PhD Thesis _MLC1

AKADEMIA GÓRNICZO-HUTNICZA im. Stanisława Staszica w Krakowie

WYDZIAŁ INśYNIERII METALI I INFORMATYKI PRZEMYSŁOWEJ KATEDRA METALOZNAWSTWA I METALURGII PROSZKÓW

ROZPRAWA DOKTORSKA

Crystal growth and plastic properties of single-crystalline complex metallic alloys mgr inŜ. Marta Lipińska-Chwałek

Promotor: Prof. dr hab. inŜ. Aleksandra Czyrska-Filemonowicz

KRAKÓW 2009

I would like to thank all the people who contributed one way or another to the realization of this thesis. Firstly, I express my deepest gratitude to my supervisor Professor dr hab. inŜ. A. Czyrska-Filemonowicz for her excellent concern as well as optimistic and motivating support during the course of my research study. I am very grateful to Professor Dr K. Urban for the possibility to work in his institute, the great hospitality and encouragements he gave me over the last years. My special thanks go to Dr M. Feuerbacher. One does not often meet people with such a charisma like you have, Mick. Thanks for your very friendly, open-minded and always supporting guidance. I am very grateful for your valuable suggestions and criticisms during my investigations and for your effort to proofread this thesis. Great thanks to all my colleagues in the Metals Group and other co-workers of the Microstructure Research Institute (IFF-8) in Forschungszentrum Jülich for the pleasant working environment; especially to Dr S. Roitsch, Dr M. Heggen, Dr S. Balanetskyy and Dr J. Barthel for being always open for discussions, ready to share your knowledge and to offer your help in various, not only scientific fields; to Mrs E.-M. Würtz, Mrs M. Schmidt and Dipl. Ing. C. Thomas for friendly and fruitful cooperation in producing the single crystals and their characterization. I acknowledge also Mrs I. Rische-Radloff, Mrs G. Wassenhofen, Mrs W. Sybertz, Mrs D. Meertens and Mr W. Pieper for the entire administrative and technical support they have offered to me; and all other colleagues which I can not name here individually - it was great pleasure to know you and to work with you. Moreover, I owe great thanks to all my colleagues from the Department of Physical and Powder Metallurgy in the AGH University of Science of Technology, especially to Mrs E. Batko, MSc M. Ziętara and MSc G. Cempura for the great help they offered to me over the last years. I thank Professor dr hab. inŜ. W. Ratuszek for proofreading this thesis and Professor dr hab. inŜ. K. Wiencek for valuable remarks on the quantitative defects analysis. My family and friends I thank for the most important support. This work would not be possible without the constant support of my husband Paweł – thank you Pawcio for your great patience and understanding, always staying by my side and making our live interesting and enjoyable. I would like to dedicate this work to my son Jeremi – you were my strongest motivation to complete it. This work was financially supported by the Polish Ministry of Science and Higher Education (project No. N N507 400835) and the 6th Framework EU Network of Excellence “Complex Metallic Alloys” (Contract No. NMP3-CT-2005-500140).

INDEX

Introduction ......................................................................................................................... 1

Theoretical part I Complex Metallic Alloys ................................................................................................... 5 I.1 The structure of CMAs ..................................................................................................5 I.2 The β-Al-Mg phase......................................................................................................12 I.2.1 The β-phase in the Al-Mg system..........................................................................12 I.2.2 The crystal structure of the β-Al-Mg phase ...........................................................14 I.3 The κ-Al-Mn-Ni phase ................................................................................................17 I.3.1 The к-phase in the Al-Mn-Ni system.....................................................................17 I.3.2 The crystal structure of the к-phase.......................................................................20 I.4 Structure defects in CMAs...........................................................................................25 II Single-crystal growth..................................................................................................... 29 II.1 Fundamentals of single-crystal growth .......................................................................29 II.2 Single-crystal growth techniques ................................................................................31 II.2.1 Bridgman technique.............................................................................................31 II.2.2 Czochralski technique..........................................................................................33 II.2.3 Self-flux growth technique ..................................................................................34 III Crystal plasticity and plastic deformation .................................................................. 37 III.1 Fundamentals of crystal plasticity .............................................................................37 III.2 Incremental tests .......................................................................................................42 III.3 Experimental testing procedure .................................................................................44 IV Characterization of structure defects by TEM ........................................................... 45 IV.1 Imaging in TEM .......................................................................................................45 IV.2 Defect-density determination ....................................................................................47

