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Determination of shapes and preferred growth directions of icosahedral quasicrystals from the 2D-sections
Journal of Alloys and Compounds 681 (2016) 532e540
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Determination of shapes and preferred growth directions of icosahedral quasicrystals from the 2D-sections Franc Zupani c*, Tonica Bon cina University of Maribor, Faculty of Mechanical Engineering, Smetanova ulica 17, SI-2000 Maribor, Slovenia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 January 2016 Received in revised form 24 April 2016 Accepted 25 April 2016 Available online 27 April 2016
The icosahedral quasicrystalline phases have been present in many alloys either as nearly regular polyhedra or dendrites. This work provides a systematic approach for the determination of shapes of polyhedra with icosahedral symmetry, and the determination of preferred growth directions of icosahedral dendrites from their 2D-sections; i.e. from the surface of polished samples. The possible shapes of polyhedra and dendrites by differently oriented intersecting planes are presented within the icosahedral stereographic triangle. © 2016 Elsevier B.V. All rights reserved.
Keywords: Quasicrystals Microstructure
1. Introduction The point group symmetry dictates the shapes of a crystal. The quasicrystals possess non-crystallographic point group symmetries having distinctive shapes from those of periodic crystals. The minimum surface energy principle determines the equilibrium shape of the crystal, while the growth morphologies depend on the growth kinetics and processing conditions [1]. There are three main regular polyhedra with the icosahedral symmetry: pentagonal dodecahedron (12 pentagonal faces, 30 edges, 20 vertices) icosahedron (20 triangular faces, 30 edges, 12 vertices) rhombic triacontahedron (30 rhombic faces, 60 edges, 32 vertices). They can preferentially grow in twofold, threefold and fivefold directions. The AleMn and AleMneSi icosahedral quasicrystals mainly formed a dendritic structure with branches extending in the preferred threefold directions [2]. Thangaraj et al. [3] established that the pentagonal dodecahedron was the shape for the AleMn quasicrystal. In the AleCueFe system, the equilibrium shape of the stable quasicrystal was the pentagonal dodecahedron. Also, quasicrystals with the icosidodecahedral morphology were observed
* Corresponding author. E-mail address: [email protected] (F. Zupani c). http://dx.doi.org/10.1016/j.jallcom.2016.04.255 0925-8388/© 2016 Elsevier B.V. All rights reserved.
[4,5]. In several alloy systems, such as Sc12Zn88 [6], AleLieCu [7], AleMgeZn [8] and ZneMgeY face-centred icosahedral alloys, the quasicrystals possessed the triacontahedral growth morphology [9]. The preferred growth direction was mainly along threefold axes and occasionally along fivefold axes [2]. There is some confusion regarding quasicrystalline shapes, especially those determined from the 2D-sections. Several methods have been used for the determination of quasicrystal shapes. The first quasicrystalline phases were metastable and were obtained during rapid solidification. Rapid solidification of dilute AleMn alloys provided substantial undercooling promoting the nucleation of small quasicrystalline particles in the melt [10]. Aluminium nucleated later and trapped the small quasicrystalline particles, retaining their shapes. Due to small sizes of quasicrystalline particles, the basic shapes were investigated using TEM. The shapes were identical when projected along higher symmetry axes like threefold and fivefold axes, while the projections along the twofold axes enabled distinction between different shapes [3]. By developing stable quasicrystalline phases, as well as new aluminium alloys with increased quasicrystal-forming ability, determination of the quasicrystalline shape from the micrographs of the polished surface has become crucial. Although the icosahedral quasicrystalline particles on the polished surface sometimes exhibited a pentagonal symmetry [11], it was not always clear, what was their exact shape, and what was the preferred growth direction. Kral et al. [12] obtained 3D-shapes of phases in steels by using computer-aided visualization of 3D-reconstructions from series of
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section images. Nowadays, sequential cross sectioning is done using focused ion beam FIB, usually within a dual beam SEM/FIB system. By the application of microanalytical techniques the 3Delemental distribution can also be obtained along with the particle shapes [13]. Such approach is feasible, but it is very timeconsuming. Direct observation of 3D-shapes of quasicrystals was done by decanting of the melt after growing single quasicrystals [14], observations of faceted microholes in slowly cooled icosahedral single quasicrystals [15] and inside solidification shrinkage microvoids [5]. 3D-shapes of quasicrystalline phases can also be obtained by deep etching and particles extraction techniques [16,17]. Over the last years, several new aluminium and magnesium alloys have been developed containing quasicrystalline phase in the as-cast cast state after conventional casting [18e22]. It is expected that the number of such alloys will increase in the future. The main aim of this work is to provide a systematic approach to identification of icosahedral quasicrystals from the polished samples, i.e. from the 2D-sections. Thus, the possible shapes of the regular pentagonal dodecahedron, icosahedron, and rhombic triacontahedron with differently oriented intersecting planes are systematically presented within the icosahedral stereographic triangle. The intersections of the intersecting plane with dendritic icosahedral quasicrystals were established to enable easier and more reliable determination of the preferred growth directions.
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directions. The arms had the length L and circular cross-section with the diameter D. The unit vector of the intersection plane passed through the dendrite centre. For example, when the arms grew in the threefold directions, then vectors 43 to 62 were chosen (see Facets_Spherical_Cartesian_Coordinates.xlsx). In the next stage, the dot products between the plane normal vector and the unit vectors of all arms were calculated. The intersecting plane was positioned at a particular distance, h, from the dendrite centre. At this distance, the arms cut by the intersecting plane were determined. For these arms, the following characteristics were calculated: (1) the longer axis of the ellipse, dl; the intersection of a plane and a tube is a cylinder; (2) the lateral distance of the centre of the intersection from the normal of the intersecting plane passing through dendrite centre, t; and (3) the orientation of the longer ellipse axis. The results for one particular case are given in Table 1. The calculation was done for four of twenty arms. The application of the whole procedure is described in detail in Section 3.4. The results for all preferred growth directions and all seven intersecting planes were depicted in the icosahedral stereographic triangle. 3. Results and discussion 3.1. Intersections of planes with pentagonal dodecahedron, icosahedrons, and triacontahedron
2. Methods As stated in the introduction, the work was focused to. a) The determination of possible shapes of the regular pentagonal dodecahedron, icosahedron, and rhombic triacontahedron with differently oriented intersecting planes and their positioning within the icosahedral stereographic triangle. b) The determination of the intersections of the plane with dendritic icosahedral quasicrystals depending on their preferred growth directions. For both tasks, a set of 62 unit vectors was defined. This set was divided into three subsets, containing 12 normal vectors to the facets of the pentagonal dodecahedron, 20 normal vectors to the facets of the icosahedron, and 30 normal vectors to the facets of the rhombic triacontahedron. The same vectors also represented the vectors indicating the preferred growth directions of dendritic arms in the fivefold, threefold and twofold axes. All these unit normal vectors in polar and Cartesian coordinates are listed in the Supplement Material (Facets_Spherical_Cartesian_Coordinates.xlsx). A software JCrystal [23] was first used for the creation of the regular pentagonal dodecahedron, icosahedron and triacontahedron, and then for the construction of the intersections between the regular polyhedra and seven intersecting planes. The first three normal vectors of the intersecting planes were parallel to the fivefold, threefold and twofold axes, the next three unit vectors were in the middle between the pairs of the previous axes (in fact, on the edges of the stereographic triangle), and the last one was inside the stereographic triangle. In the case of a fivefold intersecting plane, the unit vector 1 (v1) in Facets_Spherical_Cartesian_Coordinates.xlsx was selected. Its indices were: 0 0 1. A computer programme was written for the determination of the intersections of any plane with dendrite arms. The programme started with the definition of the unit normal vector of an intersecting plane. In the next step, a dendrite was defined with the dendrite arms extending in fivefold, threefold or twofold growth
