Python/耐烧蚀合金智能体/wiki/raw/sources/.cache/HfNbTaTiZr难熔高熵合金中的旋转形变孪晶.pdf.txt

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Rotational deformation twins in a HfNbTaTiZr refractory high entropy alloy
Oleg N. Senkov a,b,*, Frederick Meisenkothen c, Robert Wheeler a,d
a Air Force Research Laboratory, Materials and Manufacturing Directorate, Wright-Patterson AFB, OH 45433, United States b MRL Materials Resources LLC, Xenia Township, OH 45385, United States c National Institute of Standards and Technology, Gaithersburg, MD 20899, United States d UES, Inc., Dayton, OH 45432, United States
ARTICLE INFO
Keywords:
Refractory alloy
Refractory high entropy alloy
HfNbTaTiZr
Microstructure
Deformation twinning
ABSTRACT
Results of the analysis of deformation twins formed in a HfNbTaTiZr refractory high entropy alloy during
compression deformation at 20 ◦C – 600 ◦C are reported. All studied twins were formed by rotation of the matrix
crystal lattice around a 〈110〉 pole and have the common reciprocal twinning plane K2 = {001} and reciprocal
twinning direction η2 = <110>. At the same time, the twinning plane K1 and direction η1 depend on the degree
of rotation of K2 and η2 around the common 〈110〉 pole. The following rotational twinning modes have been
identified in HfNbTaTiZr: {114}<221> (38.94◦ rotation), {115}<552> (31.58◦), {116}<331> (26.54◦),
{117}<772> (22.84◦), and {118}<441> (20.04◦). A rotational mechanism of twinning, with the rotation axis
〈011〉 or 〈001〉, is proposed as an alternative to the commonly accepted shear mechanism.
1. Introduction
An equiatomic HfNbTaTiZr refractory high entropy alloy (RHEA),
which is also called the Senkov alloy [1], is currently one of the most
studied refractory high entropy alloys (RHEAs) because of its unique
mechanical properties and processability [2-5]. In particular, it has high
strength, good strain hardening behavior, excellent malleability and
good tensile ductility in a wide temperature range. Despite its body
centered cubic (BCC) crystal structure, the Senkov alloy does not have a
ductile-to-brittle transition temperature at or above -196 ◦C [6]. In fact,
the alloy can easily be cold rolled or forged to very high strains, both at
room temperature and cryogenic temperatures [4,5,7-11]. Thermomechanical processing consisting of cold working and recrystallization
annealing has been used to refine the microstructure and control
strength and ductility of the Senkov alloy [5,12,13].
Detailed microstructural analysis aided with solid solution
strengthening models indicates that the yield strength of this alloy at
room temperature is mainly controlled by the mobility of screw dislocations [14,15]. At later stages of deformation, heterogeneous microstructures consisting of slip bands, deformation twins, kink bands and
shear bands were reported [2-5,16,17]. Deformation twins in HfNbTaTiZr were first reported by Senkov et al. [2,3] after 50 % compression
deformation at 23 ◦C, 400 ◦C and 600 ◦C; however, the type of twins was
not described. Cizek et al. [18] observed {113}<332> twins during
early stages of high pressure torsion processing. The twinning mechanism occurred predominantly in grains with orientations unfavorable
for shear in {110}<111> and {112}<111> slip systems. Wang et al.
[19] reported very good tensile ductility of the Senkov alloy at -196 ◦C
and related this to the activation of {112}<111> nano-twinning
accompanied by formation of hexagonal omega-phase nano-particles
and extensive dislocation slip. Zherebtsov, et al. [7] studied the alloy
microstructure after room-temperature (RT) rolling to different levels of
the plate thickness reduction, up to 80 %. After rolling with 15 % to 40 %
thickness reduction, they observed features similar to deformation twins
but interpreted them as kink bands, because the misorientation angles
between the matrix grain and the bands were not constant along the
interfaces. Characteristic misorientation angles were reported to be 20◦
or smaller after 15 % rolling and approximately 30◦ - 50◦ after 40 %
rolling.
In the present work, we report deformation twins of different types
observed in the Senkov alloy after compression deformation at room
temperature (RT, ~22–27 ◦C), 400 ◦C and 600 ◦C. All the observed twins
have a common 〈110〉 pole with the parent grain and form by rotation of
the grain segment to a fixed angle about this pole. Contrary to deformation kinks, which also form by crystal rotation about a 〈110〉 pole and
whose formation is associated with the cooperative dislocation glide in
{112}<111> slip systems [20], the observed twins likely form by
localized cooperative shear in {110}<001> systems and follow twin
* Corresponding author.
