Crystal Growth of Quasicrystals

26 Crystal Growth of Quasicrystals An-Pang Tsai1, 2, Can Cui3 1 INSTITUT E OF M ULTIDISCIPLINA RY RESEARCH FOR ADVANCED MATERIALS, TOHOKU UNIVERSITY, SENDAI, JAPAN; 2 NA TIONAL INSTI TUTE FOR MATERIALS SCIENCE, TSUKUBA, JAPAN; 3 DEPARTME NT OF PHYSICS, CENTER FOR OPTOELECTRONIC M AT ERIALS AND DEVI CE S, ZHEIJIANG SCI-TECH UNIVERS ITY, HANGZHOU, CHINA

CHAPTER OUTLINE 26.1 Introduction to Quasicrystals ................................................................................................ 1114 26.1.1 Crystal and Periodicity ................................................................................................ 1115 26.1.2 Restriction on Symmetry by Translation................................................................... 1116 26.1.3 Discovery of QCs.......................................................................................................... 1117 26.2 Structures of QCs .................................................................................................................... 1118 26.2.1 One-dimensional Quasiperiodic Structures and the Fibonacci Sequence ............. 1118 26.2.2 Two-dimensional Quasiperiodic Structures and the Penrose Pattern ................... 1120 26.2.3 Three-dimensional Quasiperiodic Structures............................................................ 1123 26.3 Variations of QCs .................................................................................................................... 1126 26.3.1 Three-dimensional QCs ............................................................................................... 1126 26.3.1.1 Lattice Type .....................................................................................................1126 26.3.1.2 Structure of Icosahedral Clusters ..................................................................... 1127 26.3.2 Two-dimensional QCs ................................................................................................. 1128 26.3.3 Alloy Systems of Stable Phases .................................................................................. 1129 26.4 Growth Methods..................................................................................................................... 1131 26.4.1 Czochralski Method .................................................................................................... 1131 26.4.2 Bridgman Method....................................................................................................... 1132 26.4.3 Floating Zone Method................................................................................................ 1133 26.4.4 Solution Growth (Self-Flux) Method ......................................................................... 1134 26.5 Selected Results ...................................................................................................................... 1135 26.5.1 Al–Pd–Mn System ........................................................................................................ 1135 26.5.2 Al–Ni–Co System.......................................................................................................... 1138 26.5.3 Zn–Mg-RE System ........................................................................................................ 1139 26.5.4 Ag–In–Yb System......................................................................................................... 1140 26.5.5 Cd-RE and Zn–Sc Systems ........................................................................................... 1142

Handbook of Crystal Growth. http://dx.doi.org/10.1016/B978-0-444-56369-9.00026-5 Copyright © 2015 Elsevier B.V. All rights reserved.

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26.6 Growth Mechanism ................................................................................................................ 1142 26.6.1 Morphologies of QCs .................................................................................................. 1142 26.6.1.1 Stellated Polyhedron ........................................................................................ 1142 26.6.1.2 Rhombic Triacontahedron................................................................................ 1143 26.6.1.3 Pentagonal Dodecahedron .............................................................................. 1143 26.6.1.4 Decaprism........................................................................................................ 1145 26.6.2 Morphological Stability Theory ................................................................................. 1146 26.6.3 QCs upon Rapid Solidification ................................................................................... 1148 26.6.4 QCs Prepared by Classic Crystal Growth Methods................................................... 1149 26.6.5 Selection of Growth Direction ................................................................................... 1151 26.7 Concluding Remarks ............................................................................................................... 1152 Acknowledgments ........................................................................................................................... 1153 References......................................................................................................................................... 1153

26.1 Introduction to Quasicrystals Quasicrystals (QCs) are a new form of matter, which differs from crystalline and amorphous materials by exhibiting a new ordered structure, quasiperiodicity, and symmetries that are forbidden in classic crystallography (e.g., 5-fold, 10-fold, 8-fold, 12-fold). The first diffraction pattern (Figure 26.1) of a QC was reported by Shechtman et al. in 1984 [1]. Since then, several new concepts have been employed to understand the structure and stability of QCs. Soon after the discovery in the 1980s, great progress was made by theorists in understanding the framework of atomic structure and fundamental properties in structure of QCs [2]. Progress on the experimental side was made by the discovery of a series of stable QCs [3].

FIGURE 26.1 Electron diffraction pattern of a rapidly solidified Al86Mn14 alloy taken along a fivefold axis [1].

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The Nobel Prize in Chemistry of 2011 was awarded to Shechtman for the discovery of QCs. This made the name “quasicrystal” more widely known among various fields in science as well as industry; nevertheless, the QC is not well understood. The difficulty of understanding QCs arises from the fact that one must figure out the structure of a QC in reciprocal space, which is not easy to imagine in real space for those outside the field of crystallography.

26.1.1

Crystal and Periodicity

The structures of crystals are realized by Bragg’s law, by which the atoms arrange periodically in three dimensions. Bragg’s law is shown in Eqn (26.1): 2d sin q ¼ nl

(26.1)

where d and l are the lattice spacing of a crystal and wavelength of radiation, respectively; q is the incident angle of radiation; and n is any integer. Because l is fixed, diffraction is generated when q and d satisfy Bragg’s law. Equation (26.1) can be rewritten as sin q ¼ nl/2d. Here, d represents a sequence of lattice planes with equidistance, which is inversely proportional to sin q, implying that periodicity of atomic planes is requested for generating diffractions. Assuming the observation of crystal was performed with transmission electron microscopy (TEM) operating at an accelerating voltage at 200 kV, as shown in Figure 26.2(A), l would be around 0.003 nm,

FIGURE 26.2 (A) Schematic description of formation of diffraction pattern in transmission electron microscopy (TEM). (B) Real square and triangular lattices and their corresponding calculated diffraction patterns.

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which is much smaller than d; hence, sin q y q. Then, Eqn (26.1) could be simplified to q ¼ nl/2d. Note that the relationship between l, d, and q shown in Figure 26.2(A) satisfies Bragg’s law. Therefore, it could be regarded approximately that q is inversely proportional to d. On the other hand, camera constant L is the distance between the specimen and the screen where diffractions are being projected, which is a constant parameter of the microscopy. r is the distance between the transmitted peak and diffraction peaks on the screen, which is measurable from the diffraction pattern. Once r is obtained, one can easily determine d or q. Again, r is inversely proportional to d. With this in mind, let us consider a diffraction pattern generated from a square lattice, as shown in Figure 26.2(B). A sequence of lattice spacing d generated a diffraction of r ¼ d* on the diffraction pattern. Then, the other sequence of lattice spacing D, where D ¼ 2d, would generate a diffraction of r ¼ D*, where D* ¼ d*/2. Although the diffraction is reciprocal to the corresponding lattice spacing, both of them share the same property of periodicity. That is, the arrangement of diffractions generated by a crystal with periodic structure (real lattice) must be periodic in the diffraction pattern (reciprocal lattice). By the definition of Bragg’s law, the occurrence of diffractions is equivalent to periodicity, which is another name for crystals in classic crystallography. For a two-dimensional square lattice, the same periodic arrangement of diffractions would be observed along the vertical direction. As a result, diffractions form a square arrangement in the diffraction pattern analogy to its original lattice. A similar correlation must be observed in any three-dimensional lattice as well. An important feature inherent in this correlation is that the shape in the real lattice is preserved in the reciprocal lattice. Therefore, in Figure 26.2(B), if the real lattice is described by repetition of a square, then one must be able to find a square to describe the reciprocal lattice. The words “repetition” and “translation” have different meanings, but they are commonly used to describe the periodicity. Therefore, for a long time, “periodicity” was another name for a crystal, until the discovery of QCs.

26.1.2

Restriction on Symmetry by Translation

Because a crystal structure is composed of a translation of the lattice and point group of symmetry of the basis, only a number of symmetries are allowed. Restriction of symmetry due to translation is described in Figure 26.3(A). Assuming a crystal containing lattice points of A, B, C, D, and E, distances between these lattice points are commonly a. If angles between lines formed by lattice points are a, then na ¼ 360 ¼ 2p, where n is an integer that represents the times of symmetry. Let AB ¼ b; then b ¼ ma must be approved, where m must be an integer in order to satisfy translation. The equation of b ¼ 2acos a could be derived, and conclusively cos a ¼ m/2. Because m must be an integer, only a number of rotation angles are allowed, as shown in Table 26.1 Consequently, the allowed rotation symmetries n are 1, 2, 3, 4 and 6, and no rotation symmetry more than 6 is allowed. According to this derivation, fivefold

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FIGURE 26.3 Descriptions of (A) rotational symmetries compatible with lattice translation and (B) the impossibility of tiling by only using pentagons.

Table 26.1 Rotational Symmetries Compatible with Lattice Translation m

cos a [ m/2

a

n [ 360 /a

2 1 0 1 2

1 1/2 0 1/2 1

0, 2p p/3 p/2 2p/3 p

1 6 4 3 2

symmetry and its multiplicity are not allowed in the classic crystallography. This can also be easily understood in terms of tiling by using a pentagon, as shown in Figure 26.3(B), where the rhombus cavity would come out. In other words, a basis or a motif with fivefold symmetry is impossible to fill in a two-dimensional plane or threedimensional space.

26.1.3

Discovery of QCs

The electron diffraction pattern shown in Figure 26.1 was in conflict with the definition and restriction of a crystal. The electron diffraction pattern exhibits somewhat sharp diffractions arraying with tenfold symmetry, which were forbidden in crystals as mentioned above. However, the first discovered Al–Mn icosahedral QC (IQC) shown in Figure 26.1 was prepared by rapid solidification; hence, it had a highly disordered structure as evidenced by peak broadening and shifting of diffractions, as well as low thermal stability. Two promising structural models for IQC—the icosahedral glass (IG) model [4,5] and the giant crystal model [6]—were proposed to explain the disorder. IQC structures described by the IG model or a metastable state of approximants can be interpreted within a framework of typical solid structures [7]. Therefore, new

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(A)

(B)

FIGURE 26.4 (A) High-resolution transmission electron microscopy image and (B) electron diffraction pattern taken along a fivefold axis of the stable Al65Cu20Fe15 IQC [3].

structures, such as the Penrose pattern [8] or the concept of quasilattice, would have been unnecessary to describe IQC structures; hence, there was a huge controversy concerning whether IQCs really represented a new form of solid. This controversy lasted for a few years after the discovery of the first IQC. The discovery of the stable Al–Cu–Fe IQC (Figure 26.4) represented a breakthrough because its diffraction pattern and high-resolution image could not be interpreted by either the IG model or metastable states of approximants [3]. The highly ordered structure was maintained over a wide range, as large as on the order of millimeters. Consequently, stable QCs revealing sharp diffractions, as verified in a sequence of alloys, established that the QCs were a new form of solids. The International Union of Crystallography redefined the crystal in 1991 as “any solid having an essentially discrete diffraction diagram” [9]. In the new definition, the QC, which will be described next, is a crystal in high-dimensional space.