Intersections of planes with the three fundamental polyhedra having the icosahedral symmetry can produce a variety of shapes. These shapes also appear on metallographic cross-sections in microstructure consisting of icosahedral quasicrystals and metallic matrix. Let us first examine the possible shapes arising from the intersection of planes with the pentagonal dodecahedron, which has the smallest number of facets (12), and each facet is a regular pentagon. When the normal to the intersecting plane (i.e. the polished surface) is parallel to a fivefold axis, the first intersection will go through the vertices indicated by v1 (Fig. 1a), and will produce a regular pentagon. The smallest pentagon will have the size of the facet. As the intersecting plane moves downwards, the resulting pentagon grows, until its side reaches the largest value of a (1 þ 2*cos72 ) when passing through the vertices v2 (a is the edge of a pentagonal facet). When the intersecting plane lies between v2 and v20 , then a decagon appears. The regular decagon forms in the middle of the pentagonal dodecahedron (in the middle between vertices v2 and v3). Its edge has the length a (1 þ 2*cos 72 )/2. In other sections, a decagon can be interpreted as a truncated pentagon. The equivalent positions on the bottom half of the pentagonal dodecahedra are labelled by v10 and v20 . Thus, the intersecting plane will again produce a pentagon between vertices v10 and v20 . When the normal to the intersecting plane is parallel to a threefold axis, then the plane first touches the pentagonal dodecahedron in the vertex v1 (Fig. 1b). When it moves towards the vertex v2, the intersection will always be a regular triangle. The largest side will be at the vertex v2: a (1 þ 2*cos72 ). Between vertices 2 and 3, the intersection will be a hexagon. The regular hexagon just appears at the equidistant position from the vertices v2 and v3. When the normal to the intersecting plane is parallel to a twofold axis, then the first touch with the pentagonal dodecahedron will be the line of length a (Fig. 1c). When the plane moves downwards, then the intersection has a rectangular shape. When it passes through vertices v2, a square appears having a side of a(1 þ 2*cos72 ). Between the vertices v2 and v3, the intersection
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Table 1 The results of the computer programme for the case when the unit vector of the intersecting plane was parallel to one of the fivefold axis and positioned 8 units from the dendrite centre; the Cartesian coordinates of its unit vector were 0 0 1, and the intersecting plane intersected the threefold arms (length: 50 units, diameter: 4 units). The number of the threefold arm Cartesian coordinate, x Cartesian coordinate, y Cartesian coordinate, z The dot product between the normal to the intersecting plane and the dendrite arm The angle between the normal of the intersecting plane and the dendrite arm, a The length of the larger ellipse axis, dl; the arm thickness, D, was 4 units The lateral distance of the intersection, t, when the intersection plane is position at a height h ¼ 8 units from the dendrite centre Orientation of the longer ellipse axis of the cut arm
43 0.491 0.357 0.795 0.795 37.38 5.03 6.11 36
48 0.491 0.357 0.795 0.795 142.62 not cut not cut not cut
53 0.795 0.577 0.188 0.188 79.19 21.32 41.84 36
61 0.304 0.934 0.188 0.188 100.81 not cut not cut not cut
Fig. 1. Intersections through a pentagonal dodecahedron by intersecting planes whose normal is perpendicular to a) fivefold, b) threefold or c) twofold axis. The larger number close to a polygon represents the number of its edges. The vertices labelled by the same symbol (e. g. v2) are intersected by the intersecting plane at the same time. Some vertices labelled by prime; v10 represent the equivalent vertices on the bottom half of the pentagonal dodecahedron.
will have an octagonal shape. The regular octahedron with the edge of a(1 þ 2*cos72 )/2 is in the middle between the vertices v2 and v3. Between vertices v3 and v30 , the intersection will be a hexagon. This analysis showed that when a plane normal is perpendicular to one of the principal symmetry axes of the pentagonal dodecahedra, then the intersections have at least rotational symmetry of a particular axis, and in some cases, even higher. This statement can also be extended to the icosahedron and triacontahedron.
3.2. Presentation of intersections within the stereographic triangle All intersections can be conveniently illustrated within the stereographic triangle. The vertices of the triangle represent a five-, three- and two-fold axis. The following orientations exist between these axes: 37.3770 between threefold and fivefold axes; 20.9052 between the threefold and twofold axis and 31.7175 between fivefold and twofold axes. In addition, the shapes along the four other axes were examined: 1 2 3 4
e e e e
between the fivefold and threefold axes between the fivefold and twofold axes between the twofold and threefold axes and within the stereographic triangle.
The possible shapes are shown one inside another (Fig. 2a). The shapes appeared when the normal to the intersecting plane is parallel to the principal axes are the same as shown already in Fig. 1. In other cases, the polygons are either distorted or truncated, thus in addition to polygons with 3, 4, 5, 6, 8 and 10 sides. Also, polygons with 7, 9 and 11 sides can be present. The same procedure was also done for the icosahedron and
triacontahedron. The possible shapes are given in Fig. 2b, c. The analysis of all shapes showed that intersecting of all polyhedra with a plane produces triangles, pentagons, hexagons, octagons and decagons, together with their distorted and truncated variants. So, if one observe a pentagon on a metallographic crosssection, it is not possible to state that the basic shape of a quasicrystal is the pentagonal dodecahedron. For discrimination of different shapes, several characteristics should be taken into account (Table 2). These characteristics were more reliable when all particles had similar size. Fig. 3a shows a microstructure of the AleCueFe alloy with several intersected quasicrystalline particles. On the first sight, it is hard to determine the 3D-shape of particles because they can be distorted due to the growth in the preferred directions, or their growth can be distorted by the presence of other particles. Also, the particles can have different sizes thus the conditions in Table 2 do not exactly apply. However, by inspecting selected particles some shapes can be eliminated. The truncated pentagon in Fig. 3b is oriented close to a fivefold axis, but can appear at the intersection of any of the considered shapes (see Fig. 2). The particle in Fig. 3c is close to a threefold axis, but this shape can appear close to the threefold axis in the pentagonal dodecahedron, close to the twofold axis in the icosahedron, and in the orientation 1 in the triacontahedron. However, the trapezoidal shape (Fig. 3d, e) seems to be possible only if the quasicrystal has a shape of the pentagonal dodecahedron. The pentagonal dodecahedron was observed in the same alloy [5].
3.3. Intersection of a plane with dendritic arms Quasicrystals often grow in preferred directions. Consequently,
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Fig. 2. Presentation of possible 2D-shapes when intersecting planes intersect a) pentagonal dodecahedron, b) icosahedron and c) triacontahedron. Intersections along seven intersecting planes were examined; the explanation is given in the text above.
dendritic arms appear and their maximum number is equal to the number of equivalent directions in the crystallographic system. In the icosahedral system, there could be up to 12 fivefold, 20 threefold and 30 twofold arms. The number of arms does not depend on
the equilibrium shape of the icosahedral quasicrystal. The intersecting plane can cut one or more arms. The positions and dimensions of intersections depend on the orientation of the intersection plane, its distance from the centre of a dendrite and
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Table 2 Characteristics for discriminating the basic shapes of a polyhedron with icosahedral symmetry from 2D-sections (A: the edge length in the 2D cross-section, a: the side of the facet in regular pentagonal, icosahedron or triacontahedron).
