). The shear strain,
g
, applied to the disc specimens
during the HPT process can be calculated as
g
¼2
p
r
tN;(1)
where ris the radial position in the disc, Nis the number of rota-
tions, and tis the thickness of the disc. Based on Eq. (1), the
maximum shear strain applied to the materials by five rotations of
HPT can be as large as
g
¼196.
After the heavy deformation by HPT, specimens were annealed
at various temperatures ranging from 700
Cto1100
C for various
periods ranging from 20 s to 7.2 ks.
2.3. Microstructural observations
Specimens for microstructural observation were mechanically
S. Yoshida et al. / Acta Materialia 171 (2019) 201e215202
polished by using 1000e4000 grid sized fine sandpapers and then
diamond pastes (1.0e3.0
m
m) to achieve mirror-like surfaces. Af-
terward, the specimens were electrically polished in a solution of
10 vol % HClO
4
and 90 vol % C
2
H
5
OH at 30 V for 15 s at room
temperature.
Microstructural observations were performed by using an FE-
SEM (JEOL, JSM-7100/7800F) equipped with backscattered elec-
tron (BSE) and electron backscattering diffraction (EBSD) detectors.
Observations were carried out on the section perpendicular to the
normal direction of the disc at the position around 3.0 mm from the
center of the disc specimens. The observed position corresponded
to the gauge part of the tensile specimens. The working distance
was set at 3.0e10 mm for BSE and 15 mm for EBSD. The data
collection of EBSD was conducted using the software produced by
TSL solution (TSL-OIM data collection, analysis ver. 5.31), and the
OIM analysis software (Tex SEM Laboratories, ver. 7.0) was used to
analyze the collected data. Grain sizes of the specimens were
determined by the line intercept method on SEM-BSE images and
EBSD grain boundary maps showing high-angle grain boundaries
with misorientations larger than 15
.
2.4. Tensile tests
Tensile tests were performed on a universal tensile test machine
(SHIMADZU, AG-100 kN Xplus) at room temperature at a quasi-
static strain rate of 8.3 10
4
s
1
for characterizing the mechani-
cal properties of the alloys. Small-sized tensile specimens with a
gage length of 2.0 mm and a gage cross section of 1.0 0.50 mm
were cut from the annealed specimens having FCC single-phase so
that the center of the gage part coincided with the position at a
radial distance of 3.0 mm from the center of the disc samples. We
have confirmed in the previous study that the small-sized speci-
mens could give reliable stress-strain behaviors equivalent to those
obtained from standard size specimens of the same materials [22].
Due to the difficulty in attaching the extensometer or strain gauge
on the small-sized specimens, the displacement of the gage section
was precisely measured by a CCD video camera extensometer
(SVS625MFCP), and the strain was calculated by the use of a digital
image correlation (DIC) technique. Surfaces of all the tensile spec-
imens were painted with white and black inks to produce a random
dot pattern (speckle pattern) acting as markers easily tracked
during deformation for the DIC analysis. In-situ DIC measurements
were achieved from a set of strain snapshots collected automati-
cally by using a Vic-2D software [30].
3. Theoretical models
3.1. Mean field Labusch model
In conventional dislocation theory, it is considered that solute
atoms in a crystal lattice act as obstacles for dislocation motion,
mainly due to elastic fields around them. According to the solid
solution strengthening model proposed by Labusch [25], in which
dislocations are assumed to interact with multiple weak obstacles
on a slip plane (Fig. 1 (a)), the solid solution strengthening effect in
dilute alloys with FCC structure is expressed as
t
ð0KÞ
G
f
h
0
loc
2
þ
a
2
ε
2
loc
2
3
r
2
3
;(2)
where
t
(0 K) is the critical resolved shear stress (CRSS) for dislo-
cation slip at 0 K, Gis the shear modulus,
h
0
loc
is the local modulus
misfit, ε
loc
is the local atomic size misfit (local lattice distortion),
a
is a constant depending on the type of dislocation considered
(Labusch assumed
a
¼16), and
r
is the number density of the
obstacles on a slip plane. It is reasonable, in binary systems, to as-
sume
r
~c, the concentration of solutes. It is well-known that this
solid solution strengthening model can effectively explain the
strength of binary alloys and dilute multicomponent alloys unless
the concentration of solutes reaches about 20 at. %.
On the other hand, in highly-concentrated systems including
MEAs and HEAs, the crystal lattice is significantly (and heteroge-
neously) distorted everywhere because all constituent elements
have different atomic radii. Consequently, the local position of each
atom may be largely shifted from the average lattice position
determined from the average lattice constant and crystal structure,
depending on the local arrangement of elements, so that different
lattice distortions arise locally. This lattice distortion state has been
confirmed by both of experimental and calculation studies. Oka-
moto et al. [20] employed synchrotron X-ray diffraction and a first-
principle calculation to determine MSAD as a parameter for eval-
uating the level of the local lattice distortion. They predicted that Cr
and Mn made MSAD large, which implied that these two elements
had great impacts on the strengthening of Ni-based FCC solid
Fig. 1. Schematic illustrations showing distributions of obstacles around a dislocation
on a slip plane in (a) conventional dilute systems and (b) high-alloy systems such as
MEAs and HEAs, respectively. (c) Relationship between the strength of an isomorphous
system and solute concentration.