26.2 Structures of QCs The question “Where are the atoms?” was asked soon after the discovery of QCs [10]. Here, we first describe the basic concept to understand the framework of structure of QCs, then provide two real examples of structures with accurate atomic decorations.

26.2.1

One-dimensional Quasiperiodic Structures and the Fibonacci Sequence

A one-dimensional quasilattice obtained by a cut-and-projection method [11], as shown in Figure 26.5(A), is the best example to describe how a high-dimensional crystal converts to a QC. First, let us put a square lattice in two-dimensional space.

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FIGURE 26.5 Projection of a two-dimensional lattice onto a one-dimensional space with (A) an irrational slope (1/s) and (B) a rational slope (2/3) to obtain a one-dimensional quasicrystals and approximant, respectively.

Then, we introduce a set of orthogonal axes, namely rjj and rt, which are rotated by an angle of q with respect to the original coordinate system of the square lattice. rjj and rt are two projection axes, called the physical axis and complementary axis, respectively. Parallel to rjj, we introduce a strip with a width of a unit square (W) along rt. The strip is also called a “window” for projection, which contains a number of lattice points of the square lattice. The next step is let the lattice points inside the window project on to rjj, as shown in Figure 26.5. A tiling consisting of two different lengths long (L) and short (S), which are the projection of two different sides of the unit square lattice, is then obtained. If the tan a p isffiffi irrational, such as in the case in Figure 26.5(A) where a ¼ tan1(1/s) z 31.716 (s ¼ 1þ2 5 is Golden mean), a one-dimensional quasiperiodic tiling with L and S line segments is formed. Note that if a small W is used, points of projection will be less dense and L and S line segments will be larger, and vice versa. The size of W only changes the density of projection points or the lengths of L and S; as long as the slope is the same, one gets the same quasiperiodic array and length ratio, S/L ¼ 1/s. This is the property of self-similarity, and the factor of self-similarity for a quasiperiodic array is irrational. If the slope of W, m (¼S/L) approached a rational number, as shown in Figure 26.5(B), such as m ¼ 2/3, the projection points will form a periodic array with a periodicity of 2S þ 3L. The quasiperiodic array is a one-dimensional QC, and a periodic array derived by projection is an approximant crystal. By choosing m to be a continued-fraction approximant to s (m ¼ 1/1, 1/2, 2/3, 3/5.), one creates a structure with larger periods that approximate the quasiperiodic tiling (Figure 26.5, right) better and better. The sequences of L and S for the quasiperiodic structure and approximant (projected with m ¼ 1/s and m ¼ 2/3, respectively) are shown in Figure 26.5. The differences between two sequences are indicated with arrowheads, where the LS observed in quasiperiodic tiling is replaced by SL in the approximant [12]. The period of the approximant is LSLLS. The difference is due to a flipping between L and S; with regard to QCs, this flipping is called “phason flipping,” which is a sort of defect in QCs.

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Because the one-dimensional quasiperiodic structure can be derived from a twodimensional periodic structure, as shown in Figure 26.5, a QC is called a crystal in high-dimensional space. The one-dimensional quasiperiodic sequence can be obtained by a substitution rule: L / LS and S / L, which can be expressed by Eqn (26.2). L LS LSL LSLLS LSLLSLSL

(26.2)

LSLLSLLSLLS .

Each series can be expressed as Fn ¼ Fn  1 þ Fn  2, where lim

Fn1 n/N Fn

¼ 1s . The number of

total L and S for each series will be 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,.. This is the Fibonacci number or Fibonacci sequence. Note that a series of ratios of numbers, such as 1/2, 2/3, 3/5, 5/8. are corresponding to m and are used to obtain a series of approximant crystals.

26.2.2

Two-dimensional Quasiperiodic Structures and the Penrose Pattern

There are a number of two-dimensional quasiperiodic structures owning different symmetries, such as 8-fold (octagonal), 10-fold (decagonal), and 12-fold (dodecahedral) symmetries [13]. In reality, QCs with a decagonal lattice (decagonal quasicrystals, DQC) are well studied and are stable, which allowed discussion of the mechanism of crystal growth; here, we focus on the decagonal lattice. The typical decagonal lattice is described by the well-known Penrose pattern shown in Figure 26.6(A). The Penrose pattern is constructed by tiling a two-dimensional plane with two rhombic tiles, namely a “fat” tile with an angle p/5 and a “skinny” one with an angle p/10. A perfect Penrose pattern is formed only when tiling is made along so-called matching rules, which are formulated using a marking of the edge. Each rhombus has single or double arrows along the edges; rhombi of the same type have identical arrow markings. Neighboring rhombi have edges in common. To make a Penrose tiling, one fits these rhombic tiles together according to the matching rule: two rhombic tiles can be placed side by side only if coinciding edges have the same type and directions of the arrows. This leads to quasiperiodicity of the tiling. Each vertex in an infinite Penrose tiling is surrounded by one of eight combinations of tiles [14]. In an infinite Penrose tiling, the ratio between the number of fat tiles (NF) and that of skinny tiles (NS) will approach s. Any vertex out of these eight combinations will be a defect and the ratio of NF/NS will deviate from s. The Penrose pattern could be obtained by the cut-and-projection method of a five-dimensional cubic lattice onto a two-dimensional space, similar to that shown in Figure 26.5. In this case, instead of

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FIGURE 26.6 (A) Description of Penrose tiling made with two rhombi. (B) Five basis vectors used to index the Penrose lattice. (C) A description of the phason flip: lattice position exchange between A and B changes the arrangement of tiles. (D) Inflation in Penrose tiling.

a strip shown in Figure 26.5, a number of polygons, namely pentagons of windows (also called occupation domains), are used. The vertex of Penrose patterns forms a twodimensional QC, which could be assigned with five vectors: e1, e2, e3, e4, and e5, as shown in Figure 26.6(B), with an angle of intersection of p/5. Because e1 þ e2 þ e3 þ e4 ¼ e5, four vectors is enough for assignment. Every spot in its diffraction pattern can be indexed with a combination of the vectors (cos(2pj/5), sin(2pj/5))/a [15], where a is the edge length of Penrose pattern and j ¼ 1,., 5. A typical flipping of tiles, as shown in Figure 26.6(C) is also observed in the mismatching of Penrose tiling, which could be made from projection by a shift or change in size of W. Two hexagons have the same outline; both are made of two skinny and one fat tiles, but their arrangements are different. In terms of tiling, the flipping of tiles makes a significant difference. However, if we assume that the seven vertexes are all occupied by atoms, this flipping is simply made by a slight switching of atomic position from A to B (or B to A) at the interior of hexagons. This is phason flipping in a two-dimensional QC, which has been observed in real QC samples. There is a self-similarity of Penrose pattern, as shown in Figure 26.6(D). By dividing the original tile with a length ratio of 1:s, one may obtain identical tiles with a scale of 1/s with respect to the original ones. In the Penrose pattern, a larger number of vertices reveal local fivefold symmetry, and along any direction there is no periodicity of lattice spacing. The calculated diffraction pattern of the Penrose pattern reveals 10-fold symmetry and a quasiperiodic arrangement of diffractions inflated with s scaling, which closely resembles that observed by Shechtman et al. Obviously, the Penrose pattern is the key to understanding the structure of QCs. It is normally used as a template for modeling the structure of two-dimensional QCs by decorating rhombic tiles with atoms.

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FIGURE 26.7 Structural model of Al72Ni20Co8 DQC superimposed by two different atomic layers. Gray circles and black circles represent Al and transition metals (Ni or Co), respectively. A highlighted decagon composed of kites, boats, and a star is Gummelt decagon, which is used to construct the structure by overlapping rather than tiling [16].

The Penrose pattern itself is a decagonal structure with a 10-fold axis and is the simplest template to describe a DQC. A structure of a DQC, as shown in Figure 26.7 [16], is periodic along one direction and quasiperiodic in the plane perpendicular to it (i.e., the paper). Thus, it can be expressed as a periodic stacking made by different atomic layers alternatively along a decagonal axis. The period of Figure 26.7 is on the order of 0.4 nm (two different atomic layers), but it could be a multiple of this periodicity. TEM is a powerful tool for studying DQC because it can get high-resolution images along the periodic direction, which contain symmetry and the interior atomic position of clusters. The formation of stable DQCs allows one to study the structure by means of a single-grain X-ray diffraction technique; one example is shown in Figure 26.7. The structure model with atomic decoration superimposes on the Penrose pattern for Al–Ni–Co DQC. This DQC exhibits a periodicity of 0.4 nm or two atomic layers; hence, the atomic positions shown in Figure 26.7 are a projection of two different atomic layers. Highresolution TEM studies suggested a similar structure, indicating the model is reliable [17]. It should be noted that the atomic positions in Figure 26.7 are not fully occupied and there are several partial sites in the real DQC structure. Two partial sites could be treated as one atom flipping between two too-close sites, shown in Figure 26.6(C); in terms of tiling, this is the phason flipping. There are several flipping sites in real DQCs, which are essential for forming long-range quasiperiodic structures and are the origin of the diffuse scattering observed in X-ray diffraction studies. The DQC structure can be described by the overlap of decorated decagons (Grummelt decagon) [18,19] with a diameter of 0.2 nm, as highlighted by the thick lines in Figure 26.7.

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FIGURE 26.8 (A) Acute rhombus (AR) and obtuse rhombus (OR) constructed by a so-called golden rhombus, of which the ratio of diagonals is s. (B) A rhombic triacontahedron constructed by 10 AR and 10 OR and a stellated dodecahedron constructed by 20 AR. (C) Six basis vectors used to index lattice of icosahedral quasicrystals, described by an icosahedron.