Pentagon Triangle
Pentagonal dodecahedron
Icosahedron
Triacontahedron
pentagons with similar sizes; no smaller than the facet size (A > a) range of sizes (0 < A < a(1þ2cos72 )
range of sizes (0 < A < a(1þ2cos72 )
range of sizes (0 < A < a(1þ2cos72 )
Shape with two mirror rectangle planes
triangle with similar sizes; no smaller than the range of sizes (0 < A < a(1þ2cos72 ) facet size (A > a) not a rectangle, octagon (or truncated rectangle) no shape with the right angle between any two edges
300 ) inclined for a3 ¼ 79.09 . The arms in the third circle are rotated for 60 in relation to the arms in the first circle. The last 6 arms in the fourth circle are perpendicular (a4 ¼ 90 ) to the threefold axis. 3.4. 2D-model
Fig. 3. Microstructure of the alloy AleCueFe with an icosahedral quasicrystalline phase. a) The area with several intersected particles, be) the selected individual particles. The equilibrium shape of the quasicrystal in this alloy is the pentagonal dodecahedron.
from the preferred growth directions. All important information can be obtained from the stereographic projections. Fig. 4 gives the stereographic projections for the fivefold, threefold and twofold axes, showing the positions of all principal axes in the upper hemisphere. Since only one preferred growth direction is typical for a particular quasicrystal, the positions of intersections are clearer, when arms with different preferred direction are shown separately. In Fig. 5, the two-, three-, and fivefold arms are presented for the threefold projection. The patterns are different, thus, these projections can be used by identifying the preferred growth directions. It is important to find a section through a dendrite with a threefold symmetry. For example, when the arms grew preferentially in twofold directions, then the image as shown in Fig. 5a would appear. The first three arms (1, 10, 100 ) are inclined for a1 ¼ 20.91 towards the threefold axis. There are six twofold arms in the second circle (2200 ) inclined for a2 ¼ 54.73 , and again three twofold arms in the third circle (3, 30 ,
The stereographic projections do not give the exact positions and shapes of the intersected arms. It is also not necessary that the intersecting plane cuts all the arms at the same time, which may be erroneously concluded from the Fig. 5. In order to explain these points, we will use a 2D-model in Fig. 6a. This model only shows the arms in the upper hemisphere having the length, L, and uniform thickness, D. The angles between the arms and normal to the surface were taken to be the same as between the twofold arm and threefold axis in Fig. 5a. The height, h, is a distance from the dendrite centre (indicated by 0), and t is the lateral distance from the centre. A plane will intersect the dendrite in this orientation when its distance to the centre is less than h < L cos a1, where L is the arm length. When L cos a2 < h < L cos a1 then the plane will intersect arms 1 and 10 (planes I and II in Fig. 6a). The lateral distance of both intersections will be equally distant from the dendrite centre because a1 ¼ a10 : t ¼ h tg(a1). The apparent thickness of the arm dl becomes dl ¼ D/cos(a1). Index l stands for the long (major) axis of the ellipse because the intersection of the plane with the circular rod will be an ellipse in most cases. When the plane is parallel or almost parallel with the arm, then the intersection will be rectangular with one side of D and other approaching L when a come near to 90 . The shorter axis of the ellipse will always be D. The plane III will intersect arms 1, 10, 2 and 20 , while the plane IV will intersect arms 1, 10, 2, 2, 3 and 30 . The plane V is a special case, and will only cut the arms 4 and 40 , which are parallel to it, thus, it will intersect the arm through its whole length. The distance of the intersection from the symmetry centre is ti ¼ h tg(ai). For a single arm t, the distance from the symmetry centre decreases with decreasing h, and therefore, the distance to the centre decreases when the plane cuts a dendrite closer to its core. At particular h, ti increases with increasing aI, thus the distances of the intersections from the centre increase from a1 to a3. These principles can be applied to the 3D-case, for the situation already discussed in Fig. 5a: the normal to the plane is parallel to one of the threefold axis, and dendritic arms point into the twofold directions. The positions of planes (IV) are similar than in Fig. 6a and the resulting intersections are depicted in Fig. 7. In reality, it is unlikely that the normal to the intersection plane will be parallel to one of the principal rotation axis. We can present such orientations as a deviation from the higher symmetry axis. An example for a 2D-case is shown in Fig. 6b. In this image, the dendrite arms were rotated in the clockwise direction for the angle a < a1 relative to their position in Fig. 6a. As a consequence, the angles between the normal to the surface and arms 14 decrease, while the angles between the normal to the surface and arms 10 40 increase. Thus, t1 becomes smaller t10 and dL1 < dL10. Additionally, in
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Fig. 4. Stereographic projections of the twofold, threefold and fivefold axes viewed along a) a twofold, b) threefold and c) fivefold axis.
Fig. 5. Presentation of arms grown in a) twofold, b) threefold and c) fivefold preferred directions in the stereographic presentation along the threefold axis.
Fig. 6. 2-D models of cutting dendritic arms with a) The normal to the planes is parallel to the axis of symmetry; b) the normal to the planes is not parallel to the axis of symmetry.
the plane IV appears the arm 4, which was not cut by this plane in Fig. 6a. In Fig. 8a the normal to the intersection coincides with the threefold axis, while in the Fig. 8b does not. The normal to the surface is approximately in the middle of the threefold and fivefold
axes, thus, the pattern retained the vertical mirror plane. Previously equivalent arms start to differ from one another by the distances from the new plane normal and in the lengths of the major ellipse axes. The arm 30 is now closer to the plane normal, and its major axis becomes smaller than in Fig. 8a. On the other hand, the arms 30
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Fig. 7. Intersections of the plane with the twofold arms with decreasing distance from the dendrite centre. The normal to the intersection plane is parallel to the threefold axis. The positions of planes (IV) are similar than in Fig. 6a.
Fig. 8. Presentation of the intersection of the twofold axis. a) The normal to the plane is parallel to the threefold axis and b) the normal to the plane is between the threefold and fivefold arms.
and 300 are farther away, and, therefore, their longer axes increased considerably. Rotation also caused the appearance of arms 4* and 40 . 3.5. Presentation of the intersected dendritic arms in the stereographic triangle Fig. 9 shows the intersections of the arms growing in five-, three- and twofold directions in the stereographic triangles. Intersections are indicated for the same axes as in Fig. 2. Each arm in Fig. 8 is labelled by a unique number. The fivefold arms are numbered by 1e12, the threefold by 43e62 and the twofold arms by 13e42. Thus, it is possible to follow how the appearances of their cross-sections vary with a rotation of a dendrite. For example, in Fig. 9c, arm 1 is one of the fivefold axes. In the fivefold corner, this axis is in the centre of the fivefold pattern (its direction coincides with the plane normal), and d ¼ D. In the twofold corner, the arm 1 is the same inclined to the twofold axis as arm 7, and its long axis is d ¼ D/cos(31.79 ). In the threefold corner, the arm 1 has the same inclination to the threefold axis as arms 7 and 9, and its long axis is d ¼ D/cos(20.91 ). 3.6. Application Fig. 10 shows a metallographic cross-section through a dendritic icosahedral quasicrystal in an AleMneBeeCu alloy. The section has an approximate mirror plane, and the sizes of arms are different.
These facts show that the normal to the plane does not coincide with any of the principal rotation axes. However, the mirror plane indicates that the orientation lies on the line between two principal axes. Similar patterns can be seen in the stereographic triangles. Therefore, we can ask ourselves whether it is possible to determine the preferred growth directions of the dendritic arms in Fig. 10 by comparing with the patterns in Fig. 9. A comparison of Figs. 9 and 10 shows that the distribution of arm in Fig. 10 closely resembles the distribution of arms along axis “2” in Fig. 9b. Thus, we can conclude that, in this case, the dendrite arm preferentially grew in the threefold directions, and that the orientation of the plane normal was almost in the middle between five- and twofold axes. The preferred growth of icosahedral quasicrystal alloys based on AleMn system in the threefold direction was confirmed previously. However, the tendency to faceted growth and non-uniform thickness of the arms can lead to something different intersections that predicted for the cylindrical arms. Nevertheless, the resemblance is quite strong thus from these cross-sections rather good predictions are possible. In some cases, the solidification starts with the formation of the central (almost ideal) polyhedron, and after that preferred growth takes place in the preferred directions, with the formation of dendritic arms. Thus, combinations of Figs. 2 and 9 can explain the intersections. The number and angles between arms do not depend on the shape of the central polyhedron.