E-mail address: oleg.senkov.ctr@afrl.af.mil (O.N. Senkov).
Contents lists available at ScienceDirect
Acta Materialia
journal homepage: www.elsevier.com/locate/actamat
https://doi.org/10.1016/j.actamat.2024.120435
Received 26 July 2024; Received in revised form 16 September 2024; Accepted 25 September 2024
Acta Materialia 281 (2024) 120435
Available online 26 September 2024
1359-6454/© 2024 Acta Materialia Inc. Published by Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar
technologies.

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symmetry operations.
2. Experimental procedures
The Senkov alloy was prepared by vacuum arc melting of an equimolar mixture of the corresponding elements. The prepared button was
about 10 mm in thickness and 50 mm in diameter. To heal internal cast
porosity and other defects, the button was hot isostatically pressed at
1200 ◦C and 207 MPa for 1 h, then vacuum annealed at 1200 ◦C for 24 h
and slow cooled inside the furnace after turning the power off. Details of
the alloy preparation are given elsewhere [3]. The alloy composition,
determined with the use of inductively coupled plasma-optical emission
spectroscopy (ICP-OES), is given in Table 1.
Compression tests were conducted at RT, 400 ◦C and 600 ◦C using
cylindrical specimens of 3.8 mm in diameter and 6.1 mm in height,
which were extracted from a middle section of the annealed alloy button. The specimen axis was perpendicular to the bottom button surface,
which was in contact with the water-cooled copper hearth during arc
melting and solidification. The specimen surfaces were mechanically
polished, and the compression faces of the specimens were paralleled. A
computer-controlled Instrone (Instron, Norwood, MA) mechanical
testing machine outfitted with a Brew vacuum furnace and silicon carbide dies was used. The elevated temperature tests were conducted in a
vacuum of 10–3 Pa or better. The specimens were heated to each testing
temperature at a heating rate of 20 ◦C/min, soaked at the temperature
for 15 min and then deformed to a true plastic strain of 0.4 at the strain
rate of 10–3 s
-1. Room temperature tests were conducted in air using the
same deformation conditions.
The microstructure was analyzed with the use of scanning electron
microscopes (SEM) FEI Quanta 650F FEG SEM, equipped with an EDAX
Hikari electron backscatter diffraction (EBSD) detector and TSL OIM
data acquisition system, and Thermo Fisher Scientific Apreo C FEG SEM,
integrated with an Oxford Instruments Symmetry EBSD detector and
AZtec data acquisition system. Quantitative analysis of twins was conducted using AZtec Crystal v.6.0 software.
3. Classification of deformation twins and kinks
In the materials science literature, deformation twins are mostly
considered as type I twins formed by a simple shear of the crystal lattice
in a specific plane K1, called the twinning or composition plane, along a
specific shear direction η1 [21-26]. The type I twinning mechanism by
shear is summarized in Fig. 1 showing displacement of a circle to an
ellipse by a shear vector s at a unit distance above the origin. The shear
plane K1 and shear direction η1 remain undistorted during this type of
twinning. The second undistorted (but rotated by the angle 2ϕ) plane of
the simple shear (also called a reciprocal twinning plane) is denoted by
K2, and the plane perpendicular to K1 and K2 by P (plane of shear). A
reciprocal twinning direction η2 is defined as a vector, which resides in
both K2 and P planes (Fig. 1). For the type I twinning mode, K1 and η2
must have rational indices, while K2 and η1 can have irrational (general
case) or rational (compound twin mode) indices. In BCC cubic crystals, a
common example of shear twinning is the simple shear of {112} planes
in a < 111> direction, however; other types of shear twinning such as
{113}<332> and {114}<221> have also been reported [18,27,28].
During type II twinning, shear occurs on K2 planes and accumulates
in tilt walls (twin boundaries). This mechanism is accompanied by
partitioning of the rotations to both the matrix and twin, creating
symmetrical tilt across the twin boundary [26,29,30]. This means that
the type II twinning involves both crystal shear and rotation. For the
type II twinning mode, K2 and η1 have rational indices and K1 and η2 are
generally irrational. In BCC crystals, all four twinning elements can have
rational indices.
The commonly observed symmetry relations between twin and matrix are reflection across the twinning plane K1, rotation of 180◦ about
twinning direction η1, reflection in the plane normal to η1, and rotation
of 180◦ about the normal to K1 [22,23,25].