26.2.3

Three-dimensional Quasiperiodic Structures

There is only one three-dimensional quasiperiodic structure: the IQC. The simplest set of unit cells for a three-dimensional quasiperiodic structure consists of the acute and obtuse rhombohedra shown in Figure 26.8(A). These two unit cells play the same roles as the skinny and fat tiles, respectively, in the Penrose tiling. All the faces are identical rhombuses, which are a Golden rhombus with length ratio s for two diagonals. As shown in Figure 26.8(B), 10 acute and obtuse rhombohedra can be packed to form a rhombic triacotahedron (RTH) and 20 acute rhombohedra can be packed to form a stellated dodecahedron. Both polyhedrons reveal icosahedral symmetry, which shows how the unit cells can be packed to fill space. Using polyhedrons as unit cells with atomic decoration, it is easy to imagine the symmetry of structure. Similar to Penrose patterns, a three-dimensional quasiperiodic structure can be obtained by a cut-andprojection method of a six-dimensional cubic lattice onto a three-dimensional space. The typical W for the projection is a rhombic triacontahedraon shape with edge length 1/s2 times that of projected structure [14]. In the six-dimensional construction, the analog of W in Figure 26.5 is the product of a three-plane and a suitable threedimensional cross-section. The L and S line segments correspond to acute and obtuse rhombohedra, respectively. If the vectors used to describe W are a series of (m, 1, 0) vectors, then the three-dimensional quasiperiodic structure results when m / s. By taking the rational approximants m ¼ 0/1, 1/1, 1/2, 2/3., one obtains a sequence of cubic structures with larger and larger lattice constants. Two 1/1 approximants have been verified in a-AlMnSi [12] and (Al, Zn)49Mg32 [20] compounds, which are

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body-centered cubic (bcc) packing of icosahedral clusters with different shell structures. For simplicity, both the IQC and the approximant consist of icosahedral clusters, of which the arrangement is quasiperiodic for the former and periodic for the latter. Therefore, the approximants contain very useful information for building an initial structural model for IQCs. Lattice points of IQCs are normally assigned with six vectors expressed on an icosahedron—namely e1, e2, e3, e4, e5, and e6, as shown in Figure 26.8(C). A possible basis for the diffraction spots of an IQC is given by the six vectors [21]: ai ¼

1 1 1 1 1 1 ð1; s; 0Þ; ð1; s; 0Þ; ð0; 1; sÞ; ðs; 0; 1Þ; ðs; 0; 1Þ; ð0; 1; sÞ a a a a a a

(26.3)

where i ¼ 1,., 6. So far, there is only one IQC, namely i-Cd5.7Yb [22], whose structure has been completely solved [23]. A cubic-phase Cd6Yb with a space group of Im-3 and lattice parameter a ¼ 0.156 nm exists in the Cd–Yb phase diagram [24]. The structure of Cd6Yb has been determined, and it was demonstrated to be a 1/1 approximant to the IQC phase. The i-Cd5.7Yb is adjacent to the Cd6Yb phase in the phase diagram. The shell structure of the icosahedral cluster deduced from Cd6Yb is shown in Figure 26.9(A). The icosahedral cluster in Cd6Yb approximant has the following structure: the first shell is created by four Cd atoms around the cluster center; the second shell consists of 20 Cd atoms that form a dodecahedron; the third shell is an icosahedron of 12 Yb atoms, the fourth shell is a Cd icosidodecahedron obtained by placing 30 Cd atoms on the edges of

FIGURE 26.9 (A) A rhombic triacontahedral (RTH) unit with atomic decoration used to construct (B) 1/1 and (C) 2/1 approximants in Cd–Yb system, where the structure of 1/1 approximant can be described by the RTH and structure of 2/1 approximant needs a (D) decorated acute rhombus unit in addition to the RTH. There are only two linkages between two adjacent RTH are allowed: (E) b-linkage link by sharing a rhombus unit and (F) c-linkage connected by overlapping an obtuse rhombus [23].

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the Yb icosahedron, and the fifth shell is an RTH in which Cd atoms are located on 32 vertices and 60 edge centers. The i-Cd5.7Yb was studied by single-grain X-ray diffraction, by which the icosahedral cluster was demonstrated to be identical to that of the Cd6Yb approximant. The icosahedral cluster contains five atomic shells (Figure 26.9(A)) and is an RTH cluster. It is important to indicate that a well-defined chemical order presents in the RTH cluster, where Yb (only on an icosahedral shell) and Cd occupy different atomic sites. The structure of the 1/1 approximant is a bcc packing of RTH clusters, as shown in Figure 26.9(B). Furthermore, the structure of a 2/1 approximant (Figure 26.9(C)) was solved to consist of the same RTH as the 1/1 approximant, but it needs an additional unit [25]—namely, an acute rhombohedron (Figure 26.9(D)) with suitable atomic decoration to fill the cavity. In both approximants, the RTH clusters are linked to each other (Figure 26.9(E)) along the twofold (b-bond) directions by sharing a rhombus face and the three-fold direction (c-bond) with interpenetration of an obtuse rhombohedron, as shown in Figure 26.9(F). Consequently, three fundamental building units—an RTH cluster (a), an acute rhombohedron (d), and an obtuse rhombohedron (f)—are supposed to be necessary for constructing the IQC structure. Using the knowledge of linkage rules and building units, a precise structural model for i-Cd5.7Yb was proposed (Figure 26.10), which describes the structure in terms of inflation and hierarchical packing of clusters. Figure 26.10 shows the RTH center positions and their connection on a fivefold plane. Starting from the center, it can be shown that a cluster of RTH units (icosidodecahedron) is formed. The cluster of RTH forms a

FIGURE 26.10 Structural description of i-Cd5.7Yb with rhombic triacontahedral (RTH) unit. (A) A dense plane of RTH units is seen along a fivefold axis. (B) A small ball represents an RTH unit used to construct an icosidodecaherl cluster, by which a larger icosidodecahedral cluster can be built up. Ratio of edge length between large and small icosidodecahedra is s3 [23].

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large “cluster of a cluster,” which is also an icosidodecahedron but with a s3 times increase in scale. A prominent feature of the model is that almost all of the IQC structure is described by interpenetrating RTH clusters because 93.8% of the atoms belong to the RTH clusters [23]. The full structure is described by the RTH clusters, where 100% of atoms belong to the RTH clusters for the 1/1 approximant. These two structures are both described by the RTH clusters but stand at two oppositional extremes, with a difference in composition of only w2 at%. One interesting feature is that all approximants exist as stable phases in nature; they can be mathematically derived by a cut-and-projection scheme.

26.3 Variations of QCs QCs can be formed through various processes and by various techniques. The structure and the intrinsic disorder of QCs are very sensitive to the process of formation. They could be stable and metastable, depending on alloys, and they reveal a number of variations in structure. QCs can be sorted by their diffraction features and symmetries, which are easily determined from diffraction patterns.

26.3.1

Three-dimensional QCs

26.3.1.1 Lattice Type In terms of symmetry, only one three-dimensional QC has been discovered so far—IQC. This is the QC studied most extensively among all known QCs. Figure 26.11 shows the typical selected area electron diffraction (SAED) patterns taken with incidences along fivefold, threefold, and twofold symmetry axes, as well as the morphology of a single IQC. The IQC possesses a very high symmetry with the point group m-3-5. It was predicted from theory that there exist three types of Bravais lattices in three-dimensions, consistent with the icosahedral point symmetry. They correspond to primitive, body-centered, FIGURE 26.11 Electron diffraction patterns taken along the fivefold, threefold, and twofold axes (left) and a SEM image (right) of stable Al65Cu20Fe15 IQC [26].

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FIGURE 26.12 Electron diffraction patterns of (A) P-type and (B) F-type IQCs and (C) indexing of diffraction spots in their corresponding fivefold axis [3].

and face-centered hypercube in a six-dimensional hyperspace. The primitive and facecentered types have been verified in real samples. These two types of IQCs can be distinguished by extinction rules of diffractions in their twofold patterns [3,13]. Figure 26.12(A) and (B) shows partial diffraction patterns taken along twofold axes for the primitive and face-centered IQCs. Indexing is given in Figure 26.13(C), where the diffraction spot arrangement is inflated by s3 and s for primitive and face-centered type lattices, respectively. Note that face-centered type lattice has odd–even parity. Most Al-based metastable IQCs should have a face-centered lattice, but it is obscured by quenched-in disorder. For stable IQCs, Al-based and Zn–Mg-RE (where RE ¼ rare earth metals) systems have face-centered lattices and Cd–Yb groups have primitive lattices.

26.3.1.2 Structure of Icosahedral Clusters IQC can be classified into three classes according to the hierarchic structures of icosahedral clusters derived from their corresponding crystalline approximants: the Al–Mn–Si class [12], the Mg–Al–Zn class [27], and the Cd–Yb class [22]. The structures of atomic shells for the three classes are shown in Figure 26.13. The three classes are also called Mackay, Bergman, and Tsai clusters, respectively. The clusters were first recognized by Henley and Elser in two compounds, a-Mn12(Al,Si)57 [12] and Mg32(AlZn)48 [20], which are supposed to be 1/1 approximants to IQCs. The clusters have icosahedral symmetry,

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FIGURE 26.13 Atomic decoration of successive shells of atomic clusters expected in various families of icosahedral quasicrystals: (A) Mackay type, (B) Bergman type, and (C) Tsai type.

(A) Vacant Al/Si

12 Al/Si

12 Mn

30

(B) Vacant Al/Zn

(C)

4 Cd 30 Cd

12 Al/Zn

20 Cd

20 Mg

12

12Yb

which was very useful for guessing the structure of IQCs. However, the structures of most stable IQCs have not been solved; the hierarchic structures of icosahedral clusters were not determined to belong to either of the two classes, except Al5.5Li3Cu, which revealed the same clusters as Mg32(AlZn)48. Recently, a series of IQC and approximants were discovered in Na-containing intermetallic compounds [28], which belong to the Bergman type. More recently, the structure of an Al–Pd–Cr–Fe 3/2 approximant to IQC of the Al–Pd–Mn system has been solved [29], which is well described as a dense packing of two kinds of clusters: mini-Bergman and pseudo-Mackay clusters. For the group of Zn60Mg30RE10 stable IQCs, the structure of icosahedral cluster is still unknown because no appropriate approximant has been reported. Obviously, grouping the IQCs by hierarchic structures of clusters may not be clear for these two classes. On the other hand, the Tsai cluster was verified to be identical in several IQCs and approximants in Cd–Yb, Ag–In–Yb, Zn–Sc, and so on [30], as shown in Figure 26.9 and Figure 26.13(C); cluster grouping is strictly clear in this group. For convenience, the stable IQCs are grouped as Al–TM, Zn–Mg-RE, and Cd–Yb classes in this chapter.