Experimental part V Single-crystal growth of CMAs ..................................................................................... 55 V.1 Experimental procedures............................................................................................55 V.2 Growth of β-Al-Mg single crystals .............................................................................56 V.3 Growth of к-Al-Mn-Ni single crystals ........................................................................61 V.4 Discussion..................................................................................................................71 VI Plasticity of the β-Al-Mg phase .................................................................................... 73 VI.1 Macroscopic deformation behaviour .........................................................................73 VI.2 Microstructural analysis ............................................................................................79 VI.3 Discussion ................................................................................................................94 VII Plasticity of the к-Al-Mn phase .................................................................................. 99 VII.1 Macroscopic deformation behaviour ........................................................................99 VII.2 Microstructural analysis.........................................................................................106 VII.3 Discussion .............................................................................................................117 Summary...........................................................................................................................127 Conclusions .......................................................................................................................131 References .........................................................................................................................133 List of symbols and abbreviations....................................................................................139

Introduction The class of Complex Metallic Alloys (CMAs) comprises systems with giant unit cells containing up to more than a thousand atoms per cell arranged in a cluster substructure (Urban and Feuerbacher, 2004). Therefore, for many phenomena, the physical length scales in these materials are substantially smaller than the unit-cell dimension. This may give rise to unique physical properties, which are mutually exclusive in conventional materials and therefore represent a challenge for science and a possible chance for future applications of CMAs. Plasticity is a materials property of particular interest in the case of CMAs. Deformation mechanisms known from structurally simple materials are prone to failure in structures featuring giant unit cells. Due to large lattice parameters, formation of perfect dislocations in CMAs would require physically unacceptable large elastic line energies. Therefore, new mechanisms of microstructural processes are expected to appear in CMAs. Indeed, novel and energetically favourable deformation carriers were revealed in some, already investigated CMAs, e.g. in ε 6 - Al - Pd - Mn (Klein et al., 1999), c 2 -Al - Pd - Fe (Heggen, 2003 and Feuerbacher et al., 2004), or Al13 Co 4 (Heggen et al., 2007). Despite the fact, that crystallography of CMAs have been studied since several decades, their physical properties remained largely unexplored. This is mainly due to the fact that until recently high-quality single-crystalline samples of these complex materials, which are required for the determination of their intrinsic physical properties, were unavailable. The development of single-crystal growth procedures for the production of high-quality material is, hence, an inescapable requirement for further progress in the characterization of CMA properties. The purpose of the present study was, first, to elaborate growth procedures for high-quality single crystals of two selected CMA phases, β - Al - Mg and κ - Al - Mn - Ni , and subsequently, to get insight into their deformation behaviour and corresponding microstructural mechanisms. The plasticity of these materials was investigated for the first time. The use of single-crystalline materials ensures that influence of impurities, secondary phases, and grain boundaries on the material behaviour was excluded. Accordingly, only the intrinsic mechanical properties of the investigated materials were examined. The combination of macroscopic and microstructural investigations enables a versatile and detailed analysis of the plasticity of the examined phases. The work covers two different structure types, face-centred cubic, and hexagonal closepacked. The studied alloys are brittle at room temperature but show ductile deformation

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Introduction behaviour at elevated temperatures. High fracture stress and pronounced yielding behaviour are common features of the investigated materials plasticity. Thermodynamic activation parameters of the deformation processes are analyzed for both materials and the microstructural deformation behaviour is elaborated. Chapter I gives a short introduction into the class of CMA materials. Basic structural characteristics as well as the most common types of local order are described. The CMA phases of interest, the Samson phase β - Al - Mg

and recently discovered ternary

κ - Al - Mn - Ni , are characterized. Some examples of novel types of structural defect found in CMAs are also briefly discussed. Chapter II, III and IV give theoretical basics and introductions to the employed procedures of single-crystal growth, crystals plasticity and electron microscopy investigations, respectively. The

single-crystal

growth

routes

developed

experimentally

for

β - Al - Mg and

κ - Al - Mn - Ni phases and the qualitative characterization of the produced materials are described in chapter V. Chapters VI and VII address the results of macroscopic and microstructural deformation behaviour of single crystalline β - Al - Mg and κ - Al - Mn - Ni phases, respectively. A discussion of the macro- and microscopic plasticity of the investigated phases is given and a comparison of these results with the plasticity of other CMAs is made. Summary of the work and conclusions drawn from the performed investigations are given at the end.