In addition to, or instead of twinning, deformation kink bands may
form in grains oriented unfavorably for dislocation slip. Under tension or
compression stress, the grain sharply bends (kinks) towards increasing
the Schmid factor of a slip system inside the bend and deformation localizes into the kink band [20,31]. The crystal rotation during bending
Table 1
Chemical composition (in at.%) of the produced HfNbTaTiZr.
Element Hf Nb Ta Ti Zr
Composition, at.
%
20.5 ±
0.6
18.9 ±
0.6
19.7 ±
0.6
19.7 ±
0.6
21.2 ±
0.6
Fig. 1. (a) Relation of the matrix sphere and twinned ellipsoid. (b) Illustration
of twinning elements: P is plane of shear, K1 – twinning plane, η1 – twinning
direction, K2 and Kʹ
2 conjugate (or reciprocal) twinning plane before and after
shear, η2 and ηʹ
2 – conjugate (or reciprocal) twinning direction before and after
shear and s is the shear vector, which is parallel to η1. Vectors k1, k2 and kʹ
2 are
normals to the respective planes. The shear angle (between K2 and Kʹ
2) is 2ϕ.
e Disclaimer: Certain commercial equipment, instruments, or materials are
identified in this paper in order to specify the experimental procedure
adequately. Such identification is not intended to imply recommendation or
endorsement by the United Stated Air Force or by the National Institute of
Standards and Technology, nor is it intended to imply that the materials or
equipment identified are necessarily the best available for the purpose.
O.N. Senkov et al. Acta Materialia 281 (2024) 120435
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occurs about an axis that lies in the slip plane and perpendicular to the
slip direction. In a BCC structure, where slip occurs along a 〈111〉 direction within {110}, {112}, {123} or {134} slip planes, the respective
crystal rotation takes place about 〈112〉,〈110〉,〈541〉 or 〈752〉 directions.
Schematic illustration of formation of deformation kinks is given in
Fig. 2 (adapted from ref. [20]), where the conventional notation of
deformation twinning is used [22,23]. A kink forms following slip on the
plane K2 in the direction η2. Due to localized slip plane rotation, slip
becomes concentrated in region ABCD. Boundaries AD and BD orient
themselves symmetrically with respect to the grain and the kink. The
associated macroscopic deformation can formally be described as a
simple shear on the plane K1 in the direction η1 and thus the kink band
can form in the same way as a Type II deformation twin. However,
contrary to twinning, the shear for the kink band is not restricted to a
particular value, but increases uniformly as the plastic strain increases.
4. Results
4.1. Deformation behavior
True stress vs true strain compression curves of HfNbTaTiZr at RT,
400 ◦C and 600 ◦C are shown in Fig. 3. The alloy had yield stress of 925
MPa, 790 MPa and 675 MPa, respectively, at these temperatures. After
yielding, continuousstrain hardening was observed, which slowed down
with an increase in strain. Starting from about 0.12 true strain, serrated
behavior of flow stress was observed at RT and 400 ◦C. At 600 ◦C, the
serrated behavior started after the true strain of about 0.3. After deformation to a true strain of 0.4, the true stress of the alloy increased to
1410 MPa, 1365 MPa and 1250 MPa, respectively, at RT, 400 ◦C and 600 ◦C.
4.2. Deformation twinning
The microstructure of the Senkov alloy in the annealed condition and
after compression deformation at different temperatures was described
in details in earlier publications [2,3]. Formation of twins during the
deformation at RT, 400 ◦C and 600 ◦C was also reported there; however,
the types of the twins were not studied at that time. In the following
sections, the results of a detailed analysis of these twins are reported and
a rotational nature of these twins is revealed.
4.2.1. Deformation twinning at room temperature
Fig. 4 represents the microstructure of the Senkov alloy after 0.4 true
strain at room temperature. Many grains are twinned (Fig. 4a), the
Kernel average misorientation map indicates an increased dislocation
density near grain and twin boundaries (Fig. 4b), and the Schmid factor
distribution map shows activation of {112}<111> (23.7 %), {110}<
111> (13.8 %) and {123}<111> (45.9 %) slip systems (Fig. 4c). It can
be noted that the {112}<111> and {123}<111> slip systems dominate
inside grains while the {110}<111> slip system mainly operates inside
the twins. The misorientation angle distribution map shows several
distinct maxima for the neighbor grain misorientations, in particular
near 20◦, 26.5◦, 30◦ and 60◦ (Fig. 4d). The 60◦ misorientation corresponds to conventional, {112}<111> type twins, while 20.04◦, 26.53◦
and 31.58◦ misorientations are found to correspond to {118}<441>,
{116}<331> and {115}<552> type twins (as illustrated below).