S. Yoshida et al. / Acta Materialia 171 (2019) 201e215 203
solution alloys. Oh et al. [21] investigated the distribution of atomic
bonding length by combining a first-principle calculation with X-
ray absorption fine structure spectroscopy (XAFS). Their results
suggested that Cr and Mn had wider bonding length distributions
than those of other constituent elements in the CoCrFeMnNi HEA.
Even the average lattice distortion was very small, Cr and Mn could
behave as if they had large atomic radii, resulting in strengthening
by the local lattice distortion in the HEA. Based on these studies, we
expect that dislocations in the crystal lattice of HEAs and MEAs are
always affected by the elastic field caused by the local lattice
distortion state. Now, we consider such a situation as a mean field
of obstacles where dislocations are surrounded by high-density
weak obstacles having mean values of atomic size misfit and
modulus misfit(Fig. 1 (b)). So that we can replace
h
0
loc
and ε
loc
by
the average modulus misfit parameter,
h
0
ave
, and the average atomic
size misfit parameter (the average lattice distortion), ε
ave
, respec-
tively. In our hypothesis, the number density of mean obstacles
becomes a constant value,
r
~ const., near the equi-atomic
composition, independent of the kind of alloying elements. This
assumption may be reasonable when we consider an analogous of
isomorphous binary systems showing a peak or plateau of the
strength around 50 at.% concentration, where the strength is
almost independent of the solute concentration, as shown in Fig. 1
(c). Thus, Eq. (2) can be modified as
t
ð0KÞ
G
f
h
0
ave
2
þ
a
2
ε
2
ave
2
3
;(3)
Also, since
h
0
ave
is small enough compared to
a
ε
ave
in many FCC
metals (See the supplementary material S2), Eq. (3) can be
simplified as
t
ð0KÞ
G
fε
4
3
ave
:(4)
Wu et al. [31 ] already reported the shear modulus, G,and
temperature dependence of the yield strength in various kinds of
HEAs and MEAs. In this study, the friction stress,
s
0
, at room
temperature (the strength of materials excluding the effect of the
grain size) were determined from the intercepts of the Hall-Petch
relationships at room temperature we experimentally acquired,
and the experimentally obtained values of
s
0
were extrapolated
to 0 K by using the results reported by Wu et al. [31], assuming
that the Hall-Petch slope was independent of temperature in a
low-temperature region. (See the supplementary materials S3)
Afterward,
t
(0 K) was determined by dividing
s
0
at 0 K by the
Taylor factor for FCC materials having random texture (¼3.06)
because recrystallization texture of FCC metals was usually very
weak.
3.2. Estimation of average atomic size misfit parameter
Toda-Caraballo et al. [32,33] proposed an effective model for
estimating the average lattice constant, a, and the average atomic
size misfit parameter, ε
ave
, of HEAs and MEAs. In their model, the
interatomic distance between element iand first-neighboring
element jis expressed as
s
ij
¼s
2
ii
K
i
x
i
þs
2
jj
K
j
x
j
s
ii
K
i
x
i
þs
jj
K
j
x
j
;(5)
where s
ii
is the atomic size of pure elements, K
i
is the bulk modulus
of pure elements and x
i
is the concentration of the elements. The
average lattice constant of the alloy, a, is calculated as
a¼fX
ij
s
ij
x
i
x
j
;(6)
where fis a constant depending on the crystal structure assuming a
rigid sphere model (e.g., f¼ffiffiffi
2
pfor FCC). The applicability of this
model was checked by comparing the calculated average lattice
constant with the lattice parameters experimentally obtained by X-
ray diffraction (XRD) in the present study. (See the supplementary
material S4.) If a small amount (
D
x
j
i
) of element iin the average
matrix of the alloys is replaced by element j, the average lattice
constant, a, will be slightly changed to (aþ
D
a
j
i
), and this small
variation of the average lattice constant, da/dx
j
i
, can be analytically
calculated as
da
dx
j
i
¼aþ
D
a
j
i
a
D
x
j
i
:(7)
Note that the average lattice constant obtained by Eqs. (5) and
(6) is continuous and differentiable against the concentration,
which means that the materials we used are single-phase solid
solutions in the arbitrary concentration regions where any
discontinuous change in the lattice constant (by phase trans-
formation, for example) does not happen. Moreover, the average
atomic size misfit of the alloy, ε
ave
, can be obtained as
ε
ave
¼X
ij
1
a
da
dx
j
i
x
i
x
j
(8)
In the present study, we calculated the ε
ave
for the MEAs and
HEAs by using Eq. (8).
3.3. Prediction of alloys with high strength
Based on Eq. (4), the alloys with large ε
ave
are expected to show
high strength. We employed Eq. (8) for optimizing the composition
of the alloy and predicted the alloy with the highest strength (the
largest ε
ave
) among subsystems of CoCrFeMnNi HEA. The metallic
bond length of pure metals was used as the interatomic distance.