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Theoretical part

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I Complex Metallic Alloys In the following chapter, the genesis and structural complexity of the complex intermetallics is elucidated. Subsequently, the crystallographic structures of the investigated phases β - Al - Mg and κ - Al - Mn - Ni are introduced in a great detail. Special attention is devoted to the unique plastic deformation mechanisms encountered in some complex metallic alloys. The latter particularly strongly emphasizes a salient feature of CMAs in comparison to other metallic materials.

I.1 The structure of CMAs Complex Metallic Alloys (CMAs) are intermetallic compounds featuring giant unit cells comprising typically some ten to more than a thousand atoms per unit cell (Urban and Feuerbacher, 2004). Along with the high number of atoms, the size of the unit cells reaches large dimensions. Inside these cells, the atomic arrangement is in most cases organized in clusters (Urban and Feuerbacher, 2004). These clusters, which typically exhibit icosahedral symmetry, represent very compact elements of the crystal structure. The icosahedral arrangement of twelve atoms around a central atom, referred by Pauling (1955) to as “closer packing than closest packing”, leads to a high atom density within these clusters. First CMAs were discovered already in twenties of the last century, when the systematic investigations of metallic alloy phase diagrams started. In a pioneering paper published in 1923, Pauling described for the first time the X-ray diffraction study of the intermetallic compound NaCd2. However, the diffraction patterns of this crystal were too complex to index the diffraction spots by the means available at the time. The same held for a large number of the other structurally complex phases that have already been discovered in these early days. Groundbreaking work was presented by Bergman et al. in 1952 on Mg32(Zn,Al)49. This compound has the cubic space group Im 3 and the unit cell of lattice parameter 1.46 nm containing 162 atoms. Discussing its structure the authors found that new types of coordination polyhedra occur as basic structural elements. In particular, these are aggregates with noncrystallographic symmetry such as icosahedral and pentagonal-dodecagonal clusters (Bergman et al., 1952 and 1957). In 1955 Pauling was finally able to solve the structure of the NaCd2 compound. It has the cubic space group Fd 3 m . The unit cell has an edge length of 3.056 nm and contains 384 sodium and 768 cadmium atoms. Another milestone was the work of Samson (1965 and 1969), who introduced the term “giant unit cell” for CMAs. He defined the structure of the β-Al-Mg phase (Samson phase) that features a cubic Fd 3 m space group and the unit cell of lattice parameter 2.82 nm containing 1168 atoms (c.f. section I.2.2). The material structure features 23 crystallographically different atomic positions which, as a result 5

I Complex Metallic Alloys of partial disorder, produce 42 different polyhedra. The inherent disorder present in this material was identified to be a result of the tendency towards formation of the maximum number of icosahedral coordination shells. A particular feature of CMA structures is the presence of inherent disorder. There are different types of disorder observed:

• Configurational disorder results from statistically varying orientations of particular subclusters inside a given cage of atoms. This kind of disorder may be easily described by the example of the c2-Al-Pd-Fe structure featuring the space group Fm3 and a unit cell with lattice parameter 1.552 nm containing 248 atoms (Edler et al., 1998 and Sugiyama et al., 2000). Dominant structural building blocks of this phase are icosahedral cages formed by Pd atoms filled by two different cluster motifs. One of these motifs is formed by an Al-cube occurring in five different orientations, generating the configurational disorder. The average structure of the different cube arrangements is a regular pentagon-dodecahedron. The configurational disorder can also result from incompatibility problems of cluster packing when these lead to local atom displacements.

• Chemical or substitutional disorder results from fractional occupancy of certain lattice sites by different elements. The possible occupation by different elements causes a variable amount of these elements inside the crystal structure. This affects the extension of the stability range of the corresponding phase within the phase diagram. For example, in the Mg32(Al,Zn)49 phase 3 different atom sites can be occupied either by Al or by Zn atoms what leads to an extension of the Mg32(Al,Zn)49 stability range in the Al-Mg-Zn system over a wide range of values of the Zn/Al ratio (Petrov et al., 1993).

• Displacement disorder and fractional site occupation (occupation smaller than 1) are types of disorder which arise from geometrical constraints. High amount of these types of disorder exists in the Samson phase. These arise from incompatibilities in the packing of Friauf polyhedra, which are the main building blocks of the β-Al-Mg phase structure (chapter I.2). Certain vertices of adjacent polyhedra should be occupied in this material by a large atom (Mg featuring an atomic radius rMg = 0.160 nm (Stöcker (1994)) for one polyhedron and simultaneously by a small atom (Al, rAl = 0.143 nm (Stöcker (1994)) for the other. This incompatibility results in displacement disorder and fractional site occupation. For example, 8 of the 11 Al positions have occupancies between 0.10 and 0.53, and 2 of the 9 Mg positions have occupancies of 0.48 and 0.84.