Fig. 5a shows a magnified twinned grain area outlined by a red
rectangle in Fig. 4a. The {011} pole figure from the grain containing
twins reveals that the twins and the parent grain have a common [101]
pole, outlined by a red circle (Fig. 5a insert). To verify if this common
pole represents the normal direction to the plane of shear, P, we rotate
the sample in such a way that the common [101] pole becomes parallel
to the normal (Z) direction (Fig. 5b insert). After such sample rotation,
the IPF color of the grain and the twins on the IPF map becomes the same
(green), which confirms that the grain and twins now have the same
crystallographic orientation [101] in the Z direction (Fig. 5b), and the
mirror symmetry between the 〈110〉 poles from the grain and twins is
revealed (Fig. 5b insert).
If (101) is the plane of shear, then all the twinning elements (vectors
η1, η2, ηʹ
2, k1, k2 and kʹ2, see Fig. 1) on the stereographic projections with
the pole [101] in the center (0◦, parallel to the normal (Z) direction)
should be located on the 90◦ great circle (i.e. inside the (101) plane). The
discrete pole figures, Fig. 6, from the twins #1 and #2 (Fig. 5b) and the
parent grain show that the poles [010], [101], [111], [111], [121] and
[121], from both the twins and the parent grain, are in the plane (101)
and the angle between the grain and the twin poles with the same indexes is the same, ≈ 20.05◦, i.e. the twin’s crystal lattice is rotated by
20.05◦ about the [101] direction relative to the parent grain lattice.
There are also two orthogonal poles in the plane (101), which are
common to the matrix and twins. From Fig. 6, these are experimentally
determined [181]M || [181]T and [414]M || [414]T. The subscripts ‘M’
and ‘T’ indicate that the indices belong to the matrix grain and twin,
respectively. The traces of the respective (181)M || (181)T and (414)M ||
(414)T planes are shown as red dashed lines on the pole figures. Inspection of the pole figures indicates that the poles of the twins can be
derived from the poles of the matrix by reflection in these two
Fig. 2. Schematic illustration of formation of deformation kinks in (a) tension
and (b) compression. In each case, a kink ABCD forms following slip on the
plane K2 in the direction η2. The associated macroscopic deformation can
formally be described as a homogeneous shear on the plane K1 in the direction η1.
Fig. 3. True stress versus true strain deformation curves of HfNbTaTiZr at RT
(23 ◦C), 400 ◦C and 600 ◦C.
O.N. Senkov et al. Acta Materialia 281 (2024) 120435
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orthogonal planes. Therefore, by definition, one of these planes (181)M
|| (181)T is the composition plane K1 and the normal to the other plane,
[414]M || [414]T, is the shear direction η1 [21]. The grain to twin
transformation is completely described by the rotational transformation
matrix:
A(181)(414) = 1
33
⃒
⃒
⃒
⃒
⃒
⃒
32 8 1
8 31 8
1 8 32
⃒
⃒
⃒
⃒
⃒
⃒
(1)
Using the similar approach, twins 3 and 4 in Fig. 5b are identified as
{161}<313> type twins formed by 26.53◦ rotation about [101] (Fig. 7)
and for which the transformation matrix is:
A(161)(313) = 1
19
⃒
⃒
⃒
⃒
⃒
⃒
18 6 1
6 17 6
1 6 18
⃒
⃒
⃒
⃒
⃒
⃒
(2)
Fig. 8a shows the other grain with a set of twins in the HfNbTaTiZr
sample deformed at RT. The twins and the grain have a common [011]
orientation in Z direction and thus the twins are not seen on the Z IPF
map (Fig. 8b). The analysis of the pole figures from the grain and the
twins reveals that these twins were formed by 31.58◦ clockwise crystal
rotation around the [011] pole (Fig. 8c). The matrix and twin poles have
a mirror symmetry about the (511)M || (511)T and (255)M || (255)T
composition planes (Fig. 8c). The twinning elements have been identified as K2 = (011), η2 = [100], K1= (511) and η1 = [255]. The grain to
twin orientation relationship is described by the following transformation matrix:
Fig. 4. HfNbTaTiZr after RT compression deformation: (a) Inverse pole figure (IPF) map of grain orientations in compression (vertical, Y) direction, (b) Kernel
average misorientation distribution map, (c) Schmid factor distribution map in compression direction and (d) Neighbor pair (green) and random pair (orange)
misorientation angle distribution.