26.3.2

Two-dimensional QCs

So far, there are three kinds of two-dimensional QCs: decagonal, dodecahedral, and octagonal QCs, as classified by their electron diffraction patterns. Figure 26.14 shows the typical diffraction patterns of DQC along a 10-fold axis and two characteristic twofold axes, together with an scanning electron microscopy image revealing decaprism morphology. The 10-fold axis is the axis along which the diffractions are arranged periodically. Perpendicular to this axis is the quasiperiodic plane, in which the

Chapter 26 • Crystal Growth of Quasicrystals

1129

FIGURE 26.14 Electron diffraction patterns taken along a 10-fold and two twofold axes, and an SEM image of Al72Ni20Co8 decagonal quasicrystal.

diffractions are arranged in a quasiperiodic array with a 10-fold symmetry. This will be 12-fold [31] and 8-fold [32] symmetries for dodecahedral and octagonal QCs, respectively, as shown in Figure 26.15. Only DQC was found to be stable in Al–Ni–Co, Al–Cu–Co, and some Al–Pd systems.

26.3.3

Alloy Systems of Stable Phases

Stable QCs have two inherent properties: stability and a well-ordered structure. The stability allows QCs to be grown at slow cooling rates, thus enabling large single-domain QCs with high structural orders to be produced for precise measurements. Both properties are crucial for verifying that QCs are a new form of solid that differs from both crystalline and amorphous solids. The first Al–Mn QC was prepared by rapid

(A)

(B)

FIGURE 26.15 Electron diffraction patterns taken along an 8-fold (A) and a 12-fold (B) from octagonal and dodecahedral quasicrystals, respectively.

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HANDBOOK OF CRYSTAL GROWTH

solidification and hence had a high defect density and low thermal stability. QCs obtained by rapid solidification are thermodynamically metastable and thus decompose into crystalline phase(s) on heating at temperatures much lower than their melting points. Moreover, their grain sizes are a few micrometers, which is too small to precisely investigate their structures. In such samples, the structure and properties originating from their quasiperiodicity are masked by the structural disorder. This resulted in controversy regarding structural modeling for interpreting the origin of diffraction patterns observed in the early 1980s. Since 1986, several stable QCs with grain sizes as large as w100 mm and facetted morphologies have been discovered in various alloys. This greatly increased the precision of QC measurements, enabling structural analysis and investigations of the physical properties of single QCs to be performed. The stability of stable QCs has been explained in terms of electronic structures or valance concentration (e/a: electron-atom ratio) in several articles, so we do not go into detail in this chapter. The stable QCs alloys discovered to date are summarized in Table 26.2. Most stable QCs (especially IQC) are a new group of electron compounds as described by HumeRothery for intermetallic compounds in the early 1900s [33]. The e/a criterion is even stricter for IQCs than crystalline compounds because stable IQCs only form at sharp compositions with strict e/a values [34]. Unlike stable IQCs, stable DQCs only form in a few alloy systems, such as Al–Ni–Co and Al–Cu–Co systems, but their compositional ranges are much wider in each system. A phase appears in the equilibrium-phase diagram, meaning that it is thermodynamically stable. This was first noted in the Al–Li–Cu phase diagram established in 1955 [35], of which an unknown T phase was verified to be an IQC in 1986 [36]. Revisiting the phase diagram after discovery of the stable i-Al–Cu–Fe [26], it was found that when the system being mapped in 1930s [37], the IQC was observed and labeled as the j phase, as shown in Figure 26.16(A). Similarly, the Zn60Mg30Y10 IQC [38] turned out to be the so-called Z-phase [39], as shown in Figure 26.16(B), which had been reported as distinct but unidentified phase in the Zn–Mg-Y system a few years before the discovery of QCs. Table 26.2

Alloy Systems Forming Stable Icosahedral Quasicrystals

P-Type Al–Mn–Al Zn–Mg–Al Cd–Yb

Al5Li3Cu Zn70Mg20RE10 (RE:Er Ho) Cd5.7M (M:Yb Ca), Zn88Sc12, Cd88M12(M:Gd,Tm,Y) Cd65Mg20M15 (M:Yb Ca Y Ho Gd Er Tb) Zn80Mg5Sc15 In42Ag42M16 (M:Yb Ca) Au60Sn25M15(M ¼ Yb,Ca), Au51Al34Yb15 Zn74Ag10Sc16, Zn75Pd9Sc16 Zn77Fe7Sc16, Zn18Co6Sc16, Zn75NI10Sc15

F-Type Al63Cu25TM12 (TM:Fe Ru Os) Al70Pd20TM10 (TM:Mn Tc RE) Zn60Mg30RE10 (RE:Y Dy Ho Gd Er Tb) Zn74Mg19TM7 (TM:Zr Hf)

Chapter 26 • Crystal Growth of Quasicrystals

1131

FIGURE 26.16 Partial phase diagrams involving the icosahedral quasicrystal (IQC) phase in the (A) Al–Cu–Fe [37] and (B) Zn–Mg-Y [39] systems, where j and Z phases were indicated as unknown compounds and later were verified to be IQC phases.

26.4 Growth Methods Many quasicrystalline phases have been discovered as stable phases in equilibriumphase diagrams. Thus, it is possible to grow large single-grain QCs by conventional crystal growth methods for the precise measurement of crystal structure, surface structure, and physical properties.

26.4.1

Czochralski Method

The Czochralski method has been adopted to grow large single QCs of i-Al–Fe–Cu [40,41], i-Al–Pd–Mn [42,43], i-Al–Li–Cu [44], d-Al–Ni–Co [45–50], and d-Al–Cu–Co [51,52]. The crystals prepared by the Czochralski method grow from near-equilibrium conditions, so a detailed phase diagram involving the primary crystallization field of quasicrystalline phase is necessary for designing the starting composition and temperature program. A typical setup of a Czochralski furnace is illustrated in Figure 26.17(A) [42]. As for many others materials, the necking process is applied to obtain a narrow volume for selective growth of a single nucleus and eliminate dislocations in the subsequent growth. Most QCs are ternary intermetallic compounds with incongruent solidification, so relatively slow growth rates are required for growing high-quality single QCs. The typical

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HANDBOOK OF CRYSTAL GROWTH

FIGURE 26.17 (A) The schematic illustration of the Czochralski growth apparatus [42]. The crucible containing molten alloy is heated by high-frequency induction coil. A pull rod with a seed crystal is mounted axially above the crucible. (B) Single d-Al–Ni–Co synthesized by the Czochralski method on [00001] and [10-100] oriented seeds [50].

pulling rate for Al-based QCs is in the range of 0.1–10 mm/h. The diameter of the ingot is controlled by the temperature of molten solution, and the composition of the residual liquid melt in the crucible is varied during the growth. With the Czochralski method, a skilled researcher can grow single QCs of about 1 cm in diameter and several centimeters in length. Figure 26.17(B) shows photographs of the Czochralski d-Al–Ni–Co ingots [50].

26.4.2

Bridgman Method

In most cases of QC growth, the sample nucleates spontaneously rather than grows on a preset seed, because it is difficult to control the partial melting of the seed at the beginning of crystal growth. Therefore, the control of spontaneous nucleation at the early stage of crystal growth is of primary importance. A crucible with a cone-shape bottom is usually used to minimize the possibility of forming many nuclei (Figure 26.18). The conventional Bridgman furnace that the authors used has three temperature zones, as schematically illustrated in Figure 26.18. To protect sample from oxidation or evaporation, the crucible with initial materials is sealed in a quartz ampoule (Figure 26.18), which is preliminarily evacuated and filled with an inert gas. At the beginning of crystal growth, the ampoule is raised into the upper zone and kept at a high temperature for a long time to get a homogeneous molten solution. Then, the ampoule is pulled down slowly in the furnace to pass the temperature gradient zone. The QC first nucleates at the bottom of the crucible and then grows upwards when the molten solution passes the liquid–solid interface. A slow pulling speed in the range of 0.1–10 mm/h is used for single QC growth. With this method, largesize, high-quality single i-Al–Pd–Mn [53], i-Ag–In–Yb [54,55], i-Zn–Mg–Ho [56,57], d-Al–Ni–Co [43,58], and d-Al–Cu–Co [59] have been grown. For congruent melting QCs, such as i-Ag–In–Yb [55], i-Cd–Yb, and i-Ag–In–Ca, the whole ingot can be a single grain when the starting material is of stoichiometry.

Chapter 26 • Crystal Growth of Quasicrystals

1133

FIGURE 26.18 The schematic illustration of the Bridgman furnace (left), quartz ampoule encapsulated crucible (middle), and temperature profile in the furnace (right).

FIGURE 26.19 (A) The schematic illustration of the floating zone method. (B) A single i-Al–Pd–Mn grown by the floating zone method [3]. HF, high-frequency.

26.4.3

Floating Zone Method

The floating zone method is schematically illustrated in Figure 26.19(A). Before crystal growth, a polycrystalline feed rod with the same composition as the target QC is prepared by arc melting or the electrical induction method. The diameter of the feed rod is similar to that of the final single QC. The concept of this method is forming a molten zone (floating zone) in the feed rod using a high-frequency induction coil or infrared heater (halogen lamps mounted around the growth region) and moving the molten zone upwards at a speed around 0.5–1 mm/h. The crystal growth proceeds by moving the molten zone from the bottom to the upper part of the feed rod.

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HANDBOOK OF CRYSTAL GROWTH

Similar to the Czochralski method, an oriented seed can be preset at the bottom of the feed rod to control the quality and direction of the single QC. Because a planar liquid–solid interface is hard to achieve due to the inhomogeneous heating in the molten zone, the feed rod and grown rods are rotated slowly in opposite directions to eliminate the composition gradient along the radial direction in the molten zone. So far, the floating zone method has been successfully used for growing large single QCs of i-Al–Pd–Mn [3] and d-Al–Ni–Co [60,61]. Figure 26.19(B) shows a photograph of the single i-Al–Pd–Mn produced by the floating zone method [3]. The growth direction is along a twofold axis, which is controlled by a single grain seed at the right part of the image.