• Split occupation is as well caused by steric hindrances. It takes place where two lattice sites are localized too close to be occupied simultaneously. This leads to an alternative occupation of only one of these sites. The real atomic content of the unit cell is therefore usually lower than indicated by the Pearson symbol of the corresponding phase. The latter 6

I Complex Metallic Alloys discrepancy can be then characterized by a modified Pearson symbol, in which a number subtracted from the site number indicates the reduced number of atoms per unit cell. The

cP296-49 structure of the Al68Pd20Ru12 phase (Mahne and Steurer, 1996) can be used here as an example. Nowadays hundreds of intermetallic compounds are known whose structures are based on giant unit cells (Villars and Calvert, 1986 and Tamura, 1997). Their number, however, increases continuously in the course of studies of new ternary phase diagrams. For a number of years an increasing interest in the giant unit cell intermetallic compounds was arising from the structural similarity with quasicrystals. Many CMAs have been found in the close stoichiometric vicinity to the quasicrystalline phases, both featuring very similar local atomic structures, which are arranged aperiodically in quasicrystals and periodically in corresponding giant unit cell phases (Sui et al., 1997). This is why in the quasicrystal literature CMAs are frequently referred to as ”approximants” (Janot, 1994). Due to this fact, a considerable progress in the determination of the structure of many quasicrystalline phases was made on a basis of known structure of related crystalline intermetallics (Elser and Heleny, 1985 and Audier and Guyot, 1986). For example, in many quasicrystals atom clusters, e.g. the Mackay and the Bergman cluster, play a prominent role. These occur frequently also as basic structural elements in the CMA phases. Both clusters are based on a concentric shell structure with icosahedral configurations comprising 55 and 117 atoms, respectively. The shells correspond to symmetric polyhedra with atom sites located at the vertices. The structure of the Bergman cluster was discovered by Bergman et al. (1957) in the Mg32(Zn,Al)49 phase. Bergman clusters are built up by successive atom-shells as illustrated in figure I.1. One Al site in the centre (figure I.1 (a)) is surrounded by twelve atoms (Zn and Al) located at the vertices of an icosahedron (figure I.1 (b)). The subsequent shell consists of 20 Mg atoms (red) located above the centres of the triangular faces of the icosahedron, forming a pentagonal dodecahedron. Twelve Al and Zn (rZn = 0.139 nm (Stöcker (1994)) atoms are subsequently located at the centres of the pentagons (figure I.1 (c)). Sixty atoms in the following shell form a truncated icosahedron (soccer ball) what is shown in figure I.1 (d)). The next shell consists of twelve Mg atoms, arranged as shown in figure I.1 (e). The last two shells combined, comprising 72 atoms, form a cuboctahedron. This polyhedron is actually a superposition of a cube and an octahedron and features 6 square and 8 triangular faces. Due to its symmetry, the cuboctahedral shell of the Bergman cluster packed in a bcc lattice (e.g. in the Mg32(Zn,Al)49 phase) can share all atoms with other equal complexes. Therefore, despite overlapping of the Bergman clusters, neither fractional site occupation nor displacement disorder occurs in such structures (Bergman et al., 1957).

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I Complex Metallic Alloys

Figure I.1: Schematic sphere model of the successive atom-shells of the Bergman cluster (a-e) at the centre position of the body-centred Mg32(Al,Zn)49 phase unit cell (grey cubic frame). (f) Bergman clusters at centre and one corner position of the bcc unit cell (Roitsch, 2008). The corresponding atomic radii of the component elements are rAl = 0.143 nm, rMg= 0.160 nm and, rZn= 0.139 nm, respectively (Stöcker, 1994). The Mackay cluster consists of three concentric atom shells (Mackay, 1962). The first shell is an icosahedron (12 atoms) surrounding a central atom. An icosahedron is schematically shown in figure I.2 (a). The second shell is an icosidodecahedron (30 atoms), i.e. a superposition of an icosahedron and a dodecahedron, featuring 12 pentagonal and 20 triangular faces (illustrated in figure I.2 (b)). The last shell of the Mackay cluster is an icosahedron, the vertices of which are located above the pentagon centres of the icosidodecahedron, forming together a surface with 80 triangular faces.