Fig. 5. Magnified IPF maps of grain and twin orientations in normal (Z) direction, from the area inside the red box in Fig. 4a, and the respective 〈110〉 pole figures (a)
in an original sample position and (b) after the sample rotation to orient the [101] crystal direction, common to the twins and the parent grain, parallel to the Z
direction. The common [101] pole is outlined by the red circle in the 〈110〉 pole figures. The matrix-grain poles are identified by the letter M and the poles belonging
to the twins are identified by the letter T.
O.N. Senkov et al. Acta Materialia 281 (2024) 120435
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A(511)(255) = 1
27
⃒
⃒
⃒
⃒
⃒
⃒
23 10 10
10 25 2
10 2 25
⃒
⃒
⃒
⃒
⃒
⃒
(3)
4.2.2. Deformation twinning at 400 ◦C
Fig. 9 illustrates deformed grains in HfNbTaTiZr after 0.4 true strain
at 400 ◦C. Deformation twins and kinks are seen inside the grains
(Fig. 9a,b), The Kernel average misorientation map indicates an
increased dislocation density near grain and interface boundaries, as
well as inside the rotated regions (Fig. 9c). The disorientation angle
distribution map shows distinct maxima near 23◦ and 31◦, as well as a
small maximum near 51◦, for the neighbor pixel pair misorientations
(Fig. 9d), which correspond to {117}<772> (2ϕ=22.84◦), {115}<552>
(2ϕ=31.58◦) and {113}<332> (2ϕ=50.4◦) type twins. There is also a
wide neighbor-pair misorientation distribution in the 2ϕ range between
10◦ and 20◦, which is likely characteristic of deformation kink bands.
(See also Section 1 of Supplementary Materials for additional details.)
Fig. 10a shows a magnified twinned grain area outlined by a
Fig. 6. Discrete (scattered) pole figures from twins #1 and #2 (red spots) shown in Figure 5b and the parent grain (blue spots). Both twins have the same crystallographic orientation. The poles, which are common to the twins and the grain, are outlined by red circles. Indexes from the representative twin poles are
underlined. Two orthogonal dashed lines crossing the centers of the stereographic projections represent the traces of the (181)M || (181)T and (414)M || (414)T
composition (mirror) planes. The twinning elements are: P = (101), K2 = (101), η2 = [010], K1= (181), η1 = [414], 2ϕ = 20.05◦ and s = 0.354.
Fig. 7. Discrete (scattered) pole figures from twins #3 and #4 shown in Figure 5b (red spots) and the parent grain (blue spots). The poles, which are common to the
twin and the grain, are outlined by red circles. Indexes from the representative twin poles are underlined. Two orthogonal dashed lines crossing the centers of the
stereographic projections represent the traces of the (161)M || (161)T and (313)M || (313)T composition (mirror) planes. The twinning elements are: P = (101), K2 =
(101), η2 = [010], K1= (161), η1 = [313], 2ϕ = 26.53◦ and s = 0.471.
O.N. Senkov et al. Acta Materialia 281 (2024) 120435
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Fig. 8. (a,b) IPF orientation maps of twins and the parent grain in (a) vertical (Y) direction and (b) normal (Z) direction. The IPF color code is shown at the top right
corner of figure (b). (c) Discrete (scattered) pole figures from the twins shown in (a) (red spots) and the parent grain (blue spots). The poles, which are common to the
twins and the grain, are outlined by red circles. Indexes from the representative twin poles are italicized and underlined. Two orthogonal dashed lines crossing the
centers of the stereographic projections represent the traces of the (511)M || (511)T and (255)M || (255)T composition (mirror) planes. The twinning elements are: P =
(011), K2 = (011), η2 = [100], K1= (511), η1 = [255], 2ϕ = 31.58◦ and s = 0.566.
Fig. 9. (a, b) IPF maps of grains and twins in (a) normal (Z) and vertical (Y) directions, (c) Kernel average misorientation map and (d) disorientation angle distributions between neighbor pixel pairs (blue) and random pixel pairs (orange). The compression direction is vertical.
O.N. Senkov et al. Acta Materialia 281 (2024) 120435
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rectangle in Fig. 9a. The twins have a red IPF color indicating that one of
their 〈001〉 directions is parallel to the Z direction. The analysis of the
pole figures indicates that the twins and the grain have a common [101]
pole and other twin poles experience ~31.6◦ counterclockwise rotation
about the [101] pole relative to the respective crystallographic poles of
the parent grain. Bringing the common [101] pole to the center of the
stereographic projection (i.e. parallel to Z direction) results in the same
(green) IPF color of the grain and the twins (Fig. 10b), confirming the
same crystallographic orientation of these constituents in Z direction
after the respective sample rotation.