26.4.4

Solution Growth (Self-Flux) Method

There exists a field that the QC is in equilibrium with the melt in the phase diagrams in many QC alloys; hence, the self-flux method is applicable. It is possible to extract single grains from molten solution at high temperatures if an appropriate starting composition and a suitable decanting temperature are selected. So far, the self-flux method has been widely applied for growing single QCs of i-Al–Cu–Fe [62], i-Al–Pd–Mn [63], i-Al–Cu–Ru [64], i-Al–Pd-RE [65,66], i-Ag–In–Yb [67], i-Zn–Mg-RE (RE ¼ Y and rare earth) [68], i-Zn–Sc [69], i-Cd-RE [70], and d-Al–Ni–Co [43,71] by cooling an off-stoichiometric melt that intersects the liquids associated with the QC phase. Figure 26.20 shows a schematic illustration of process for the self-flux method applied in several QCs. Special care in the process is that a mesh fixed at a position above the alloy for decanting the melt at the end of crystal growth. Normally, the melt is homogenized at high temperatures and then cooled down slowly at a rate of 0.1–3  C/h. In some cases, to control the nucleation process and reduce the number of nuclei, a cold finger attached at the bottom of the crucible is introduced in a modified setup [72]. At a certain temperature before a second phase solidifies, the ampoule containing the single QCs and melt is removed from the furnace and turned upside down into a centrifuge to spin off the remaining liquid phase. There is an advantage to this method, in that the QCs grown always exhibited a natural facet, as shown in Figure 26.21, which enable one to index the orientation of QC quickly.

FIGURE 26.20 The schematic illustration of the self-flux method.

Chapter 26 • Crystal Growth of Quasicrystals

(A)

(B)

1135

(C)

FIGURE 26.21 Single quasicrystals of (A) i-Al–Pd–Mn [63], (B) i-Ag–In–Yb [67], and (C) d-Al–Ni–Co [71] synthesized by the self-flux method.

26.5 Selected Results As shown in Section 26.3 and Table 26.2, QCs are stable in some alloy systems and exist in the phase diagrams. Moreover, most stable QCs reveal high structural perfection, for which one may expect production of high-quality single-grain samples. However, in reality, this is still a technical challenge because of the following reasons. (1) Most stable QCs are ternary intermetallic compounds. This gives a complexity to the solute partition between melts and solids upon solidification. (2) Most QCs form through complex peritectic reactions. That is, they do not crystallize primarily in the melts. (3) Some QCs consist of elements such as Zn or Mg, which have low melting temperatures and high vapor pressures, thus creating technical difficulty in crystal growth. These problems are also inherent in many binary and ternary crystalline intermetallic compounds. Hence, the successful growth of single crystal requires applying the proper method, designating the alloy composition, and maintaining precise control of the temperature program. The single crystal growth processes developed for a number of QCs using various techniques in different alloys systems are summarized in Table 26.3 A number of cases are described in the following sections.

26.5.1

Al–Pd–Mn System

The temperature gap between the L (liquid) and IQC phase in Al–Pd–Mn system is around 20  C, which is the smallest among Al-base QC alloys and can be easily bypassed by supercooling during crystal growth. Hence, only the single grains of i-Al–Pd–Mn with sizes on the order of centimeters have been easily grown by the Czochralski method [42,43], Bridgman method [53], floating zone method [3], and self-flux method [63]. Yokoyama et al. reported a partial isothermal phase diagram, in which the IQC phase precipitates as a primary phase in the range of 3–9 at% Mn and 18–25 at% Pd [42]. Based on the phase diagram, a large single IQC with diameter of 10 mm and length of 50 mm could be grown with an initial composition of Al73.5Pd19.7Mn6.8 at 870  C by the Czochralski method [42]. Go¨decke and Lu¨ck [73] reported the refined phase diagram shown in Figure 26.22, where the IQC phase solidifies from melt in range of about 71–78 at% Al, 15–22 at% Pd, and 4–10 at% Mn. A vertical cut section at 20 at% Pd content

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HANDBOOK OF CRYSTAL GROWTH

Table 26.3 Single Crystal Growth Process Developed for a Number of Quasicrystals Using Various Techniques in Different Alloys Systems Alloys i-Al–Fe–Cu

Techniques Czochralski Self-flux

i-Al–Pd–Mn

Czochralski Bridgman Floating zone Self-flux

i-Al–Li–Cu i-Al–Cu–Ru

Czochralski Self-flux

i-Al–Pd-RE

Self-flux

i-Ag–In–Yb

Bridgman

Self-flux i-Zn–Mg-RE

Bridgman

Self-flux i-Zn–Sc

Self-flux

i-Cd-RE

Self-flux

d-Al–Ni–Co

Czochralski

Conditions (Initial Composition, Growth Temperature, Growth Rate) 

Al57.7Cu37.7Fe3.5Si1.1, 800 C, 0.2 mm/h Al57.7Cu37.7Fe3.5Si1.1, 800  C, 0.18 mm/h Al60Cu36Fe4, cooled from 900 to 780  C at 1  C/h and decanted Al73.5Pd19.7Mn6.8, 870  C Al72.4Pd20.5Mn7.1, 890  C 0.5  C/h Al72Pd19Mn9, 1 mm/h Al73Pd19Mn8, cooled from 875  C to Td (835–870  C) in a duration of 120 h Al63Li25Cu12, 600  C, 0.3 mm/h Al62Cu34.5Ru3.5, cooled from 1050 to 800  C at 2  C/h and decanted Al75Pd20Re5, cooled from 1050 to 900  C at 1  C/h and decanted Al93xPdxRe7 (x ¼ 15–22), cooled from 1025 to 900  C at 0.5  C/h and decanted Ag42In42Yb16, upper temperature 800  C, 0.2 mm/h Ag42In42Yb16, upper temperature 800  C, 0.5–0.8 mm/h Ag41In44Yb15, cooled from 635 to 610  C at 1  C/h and decanted Zn46Mg51Ho3, upper temperature 680  C, 0.2 mm/h Zn62.8Mg33.6Ho3.6, upper temperature 750  C, 0.2 mm/h Zn46Mg51RE3, cooled from 650 to 480  C at 2  C/h and decanted Zn100xScx (x ¼ 2–4), cooled from 800 to 480  C at 5–10  C/h and decanted Cd99.2RE0.8 (RE ¼ Y, Gd–Dy) or Cd99.4RE0.6 (RE ¼ Ho–Tm), cooled from 455 to 335  C at 2  C/h and decanted Al75Ni14.5Co10.5, 7 mm/h Al83Ni9Co8, 927  C, 1 mm/h Al75.6–78.5Ni11.0–14.0Co7.5–13.4, 1050 or 1100  C, 0.2–0.5 mm/h Al75Ni14.5Co10.5, 1–2 mm/h Al77Ni17Co6, 0.15 mm/h

Size

References

F ¼ 4 mm, L ¼ 40 mm F ¼ 2 mm, L ¼ 100 mm 5 mm

[40] [41] [62]

F ¼ 10 mm, L ¼ 50 mm F ¼ 8 mm, L ¼ 50 mm 1.5  0.5  0.5 cm3 F ¼ 12 mm, L ¼ 50 mm 10 mm

[42] [43] [53] [3] [63]

F ¼ 1–5 mm, L ¼ 60 mm 5 mm

[44] [64]

5 mm

[65]

5 mm

[66]

F ¼ 10 mm

[54]

F ¼ 11 mm, L ¼ 30 mm

[55]

10 mm

[67]

0.5 cm3

[56]

1.5  0.8  0.2 mm3

[57]

8 mm

[68]

0.5 cm3

[69]

1 mm

[70]

F ¼ 8 mm, L ¼ 30 mm F ¼ 1–7 mm, L ¼ 60 mm 1 cm3

[45] [46] [47,48]

F ¼ 10 mm, L ¼ 60 mm Centimeters

[49] [50]

Chapter 26 • Crystal Growth of Quasicrystals

1137

Table 26.3 Single Crystal Growth Process Developed for a Number of Quasicrystals Using Various Techniques in Different Alloys Systems—cont’d Alloys

Techniques Bridgman Floating zone Self-flux

d-Al–Cu–Co

Czochralski Bridgman

Conditions (Initial Composition, Growth Temperature, Growth Rate) Al77Ni12.5Co10.5, upper temperature 1200  C, 1 mm/h Al72Ni12Co16, 0.5 mm/h Al77.0Ni10.5Co12.5, slowly cooled from 1200 to 1000  C and decanted Al77.0Ni10.5Co12.5, slowly cooled to 1010  C and decanted Al63.9Cu28.2Co7.9, 900  C, 0.3 mm/h Al66.0Cu26.0Co8.0, 0.1 mm/h Al65Cu27.5Co7.5, upper temperature 1100  C, 0.2 mm/h

Size 3–4 cm

References 3

[43]

1 cm3 1 cm3

[60,61] [71]

1–2 cm3

[43]

F ¼ 1–5 mm, L ¼ 30 2 g, needle shape F ¼ 5 mm, L ¼ 10 mm

[51] [52] [59]

F, diameter; L, length.

FIGURE 26.22 (A) Phase diagram of the Al–Pd–Mn system involves the solidification area of the icosahedral quasicrystal phase. (B) A vertical cut section at 20 at% Pd content, as highlighted by the dotted line in (A) [74].

(highlighted by the dotted line in Figure 26.22(A)) is shown in Figure 26.22(B), in which the L þ I refers to the area that the IQC phase is in equilibrium with liquid phase, with I being the stability range of the IQC phase. Based on the phase diagram, Feuerbacher et al. prepared high-quality single i-Al–Pd–Mn from a melt of Al72.4Pd20.5Mn7.1 using the Czochraski method [43]. Single grain with twofold or fivefold orientation was used as seeds and dipped into the melt at 890  C. After wetting the seed, the temperature of the melt was increased by about 10  C to grow a thin neck of about 1 cm in length, followed by a decrease in temperature to increase the ingot diameter. The final temperature for constant diameter growth was about 880  C. Further progress for single IQC growth in a Al–Pd–Mn system was made using a selfflux method developed by Fisher et al. [63]. The optimal initial composition is Al73Pd19Mn8, which is close to the composition of the IQC phase (Al71Pd21Mn8). The temperature profile is as follows: heat to 1100  C to get a homogeneous melt, rapidly cool

1138

HANDBOOK OF CRYSTAL GROWTH

down to 875  C, slow cool (approximately 120 h) from 875 to Td (835  C < Td < 870  C), and decant the residual liquid phase. By changing the decanting temperature, single grains with a mass of several to several tens of grams have been grown. The obtained single grains exhibited pentagonal dodecahedral facetted morphology (Figure 26.21(A)).