Figure I.2: Schematic illustrations of the icosahedron (a) and icosidodecahedron (b) 8


Atom clusters in CMAs are arranged according to usual crystallographic Bravais lattices. Therefore, the long-range orientational order of these materials is determined by symmetry operations known from structurally simple materials. Nearly all lattice types can be found within the CMAs structures. There exists cubic CMAs as Samson phase from the Al-Mg system (fcc, 1168 atoms/unit cell, a = 2.82 nm) (Samson, 1965 and 1969), Bergman phase, Mg32(Zn,Al)49 system (bcc, 162 atoms/unit cell, a = 1.42 nm) (Bergman et al., 1957), and the phase c2-Al-Pd-Fe (fcc, 248 atoms/unit cell, a = 1.552 nm) (Edler et al., 1998 and Sugiyama et al., 2000). Examples for orthorhombic phases are Al13Co4 with 102 atoms per unit cell and lattice parameters a = 0.82 nm, b = 1.23 nm and c = 1.45 nm (Grin et al., 1994) or the class of Taylor phases, i.e. T-Al-Mn with 156 atoms in its unit cell featuring lattice constants a = 1.47 nm, b = 1.25 nm, c = 1.26 nm (Taylor, 1960, Hiraga et al., 1993) and ternary extensions with Pd, Fe, Cr, and Ni (Balanetskyy 2007 and Balanetskyy et al., 2008). The phases κ-Al-Mn-Ni or κ-Al-Mn-Fe with 227 atoms per unit cell and lattice parameters a = 1.76 and c = 1.25 nm (Marsh, 1998 and Balanetskyy et al., 2008) and the µ-phase from the Al-Mn system containing 563 atoms per unit cell with lattice constants a = 1.998 and c = 2.467 nm (Shoemaker et al., 1989) represent phases with hexagonal unit cells. Also monoclinic structures can be found within CMAs, e.g. Al13Fe4 featuring 102 atoms per unit cell of lattice parameters a = 1.55 nm, b = 0.81 nm, c = 1.25 nm and β = 107.7° (Grin et al., 1994a). The lattice parameters of CMAs are about one order of magnitude larger than the interatomic distances. The length scale on which elementary physical processes take place in these materials may be, therefore, largely governed by the cluster substructure rather than by the geometry of the Bravais lattice. As a result, the macroscopic physical properties of these materials might be influenced by an interplay between these two length scales, e.g. between the next-neighbour scale and the translational invariant distances. A novel plastic deformation mechanism recently discovered in ε-Al-Pd-Mn (see chapter I.4), an orthorhombic CMA with 320 atoms per unit cell, provides direct evidence for this interplay. The ε-Al-Pd-Mn phase belongs to the family of structurally related CMAs called ε-phases existing in different alloy systems. These are found for example in the systems Al-Pd-(Mn, Fe, Rh, Re, Ru, Co, Ir) and the Al-Rh-(Ru, Cu, Ni) (Audier et al., 1993, Klein et al., 1996, Yurechko et al., 2001 and 2004, Balanetskyy et al., 2004 and 2004a). They are denoted εl (l = 6, 16, 22, 28, 34) according to the index of the strong (0 0 l ) diffraction spot corresponding to the interplanar spacing of about 0.2 nm occurring in all of those phases (Yurechko et al., 2001 and 2004, Balanetskyy et al., 2004). All the εl-phases in the respective alloy systems feature identical lattice parameters along crystallographic [1 0 0] and [0 1 0] directions but different

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I Complex Metallic Alloys [0 0 1] lattice constant. Basic structural building blocks of the ε-phases are pseudo-Mackay clusters (figure I.3) arranged in columns along the [0 1 0] direction (Boudard et al., 1996). Viewed along this direction, the structures can be described using a tiling representation, which employs two different elements. One is a flattened hexagon and the second, representing a phason line (Klein et al., 1996), consists of a nine-edged banana-shaped polygon combined with a pentagon (dark grey tiles in figure I.4 (b)). In this approach, the vertices of the tiles are located on the centres of the cluster columns as it is shown on the example of the ε6 structure illustrated in figure I.3.