Scattered-orientation pole figures from the selected twins and the
parent grain after bringing the common [101] pole to the center of the
stereographic projection are shown in Fig. 10c. The poles from the grain
are blue and those from the twins are red. The pole spots are rather wide,
which indicates crystallographic orientation spread, both inside the
grain and twins. This was likely due to continuous deformation after the
twin formation. Two composition planes, (151)M || (151)T and (525)M ||
(525)T, providing the mirror symmetry between the twin and parent
grain crystallographic orientations, are clearly identified on the pole
figures. The twinning elements have been identified as P = (101), K2 =
(101), η2 = [010], K1= (151) and η1 = [525]. The grain-twin transformation is described by the transformation matrix:
A(151)[525] = 1
27
⃒
⃒
⃒
⃒
⃒
⃒
25 10 2
10 23 10
2 10 25
⃒
⃒
⃒
⃒
⃒
⃒
(4)
4.2.3. Deformation twinning at 600 ◦C
Fig. 11a and Fig. 11b show IPF maps, in the vertical (Y) and normal
(Z) directions, respectively, of deformation twins and the parent grain
after 0.4 true plastic strain at 600 ◦C. The microstructure of a larger
sample area is given in Section 2 of Supplementary Materials. Scatteredorientation pole figures from the grain and twins are shown in Fig. 11c.
The twins have a [001] orientation and the grain has an approximate
[223] orientation in Y direction, and they all have a common [110]
orientation in Z direction. The crystal lattice of the twins is rotated
clockwise by 44.0◦ about [110] relative to the grain crystal lattice. A
mirror symmetry of the twin and grain poles relative to two orthogonal
crystallographic planes, (227)M || (227)T and (774)M || (774)T, is clearly
identified. The traces of these composition planes are shown as dashed
lines crossing the centers of the stereographic projections (Fig. 11c).
Based on this analysis, the twinning elements have been identified as P =
(110), K2 = (110), η2 = [001], K1= (227), η1 = [774]. The grain to twin
transformation is described by the transformation matrix:
A(22 7)[774] = 1
57
⃒
⃒
⃒
⃒
⃒
⃒
49 8 28
8 49 28
28 28 41
⃒
⃒
⃒
⃒
⃒
⃒
(5)
5. Discussion
Different types of the deformation twins reported in this work for the
Senkov alloy after compression deformation at RT, 400 ◦C and 600 ◦C
are summarized in Table 2. They all have the reciprocal twin plane K2 =
{110} and the reciprocal twin direction η2 = 〈001〉. Their relationship to
the parent grain can be interpreted asrotation to a fixed angle 2ϕ about a
〈110〉 axis that lies in K2 and is perpendicular to η2. The composition
(mirror) plane, K1 is then identified as the plane containing the rotation
axis 〈110〉 and having the angle of 90◦+ ϕ with K2. The second
composition plane (Н1), which is perpendicular to η1, is identified as the
plane containing the rotation axis 〈110〉 and the vector k1, which is
perpendicular to the plane K1 (Fig. 1).
Rotation of a BCC crystal about a 〈110〉 axis generally occurs during
dislocation glide in <111> directions in {112} slip planes, and this is
Fig. 10. (a,b) IPF maps of the selected twins and the parent grain in (a) original sample position and (b) after rotating the sample and positioning the common [101]
pole parallel to Z direction. The color coded IPF is shown on the top of figure (b). (c) Scattered-orientation pole figures from the studied region, with [101] parallel to
Z. Two orthogonal dashed lines crossing the centers of the stereographic projections represent the traces of the (151)M || (151)T and (525)M || (525)T composition
(mirror) planes. The poles from the twins are red and their indexes are underlined, and the poles from the parent grain are blue. The twinning elements are: P =
(101), K2 = (101), η2 = [010], K1= (151), η1 = [525], 2ϕ = 31.58◦ and s = 0.566. HfNbTaTiZr sample after compression deformation at 400 ◦C.