The interatomic distance, s
ii
, and bulk modulus, K
i
, used for the
calculation were presented in Table S2 in the supplementary ma-
terials (S5) [34]. The calculation was performed on all possible
elemental combinations by changing the concentration of each
element, x
i
, by 0.2 at. %.
4. Results
4.1. Alloys with large average atomic size misfit parameters
The examples of the calculated results showing large average
atomic size misfit parameters, ε
ave
, are summarized in Table 1.It
should be noted that the model we employed did not take the
phase stability of the alloys into account, and we need to exclude
Table 1
Predicted alloys with large average atomic size misfit pa-
rameters, ε
ave
.
Alloy ε
ave
Cr
42
Mn
6
Fe
4
Co
6
Ni
42
0.02449
Cr
40
Mn
8
Fe
4
Co
8
Ni
40
0.02448
Cr
40
Mn
10
Co
10
Ni
40
0.02443
Cr
46
Co
10
Ni
44
0.02426
Cr
40
Fe
12
Co
10
Ni
38
0.02418
Co
20
(CrNi)
80
0.02366
S. Yoshida et al. / Acta Materialia 171 (2019) 201e215204
alloys showing multiple phases from the lists based on existing
experimental reports of phase diagrams. In addition to five equi-
atomic subsystems of the CoCrFeMnNi HEA, i.e., CoCrFeNi,
CoFeMnNi, CoFeNi, CoMnNi, FeMnNi, and two non-equi-atomic
subsystems (CoNi)
(100-x)
Cr
x
(x ¼5, 20 at. %), we selected
Co
20
(CrNi)
80
with ε
ave
¼0.02366, which was very close to the
highest value among calculated results, for the subsequent exper-
iments since it was expected that the alloys could exhibit FCC
Fig. 2. Representative SEM-BSE images of the studied alloys heavily deformed by HPT and subsequently annealed under different conditions. FeMnNi annealed at (a) 850 C for
600 s and (b) 750 C for 120 s. CoMnNi annealed at (c) 850 C for 1.2 ks and (d) 750 C for 30 s. CoFeNi annealed at (e) 90 0 C for 600 s and (f) 750 C for 1.2 ks. (CoNi)
80
Cr
20
annealed
at (g) 1000 C for 600 s and (h) 800 C for 30 s (CoNi)
95
Cr
5
annealed at (i) 900 C for 600 s and (j) 750 C for 30 s. CoFeMnNi annealed at (k) 900 C for 30 s and (l) 750 C for 30 s.
CoCrFeNi annealed at (m) 1100 C for 1.2 ks and (n) 800 C for 30 s and Co
20
(CrNi)
80
annealed at (o) 850 C for 7.2 ks and (p) 850 C for 20 s. Mean grain sizes counting twin
boundaries (d) are indicated.
(Continued): Representative SEM-BSE images of the studied alloys heavily deformed by HPT and subsequently annealed under different conditions. FeMnNi annealed at (a) 850C
for 600 s and (b) 750 C for 120 s. CoMnNi annealed at (c) 850 C for 1.2 ks and (d) 750 C for 30 s. CoFeNi annealed at (e) 900 C for 600 s and (f) 750 C for 1.2 ks. (CoNi)
80
Cr
20
annealed at (g) 1000 C for 600 s and (h) 80 0 C for 30 s (CoNi)
95
Cr
5
annealed at (i) 900 C for 60 0 s and (j) 750 C for 30 s. CoFeMnNi annealed at (k) 900 C for 30 s and (l) 750 C for
30 s. CoCrFeNi annealed at (m) 1100 C for 1.2 ks and (n) 800 C for 30 s and Co
20
(CrNi)
80
annealed at (o) 850 C for 7.2 ks and (p) 850 C for 20 s. Mean grain sizes counting twin
boundaries (d) are indicated.
S. Yoshida et al. / Acta Materialia 171 (2019) 201e215 205
single-phase according to a previous report on phase equilibria in
Co-Cr-Ni ternary systems [35].
4.2. Microstructures after annealing process
Representative microstructures obtained by SEM-BSE observa-
tions of the studied alloys heavily deformed by HPT and subse-
quently annealed are shown in Fig. 2(aep). Fully-recrystallized
microstructures with various mean grain sizes were obtained. It
was also confirmed by the use of XRD and EBSD that all the spec-
imens used in the present study showed FCC single-phase. Note
that specimens annealed at low temperature and showed second
phases were not used for the proceeding experiments. (See the
supplementary materials S6.) Large numbers of annealing twins
were observed in most alloys. In some alloys (e.g., CoFeNi (e, f)), the
density of annealing twins was relatively small possibly due to the
effect of stacking fault energy. Mean grain sizes counting twin
boundaries (d) are indicated in the figures. Ultra-fine grained (UFG)
microstructures with mean grain size smaller than 1
m
mwere
attained by changing the annealing temperature and holding time,
which suggested high-density nucleation sites for recrystallization
were introduced by the HPT process. It was also suggested in some
Fig. 2. (continued).