26.5.2

Al–Ni–Co System

Among the DQCs found to date, it is easy to grow a large single grain in d-Al–Ni–Co (Al72Ni12Co16). As shown in Table 26.3, single d-Al–Ni–Co grains with a size on the order of centimeters have been synthesized via the Czochralski method [45–50], Bridgman method [43], floating zone method [60,61], and self-flux method [43,71]. The phase diagram of the Al–Co–Ni alloy system has been studied by several groups [75,76]. Figure 26.23 shows a partial phase diagram along Al72.5Co27.5xNix (x ¼ 5–20 at%) isopleths involving the primary crystallization field of the quasicrystalline phase [75]. The d-Al–Ni–Co is an incongruent melting phase, existing in a large composition range compared to the IQCs in Al-based alloys. L þ D shows the area that the DQC phase is in equilibrium with liquid phase and D is the stability range of DQC phase. Sato et al. were the first to grow centimeter-sized single d-Al–Ni–Co by the floating zone method [60]. Single grains with a volume of 1 cm3 have been obtained using starting polycrystalline rods with a composition of Al72Ni12Co16 at a growth rate of 0.5 mm/h. Because the d-Al–Ni–Co phase is in equilibrium with the Al-enriched liquid phase in a large composition range in the phase diagram, it is easy to grow single d-Al–Ni–Co from Al-rich molten solution by the self-flux method. Fisher et al. [71] and Feuerbacher et al. [43] have prepared centimeter-sized single d-Al–Ni–Co from an initial melt of Al77.0Ni10.5Co12.5. The most successful growth of centimeter-sized single d-Al–Ni–Co has been carried out using the Czochralski method [45–50]. The details have been studied by Gille et al. [47,48,50]. With this method, high-quality single grains of centimeter size have been FIGURE 26.23 A partial phase diagram along Al72.5Co27.5xNix (x ¼ 5–20 at%) isopleths involving the primary crystallization field of the decagonal quasicrystal phase [75].

Chapter 26 • Crystal Growth of Quasicrystals

1139

synthesized form Al-rich melts in the range of approximately 75.6–78.5 at% Al, 11.0–14.0 at% Ni, and 7.5–13.4 at% Co. During the process, the seed was dipped into the melt at 1050  C. The crystal growth was performed using a very slow pulling rate (0.1–0.5 mm/day) as well as rotation of the crucible and the growing ingot in opposite directions. The diameter of the ingot is mainly controlled by the temperature of the melt. Thus, after having increased the diameter of the ingot to about 10–15 mm at a certain temperature, a decrease in temperature of 0.1–0.8 K/h is applied to keep the diameter constant. The morphology of the final ingot is dependent on the orientation of the seed QC. Circular cross-sections of the growing ingot were obtained when d-Al–Ni–Co seeds of [00001] orientation were used, whereas with other growth directions elliptical shapes of the growth interface occurred due to the anisotropic growth, with much faster growth rates along the periodic direction of the 10-fold axis. The single d-Al–Ni–Co grown on an oriented seed shows a clear faceting corresponding to the 10-fold or 2-fold symmetry, as shown in Figure 26.17(B). The unoriented single d-Al–Ni–Co has facets of 32 types, with different inclination of planes toward the 10-fold axis [50].

26.5.3

Zn–Mg-RE System

The i-Zn–Mg-RE system (Zn60Mg30RE10, RE ¼ Y and rare earth) is a family of stable IQCs that contains localized 4f electrons of rare earth elements, which enable intrinsic magnetism inherent in quasiperiodic lattices [39,68]. The growth of single IQCs in this system allows one to study the magnetic properties of a quasiperiodic structure. Langsdorf et al. has performed single crystal growth in a Zn–Mg-Y system using a liquid-encapsulated top-seeded solution growth method [77,78], in which a LiCl–KCl mixture was used as liquid encapsulant to prevent the evaporation of Zn and Mg components. With this method, single grains of iZn–Mg-Y with dimensions of a few millimeters in size can be obtained. Sato et al. reported the growth of a large single i-Zn–Mg–Ho of 0.5 mm3 in volume with the Bridgman method, in which the vapor can be confined within a small closed crucible [56]. However, considering the phase diagram of the Zn–Mg-RE systems, the most suitable method for single i-Zn–MgRE growth is the self-flux method from Mg-rich melts [68]. Figure 26.24(A) shows a pseudobinary section of the Zn–Mg-Y phase diagram for the section of Zn40þ2yMg603yYy determined by Langsdorf et al. [77,78], in which the IQC phase is in equilibrium with the liquid in a wide range (Q þ melt), being easy to grow single IQCs from molten solution via the self-flux method. Other Zn–Mg-RE systems have similar phase diagrams as that of Zn–Mg-Y. Based on the phase diagram, the optimal starting composition for crystal growth with the self-flux method has been adjusted to be Zn46Mg51RE3 [68,77,78]. An exception is the Zn–Mg–Tb system, in which the primary solidification surface has shifted in composition and the initial composition is set to Zn40Mg57.4Tb2.6 [68]. Ta (or Mo) tubes are used in the experiment to prevent attack from Mg and rare earth elements. A perforated Ta strainer is incorporated into the Ta tube for decanting the remaining liquid melt. Following this procedure and the temperature program shown in Ref. [68], one can produce single IQCs with pentagonal

1140

HANDBOOK OF CRYSTAL GROWTH

FIGURE 26.24 (A) Pseudo-binary cut of the Zn–Mg-Y phase diagram for the section of Zn40þ2yMg603yYy [77,78]. The two-phase region in which the icosahedral quasicrystal phase is in equilibrium with the liquid is labeled Q þ melt. The arrow shows the starting composition for crystal growth with the self-flux method. (B) Photograph of a single i-Zn–Mg–Dy (w2 mm in size) grown by self-flux method.

dodecahedral facetted morphology of up to centimeters in size (Figure 26.24(B)). The composition of the final obtained single i-Zn–Mg-RE is around Zn57Mg34RE9.

26.5.4

Ag–In–Yb System

The i-Ag–In–Yb system (Ag42In42Yb16) has been discovered based on the stable binary i-Cd–Yb (Cd5.7Yb) by replacing one half of the Cd with Ag and the other half of Cd with In [22,74]. In addition to the IQC phase, two kinds of periodic cubic structural approximants (the 1/1 Ag40In46Yb14 approximant and 2/1 Ag41In44Yb15 approximant) have also been discovered [54]. It is thus possible to carry out comparative investigation of IQC and approximants, which have similar building blocks but different long range orders [23]. The Ag–In–Yb can be regarded as a pseudo-binary system, and its partial phase diagram involving IQC phase and approximants is similar as that of the Cd–Yb system (Figure 26.25). The i-Ag–In–Yb is a congruently melting compound and is suitable for growing a large single grain with the Bridgman method [54,55], as shown in Figure 26.18. The sample is kept heating at 800  C for 2 days to enable the homogeneous melting of the elements; then, the temperature of the upper zone is decreased to 627  C for crystal growth. In the experiment, a growth rate of w0.8 mm/h produces high-quality single iAg–In–Yb. At a growth rate above 5 mm/h, the possibility of obtaining a polycrystalline sample increases evidently. Figure 26.26(A) shows the macrographs of a sample grown with a starting composition of Ag42In42Yb16 at a growth rate of 0.8 mm/h. The size of the sample is 1.1 cm in diameter

Chapter 26 • Crystal Growth of Quasicrystals

1141

FIGURE 26.25 A partial phase diagram of the temperature-composition section along the composition line Ag26þxIn742xYbx [54].

(A)

(B)

(C)

FIGURE 26.26 Macrograph of (A) appearance and (B) naturally cleaved fracture surface for the Ag42In42Yb16 sample [55]. (C) Back Laue diffraction patterns obtained from single grain of (A, B).

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and over 3 cm in length. From the fragment, one can clearly observe the shiny part corresponding to the single grain region. Figure 26.26(B) shows the mirror-like naturally cleaved fracture surface of the same sample, which has luster and is smooth but not absolutely flat. The single grain is chemically homogeneous to be around Ag42In42Yb16 and structurally uniform, as confirmed by the back-reflection Laue X-ray diffraction patterns shown in Figure 26.26(C). The clear diffraction patterns with two-, three-, and fivefold symmetries indicate a highly ordered IQC structure over a long range of centimeter order. All the single-grain IQCs (about 20 samples) synthesized by the Bridgman method grew nearly along a twofold axis, according to the Laue diffraction experiment.

26.5.5

Cd-RE and Zn–Sc Systems

Two types of stable binary QCs—i-Cd-RE (RE ¼ Gd to Tm, Y) and i-Zn–Sc (Zn88Sc12)— were discovered by Goldman and Canfield et al. via re-examination of the existing binary phase diagrams at compositions close to the approximant phase [69,70]. Both types of IQCs contain the same basic structural unit of an RTH cluster as that of i-Cd–Yb. However, the newly discovered i-Cd-RE QCs contain localized magnetic moments, which are different from the previously discovered nonmagnetic i-Cd–Yb. These binary QCs are obtained at compositions near the known crystalline approximants by the solution growth decanted at high temperatures and normally exhibited high disorder in structures.

26.6 Growth Mechanism Because QCs are produced by both rapid and slow solidification, their morphologies and growth mechanism can be qualitatively understood in terms of classical solidification theory. However, due to the complexity of structure and phase diagrams, difficulties arose in the course of experiments. Short of structural information, there were only few reports dealing with growth mechanism. Recently, great progress on structural analysis has been made for i-Cd–Yb, which is helpful for understanding growth mechanism of QCs. Surface studies on QCs also provided crucial information for understanding growth morphology and stability. Indeed, understanding the stability and morphologies of QCs surfaces allows insight into the growth mechanism of QCs. Here, the morphologies of QCs will first be described; then, on the basis of the morphologies, phase diagrams, and surface studies, the growth mechanism will be discussed in terms of solidification theory.