O.N. Senkov et al. Acta Materialia 281 (2024) 120435
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one of the widely accepted mechanisms of formation of deformation
kinks (see Fig. 2 and the related text). However, crystallographic analyses of the reported features allowed us to confirm that these are
deformation twins rather than kinks. Indeed, a deformation kink forming by rotation about 〈110〉 should have K2 = {112} and η2 = <111>,
which, during kinking, rotate to the positions K2’ and η2’, respectively,
and the rotation angle 2ϕ generally increases with increasing the plastic
deformation inside the kink band [20,32–34]. Additionally, K2 and K2’,
as well as η2 and η2’, should have a reflection (mirror) symmetry across
the kink boundary K1. In other words, the direction η1, which is normal
to the rotation axis 〈110〉 and lays in plane K1, should have the same
angle ϕ with K2 ={112}M and K2’ ={112}KB. (The subscript ‘KB’ indicates that the respective plane belongs to the kink band). None of these
agrees with the experiment. In particular, none of the planes, which are
parallel to the rotation axis 〈110〉 and laying between K2 ={112}M and
K2’ ={112}KB provides a reflection symmetry between the rotated band
and the parent grain. Instead, all the bands reported here have K2 =
{110} and η2 = 〈001〉, they all have a reflection symmetry with the
parent grain across the band boundary K1 and two-fold rotation
symmetry about the direction η1 and thus these bands follow twin laws
and they should be considered as deformation twins. That is, in these
deformation twins K1 has the same angular distance 90◦- ϕ with K2 =
{110}M and K2’ = {110}T, which have the angular distance 2ϕ. The
observed small variations in 2ϕ for the same twin can be explained by
the accumulation of dislocations near twin boundaries during continuous deformation of the parent grain and the twin after the twin formation at earlier stages of deformation.
It is worth noting that the crystallographic analysis of the observed
deformation twins in the Senkov alloy indicates that they can be interpreted as rotation twins rather than shear twins by expanding the current definition of rotation twins. The difference between the shear and
rotation twins was discussed by Cahn [35] and the current definition of
rotation twins was first given by Barrett and Massalski [36] (p. 406):
“Crystals are rotation twins if a two-, three-, four- or six-fold rotation
about a twinning axis produces the orientation of the other. The rotation
axis lies either in the twinning plane or normal to it and is not a symmetry element of the individual crystals.” The fundamental difference
between the shear and rotational twins lies in the formation mechanism.
Rotational twins involve a rotation of crystal domains, while shear twins
involve shearing or sliding of crystal planes. Thus, rotational twins are
characterized by a rotation axis and angle, while shear twins are characterized by a shear plane and direction. A typical example of a rotation
twin in a cubic crystal is a twin formed by 60◦ rotation about 〈111〉 axis.
In a simple cubic metal this twin is crystallographically similar to a shear
twin <111>{112}; however, this is not the case in alloys having complex ordered cubic structures.
Our crystallographic analysis shows that formation of rotational
twins in cubic crystals is not limited by a 180◦, 120◦, 90◦ or 60◦ rotation
about the twinning axis. In particular, a much larger number of discrete
rotation angles about a 〈110〉 axis can produce twin orientations.
Indeed, the 〈110〉 zone in a cubic crystal contains two planes of
Fig. 11. A HfNbTaTiZr sample after deformation at 600 ◦C: (a,b) IPF maps of twins and the parent grain orientations in (a) Y (vertical) and (b) Z (normal) directions.
The twins are not seen on the Z IPF map indicating that both the grain and the twins have the same orientation in Z direction. (c) Scattered-orientation pole figures
from the region outlined in (a). The common poles are circled. The poles from the twins are red and their indexes are underlined, and the poles from the parent grain
are blue. The twins’ poles are rotated 44◦ clockwise about the common [110] pole relative to the respective poles from the matrix grain. Two orthogonal dashed lines
crossing the centers of the stereographic projections represent the traces of the (227)M || (227)T and (774)M || (774)T composition (mirror) planes. The twinning
elements are: P = (110), K2 = (110), η2 = [001], K1= (227), η1 = [774], 2ϕ = 44.0◦ and s = 0.808.
Table 2
Deformation twins identified in HfNbTaTiZr after compression deformation at
23 ◦C, 400 ◦C and 600 ◦C.
T, ◦C K1 η1 K2 η2 P 2ϕ (◦) s
23 (181) [414] (101) [010] (101) 20.05 0.354
23 (161) [313] (101) [010] (101) 26.53 0.471
23 (511) [255] (011) [100] (011) 31.58 0.566
400 (171) [727] (101) [010] (101) 22.84 0.404
400 (151) [525] (101) [010] (101) 31.58 0.566
600 (227) [774] (110) [001] (110) 44.0 0.808
O.N. Senkov et al. Acta Materialia 281 (2024) 120435
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symmetry, {001} and {110}. Accordingly, the crystallographic direction
<u u w> is a reflection of the direction <u u w> in the plane {110} and
the direction <w w 2u>, which is perpendicular to 〈110〉 and <u u w>, is
a reflection of the direction < w w 2u> in the plane {001}. These four
crystallographic directions, <u u w>, <u u w>, <w w 2u> and <w w
2u>, as well as normals 〈001〉 and <110> to the respective {001} and
{110} planes, are all perpendicular to 〈110〉. Moreover, the angular
distance ϕ between <u u w> and <110>, <u u w> and <110>, <w w
2u> and 〈001〉, and <w w 2u> and 〈001〉 is the same and is given by Eq.