S. Yoshida et al. / Acta Materialia 171 (2019) 201e215206
previous studies that significant grain refinement could be ach-
ieved in HEAs due to the slow kinetics of grain growth derived from
their sluggish diffusion nature [36,37]. However, in some alloys
(e.g., CoFeNi (e, f)), the grain size after annealing was relatively
large indicating that there can be elemental effects on the micro-
structure evolution. A systematic investigation on recrystallization
and grain growth behaviors in the present MEAs and HEA has been
conducted in our on-going study.
4.3. Mechanical properties and Hall-Petch relationship
Fig. 3(aeh) show engineering stress-strain curves of the studied
alloys with various mean grain sizes, obtained by the tensile tests at
room temperature. In all the materials, the yield strength increased
with decreasing the mean grain size. This is the well-known Hall-
Petch effect [38,39] in which the yield stress (
s
Y
) of polycrystalline
materials is described as
s
Y
¼
s
0
þkd
1
2
;(9)
where
s
0
is the friction stress, kis the constant (the Hall-Petch
slope), and dis the mean grain size of the material. It was note-
worthy, moreover, that many of the specimens with UFG micro-
structures exhibited discontinuous yielding, as has been previously
reported in other UFG materials with FCC crystal structure [40e43].
Generally, there is a trade-off relationship between strength and
tensile ductility in metallic materials, and most of the alloys we
used also showed the same tendency. However, interestingly, some
of the alloys (e.g., (a) FeMnNi, (e) (CoNi)
95
Cr
5
, etc.) were found not
to follow the trade-off relationship of strength and ductility. This
might be related to the change in deformation mechanisms leading
to different work-hardening behaviors depending on the mean
grain size, which has been under investigation.
Hall-Petch relationships obtained from the tensile tests of the
alloys studied in the present study are shown in Fig. 4(aeh). Fairly
Fig. 3. Engineering stress-strain curves of the studied alloys with various mean grain sizes, obtained by the tensile tests at room temperature. (a) FeMnNi, (b) CoMnNi, (c) CoFeNi,
(d) (CoNi)
80
Cr
20
, (e) (CoNi)
95
Cr
5
, (f) CoFeMnNi, (g) CoCrFeNi, and (h) Co
20
(CrNi)
80
, respectively, all heavily deformed by HPT and subsequently annealed under different conditions to
achieve various grain sizes.
(Continued): Engineering stress-strain curves of the studied alloys with various mean grain sizes, obtained by the tensile tests at room temperature. (a) FeMnNi, (b) CoMnNi, (c)
CoFeNi, (d) (CoNi)
80
Cr
20
, (e) (CoNi)
95
Cr
5
, (f) CoFeMnNi, (g) CoCrFeNi, and (h) Co
20
(CrNi)
80
, respectively, all heavily deformed by HPT and subsequently annealed under different
conditions to achieve various grain sizes.
S. Yoshida et al. / Acta Materialia 171 (2019) 201e215 207
precise and reliable Hall-Petch curves could be obtained after col-
lecting a number of data plots from the present experiments using
specimens with wide ranges of the mean grain sizes. The extra-
hardening phenomenon, in which the Hall-Petch slope became
larger in fine grain-size regions than that in coarse grain-size re-
gions, was observed. The extra hardening phenomenon has been
reported in other UFG materials [44,45], whereas the reason is still
unclear. The friction stress,
s
0
of each alloy was obtained by fitting
the data plots with Eq. (9), and the fitting lines are presented as
solid lines in Fig. 4. Young's modulus, E, Shear modulus, G, obtained
s
0
,k, and the grain size ranges of the alloys used for fitting are
summarized in Table 2 together with the values for pure Ni, Ni-
40Co, CoCrNi MEA, and CoCrFeMnNi HEA previously reported
[22,31,46,47]. All experimentally obtained values of the friction
stress at room temperature for the present alloys are compared in
Fig. 5, where the
s
0
values were divided by Young's modulus, E,to
remove the effect of elastic modulus. It was confirmed that
Co
20
(CrNi)
80
showed the highest friction stress (
s
0
¼280 MPa,
s
0
/
E¼1.17 10
3
) among the alloys. It should be noted that the in-
crease in the number of alloying elements does not simply result in
an increase in the strength (
s
0
).
5. Discussion
5.1. Effect of lattice distortion
Fig. 6 (a) shows a relationship between the average atomic size
misfit (lattice distortion) calculated by the Toda-Caraballo model
and the strength of the alloys at 0 K normalized by the shear
modulus (G). The experimentally obtained values of
s
0
were
extrapolated to 0 K by using the data of various FCC HEAs and MEAs
reported by Wu et al. [31] (See the supplementary materials S7 for
the details). Afterward,
t
(0 K) was determined by dividing
s
0
at 0 K
by the Taylor factor for FCC materials having random texture (¼
3.06). The solid black line is a fitting line based on the mean field
Labusch model (Eq. (4)) assuming that elements are nearly
randomly distributed in the alloys. It is clear that Eq. (4) fits well
with the normalized strength with a correlation of more than 97%.