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Fig. 9. Presentation of intersections of dendritic icosahedral quasicrystals within the stereographic triangle. Preferred growth directions were: a) twofold, b) threefold and c) fivefold. Seven intersection planes were examined; the same as in Fig. 2 and explained in Section 3.2 in detail.
4. Conclusions In this work, the possible shapes of the principal polyhedra with the icosahedral symmetry with the differently oriented intersecting planes (with the polished surface) were systematically depicted within the icosahedral stereographic triangle.
The same procedure was carried out for the icosahedral dendrite with different preferred growth directions. A methodological observation of the shapes of particles on the polished surface can give us a clue about the principal shape of the icosahedral polyhedron, and the information about the deviations from its ideal shape.
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[5] [6]
[7]
[8]
[9] Fig. 10. The cross-section through a dendritic icosahedral quasicrystal (alloy AleMneBeeCu). The shape and distribution of arms indicate the preferred growth in threefold directions, and the orientation of the intersecting plane is close to the direction 2 in Fig. 9b (between fivefold and twofold axes). The arms are labelled in the same way as in Fig. 9b.
[10]
[11]
[12]
When an icosahedral quasicrystal exhibits dendritic growth morphology, then it is possible to determine the preferred growth direction from the 2D-sections. The approach shown in the article cannot be taken as selfsufficient and standalone, but should be used as a complementary method to other techniques for determination of crystal shapes. Nevertheless, the approach can be extended to any polyhedron and any dendrite.
[13]
[14]
[15]
[16]
Acknowledgements This work was partly financed by the research programme P2d0120 (Slovenian Research Agency e ARRS).
[17]
[18]
Appendix A. Supplementary data [19]
Supplementary data related to this article can be found at http:// dx.doi.org/10.1016/j.jallcom.2016.04.255. References [1] K. Chattopadhyay, N. Ravishankar, R. Goswami, Shapes of quasicrystals, Prog. Cryst. Growth Charact. Mater. 34 (1997) 237e249. [2] K.F. Kelton, Quasi-crystals e structure and stability, Int. Mater. Rev. 38 (1993) 105e137. [3] N. Thangaraj, G.N. Subbanna, S. Ranganathan, K. Chattopadhyay, Electronmicroscopy and diffraction of icosahedral and decagonal quasi-crystals in aluminum manganese alloys, J. Microsc.-Oxf. 146 (1987) 287e302. [4] A.P. Tsai, A. Inoue, T. Masumoto, Preparation of a new Al-Cu-Fe quasicrystal
[20]
[21]
[22]
[23]
with large grain sizes by rapid solidification, J. Mater. Sci. Lett. 6 (1987) 1403e1405. T. Boncina, B. Markoli, I. Anzel, F. Zupanic, Phases in a quasicrystalline alloy Al(64,4)Cu(23,5)Fe13,1, Mater. Tehnol. 41 (2007) 271e277. P.C. Canfield, M.L. Caudle, C.S. Ho, A. Kreyssig, S. Nandi, M.G. Kim, X. Lin, A. Kracher, K.W. Dennis, R.W. McCallum, A.I. Goldman, Solution growth of a binary icosahedral quasicrystal of Sc12Zn88, Phys. Rev. B 81 (2010) 4. B. Dubost, J.M. Lang, M. Tanaka, P. Sainfort, M. Audier, Large AlCuLi single quasi-crystals with triacontahedral solidification morphology, Nature 324 (1986) 48e50. K. Sato, H. Uchiyama, Y. Takahashi, I. Kanazawa, R. Suzuki, T. Ohdaira, T. Takeuchi, T. Mizuno, U. Mizutani, Positron-annihilation studies of rhombic triacontahedral-type icosahedral quasicrystals and their 1/1 and 2/1 approximants in the Al-Mg-Zn alloy system, Phys. Rev. B 64 (2001) 8. N. Tamura, M. Beyss, K. Urban, Triacontahedral growth morphology in icosahedral Zn-Mg-Y alloy, Philos. Mag. Lett. 74 (1996) 89e97. D. Shechtman, I. Blech, D. Gratias, J.W. Cahn, Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53 (1984) 1951e1953. D.Q. Wan, G. Yang, S. Chen, M. Zhu, Y.H. Zhou, Growth morphology and evolution of quasicrystal in as-solidified Y-rich Mg-Zn-Y ternary alloys, Rare Metals 26 (2007) 435e439. M.V. Kral, M.A. Mangan, G. Spanos, R.O. Rosenberg, Three-dimensional analysis of microstructures, Mater. Charact. 45 (2000) 17e23. F. Lasagni, A. Lasagni, E. Marks, C. Holzapfel, F. Mucklich, H.P. Degischer, Three-dimensional characterization of ‘as-east’ and solution-treated AlSi12(Sr) alloys by high-resolution FIB tomography, Acta Mater. 55 (2007) 3875e3882. I.R. Fisher, M.J. Kramer, T.A. Wiener, Z. Islam, A.R. Ross, T.A. Lograsso, A. Kracher, A.I. Goldman, P.C. Canfield, On the growth of icosahedral Al-Pd-Mn quasicrystals from the ternary melt, Philos. Mag. B Phys. Condens. Matter Stat. Mech. Electron. Opt. Magn. Prop. 79 (1999) 1673e1684. C. Beeli, H.U. Nissen, Growth-morphology of Icosahedral Al-Mn-Pd single quasi-crystals, Philos. Mag. B Phys. Condens. Matter Stat. Mech. Electron. Opt. Magn. Prop. 68 (1993) 487e512. T. Boncina, B. Markoli, I. Anzel, F. Zupanic, Metallographic techniques for the characterization of quasicrystalline phases in aluminium alloys, Z. Krist. 223 (2008) 747e750. F. Zupani c, T. Bon cina, N. Rozman, B. Markoli, Morphologies of icosahedral quasicrystals in Al-Mn-Be-(Cu) alloys, in: Quasicrystals: Types, Systems, and Techniques, Nova Science Publishers, Hauppauge, N.Y., 2011, pp. 195e217. Y. Yang, K. Zhang, M.L. Ma, J.W. Yuan, Microstructure and phase compositions of as-cast Mg-3.9Zn-0.6RE (Gd, Y) alloy with different Gd/Y ratios, Rare Metals 34 (2015) 160e163. Y. Tian, H. Huang, G.Y. Yuan, W.J. Ding, Microstructure evolution and mechanical properties of quasicrystal-reinforced Mg-Zn-Gd alloy processed by cyclic extrusion and compression, J. Alloys Compd. 626 (2015) 42e48. L. Yang, Y.H. Zhao, H. Hou, X.M. Yang, Effects of Y on microstructures and properties of Mg-Zn-Y alloys prepared by conventional solidification, Rare Metal Mater. Eng. 43 (2014) 1754e1758. N. Strekelj, I. Naglic, G. Klancnik, A. Nagode, B. Markoli, Microstructural changes in quasicrystalline Al - Mn - Be - Cu alloy after various heat treatments, Int. J. Mater. Res. 106 (2015) 342e351. T. Oz, E. Karakose, M. Keskin, Impact of beryllium additions on thermal and mechanical properties of conventionally solidified and melt-spun Al-4.5 wt.% Mn-x wt.%Be (x¼0, 1, 3, 5) alloys, Mater. Des. 50 (2013) 399e412. JCrystal, Crystal Shape Editor/Viewer 1.0.5, ©JCrystalSoft, 2011, 2005e2011, http://www.jcrystal.com/jcrystal.html.
