23. - 25. 5. 2012, Brno, Czech Republic, EU
ANALYSIS OF NEAR-COINCIDENCE SITE LATTICE BOUNDARY FREQUENCY IN AZ31
MAGNESIUM ALLOY
Andriy Ostapovets, Peter Molnár, Aleš Jäger, Pavel Lejček
Institute of Physics ASCR, Na Slovance 2, Prague, Czech Republic, e-mail: [email protected]
Abstract
The grain boundary misorientation distribution and distribution of coincident site lattice boundaries are
reported for the case of magnesium-based AZ31 alloy processed by equal-channel angular pressing. The
experimental data were collected by electron backscatter diffraction. It is shown that the most frequent Σ15b
and Σ17a coincident site lattice boundaries correspond to twin boundaries. Other frequent coincident site
lattice boundaries are Σ15a and Σ13a and correspond to the local maximum around 30° on grain
misorientation distribution.
Keywords:
Equal-channel angular pressing, electron back scattered diffraction, coincident site lattice
1. INTRODUCTION
Concept of coincidence site lattice (CSL) is often used for classification of grain boundaries because it allows
prediction of boundaries, which potentially have special properties. The CSL model is geometrical
construction consisted of coincidence sites of two superimposed lattices [1]. A grain boundary can be
characterized by CSL generated from lattices on both sides of boundary. The CSL misorientations are
characterized by the parameter Σ, which is equal to the ratio of CSL and crystal unit cell volumes. In
comparison with cubic materials, hexagonal ones have some specifics, e.g. the smallest value of Σ is 7 in
contrast to 3 in cubic materials, and even Σ values are also possible. The CSL boundaries often have low
energy [1]. It is also usually accepted that CSL boundaries can improve properties of materials because they
have better cracking and corrosion resistance [2-4] than random boundaries. In the case of materials with
2
hexagonal structure ideal coincidence can only exist for rational values of squared axial ratio (c/a) , except
for several misorientations obtained by rotation about < 0001> axis. The sets of CSL (Σ) are different for
different axial ratios c/a. In practice, near-CSL configurations are considered for the case of hexagonal
materials i.e. experimental axis ratio is approximated by close rational value [5]. There is no CSL with low Σ
and misorientation axis different from < 0001> for magnesium and its alloys with c/a=1.624. Hence the nearCSL boundaries are considered which are generated by lattices with close values of c/a.
Previously authors reported distribution of CSL boundaries in pure magnesium single crystals processed by
equal-channel angular pressing (ECAP) [6]. It was suggested that frequent Σ13a and Σ15a boundaries can
correspond to the typically maximum observed around 30° on grain misorientation distribution. The evolution
of grain boundary misorientation distribution and CSL boundary distribution during ECAP was modeled in
paper [7]. The misorientation scheme was proposed which is based on possible interaction of dislocations
belonging to different slip systems. It produces specific grain misorientations depending on relative activity of
slip systems. In general the including of misorientation scheme in the visco-plastic self-consistent model
improve the model prediction. However the 30° misorientation maximum containing large number of CSL
boundaries was not reproduced precisely in [7]. This can be connected with properties of CSL boundaries
which were not included in the model. Hence the further study of CSL boundaries and their properties can
give new information about material behaviour.
23. - 25. 5. 2012, Brno, Czech Republic, EU
The aim of the present work is to report grain boundary misorientation distribution and distribution of CSL
boundaries in Mg-based AZ31 alloy processed by ECAP. The frequency of CSL boundaries is discussed.
Deviations from CSL are analyzed for the most frequent CSL boundaries which allows to elucidate their
prospective special properties.
2. METHODS AND MATERIALS
2.1 Experimental
AZ31 alloy with nominal composition Mg-3wt%Al-1wt%Zn was used for our research. Billets with dimensions
mm were machined from as-rolled plate. The ECAP die with inner angle of Φ=90° and outer
anle Ψ=45° was used. The billets were processed by two ECAP passes at 200°C using route A. The route A
does not include rotations about longitudinal axis of the sample. The billets were inserted into pre-heated die
and exposed during 3 min in the die before extrusion. The electron backscatter diffraction (EBSD) data
was collected by Dual-Beam microscope FEI Quanta 3D FEG. The TSL 5.3 OIM analysis software was used
in order to extract data about grain boundary misorientations. Additional analyses were performed in order to
obtain CSL boundary frequency and distribution of boundaries near to CSL configurations.