The consequence indicates that crystal lattices of equi-atomic
multicomponent alloys are entirely distorted due to the differ-
ence of atomic sizes of the constituent alloying elements, and dis-
locations in such crystal lattices are considered to be always
affected by the elastic fields originated from the lattice distortion
Fig. 3. (continued).
S. Yoshida et al. / Acta Materialia 171 (2019) 201e215208
state, i.e., dislocations are surrounded by the high-density weak
obstacles with average atomic size misfit and modulus misfit (as
illustrated in Fig. 1 (b)).
It is interesting that the datum point of the Co
20
(CrNi)
80
alloy,
which showed the highest normalized strength (
t
(0 K)/
G¼2.27 10
3
), follows the tendency as well as other MEAs in
Fig. 6 (a). This means that the subsystems of CoCrFeMnNi HEA with
higher strength could be successfully predicted by the mean field
Labusch model combining the Toda-Caraballo model. Therefore, we
believe that the prediction method used in the present study would
be effective for designing new alloys with higher strength.
In Fig. 6 (b), the data of some dilute alloys (Ag alloys, Cu alloys,
etc.), which were obtained from previous literature [48e50], are
plotted together with the present experimental data shown in Fig. 6
(a). The shear modulus of the dilute alloys were roughly estimated
by the simple rule of mixture for the values of constituent pure
elements, and the effect of the grain size was not excluded because
of the lack of data. In addition, the average atomic size misfit pa-
rameters of the dilute alloys that reflect “average”lattice distortion
were determined by Eq. (8), which was different from the definition
of the local atomic size misfit parameters reflecting “local”lattice
distortion (ε
loc
¼
1
ada
dc
), which has been usually used for the con-
ventional Labusch model (Eq. (2)). Thus, we can simply compare
the strength of different materials as a function of the average
atomic size misfit (average lattice distortion). It is very interesting
that values of the average atomic size misfits for the dilute alloys
distribute in a wide range up to 0.01, which is comparable to the
distribution of the misfit values for the MEAs and HEA, although it
has been believed that the reason for the high strength of HEAs is
due to the “severe”lattice distortion effect. In fact, the atomic size
differences in some dilute alloys (8% in Ag-Zn and 12% in Al-Mg
pairs, for instance) are much larger than that in CoCrFeMnNi HEA
(3% in the maximum for a Cr-Ni pair). This means that the local
lattice distortion in dilute alloys can be much larger than that in
HEAs and MEAs. However, the density of obstacle is very small in
dilute alloys, so that similar average atomic size misfit parameters
are obtained after averaging the local lattice distortions in an
overall slip plane based on Eq. (8). It should be noted, on the other
Fig. 4. Hall-Petch relationships of the alloys studied: (a) FeMnNi, (b) CoMnNi, (c) CoFeNi, (d) (CoNi)
80
Cr
20
, (e) (CoNi)
95
Cr
5
, (f) CoFeMnNi, (g) CoCrFeNi, and (h) Co
20
(CrNi)
80
. The solid
black lines are fitting lines based on Eq. (9) in the grain size ranges shown in Table 2. Concrete Hall-Petch equations obtained are shown in the figures.
(Continued): Hall-Petch relationships of the alloys studied: (a) FeMnNi, (b) CoMnNi, (c) CoFeNi, (d) (CoNi)
80
Cr
20
, (e) (CoNi)
95
Cr
5
, (f) CoFeMnNi, (g) CoCrFeNi, and (h) Co
20
(CrNi)
80
.
The solid black lines are fitting lines based on Eq. (9) in the grain size ranges shown in Table 2. Concrete Hall-Petch equations obtained are shown in the figures.
S. Yoshida et al. / Acta Materialia 171 (2019) 201e215 209
hand, that the strength of the MEAs and HEA is found to be much
higher than that of the dilute alloys even at an equivalent average
lattice distortion (average atomic size misfit) in Fig. 6 (b). This
suggests that the high strength of HEAs and MEAs cannot be simply
explained only by the effect of lattice distortion.
5.2. Strengthening by element-element interaction
In conventional dislocation theory, dislocations in crystal lat-
tices of random dilute alloys are thought to interact mainly with
elastic fields around solute atoms caused by the effect of atomic size
misfit and modulus misfit (as illustrated in Fig. 7 (a)), resulting in
solid-solution strengthening. In this case, the local potential energy
for a dislocation segment in dilute alloys,
D
E
DA
, can be given as the
superposition of the local Peierls potential energy in the solvent
pure metal,
D
E
PM
, and the energy contribution of lattice distortion
caused by the solute atom,
D
E
LD
,(Fig. 7 (c)). The average potential
energy along the dislocation line, <
D
E
DA
>, can be obtained as a
composite of
D
E
PM
and
D
E
LD
(
D
E
DA
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
D
E
PM
Þ
2
þð
D
E
LD
Þ
2
q)(Fig. 7
(e)). (See the supplementary material S8 for the mathematical
background of the additivity.) Due to the contribution of
D
E
LD
,
dislocations have to overcome potential energy (
D
E
DA
) larger than
the case of pure metals (
D
E
PM
) resulting in strengthening. It should
be noted that these discussions can be valid if solid solutions are
“random”and solute-solute interaction is negligible.