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Determination of shapes and preferred growth directions of icosahedral quasicrystals from the 2D-sections Franc Zupani c*, Tonica Bon cina University of Maribor, Faculty of Mechanical Engineering, Smetanova ulica 17, SI-2000 Maribor, Slovenia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 January 2016 Received in revised form 24 April 2016 Accepted 25 April 2016 Available online 27 April 2016
The icosahedral quasicrystalline phases have been present in many alloys either as nearly regular polyhedra or dendrites. This work provides a systematic approach for the determination of shapes of polyhedra with icosahedral symmetry, and the determination of preferred growth directions of icosahedral dendrites from their 2D-sections; i.e. from the surface of polished samples. The possible shapes of polyhedra and dendrites by differently oriented intersecting planes are presented within the icosahedral stereographic triangle. © 2016 Elsevier B.V. All rights reserved.
Keywords: Quasicrystals Microstructure
1. Introduction The point group symmetry dictates the shapes of a crystal. The quasicrystals possess non-crystallographic point group symmetries having distinctive shapes from those of periodic crystals. The minimum surface energy principle determines the equilibrium shape of the crystal, while the growth morphologies depend on the growth kinetics and processing conditions [1]. There are three main regular polyhedra with the icosahedral symmetry: pentagonal dodecahedron (12 pentagonal faces, 30 edges, 20 vertices) icosahedron (20 triangular faces, 30 edges, 12 vertices) rhombic triacontahedron (30 rhombic faces, 60 edges, 32 vertices). They can preferentially grow in twofold, threefold and fivefold directions. The AleMn and AleMneSi icosahedral quasicrystals mainly formed a dendritic structure with branches extending in the preferred threefold directions [2]. Thangaraj et al. [3] established that the pentagonal dodecahedron was the shape for the AleMn quasicrystal. In the AleCueFe system, the equilibrium shape of the stable quasicrystal was the pentagonal dodecahedron. Also, quasicrystals with the icosidodecahedral morphology were observed
* Corresponding author. E-mail address: [email protected] (F. Zupani c). http://dx.doi.org/10.1016/j.jallcom.2016.04.255 0925-8388/© 2016 Elsevier B.V. All rights reserved.
[4,5]. In several alloy systems, such as Sc12Zn88 [6], AleLieCu [7], AleMgeZn [8] and ZneMgeY face-centred icosahedral alloys, the quasicrystals possessed the triacontahedral growth morphology [9]. The preferred growth direction was mainly along threefold axes and occasionally along fivefold axes [2]. There is some confusion regarding quasicrystalline shapes, especially those determined from the 2D-sections. Several methods have been used for the determination of quasicrystal shapes. The first quasicrystalline phases were metastable and were obtained during rapid solidification. Rapid solidification of dilute AleMn alloys provided substantial undercooling promoting the nucleation of small quasicrystalline particles in the melt [10]. Aluminium nucleated later and trapped the small quasicrystalline particles, retaining their shapes. Due to small sizes of quasicrystalline particles, the basic shapes were investigated using TEM. The shapes were identical when projected along higher symmetry axes like threefold and fivefold axes, while the projections along the twofold axes enabled distinction between different shapes [3]. By developing stable quasicrystalline phases, as well as new aluminium alloys with increased quasicrystal-forming ability, determination of the quasicrystalline shape from the micrographs of the polished surface has become crucial. Although the icosahedral quasicrystalline particles on the polished surface sometimes exhibited a pentagonal symmetry [11], it was not always clear, what was their exact shape, and what was the preferred growth direction. Kral et al. [12] obtained 3D-shapes of phases in steels by using computer-aided visualization of 3D-reconstructions from series of
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section images. Nowadays, sequential cross sectioning is done using focused ion beam FIB, usually within a dual beam SEM/FIB system. By the application of microanalytical techniques the 3Delemental distribution can also be obtained along with the particle shapes [13]. Such approach is feasible, but it is very timeconsuming. Direct observation of 3D-shapes of quasicrystals was done by decanting of the melt after growing single quasicrystals [14], observations of faceted microholes in slowly cooled icosahedral single quasicrystals [15] and inside solidification shrinkage microvoids [5]. 3D-shapes of quasicrystalline phases can also be obtained by deep etching and particles extraction techniques [16,17]. Over the last years, several new aluminium and magnesium alloys have been developed containing quasicrystalline phase in the as-cast cast state after conventional casting [18e22]. It is expected that the number of such alloys will increase in the future. The main aim of this work is to provide a systematic approach to identification of icosahedral quasicrystals from the polished samples, i.e. from the 2D-sections. Thus, the possible shapes of the regular pentagonal dodecahedron, icosahedron, and rhombic triacontahedron with differently oriented intersecting planes are systematically presented within the icosahedral stereographic triangle. The intersections of the intersecting plane with dendritic icosahedral quasicrystals were established to enable easier and more reliable determination of the preferred growth directions.
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directions. The arms had the length L and circular cross-section with the diameter D. The unit vector of the intersection plane passed through the dendrite centre. For example, when the arms grew in the threefold directions, then vectors 43 to 62 were chosen (see Facets_Spherical_Cartesian_Coordinates.xlsx). In the next stage, the dot products between the plane normal vector and the unit vectors of all arms were calculated. The intersecting plane was positioned at a particular distance, h, from the dendrite centre. At this distance, the arms cut by the intersecting plane were determined. For these arms, the following characteristics were calculated: (1) the longer axis of the ellipse, dl; the intersection of a plane and a tube is a cylinder; (2) the lateral distance of the centre of the intersection from the normal of the intersecting plane passing through dendrite centre, t; and (3) the orientation of the longer ellipse axis. The results for one particular case are given in Table 1. The calculation was done for four of twenty arms. The application of the whole procedure is described in detail in Section 3.4. The results for all preferred growth directions and all seven intersecting planes were depicted in the icosahedral stereographic triangle. 3. Results and discussion 3.1. Intersections of planes with pentagonal dodecahedron, icosahedrons, and triacontahedron
2. Methods As stated in the introduction, the work was focused to. a) The determination of possible shapes of the regular pentagonal dodecahedron, icosahedron, and rhombic triacontahedron with differently oriented intersecting planes and their positioning within the icosahedral stereographic triangle. b) The determination of the intersections of the plane with dendritic icosahedral quasicrystals depending on their preferred growth directions. For both tasks, a set of 62 unit vectors was defined. This set was divided into three subsets, containing 12 normal vectors to the facets of the pentagonal dodecahedron, 20 normal vectors to the facets of the icosahedron, and 30 normal vectors to the facets of the rhombic triacontahedron. The same vectors also represented the vectors indicating the preferred growth directions of dendritic arms in the fivefold, threefold and twofold axes. All these unit normal vectors in polar and Cartesian coordinates are listed in the Supplement Material (Facets_Spherical_Cartesian_Coordinates.xlsx). A software JCrystal [23] was first used for the creation of the regular pentagonal dodecahedron, icosahedron and triacontahedron, and then for the construction of the intersections between the regular polyhedra and seven intersecting planes. The first three normal vectors of the intersecting planes were parallel to the fivefold, threefold and twofold axes, the next three unit vectors were in the middle between the pairs of the previous axes (in fact, on the edges of the stereographic triangle), and the last one was inside the stereographic triangle. In the case of a fivefold intersecting plane, the unit vector 1 (v1) in Facets_Spherical_Cartesian_Coordinates.xlsx was selected. Its indices were: 0 0 1. A computer programme was written for the determination of the intersections of any plane with dendrite arms. The programme started with the definition of the unit normal vector of an intersecting plane. In the next step, a dendrite was defined with the dendrite arms extending in fivefold, threefold or twofold growth