2.2 Methods of analysis
The list of CSL and near-CSL boundaries for AZ31 alloy is presented in Table 1. Procedure described in [4]
is used to obtain frequency of CSL boundaries from experimental data. Due to symmetry each grain
boundary misorientation can be described by a set of different axis/angle pairs. In order to avoid ambiguity
the only pair of the smallest possible misorientation angle and corresponding axis lying in the main
crystallographic triangle is considered. Brandon criterion is used to define possible deviation Δθ from CSL
configuration [8]:
Δθ < 15
o
Σ
(1)
Table 1. CSL and near-CSL misorientations for AZ31 (c/a=1.624)
c/a
axis
angle

1
any
0
any
〈
〉
7
21.79
any
9
〈 ̅̅ 〉
56.25
1.620
10
〈 ̅ 〉
78.46
1.633
11
〈 ̅ 〉
62.96
1.633
〈
〉
13a
27.8
any
13b
〈 ̅ 〉
85.59
1.620
14
〈 ̅ 〉
44.42
1.633
15a
〈 ̅̅ 〉
29.93
1.620
15b
〈 ̅̅ 〉
86.18
1.620
17a
〈 ̅̅ 〉
86.63
1.633
17b
〈 ̅ 〉
49.68
1.620
17c
〈 ̅̅ 〉
49.68
1.604
18a
〈 ̅ 〉
63.62
1.613
18b
〈 ̅̅ 〉
70.53
1.633
〈
〉
19
13.70
any
21
〈 ̅ 〉
73.40
1.620
23a
〈 ̅̅ 〉
55.80
1.643
23b
〈 ̅̅ 〉
34.30
1.604
23c
〈 ̅̅ 〉
77.44
1.620
23d
〈 ̅ 〉
34.30
1.620
25a
〈 ̅̅ 〉
63.90
1.620
25b
〈 ̅ 〉
23.07
1.633
23. - 25. 5. 2012, Brno, Czech Republic, EU
For each Σ there is distribution of grain boundaries, which satisfy Brandon criterion (1). These boundaries
are slightly inclined from CSL or near-CSL configuration listed in Table 1. It is interesting to study distribution
of Δθ for each Σ. One can expect that this distribution is not uniform if considered CSL boundary has special
properties. The maximum has to be observed for small Δθ angles if trend exists to set boundary in the exact
CSL configuration, for instance due to its low energy. It is necessary to know the random distribution in order
to compare it with experimental data. The random distribution around CSL configuration can be found in a
similar way as it was done for random misorientation angle distributions [9]. The brief description of the
approach is following. It is convenient to perform the analysis in Rodrigues space. In this space rotations are
presented by Rodrigues vectors r:
r = ntan(ω / 2)
(2)
where n is rotation axis and ω is rotation angle. The important properties of Rodrigues space is that all points
that are at the same angular distance from r as from identity rotation 0 lie on two planes perpendicular to r at
the distances from 0 equal to tan(ω/4) and cot(ω/4). For objects with symmetry each orientation is
represented by a set of symmetrically equivalent points. A polyhedron filled with points that are closer to 0
than to any other point symmetrically equivalent to 0 is called the fundamental zone. If misorientation
distribution is uniform, the number of misorientations with angle ω is proportional to the area part of sphere
with radius tan(ω/2) inside fundamental zone.
In the case if the CSL structure is taken as reference misorientation, the fundamental zone can be
-1
1
constructed by consideration of rotations rΣ, which correspond to rotation matrix (SiMΣSj )MΣ , where Si is
matrix of symmetry operation, MΣ is rotation matrix for CSL configuration. In this stage it is better to use
matrix representation for calculation of rΣ in order to avoid uncertainty. It is due to the fact that some of Si
represent 180 degrees rotations and it is presented in Rodrigues space by vector of infinite length.
3. RESULTS AND DISCUSSION
The diagram on Fig.1 shows misorientation angle distribution for AZ31 alloy after two ECAP passes at 200°.