Similar to the dilute alloys, an effect of “severe”lattice distortion
seems to contribute to strengthening in HEAs and MEAs, and we
have considered the effect of lattice distortion using the mean field
Labusch model. However, the result shown in Fig. 6 (b) indicated
that the “severe”lattice distortion was not only the reason for the
high strength in high-alloy systems, but there also seemed to be an
additional strengthening mechanism, since the strength of high-
alloy systems was much higher than that of dilute systems even
at an equivalent level of the average lattice distortion (average
atomic size misfit). Here, we propose a new strengthening mech-
anism by element-element interaction originated from weak
elemental inhomogeneity (deviation from “random”state) in equi-
atomic or near-equi-atomic alloys for understanding the additional
strengthening mechanism suggested by the results obtained in the
present study.
In general, elements are not always distributed randomly but
inhomogeneously to minimize elastic and electromagnetic inter-
action energy between constituent elements in an equilibrium
state. In case of that there is an attractive interaction between el-
ements, for instance, they tend to form order by periodically
Fig. 4. (continued).
S. Yoshida et al. / Acta Materialia 171 (2019) 201e215210
arranging different elements. If the periodicity persists in a scale of
several atoms, the phenomenon is called short-range ordering
(SRO). On the other hand, if the periodicity persists in a scale of
grains, the phenomenon is called long-range ordering (LRO), where
crystal structure can be described by defining sublattices. In case of
that there is a repulsive interaction between elements, same the
kinds of elements tend to form clusters in a solid solution, which is
called clustering. Particularly, the case of clustering persisting in a
scale of several atoms is called short-range clustering (SRC).
Although detecting weak atomic inhomogeneities (SRO and SRC) by
using conventional methods (such as STEM-EDX, 3D atom probe
tomography, etc.) is challenging, it has been reported in some HEAs
and MEAs with weak SRO and SRC. For example, Zhang et al. [51]
employed EXAFS and found that there is weak SRO in Cr-Ni and Cr-
Co pairs in CoCrNi MEA. In some HEAs containing Cr or Cu, clus-
tering has been reported [52,53]. In addition, the existence of SRO
in many MEAs has been predicted theoretically, for example by
Yuge et al. [54]. Since various kinds of elements with higher con-
centration are mixed in high-alloy systems, the interaction be-
tween constituent elements (element-element interaction) in
concentrated solid solutions can be inherently diverse.
These inhomogeneities are responsible for the strength of solid
solutions, which is not simply because it often increases the value
of ε
ave
. When a dislocation passes through the inside of ordered or
clustered regions, the most stable arrangement of elements (SRO or
SRC state) can be destroyed by shifting one atomic plane along
Burger's vector which causes extra resistance to the dislocation
motion. For example, now, we consider an FCC crystal lattice of a
high-alloy system containing an edge dislocation illustrated in
Fig. 7 (b), where neighbors of the atom A (of an element) are the
elements 1, 2, and 3 in the initial state, and it is assumed that there
are attractive interaction between atom A and element 1 (SRO
state) and repulsive interaction between atom A and element 4
(SRC state). After a dislocation passing through, the neighbors of
atom A changes to the elements 2, 3, and 4 shifting one atomic
plane along Burger's vector, where atom A leaves element 1 and
meets element 4. In such a case, extra activation energy,
D
E
ee
,
essentially derived from the change in bonding energy and lattice
strain energy caused by the change of the local arrangement of
elements, should be required for dislocations to move in the crystal
lattice in addition to
D
E
PM
and
D
E
LD
, as shown in Fig. 7 (d). It should
be noted that we consider the interaction with only first-
neighboring elements to simplify the discussion, but in some
elemental combinations higher-order neighbors should be taken
into account [55]. Similar consideration was formulated for the
dislocation cross slip in FCC binary alloys with various solute-
concentration by N€
ohring and Curtin, recently [55,56].
In dilute alloy systems, the change in the atomic arrangement
around solute atoms before and after a dislocation passes through
rarely happens because the majority of atoms are the solvent
element, and the effect of the inhomogeneity of solid solution on the
strength can be limited. In this case, it is enoughto consider only the
effect of lattice distortion, as Labusch did [25]. Exceptions are the
alloys showing relatively strong solute-solute interaction and SRO or
SRC such as Al-Mg alloys [57], showing higher normalized stress
comparable to HEAs and MEAs in Fig. 6 (b). On the other hands, in
high-alloy systems, since the concentration of alloying elements is
high, even weak element-element interaction
*
(SRO or SRC) can
have a significant impact on the strength. (
*
We named the inter-
action among alloying elements as element-element interaction,
different from “solute-solute interaction”, because we cannot
distinguish between solutes and solvent in HEAs and MEAs.) The
destruction of the stable arrangement of elements would lead to
larger fluctuation of local potential energy for a dislocation segment
in high-alloy systems,
D
E
HA
¼
D
E
PM
þ
D
E
LD
þ
D
E
ee
,(Fig. 7 (d)). By
averaging them along the dislocation line, we obtain the average
potential energy for a dislocation in high-alloy systems,
D
E
DA
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
D
E
PM
Þ
2
þð
D
E
LD
Þ
2
þð
D
E
ee
Þ
2
q,(Fig. 7 (f)), which is larger than that
Table 2
Young's modulus determined from the stress-strain curves, Shear modulus extracted from Ref. [27], friction stress at room temperature, Hall-Petch slope determined by the
Hall-Petch relationship, grain size range used for fitting of Eq. (9), and extrapolated strength at 0 K (
t
(0 K)) based on Ref. [31 ] for the studied MEAs, CoCrFeMnNi HEA and pure
Ni.