Intersections of planes with the three fundamental polyhedra having the icosahedral symmetry can produce a variety of shapes. These shapes also appear on metallographic cross-sections in microstructure consisting of icosahedral quasicrystals and metallic matrix. Let us first examine the possible shapes arising from the intersection of planes with the pentagonal dodecahedron, which has the smallest number of facets (12), and each facet is a regular pentagon. When the normal to the intersecting plane (i.e. the polished surface) is parallel to a fivefold axis, the first intersection will go through the vertices indicated by v1 (Fig. 1a), and will produce a regular pentagon. The smallest pentagon will have the size of the facet. As the intersecting plane moves downwards, the resulting pentagon grows, until its side reaches the largest value of a (1 þ 2*cos72 ) when passing through the vertices v2 (a is the edge of a pentagonal facet). When the intersecting plane lies between v2 and v20 , then a decagon appears. The regular decagon forms in the middle of the pentagonal dodecahedron (in the middle between vertices v2 and v3). Its edge has the length a (1 þ 2*cos 72 )/2. In other sections, a decagon can be interpreted as a truncated pentagon. The equivalent positions on the bottom half of the pentagonal dodecahedra are labelled by v10 and v20 . Thus, the intersecting plane will again produce a pentagon between vertices v10 and v20 . When the normal to the intersecting plane is parallel to a threefold axis, then the plane first touches the pentagonal dodecahedron in the vertex v1 (Fig. 1b). When it moves towards the vertex v2, the intersection will always be a regular triangle. The largest side will be at the vertex v2: a (1 þ 2*cos72 ). Between vertices 2 and 3, the intersection will be a hexagon. The regular hexagon just appears at the equidistant position from the vertices v2 and v3. When the normal to the intersecting plane is parallel to a twofold axis, then the first touch with the pentagonal dodecahedron will be the line of length a (Fig. 1c). When the plane moves downwards, then the intersection has a rectangular shape. When it passes through vertices v2, a square appears having a side of a(1 þ 2*cos72 ). Between the vertices v2 and v3, the intersection
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Table 1 The results of the computer programme for the case when the unit vector of the intersecting plane was parallel to one of the fivefold axis and positioned 8 units from the dendrite centre; the Cartesian coordinates of its unit vector were 0 0 1, and the intersecting plane intersected the threefold arms (length: 50 units, diameter: 4 units). The number of the threefold arm Cartesian coordinate, x Cartesian coordinate, y Cartesian coordinate, z The dot product between the normal to the intersecting plane and the dendrite arm The angle between the normal of the intersecting plane and the dendrite arm, a The length of the larger ellipse axis, dl; the arm thickness, D, was 4 units The lateral distance of the intersection, t, when the intersection plane is position at a height h ¼ 8 units from the dendrite centre Orientation of the longer ellipse axis of the cut arm
43 0.491 0.357 0.795 0.795 37.38 5.03 6.11 36
48 0.491 0.357 0.795 0.795 142.62 not cut not cut not cut
53 0.795 0.577 0.188 0.188 79.19 21.32 41.84 36
61 0.304 0.934 0.188 0.188 100.81 not cut not cut not cut
Fig. 1. Intersections through a pentagonal dodecahedron by intersecting planes whose normal is perpendicular to a) fivefold, b) threefold or c) twofold axis. The larger number close to a polygon represents the number of its edges. The vertices labelled by the same symbol (e. g. v2) are intersected by the intersecting plane at the same time. Some vertices labelled by prime; v10 represent the equivalent vertices on the bottom half of the pentagonal dodecahedron.
will have an octagonal shape. The regular octahedron with the edge of a(1 þ 2*cos72 )/2 is in the middle between the vertices v2 and v3. Between vertices v3 and v30 , the intersection will be a hexagon. This analysis showed that when a plane normal is perpendicular to one of the principal symmetry axes of the pentagonal dodecahedra, then the intersections have at least rotational symmetry of a particular axis, and in some cases, even higher. This statement can also be extended to the icosahedron and triacontahedron.
3.2. Presentation of intersections within the stereographic triangle All intersections can be conveniently illustrated within the stereographic triangle. The vertices of the triangle represent a five-, three- and two-fold axis. The following orientations exist between these axes: 37.3770 between threefold and fivefold axes; 20.9052 between the threefold and twofold axis and 31.7175 between fivefold and twofold axes. In addition, the shapes along the four other axes were examined: 1 2 3 4
e e e e
between the fivefold and threefold axes between the fivefold and twofold axes between the twofold and threefold axes and within the stereographic triangle.
The possible shapes are shown one inside another (Fig. 2a). The shapes appeared when the normal to the intersecting plane is parallel to the principal axes are the same as shown already in Fig. 1. In other cases, the polygons are either distorted or truncated, thus in addition to polygons with 3, 4, 5, 6, 8 and 10 sides. Also, polygons with 7, 9 and 11 sides can be present. The same procedure was also done for the icosahedron and
triacontahedron. The possible shapes are given in Fig. 2b, c. The analysis of all shapes showed that intersecting of all polyhedra with a plane produces triangles, pentagons, hexagons, octagons and decagons, together with their distorted and truncated variants. So, if one observe a pentagon on a metallographic crosssection, it is not possible to state that the basic shape of a quasicrystal is the pentagonal dodecahedron. For discrimination of different shapes, several characteristics should be taken into account (Table 2). These characteristics were more reliable when all particles had similar size. Fig. 3a shows a microstructure of the AleCueFe alloy with several intersected quasicrystalline particles. On the first sight, it is hard to determine the 3D-shape of particles because they can be distorted due to the growth in the preferred directions, or their growth can be distorted by the presence of other particles. Also, the particles can have different sizes thus the conditions in Table 2 do not exactly apply. However, by inspecting selected particles some shapes can be eliminated. The truncated pentagon in Fig. 3b is oriented close to a fivefold axis, but can appear at the intersection of any of the considered shapes (see Fig. 2). The particle in Fig. 3c is close to a threefold axis, but this shape can appear close to the threefold axis in the pentagonal dodecahedron, close to the twofold axis in the icosahedron, and in the orientation 1 in the triacontahedron. However, the trapezoidal shape (Fig. 3d, e) seems to be possible only if the quasicrystal has a shape of the pentagonal dodecahedron. The pentagonal dodecahedron was observed in the same alloy [5].
3.3. Intersection of a plane with dendritic arms Quasicrystals often grow in preferred directions. Consequently,
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Fig. 2. Presentation of possible 2D-shapes when intersecting planes intersect a) pentagonal dodecahedron, b) icosahedron and c) triacontahedron. Intersections along seven intersecting planes were examined; the explanation is given in the text above.
dendritic arms appear and their maximum number is equal to the number of equivalent directions in the crystallographic system. In the icosahedral system, there could be up to 12 fivefold, 20 threefold and 30 twofold arms. The number of arms does not depend on
the equilibrium shape of the icosahedral quasicrystal. The intersecting plane can cut one or more arms. The positions and dimensions of intersections depend on the orientation of the intersection plane, its distance from the centre of a dendrite and
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Table 2 Characteristics for discriminating the basic shapes of a polyhedron with icosahedral symmetry from 2D-sections (A: the edge length in the 2D cross-section, a: the side of the facet in regular pentagonal, icosahedron or triacontahedron).