The distribution has three maxima. The first one corresponds to low angle grain boundaries with
misorientations up to 5°, the second maximum is observed at misorientations about 30° and the third
maximum occurs at misorientations about 85°. The main contribution to the last maximum is from {1012} twin
boundaries with misorientation 86˚ about < 1210 > axis (See Table. 1).
number fraction
0,10
0,05
0,00
0
20
40
60
80
misorientation angle, deg
Fig.1 Misorientation angle distribution in AZ31 alloy after 2 ECAP passes at 200˚C.
23. - 25. 5. 2012, Brno, Czech Republic, EU
CSL boundary distribution is shown in Fig2(a). The most frequent Σ15b and Σ17a correspond to
{1012} twin
boundaries. These CSL are close to the twinning configuration and are misoriented ~0.45˚ relative to each
other. Both are produced by lattices with different c/a ratio, close to that c/a=1.624 of magnesium. The next
frequent CSL boundaries are Σ15a, Σ13a, Σ13b, Σ7 and Σ18b. In contrast to the previously reported case of
Mg single crystals [6,7] the portion of Σ15a boundaries are larger than Σ13a boundaries. Both Σ15a and
Σ13a correspond to the 30˚ maximum in the misorientation angle distribution and they remarkably contribute
to this maximum. The Σ13b boundaries lie inside 85˚ maximum together with twin boundaries. Relatively
high frequency of Σ18b boundaries can be explained if one supposes that they occur from deformed twin
boundaries. The twin boundaries always contain large number of defects after course of intensive
deformation through ECAP. It can be concluded from EBSD, which show that twin boundaries are not
straight-line. Therefore they are often inclined from their ideal orientation. Consequently some of twin
boundaries can have relatively large inclination and reach the Σ18b configuration. The frequency of Σ7
boundary is noticeably smaller in polycrystalline AZ31 than in magnesium single crystals.
Fig.2 (a) Number fraction of CSL bounaries in AZ31 alloy after 2 ECAP passes at 200˚C. The distribution of
boundaries inclined from CSL by angle Δθ, which satisfied Brandon criterion. The red and black lines
represent uniform random distribution and experimental distribution, respectively for (b) Σ13a (b), (c) Σ15
and (d) Σ17a. The curves are normalized in a way that area under curve is equal to 1.
The CSL boundaries often have special properties (e.g. low energy) in comparison with random boundaries.
In this case it can be expected that smaller deviation from exact CSL configuration are more probable than
higher ones. The Figs. 2(b-d) shows normalized frequency of grain boundary occurrence versus inclination
angle Δθ from CSL configuration for several CSL boundaries. The value of Δθ=0 corresponds to the exact
CSL configuration. Red curves show the random distribution of inclinations and the black curves correspond
to the experimental data. The data are normalized in such a way that area under curves is equal to 1. The
trend to reorient boundaries to the CSL position is observed for Σ15a and Σ17a boundaries because
experimental curves lie higher than theoretical random curves at small Δθ for these CSL. However the
23. - 25. 5. 2012, Brno, Czech Republic, EU
experimental curve practically coincides with curve for random distribution at small Δθ angles in the case of
Σ13a boundary. The maximum of black curve is observed at Δθ=2.2° in Fig.2(b). It means that preferable
misorientation angle of Σ13a boundaries is different from 27.8°. This is unexpected result because Σ13a has
< 0001> misorientation axis and hence belong to the true CSL boundary with coincidence independent on
c/a ratio of lattice.
4. CONCLUSIONS
The analysis of CSL boundary frequency was performed for AZ31 magnesium alloy processed by ECAP at
200°C by two passes. It was found that the most frequent CSL are Σ15b and Σ17a twin boundaries followed
by Σ15a, Σ13a, Σ13b, Σ7 and Σ18b boundaries. The frequency of Σ7 and Σ13a boundaries is lower in
comparison with the previously reported case of magnesium single crystals. Frequencies of the Σ17a and
Σ15a boundaries are increased with decreasing deviation angle Δθ from exact CSL configuration. However
Σ13a has distribution close to the random one without preference of small Δθ angles.
ACKNOWLEDGEMENT
The financial support of the Academy of Sciences of Czech Republic (KAN300100801), Czech
Science Foundation (P108/12/P054) and Grant Agency of AS CR ( IAA100100920) is gratefully
acknowledged.
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