26.6.1

Morphologies of QCs

26.6.1.1 Stellated Polyhedron In most of the early experiments, QCs synthesized by rapid solidification revealed highly dendritic morphologies. Visible but distorted pentagon dodecahedra around w0.01 mm

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FIGURE 26.27 An SEM image of melt-quenched A94Mn6 alloy after etching treatment. Icosahedral quasicrystal reveals a stellated dodecahedral form.

were observed in melt-quenched AlMnSi [79,80]. For alloys containing lower contents of Mn (below 8 at%), a stellated polyhedron as shown in Figure 26.27 appeared, resembling one of the polyhedrons in the three-dimensional quasiperiodic lattice; this is shown in Figure 26.8(B), as composed of 20 acute rhombohedra and exhibiting icosahedral symmetry. The beautiful stellated polyhedral morphologies are only observed in low Mncontent Al alloys because of the formation of a small number of nuclei in the melt upon solidification, which allowed equaxial growth of IQC grains. In view of the morphologies, it seems that the preferential growth direction is along the threefold direction.

26.6.1.2 Rhombic Triacontahedron One of most beautiful morphologies observed is the conventional solidification state of i-Al–Li–Cu (Al5.5Li3Cu), which exhibits a form of a rhombic triacontahedron [81,82], as shown in Figure 26.28. This is the other polyhedron existing in a three-dimensional quasiperiodic lattice, which is consisted of 10 acute rhombohedra and 10 obtuse rhombohedra.

26.6.1.3 Pentagonal Dodecahedron The pentagonal dodecahedron is the morphology most often observed in IQCs. Clear morphologies were observed in conventionally solidified Al–Cu–Fe and Ga–Mg–Zn alloys. Similar morphologies with grain sizes of w1 mm were also observed in most stable IQCs, such as i-Al–Cu–Ru, i-Al–Pd-RE, i-Zn–Mg-RE, and i-Ag–In–Yb. As mentioned previously, except for i-Cd–Yb and i-Ag–In–Yb, all stable IQCs form through peritectic reactions at stoichiometric compositions. Figure 26.29(A) shows a faceted sphere morphological form in an as-solidified Al65Cu20Fe15 alloy. According to the phase diagram, the b and l phases are the nucleation sites of IQC phase. It was recognized that the

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FIGURE 26.28 An SEM image for an isolated single icosahedral quasicrystal with a rhombic triacontahedral morphology in Al–Li–Cu alloy [82].

(A)

(B)

FIGURE 26.29 (A) A facetted sphere and (B) a dodecahedral morphological forms of b and icosahedral quasicrystal phases in an arc-melting Al65Cu20Fe15 alloy [13].

faceted sphere is the b phase on which facetted planes can be indexed. Many steps surrounding the facet planes are clearly observed, which is a fingerprint of peritectic reactions. The [110] axis has been found to be coincident with one of the fivefold axes of the IQC phase. Hence, we may expect that the [110] plane would grow and the [100] and [111] would degenerate [13]. Therefore, the facetted sphere shown in Figure 26.29(A) is a midway stage of crystal growth; at a sufficiently slowing cooling rate, the b phase would be fully consumed to form a dodecahedral IQC phase, as shown in Figure 26.29(B). According to the morphology, it is clear that the IQCs in this case are not dendritic growth along three-fold axes. Instead, exhibition of larger and flat pentagonal planes indicates planar growth of IQCs along fivefold axes in melts upon slow cooling.

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Surface studies on the i-Ag–In–Yb and i-Al–Pd–Mn show that fivefold planes have a flat structure—enough for obtaining atomic resolution [83]. This is evidence that the fivefold planes are relatively stable, terminating at planes containing relatively high concentrations of Yb (40 at%) for i-Ag–In–Yb (16 at% Yb) and Al (w80 at%) for i-Al–Pd–Mn (70 at% Al), with respect to their bulk compositions. According to structure models, the atomic densities r of twofold planes are highest (r2 > r5 > r3) in these two IQCs. However, Yb and Al, which have relatively low surface energies [84] with respect to the other two constituent elements, concentrated at fivefold planes would sufficiently lower the surface energy of fivefold planes. Especially upon crystal growth produced by solution growth process, the lower surface energies would reduce solid–liquid interfacial energies; consequently, a pentagonal dodecahedron would form.

26.6.1.4 Decaprism Figure 26.30 shows columnar decagonal solidification morphologies of single grains in a conventionally solidified Al70Ni15Co15 alloy. The DQC phase possesses a twodimensional quasiperiodic plane stacked periodically and the decaprism morphology reflects the crystallographic symmetry of the DQC itself. Because the DQC forms in a wide composition range in the Al–Ni–Co system, this morphology was universally observed in several alloys. The elongated prismatic axis demonstrates that the growth is fastest along the 10-fold axis and slowest along twofold directions. Figure 26.30(A) shows the beginning of columnar growth for DQC, where a number of grains are growing along the same 10-fold direction. The fronts of columns are basically spherical, but many facets inclined to the 10-fold axis on the fronts are visible. These facets correspond to planes relating to the periodic and quasiperiodic directions; they indicate the existence of dense atomic layers in DQC on net planes (lattice planes) inclined to the 10-fold axis and are related to strong Bragg reflections. Because they link

(A)

(B)

FIGURE 26.30 A columnar decaprismatic solidification morphology of single grains of the decagonal quasicrystal in an Al70Ni15Co15 alloy. (A) A faceted column and (B) a decaprismatic solidification morphologies of single grains of decagonal quasicrystal in an Al70Ni15Co15 alloy

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the quasiperiodic and periodic directions, these inclined net planes may play a crucial role in the growth and stabilization of DQC. Evidently, the atomic layers associated with these planes are not completely flat and exhibit some degree of corrugation [85]. However, this is supposed to be observed only in the very beginning of crystal growth, where the crystal is small enough and interfacial energy still works. In Figure 26.30(B), many wrinkle-like lines are clearly seen at the twofold planes, which run across on different twofold planes along the twofold directions. This is a sign that crystal growth in DQCs is planar growth along the 10-fold axis. Surface studies on DQCs also revealed that the tenfold plane is stable because surface preparation was easy to prepare and large terraces were easy to obtain for 10-fold planes. On the other hand, there were always strong corrugations at twofold planes. One might be able to predict the morphology of decaprism for QDCs from their surface stabilities.

26.6.2

Morphological Stability Theory

Because most stable QCs available for studying crystal growth are ternary alloys and have complicated phase diagrams, initial solidification processes are not easy to study. However, crystal growth for QCs could be qualitatively figured out in terms of the classical solidification theory. According to the solidification theory, the solidification morphology of a crystalline grain is determined by the stability of the solid–liquid interface. When the interface changes from stable to unstable, the morphology of crystalline grains change from equaxial grains with a planar interface to cellular grains, and then from cellular grains to dendritic grains on the basis of the constitutional undercooling criterion. Generally, the limit of constitutional undercooling can be expressed in its usual form [86]: G=R > mC0 ðk  1Þ=D

(26.4)

where G is the interface temperature gradient, m is the liqudus slope, R is the rate of interface movement, C0 is the initial bulk solute concentration, k is the partition coefficient (the ratio of solute concentration in solid to that in liquid), and D is the diffusion coefficient. If the ratio of G/R is smaller than mC0(k  1)/Dk, interface instability will occur. It is clear that in a relatively low solute concentration, a high thermal gradient must be imposed in order to suppress interface instability and cellular dendritic growth. By taking account of the interfacial energy, an absolute interface stability criterion has been derived in a dilute alloy [87].  A ¼ k2 Tm GR ðk  1ÞmC0 D

(26.5)

where A is an interface stability parameter (being 1 for a stable interface), G ¼ s/DH is the surface tension constant, s is the interfacial energy, and DH is the latent heat. The criterion is especially available at high solidification rates. The higher the rate of interface movement, the less time there is for lateral diffusion of solute: long-wavelength perturbations at the solid–liquid interface do not have enough time to form. On the other hand, short-wavelength perturbations will be suppressed by surface tension. Thus, for a

Chapter 26 • Crystal Growth of Quasicrystals

1147

liquid with given bulk solute concentrations, there is a solidification rate above which a planar interface is always stable. At a high solidification rate, stable planar growth can occur at concentrations far greater than those predicted by the constitutional supercooling criterion. Therefore, the absolute interface stability criterion is a good approximation at high solidification rates (e.g., the melt-quenched process), whereas the constitutional supercooling criterion is a good one at low solidification rates (e.g., normal crystal growth processes) [88]. As an illustration, a metastable equilibrium pseudo-binary phase diagram of the IQC phase and a-Al face-centered cubic (fcc) phase in the Al–Mn system is shown in Figure 26.31(A) [89]. A schematic microstructure dependence on growth velocity, R, and composition, C0, was constructed and is shown in Figure 26.31(B) [89]. At small C0 when R is small, the interface is stable and planar. This regime is represented by the lower left corner of the diagram. As the growth rate of a dendritic structure approaches that for absolute interface stability, it is anticipated that

FIGURE 26.31 (A) A schematic of the metastable phase equilibria between the icosahedral quasicrystal (i) and fcc Al phases. (B) The dependence of microstructure on the growth rate R and the composition C0 [89].

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the microstructure will change from dendritic to cellular before becoming a planar front. At a very large R (A > 1), the instability is suppressed and a planar front is retained, corresponding to the upper left corner of the diagram in Figure 26.31(B).