Pentagon Triangle
Pentagonal dodecahedron
Icosahedron
Triacontahedron
pentagons with similar sizes; no smaller than the facet size (A > a) range of sizes (0 < A < a(1þ2cos72 )
range of sizes (0 < A < a(1þ2cos72 )
range of sizes (0 < A < a(1þ2cos72 )
Shape with two mirror rectangle planes
triangle with similar sizes; no smaller than the range of sizes (0 < A < a(1þ2cos72 ) facet size (A > a) not a rectangle, octagon (or truncated rectangle) no shape with the right angle between any two edges
300 ) inclined for a3 ¼ 79.09 . The arms in the third circle are rotated for 60 in relation to the arms in the first circle. The last 6 arms in the fourth circle are perpendicular (a4 ¼ 90 ) to the threefold axis. 3.4. 2D-model
Fig. 3. Microstructure of the alloy AleCueFe with an icosahedral quasicrystalline phase. a) The area with several intersected particles, be) the selected individual particles. The equilibrium shape of the quasicrystal in this alloy is the pentagonal dodecahedron.
from the preferred growth directions. All important information can be obtained from the stereographic projections. Fig. 4 gives the stereographic projections for the fivefold, threefold and twofold axes, showing the positions of all principal axes in the upper hemisphere. Since only one preferred growth direction is typical for a particular quasicrystal, the positions of intersections are clearer, when arms with different preferred direction are shown separately. In Fig. 5, the two-, three-, and fivefold arms are presented for the threefold projection. The patterns are different, thus, these projections can be used by identifying the preferred growth directions. It is important to find a section through a dendrite with a threefold symmetry. For example, when the arms grew preferentially in twofold directions, then the image as shown in Fig. 5a would appear. The first three arms (1, 10, 100 ) are inclined for a1 ¼ 20.91 towards the threefold axis. There are six twofold arms in the second circle (2200 ) inclined for a2 ¼ 54.73 , and again three twofold arms in the third circle (3, 30 ,
The stereographic projections do not give the exact positions and shapes of the intersected arms. It is also not necessary that the intersecting plane cuts all the arms at the same time, which may be erroneously concluded from the Fig. 5. In order to explain these points, we will use a 2D-model in Fig. 6a. This model only shows the arms in the upper hemisphere having the length, L, and uniform thickness, D. The angles between the arms and normal to the surface were taken to be the same as between the twofold arm and threefold axis in Fig. 5a. The height, h, is a distance from the dendrite centre (indicated by 0), and t is the lateral distance from the centre. A plane will intersect the dendrite in this orientation when its distance to the centre is less than h < L cos a1, where L is the arm length. When L cos a2 < h < L cos a1 then the plane will intersect arms 1 and 10 (planes I and II in Fig. 6a). The lateral distance of both intersections will be equally distant from the dendrite centre because a1 ¼ a10 : t ¼ h tg(a1). The apparent thickness of the arm dl becomes dl ¼ D/cos(a1). Index l stands for the long (major) axis of the ellipse because the intersection of the plane with the circular rod will be an ellipse in most cases. When the plane is parallel or almost parallel with the arm, then the intersection will be rectangular with one side of D and other approaching L when a come near to 90 . The shorter axis of the ellipse will always be D. The plane III will intersect arms 1, 10, 2 and 20 , while the plane IV will intersect arms 1, 10, 2, 2, 3 and 30 . The plane V is a special case, and will only cut the arms 4 and 40 , which are parallel to it, thus, it will intersect the arm through its whole length. The distance of the intersection from the symmetry centre is ti ¼ h tg(ai). For a single arm t, the distance from the symmetry centre decreases with decreasing h, and therefore, the distance to the centre decreases when the plane cuts a dendrite closer to its core. At particular h, ti increases with increasing aI, thus the distances of the intersections from the centre increase from a1 to a3. These principles can be applied to the 3D-case, for the situation already discussed in Fig. 5a: the normal to the plane is parallel to one of the threefold axis, and dendritic arms point into the twofold directions. The positions of planes (IV) are similar than in Fig. 6a and the resulting intersections are depicted in Fig. 7. In reality, it is unlikely that the normal to the intersection plane will be parallel to one of the principal rotation axis. We can present such orientations as a deviation from the higher symmetry axis. An example for a 2D-case is shown in Fig. 6b. In this image, the dendrite arms were rotated in the clockwise direction for the angle a < a1 relative to their position in Fig. 6a. As a consequence, the angles between the normal to the surface and arms 14 decrease, while the angles between the normal to the surface and arms 10 40 increase. Thus, t1 becomes smaller t10 and dL1 < dL10. Additionally, in
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Fig. 4. Stereographic projections of the twofold, threefold and fivefold axes viewed along a) a twofold, b) threefold and c) fivefold axis.
Fig. 5. Presentation of arms grown in a) twofold, b) threefold and c) fivefold preferred directions in the stereographic presentation along the threefold axis.
Fig. 6. 2-D models of cutting dendritic arms with a) The normal to the planes is parallel to the axis of symmetry; b) the normal to the planes is not parallel to the axis of symmetry.
the plane IV appears the arm 4, which was not cut by this plane in Fig. 6a. In Fig. 8a the normal to the intersection coincides with the threefold axis, while in the Fig. 8b does not. The normal to the surface is approximately in the middle of the threefold and fivefold
axes, thus, the pattern retained the vertical mirror plane. Previously equivalent arms start to differ from one another by the distances from the new plane normal and in the lengths of the major ellipse axes. The arm 30 is now closer to the plane normal, and its major axis becomes smaller than in Fig. 8a. On the other hand, the arms 30
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Fig. 7. Intersections of the plane with the twofold arms with decreasing distance from the dendrite centre. The normal to the intersection plane is parallel to the threefold axis. The positions of planes (IV) are similar than in Fig. 6a.
Fig. 8. Presentation of the intersection of the twofold axis. a) The normal to the plane is parallel to the threefold axis and b) the normal to the plane is between the threefold and fivefold arms.
and 300 are farther away, and, therefore, their longer axes increased considerably. Rotation also caused the appearance of arms 4* and 40 . 3.5. Presentation of the intersected dendritic arms in the stereographic triangle Fig. 9 shows the intersections of the arms growing in five-, three- and twofold directions in the stereographic triangles. Intersections are indicated for the same axes as in Fig. 2. Each arm in Fig. 8 is labelled by a unique number. The fivefold arms are numbered by 1e12, the threefold by 43e62 and the twofold arms by 13e42. Thus, it is possible to follow how the appearances of their cross-sections vary with a rotation of a dendrite. For example, in Fig. 9c, arm 1 is one of the fivefold axes. In the fivefold corner, this axis is in the centre of the fivefold pattern (its direction coincides with the plane normal), and d ¼ D. In the twofold corner, the arm 1 is the same inclined to the twofold axis as arm 7, and its long axis is d ¼ D/cos(31.79 ). In the threefold corner, the arm 1 has the same inclination to the threefold axis as arms 7 and 9, and its long axis is d ¼ D/cos(20.91 ). 3.6. Application Fig. 10 shows a metallographic cross-section through a dendritic icosahedral quasicrystal in an AleMneBeeCu alloy. The section has an approximate mirror plane, and the sizes of arms are different.
These facts show that the normal to the plane does not coincide with any of the principal rotation axes. However, the mirror plane indicates that the orientation lies on the line between two principal axes. Similar patterns can be seen in the stereographic triangles. Therefore, we can ask ourselves whether it is possible to determine the preferred growth directions of the dendritic arms in Fig. 10 by comparing with the patterns in Fig. 9. A comparison of Figs. 9 and 10 shows that the distribution of arm in Fig. 10 closely resembles the distribution of arms along axis “2” in Fig. 9b. Thus, we can conclude that, in this case, the dendrite arm preferentially grew in the threefold directions, and that the orientation of the plane normal was almost in the middle between five- and twofold axes. The preferred growth of icosahedral quasicrystal alloys based on AleMn system in the threefold direction was confirmed previously. However, the tendency to faceted growth and non-uniform thickness of the arms can lead to something different intersections that predicted for the cylindrical arms. Nevertheless, the resemblance is quite strong thus from these cross-sections rather good predictions are possible. In some cases, the solidification starts with the formation of the central (almost ideal) polyhedron, and after that preferred growth takes place in the preferred directions, with the formation of dendritic arms. Thus, combinations of Figs. 2 and 9 can explain the intersections. The number and angles between arms do not depend on the shape of the central polyhedron.