Alloy E[GPa] G[GPa]
s
0
[MPa] k[
m
m
1/2
]d[
m
m]
t
(0 K) [MPa] Ref.
Pure Ni 199 76 21.8 180 30.0e385 13.5 [27,31,46]
Ni-40Co 213 86 51.9 181 1.60e500 43.5 [22,27]
FeMnNi 182 73 119 381 0.848e5.52 103 [27]
CoMnNi 190 77 85.5 296 0.719e2.68 103 [27]
CoFeNi 161 60 62.6 366 5.64e68.3 83.5 [27]
CoCrNi 226 87 218 265 0.286e111 183 [22,27]
(CoNi)
80
Cr
20
223 86 167 266 0.256e5.35 113 [27]
(CoNi)
95
Cr
5
219 85 30.2 240 0.895e8.29 60.1 [27]
CoFeMnNi 186 77 108 282 0.431e3.48 101 [27]
CoCrFeNi 210 82 123 276 0.553e32.2 152 [27]
Co
20
(CrNi)
80
240 87 280 168 0.497e1.98 203 [27]
CoCrFeMnNi 202 80 125 494 4.40e155 149 [27,31,47]
Fig. 5. Friction stresses (
s
0
) of various alloys at room temperature (R. T.). The values of
friction stress were extracted from the Hall-Petch relationship shown in Fig. 4 and
Table 2, and then normalized by Young's modulus (E) to eliminate the effect of elastic
modulus.
S. Yoshida et al. / Acta Materialia 171 (2019) 201e215 211
(<
D
E
DA
>) in dilute alloys even having similar average lattice dis-
tortions to the high-alloy systems, resulting in an additional
strengthening effect in high-alloy systems. According to the previ-
ous studies, the estimated order of magnitude for the local free
energy change by element-element interaction,
D
E
ee
, in HEAs and
MEAs (~10
0
e10
1
meV/Å)[16 ] is comparable to the Peierls potential
energy in FCC metals,
D
E
PM
(~1.2 meV/Åin pure Cu for instance
Ref. [58]), and also a comparable to the energy contribution of lattice
distortion,
D
E
LD
(~10
1
meV/Å)[59]. Thus, the element-element
interaction can give a significant impact on the total potential en-
ergy for dislocation motion. The importance of element-element
interaction was also demonstrated by Zhang et al. [60] based on
their theoretical modeling of SRO in HEAs, very recently. This effect
has been omitted in conventional dislocation theory, but the present
results suggest that the strengthening the effect of the element-
element interaction cannot be neglected in high-alloy systems like
HEAs and MEAs because of particularly higher concentrations of
alloying elements.
Moreover, the effect of element-element interaction can be
significant especially in regions with larger lattice distortion, since
elemental combination with a larger change in free energy tends to
result in a larger change in an interatomic spacing [61]. Accordingly,
the strength of high-alloy systems is more sensitive to the lattice
distortion taking the element-element interaction into account.
This might be the reason why the slope of the fitting line for Eq. (4)
in Fig. 6(a) and (b) becomes larger than that for conventional dilute
systems.
A similar idea has been employed to understand the sluggish
diffusion kinetics of HEAs. Tsai et al. [16] considered the change in
the local interaction energy of elements when a vacancy migrated
to rearrange neighboring atomic configurations around the va-
cancy, and found that the standard deviation of the change in the
local potential energy in CoCrFeMnNi HEA was much larger than
those in austenitic stainless steels and other pure metals. Finally,
they concluded that the larger fluctuation of local potential energy
led to the slower vacancy migration kinetics compared to con-
ventional dilute systems and pure metals, i.e., the sluggishdiffusion
kinetics in HEAs.
Consequently, we believe that the migration of crystal defects
(vacancies (0-dimensional) and dislocations (1-dimensional))
accompanying with the local element-element interaction plays an
important role to determine the essential nature of high-alloy
systems such as MEAs and HEAs, which leads to inhibiting the
migration of crystal defects compared to conventional dilute alloys
and pure metals, resulting in, for example, the significant
strengthening and the sluggish diffusion kinetics corresponding to
the migration of dislocations and vacancies, respectively.