(6):
2ϕ<110> = arcos[( 2u2 − w2) / (
2u2 + w2)] (6)
Therefore, counterclockwise rotation about 〈110〉 of one part of the
crystal relative to the other (matrix) part by the angle 2ϕ produces a
twin pair with η2 = 〈001〉, K2 = {110}, η1 = <u u w>M || <u u w > T and
K1 = {w w 2u}M || {w w 2u}T. In this configuration, (u u w)M || (u u w)T
and (w w 2u)M || (w w 2u)T become mirror (composition) planes.
Using a similar approach, one can show that rotational twins can also
be produced by rotation about the 〈100〉 axis to a fixed 2ϕ angle given by
Eq. (7):
2ϕ<100> = arcos[( v2 − w2) / (
v2 + w2)] (7)
Here v and w are integers. The twinning elements for these rotational
twins are:
η2 = 〈001〉, K2 = {010}, η1 = <0 v w>M // <0 v w>T and K1 = {0 w
v}M // {0 w v}T.
A few twin configurations predicted from this analysis, as well as
their twinning elements, are given in Table 3. The experimentally
determined twins, given in Table 2, also follow the above-described
rules of rotation about a 〈011〉 axis. Simulated stereographic projections of a few rotational twins identified in this work can be found in
Section 3 of Supplementary Materials.
It should be noted that these rotational twins are equivalent to the
shear twins having the same twinning elements only for crystals with a
simple cubic structure. In the case of a complex cubic structure these
modes of twinning can become different as twins produced by shear
require extensive atom shuffling inside the cubic cells (sphere to ellipse
transformation during simple shear, Fig. 1, may result in different
relative locations of atoms inside the cell), while rotational twins do not
require atom shuffling (no relative displacements of atoms occur inside
the rotated crystal).
6. Summary and conclusion
Numerous deformation twins have been observed in a HfNbTaTiZr
refractory high entropy alloy after compression deformation at room
temperature, 400 ◦C and 600 ◦C. All these twins have been identified as
rotational twins, which were formed by rotation of the parent grain
about a 〈110〉 axis by 20.05◦, 22.84. 26.53◦, 31.58◦ or 44.0◦ This is the
first report of these types of rotational deformation twins. Transformation matrices describing orientation relationships between the
parent grain and the twins, and the twin composition planes were also
reported.
The conditions required to form twin configurations in cubic crystals
by the crystal rotation about 〈110〉 or 〈100〉 axes have been determined
and a few rotational twins and the respective twinning elements have
been identified. For the simple cubic crystals, these rotational twins are
equivalent to the shear twins having the same twinning elements.
However, in the case of a complex, ordered cubic structure these modes
of twinning can become different, as twins produced by shear require
extensive atom shuffling inside the cubic cells to hold mirror symmetry,
while rotational twins do not require atom shuffling.
CRediT authorship contribution statement
Oleg N. Senkov: Writing – review & editing, Writing – original draft,
Validation, Project administration, Investigation, Formal analysis, Data
curation, Conceptualization. Frederick Meisenkothen: Writing – review & editing, Validation, Investigation, Formal analysis. Robert
Wheeler: Writing – review & editing, Validation, Formal analysis.
Declaration of competing interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.
Acknowledgements
Work by O.N. Senkov and R. Wheeler was funded by the Metallic
Materials and Processing Program at the Air Force Research Laboratory
through the on-site contracts FA8650–21-d-5270 and FA8650–18-C5291 managed, respectively, by MRL Materials Resources LLC, Xenia,
OH, USA and UES, Inc., Dayton, OH, USA. Numerous discussions with
Drs. S. Kuhr, B. McArthur and T. Butler are greatly appreciated.
Supplementary materials
Supplementary material associated with this article can be found, in
the online version, at doi:10.1016/j.actamat.2024.120435.
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Rotational axis Rotational angle, 2ϕ (◦) K1 η1 K2 η2
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<110> 38.94 {114} <221> {110} <001>
<110> 50.4 {113} <332> {110} <001>
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