26.6.3

QCs upon Rapid Solidification

In earlier morphological investigations, the IQC grains were found to possess rounded outlines embedded in a matrix of fcc Al [90]. The IQC grains grew dendritically with arm directions along threefold axes. In some cases, facetted outlines with rather straight boundaries between the Al crystal and IQC phase and consequently pentagon dodecahedral morphology were found [80]. Now, the pentagonal dodecahedron as a principal morphology, found in rapidly solidified alloys as well as in most stable IQCs except Al–Li–Cu, is a consensus. However, depending on the preparation conditions, the morphology could be dendritic arms or pentagonal dodecahedra. This has been clarified by an investigation on melt-quenched Al–Pd–Cr alloy on the basis of criterion of the interface stability in alloy solidification [89]. Figure 26.32 shows the TEM electron micrographs of melt-quenched Al72Pd25Cr3 alloy, revealing a mixed structure consisting of IQC and DQC phases. The IQC grains show an elongated columnar structure with a flat plane perpendicular to the fivefold axis; they are surrounded by the DQC phase in Figure 26.32(A). Here, the two IQC grains show the same twofold diffraction pattern with the same orientation, showing one of the fivefold directions that coincide with the planar growth direction (indicated with arrows) of the columnar structure. The results strongly suggest that the IQC predominantly grows along the fivefold direction during rapid solidification. The diffraction patterns along the twofold direction taken from the regions near the interface between the IQC (A, C) and DQC (B, D) are compared in Figure 26.32. One of fivefold directions of the IQC coincides with the 10-fold direction of their DQC. The angle between two adjacent 10-fold axes is 63.43 , which is equal to the angle

(A)

(B)

FIGURE 26.32 Transmission electron microscopy images of a melt-quenched Al72Pd25Cr3 alloy taken along the coincident twofold axes of the decagonal quasicrystal (DQC) and the icosahedral quasicrystal (IQC) phases. (A) Each DQC phase grain is observed to be growing with its tenfold axis along one of fivefold axes of the parent IQC phase. (B) Two DQC grains grow with their tenfold axes along two different fivefold axes of the parent IQC phase [89].

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1149

between two fivefold axes on the twofold diffraction pattern or two adjacent fivefold axes of an icosahedron. The same orientation relationship was also confirmed in the different regions of the sample, as shown in Figure 26.32(B), where two DQC grains with angle of intersection of 63.43 nucleated from the same IQC grain. In all cases, the IQC phase grows along the fivefold direction, and the DQC nucleated out of the IQC and grow along the 10-fold direction, which coincides with one of the fivefold directions of the IQC. The average compositions for IQC and DQC are Al67Pd29Cr4 and Al73Pd25Cr2, respectively. IQC has higher Pd and Cr contents and, according to the phase diagrams, the solidification temperature for IQC would be higher than that of DQC. During solidification, the IQC nucleates and grows first, rejecting the excess Al into melt. This process then slows down the growth of the IQC and favors the nucleation of an Al-richer DQC. If the moving rate of the solid–liquid interface was fast enough to suppress the diffusion of Al at the interface, the formation of DQC would be suppressed and IQC with planar growth would be completed. In Figure 26.32(A), in the upper part, a horizon line indicates that the two IQC grains had the same front of the solid–liquid interface in the beginning, but it broke down upon solidification due to either the slow-moving rate of the interface or fast solute diffusion across the interface. Consequently, the dendritic growth would occur as shown in Figure 26.31. For the same alloy but at a higher solidification rate, planar growth is still expected [89].

26.6.4

QCs Prepared by Classic Crystal Growth Methods

The QCs produced by these processes have a common feature that they all revealed planar growth upon solidification. Because the solidification rates are all very small and the moving rate of the solid–liquid interface R would be small, and in terms of the constitutional supercooling criterion, planar growth occurs if the temperature gradient is not too large at the interface. This condition corresponds to the lower left part in Figure 26.31(B). The large pentagonal and decagonal planes observed in IQCs and DQCs prepared by the solution growth process are evidence of the planar growth. The planar growth is along the fivefold axis for the former and is along the 10-fold axis for the latter. The planar growth is also observed in several QCs prepared by the Bridgman and floating zone methods. Figure 26.33(A) is a longitudinal microstructure near the solidification interface for a single DQC prepared by the floating zone method at a growth rate of 0.5 mm/h, starting with a raw material of Al72Ni12Co16 [91]. Phases aligned in the form of rods near the solidification interface consisted of fcc Al, Al3Ni2, Al9Co2, and the DQC phase, although only the DQC was detected from the grown crystal and feed rod. These results indicate that at the steady state, a single DQC grain grows. When the heating power of the furnace is turned off, the melt starts to solidify from the solid–liquid interface, forming the four phases in the rod configurations. Therefore, the present quenching rate is high enough to preserve the solidification interface during the steady-state single crystal growth.

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FIGURE 26.33 (A) Optical microstructure of a longitudinal cross-section of an Al72Ni12Co16 alloy prepared by the floating zone method along the growth direction near the quenched molten zone. Note that the solidification interface is flat. (B) Composition distribution of Al, Ni, and Co along the longitudinal (growth) direction of the same sample [91].

Similar to crystalline solids, the flat quenched interface implies that planar growth proceeded, which is a prerequisite for single crystal growth of QCs. Figure 26.33(B) shows the chemical composition distribution from the grown crystal to the feed rod through the quenched molten zone. At the initial growth stage, the crystal starts to grow, with Al and Co contents that are lower than those of the feed rod and Ni content that is the opposite. As the crystal growth proceeds, the composition becomes close to that of the feed rod and reaches the steady state. Solute redistribution at this steady state can be clearly seen at the quenched solidification interface in Figure 26.33(B). Al is significantly enriched in front of the solidification interface by 6.87 at%; conversely, Co and Ni are depleted by 5.95 at% and 0.93 at%, respectively, which indicates that the DQC is incongruent at Al72Co16Ni12. The melt with this composition is considered to play the role of flux for the crystal growth of the incongruent alloy. Interestingly, the flux was spontaneously produced during the crystal growth. The solute partition ratios are supposed to be close to the equilibrium state. Due to significant solute redistribution, the single DQCs was only could grow at the lower growth rate, R. It is noted that spontaneous growth direction is parallel to the 10-fold axis. Similar planar growth with similar solute partition ratios was also observed for

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FIGURE 26.34 Optical microstructure of a cross-section near the solidification interface of Al72Pd19.5Mn8.5 grown by the Bridgman method at a growth rate of 0.2 mm/h.

an IQC prepared by the floating zone method starting with Al72Pd19.5Mn8.5 and with a growth rate of 1.0 mm/h, in which the spontaneous growth direction was parallel to a twofold axis. Figure 26.34 is the microstructure of a cross-section across the solidification interface for a single IQC grown by the Bridgman method at 0.2 mm/h starting with Al72Pd19.5Mn8.5. The front of solidification interface is slightly curved due to the surface tension of liquid and the effect of the crucible, but basically it is planar growth parallel to the twofold axis.

26.6.5

Selection of Growth Direction

The previously mentioned processes are common when using slow cooling upon solidification, but they differ in temperature gradient, growth direction, and crucible. In the floating zone, Czochralski, and solution growth process, QCs were grown in liquid without effects from the furnace; hence, they revealed different degrees of facet. In the solution growth process, because the overall furnace was cooled homogenously, the temperature gradient at the solid–liquid interface is supposed to be very small. Isotropic growth of IQCs and anisotropic growth of DQCs in the liquids are expected according to their structure symmetries. In this case, the growth direction will be determined by the stability of atomic plane (i.e., the atomic structure of QCs themselves). According to structure models and surface studies [92,93], densities r of the atomic planes are in the order of r2 > r5 > r3 for i-Ag–In–Yb and i-Al–Pd–Mn (or i-Al–Cu–Fe). Actually, the difference between r5 and r2 is very small and the concentration of elements with lower surface energies, such as Yb in i-Ag–In–Yb and Al in i-Al–Cu–Fe, are relatively higher on fivefold planes. In other words, concentration fluctuation of Yb or Al is the largest along the fivefold axis, and this scenario has been successful for interpreting the surface structures of fivefold planes for IQCs, where large terraces with nonperiodic step heights are observed [92,93]. The lower surface energy of the fivefold plane induces lower interfacial energy of the solid–liquid interface; this is a plausible reason why the fivefold plane is a steady state for growth, and consequently pentagonal dodecahedra of a few millimeters in size were observed in these two systems. This is also the case for i-Zn–Mg-RE.

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FIGURE 26.35 Schematics of (A) dodecahedrons and (B) icosahedrons used for describing the favorable direction upon unidirectional growth of icosahedral quasicrystals.

However, when temperature gradient was employed in crystal growth (e.g., floating zone and Bridgman methods) for IQCs, the growth direction was always along the direction parallel to a twofold axis. Two reasons are considerable for the conflict with the solution growth process. First, with the temperature gradient existing near the solid–liquid interface, the stability of the interface will be dominated by both solute distribution and interfacial energy. As a result, planar growth will prefer a direction with less concentration fluctuation (homogeneous) and, on an atomic plane, lower surface energy or higher atomic density. The second reason is explained by icosahedral symmetry of the IQCs. Figure 26.35 shows a schematic pentagonal dodecahedron along a fivefold axis (Figure 26.35(A)) and projection of icosahedron onto a twofold axis (Figure 26.35(B)), where twofold and fivefold axes are indicated with arrows. If there is a temperature gradient at the interface, the growth of IQCs will be constrained to unidirectional; six different fivefold planes with lower surface energy will be competitive for growth and only one of them will be selected, with the rest being suppressed (Figure 26.35(A)). With respect to this as shown in Figure 26.35(B), two adjacent fivefold planes complementarily grow; consequently, growth along a twofold axis will be energetically favored. On the other hand, due to strong anisotropy, the DQCs reveal a common feature of growth direction regardless of the crystal growth process.

26.7 Concluding Remarks After a general introduction of QCs, including structural properties, structural models, and alloy systems, the crystal growth of QCs (as studied so far) was reviewed. Solidification theories, such as constitutional supercooling, and absolute interface stability criteria were employed to understand the growth mechanism of QCs. The solid–liquid interface stability and growth direction of QCs were interpreted in terms of structural models and surface studies. It is clear now that more than a few QCs are stable and exist in the phase diagrams as equilibrium phases. All stable IQCs could be regarded as a new group of intermetallic

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compounds that form at very strict compositions and normally have a special value of e/a. On the other hand, the formation of stable DQCs is more flexible in composition. For the former, the strict composition is an indication of the high structural stability, which favors crystal growth of IQCs in a given process. For the latter, flexibility in composition makes growing single grains easy, but their precise compositions are not controllable. Although the crystal structure of QCs is very complex, the quality of single grains of QC seems to be better than well-known intermetallic compounds with simple structures, as evidenced by the occurrence of a large number of sharp diffractions. Due to the success in growing high-quality single grains of QCs, great progress in the structure, physical, and chemical properties and surfaces have been achieved. This progress can be used to understand the mechanisms of crystal growth for QCs.

Acknowledgments CC acknowledges the support of the National Natural Science Foundation of China (No.11074220) and Qianjiang Talent Program of Zhejiang Province (2013R10060).

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