6. Conclusion
We have experimentally studied the mechanism and essential
nature of strengthening in various equi-atomic multicomponent
systems (high entropy alloys and medium entropy alloys)
composed of Co, Cr, Fe, Mn, and Ni. Fully-recrystallized specimens
with FCC single-phase microstructures with various grain sizes
could be successfully obtained in various MEAs by HPT and sub-
sequent annealing. Systematic tensile tests confirmed the accurate
and reliable Hall-Petch relationships, where extra-hardening phe-
nomena accompanying with discontinuous yielding in their stress-
strain curves were observed in the ultrafine-grained alloys. The
strengthening mechanisms in high-alloy systems were discussed
by comparing the friction stress experimentally obtained with the
theoretical models. The important results obtained in the present
study are as follows:
(1) Co
20
(CrNi)
80
was predicted as the alloy showing one of the
highest strength among the subsystems of CoCrFeMnNi HEA
by the consideration using the Toda-Caraballo model and the
mean field Labusch model, the latter of which was a theo-
retical model modifying the original Labusch model for dilute
alloys. It was experimentally confirmed that the Co
20
(CrNi)
80
alloy showed the highest friction stress (
s
0
¼280 MPa)
among the MEAs investigated in the present study.
(2) The experimental values of the friction stress for the HEA and
MEAs were found to fit with the mean field Labusch model
very well, indicating that the strength of the alloys was
closely correlated with their distorted crystal lattices acting
as high-density obstacles for dislocation motion. It was also
found that the average lattice distortion in dilute alloys could
Fig. 6. (a) Relationship between the average atomic size misfit parameter (the average
lattice distortion) calculated by the Toda-Caraballo model and the strength (the friction
stress at 0 K estimated by Eq. (10) and divided by the Taylor factor of random texture)
normalized by shear modulus (G) of the alloys studied in the present study. The data of
pure Ni [27,31,46], Ni-40Co [22,27,31], CoCrNi [22,27,31], and CoCrFeMnNi HEA
[27,31,47] are also plotted in the figure. The solid black line is a fitting line determined
based on the mean field Labusch model (Eq. (4)). (b) The same plot as (a), including the
data of various dilute alloys (Ag-Al, Ag-Zn, Al-Mg, Cu-Mn, Cu-Ni, Cu-Zn) [48,49] and
Fe-Mn-Si-Al TWIP steel [50]) obtained from previous literature.
S. Yoshida et al. / Acta Materialia 171 (2019) 201e215212
be comparable to that in MEAs and HEAs although the higher
lattice distortion in HEAs had been believed to be the reason
for the higher strength than dilute alloys.
(3) A strengthening mechanism due to element-element inter-
action was proposed, as an additional mechanism operating
in high-alloy systems. Considering the atomic rearrangement
mechanism, which was also considered for the slow vacancy
migration kinetics in CoCrFeMnNi HEA, we think that the
inhibition of the movement of crystal defects (such as dis-
locations and vacancies) due to the element-element inter-
action can have a significant impact on the properties (such
as strength and diffusivity) in high-alloy systems like HEAs
and MEAs.
Acknowledgments
This work was financially supported by the Elements Strategy
Initiative for Structural Materials (ESISM), the Grant-in-Aid for
Scientific Research on Innovative Area “High Entropy Alloys”(No.
18H05455), the Grant-in-Aid for Scientific Research (S) (No.
15H05767) and the Grant-in-Aid for JSPS Research Fellow (No.
18J20766), all through the Ministry of Education, Culture, Sports,
Science and Technology (MEXT), Japan. The supports are gratefully
appreciated. The authors also would like to thank Prof. W. Curtin of
Ecole Polytechnique F
ed
erale de Lausanne (EPFL), Switzerland, for
valuable comments and suggestions from him.
Fig. 7. Schematic illustrations showing changes in the arrangement of neighboring atoms caused by an edge dislocation slip in FCC crystal lattices of (a) a dilute solid solution and
(b) a high-alloy system like HEAs and MEAs, respectively. Local Peierls potential energy profiles around a dislocation segment* are schematically illustrated in (c) for the dilute
system (corresponding to (a)) and (d) for the high-alloy system (corresponding to (b)). (* We consider that the potential is different at segment by segment even along one
dislocation line since the arrangement of atoms also differs along the dislocation.)
D
E
DA
(blue line in (c)) and
D
E
HA
(red line in (d)) are the local Peierls potential energy for a
dislocation segment in dilute and high-alloy systems, respectively.
D
E
PM
,
D
E
LD
and
D
E
ee
shown in (c) and (d) indicate the energy contribution for
D
E
DA
and
D
E
HA
by lattice friction in
pure FCC metals, lattice distortion and the effect of element-element interaction, respectively. Distributions of the average total potential energies for dislocation ((<
D
E
DA
>for dilute
alloys and <
D
E
HA
>for high alloys), which were obtained by averaging the local potential energies shown in (c) and (d) along the dislocation line, are shown in (e) and (f),
respectively. The angle brackets indicate the average values of physical quantities. Equations for convoluting <
D
E
DA
>and <
D
E
HA
>are inserted in the figures. (For interpretation of
the references to color in this figure legend, the reader is referred to the Web version of this article.)
S. Yoshida et al. / Acta Materialia 171 (2019) 201e215 213
Appendix A. Supplementary data
Supplementary data to this article can be found online at
https://doi.org/10.1016/j.actamat.2019.04.017.
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