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Fig. 9. Presentation of intersections of dendritic icosahedral quasicrystals within the stereographic triangle. Preferred growth directions were: a) twofold, b) threefold and c) fivefold. Seven intersection planes were examined; the same as in Fig. 2 and explained in Section 3.2 in detail.
4. Conclusions In this work, the possible shapes of the principal polyhedra with the icosahedral symmetry with the differently oriented intersecting planes (with the polished surface) were systematically depicted within the icosahedral stereographic triangle.
The same procedure was carried out for the icosahedral dendrite with different preferred growth directions. A methodological observation of the shapes of particles on the polished surface can give us a clue about the principal shape of the icosahedral polyhedron, and the information about the deviations from its ideal shape.
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[5] [6]
[7]
[8]
[9] Fig. 10. The cross-section through a dendritic icosahedral quasicrystal (alloy AleMneBeeCu). The shape and distribution of arms indicate the preferred growth in threefold directions, and the orientation of the intersecting plane is close to the direction 2 in Fig. 9b (between fivefold and twofold axes). The arms are labelled in the same way as in Fig. 9b.
[10]
[11]
[12]
When an icosahedral quasicrystal exhibits dendritic growth morphology, then it is possible to determine the preferred growth direction from the 2D-sections. The approach shown in the article cannot be taken as selfsufficient and standalone, but should be used as a complementary method to other techniques for determination of crystal shapes. Nevertheless, the approach can be extended to any polyhedron and any dendrite.
[13]
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Acknowledgements This work was partly financed by the research programme P2d0120 (Slovenian Research Agency e ARRS).
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Appendix A. Supplementary data [19]
Supplementary data related to this article can be found at http:// dx.doi.org/10.1016/j.jallcom.2016.04.255. References [1] K. Chattopadhyay, N. Ravishankar, R. Goswami, Shapes of quasicrystals, Prog. Cryst. Growth Charact. Mater. 34 (1997) 237e249. [2] K.F. Kelton, Quasi-crystals e structure and stability, Int. Mater. Rev. 38 (1993) 105e137. [3] N. Thangaraj, G.N. Subbanna, S. Ranganathan, K. Chattopadhyay, Electronmicroscopy and diffraction of icosahedral and decagonal quasi-crystals in aluminum manganese alloys, J. Microsc.-Oxf. 146 (1987) 287e302. [4] A.P. Tsai, A. Inoue, T. Masumoto, Preparation of a new Al-Cu-Fe quasicrystal
[20]
[21]
[22]
[23]
with large grain sizes by rapid solidification, J. Mater. Sci. Lett. 6 (1987) 1403e1405. T. Boncina, B. Markoli, I. Anzel, F. Zupanic, Phases in a quasicrystalline alloy Al(64,4)Cu(23,5)Fe13,1, Mater. Tehnol. 41 (2007) 271e277. P.C. Canfield, M.L. Caudle, C.S. Ho, A. Kreyssig, S. Nandi, M.G. Kim, X. Lin, A. Kracher, K.W. Dennis, R.W. McCallum, A.I. Goldman, Solution growth of a binary icosahedral quasicrystal of Sc12Zn88, Phys. Rev. B 81 (2010) 4. B. Dubost, J.M. Lang, M. Tanaka, P. Sainfort, M. Audier, Large AlCuLi single quasi-crystals with triacontahedral solidification morphology, Nature 324 (1986) 48e50. K. Sato, H. Uchiyama, Y. Takahashi, I. Kanazawa, R. Suzuki, T. Ohdaira, T. Takeuchi, T. Mizuno, U. Mizutani, Positron-annihilation studies of rhombic triacontahedral-type icosahedral quasicrystals and their 1/1 and 2/1 approximants in the Al-Mg-Zn alloy system, Phys. Rev. B 64 (2001) 8. N. Tamura, M. Beyss, K. Urban, Triacontahedral growth morphology in icosahedral Zn-Mg-Y alloy, Philos. Mag. Lett. 74 (1996) 89e97. D. Shechtman, I. Blech, D. Gratias, J.W. Cahn, Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett. 53 (1984) 1951e1953. D.Q. Wan, G. Yang, S. Chen, M. Zhu, Y.H. Zhou, Growth morphology and evolution of quasicrystal in as-solidified Y-rich Mg-Zn-Y ternary alloys, Rare Metals 26 (2007) 435e439. M.V. Kral, M.A. Mangan, G. Spanos, R.O. Rosenberg, Three-dimensional analysis of microstructures, Mater. Charact. 45 (2000) 17e23. F. Lasagni, A. Lasagni, E. Marks, C. Holzapfel, F. Mucklich, H.P. Degischer, Three-dimensional characterization of ‘as-east’ and solution-treated AlSi12(Sr) alloys by high-resolution FIB tomography, Acta Mater. 55 (2007) 3875e3882. I.R. Fisher, M.J. Kramer, T.A. Wiener, Z. Islam, A.R. Ross, T.A. Lograsso, A. Kracher, A.I. Goldman, P.C. Canfield, On the growth of icosahedral Al-Pd-Mn quasicrystals from the ternary melt, Philos. Mag. B Phys. Condens. Matter Stat. Mech. Electron. Opt. Magn. Prop. 79 (1999) 1673e1684. C. Beeli, H.U. Nissen, Growth-morphology of Icosahedral Al-Mn-Pd single quasi-crystals, Philos. Mag. B Phys. Condens. Matter Stat. Mech. Electron. Opt. Magn. Prop. 68 (1993) 487e512. T. Boncina, B. Markoli, I. Anzel, F. Zupanic, Metallographic techniques for the characterization of quasicrystalline phases in aluminium alloys, Z. Krist. 223 (2008) 747e750. F. Zupani c, T. Bon cina, N. Rozman, B. Markoli, Morphologies of icosahedral quasicrystals in Al-Mn-Be-(Cu) alloys, in: Quasicrystals: Types, Systems, and Techniques, Nova Science Publishers, Hauppauge, N.Y., 2011, pp. 195e217. Y. Yang, K. Zhang, M.L. Ma, J.W. Yuan, Microstructure and phase compositions of as-cast Mg-3.9Zn-0.6RE (Gd, Y) alloy with different Gd/Y ratios, Rare Metals 34 (2015) 160e163. Y. Tian, H. Huang, G.Y. Yuan, W.J. Ding, Microstructure evolution and mechanical properties of quasicrystal-reinforced Mg-Zn-Gd alloy processed by cyclic extrusion and compression, J. Alloys Compd. 626 (2015) 42e48. L. Yang, Y.H. Zhao, H. Hou, X.M. Yang, Effects of Y on microstructures and properties of Mg-Zn-Y alloys prepared by conventional solidification, Rare Metal Mater. Eng. 43 (2014) 1754e1758. N. Strekelj, I. Naglic, G. Klancnik, A. Nagode, B. Markoli, Microstructural changes in quasicrystalline Al - Mn - Be - Cu alloy after various heat treatments, Int. J. Mater. Res. 106 (2015) 342e351. T. Oz, E. Karakose, M. Keskin, Impact of beryllium additions on thermal and mechanical properties of conventionally solidified and melt-spun Al-4.5 wt.% Mn-x wt.%Be (x¼0, 1, 3, 5) alloys, Mater. Des. 50 (2013) 399e412. JCrystal, Crystal Shape Editor/Viewer 1.0.5, ©JCrystalSoft, 2011, 2005e2011, http://www.jcrystal.com/jcrystal.html.