Figure I.3: Schematic illustration of the ε6-Al–Pd–Mn structure (rAl = 0.143 nm, rPd = 0.137 nm, rMn = 0.130 nm (Stöcker, 1994). The so called I-layer is shown (Beraha et al., 1997). The tenfold units are Mackay icosahedra. The entire atomic structure of one of these icosahedra is shown below the black rectangle marking the projected ε6 unit cell. The flattened hexagons of the tiling representation are outlined by connecting the centers of Mackay icosahedra (Urban and Feuerbacher, 2004). Figure I.4 depicts tiling representations of the phases ε6 (a) and ε28 (b). These are closely related phases featuring the orthorhombic space groups Pnma and C2mm, respectively. The lattice parameters a = 2.3541 nm and b = 1.6566 nm are equal for both materials and c-lattice parameter equals 1.2339 nm in ε6 and 5.7 nm in ε28 (Boudard et al., 1996, Edler, 1997). The structure of ε6 can be represented making use of two types of flattened hexagon arranged in 10

I Complex Metallic Alloys rows of alternating orientations. In ε28 additionally periodic [0 0 1] stacking of phason planes (rows of phason lines aligned along the [1 0 0] direction) in distances of 2.9 nm are present. The phases ε6 and ε28 are frequently denoted in the literature as ξ’ and Ψ, respectively.

Figure I.4: Schematic representation of the phases ε6 (a) and ε28 (b) viewed along [ 0 10 ] . The grey rectangles indicate the unit cells of the respective phases. A single phason line is marked dark grey in (b). (Feuerbacher and Heggen, 2006)

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I.2 The β-Al-Mg phase I.2.1 The β-phase in the Al-Mg system Al-Mg alloys are among the most familiar materials and have been extensively investigated starting from the beginning of the XX century. One of the first equilibrium phase diagram of Al-Mg alloys has been given by Hanson et al. already in 1920. The existence of the β-phase in this system was established by Riederer in 1936. Structural investigations of this phase were firstly performed by Perlitz in 1944, who determined its approximate composition as Al3Mg2. Further investigations carried out throughout the last several decades had led to the confirmation of the previous diagram and introduction of some new data. In the present work the assessments performed by Murray in 1988 (figure I.5) with some corrections made on a basis of experiments described elsewhere (Lipińska, 2005 and Lipińska-Chwałek et al., 2007) will be used.

Figure I.5: Al-Mg Phase Diagram assessed by J.L. Murray, 1988.

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I Complex Metallic Alloys According to that work, β-Al-Mg is a stable phase from its congruent melting point at 451 °C and 38.5 at.% Mg down to the low temperature. The stability range of the β phase extends from 38.5 up to 40.3 at.% Mg. On the Al-rich side, the β phase coexists with the face centred cubic α-phase, which is a solid solution of Mg in Al. The (β+α) eutectic formation takes place in Al-38.2%Mg alloy in 450°C, which is almost consistent with the congruent melting point of the β phase. Part of the Al-rich boundary of the single-phase β region has been drawn in figure I.5 as a dashed line because of evidence (Schwellinger et al., 1979, Timm and Warlimont, 1980) that the β phase of composition 37.5 at.% Mg undergoes a structural transformation at approximately 240°C. In very recent investigations performed on the Al-Mg phase diagram in the vicinity of the stability range of β-Al-Mg by Feuerbacher et al. (2007), the occurrence of this transition was confirmed and assigned to the formation of the lowtemperature phase, β’-Al-Mg. The temperature of this transformation was found, however, to decrease strongly with the increasing Mg content of the alloy. The crystal structure of β’-AlMg was identified as rhombohedral (space group R3m) with lattice parameters a = 1.9968 nm and c = 4.89114 nm. According to these results, a metastable high-temperature β phase is kinetically stabilized at temperatures below this phase transformation (Feuerbacher et al., 2007). On the Mg-rich side, the β phase coexists with γ-Al12Mg17, claimed to be isostructural with α-Mn, and with the line compound R that forms below 350 °C in a peritectoid reaction between β and γ, and transforms back to β + γ below approximately 320 °C. In the initial experiments (Lipińska-Chwałek et al., 2007 and Lipińska, 2005), performed to probe the Al-Mg phase diagram in the vicinity of the β phase, the phase diagram of Murray (1988) was basically confirmed, except for three differences which have been detected. Signatures of a high-temperature transformation at about 430 °C and a low-temperature transformation in the range from 230 to 260 °C were observed. The homogeneity range of the β phase was identified to extend by 1.1% further to the Al-rich side.

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I.2.2 The crystal structure of the β-Al-Mg phase A complete structural model of the β-Al-Mg phase was developed by Samson in 1965 and therefore the material is frequently referred as “Samson phase”. According to his work, the β phase possesses the space group Fd 3 m and is based on a unit cell with an edge length of 2.82 nm. It comprises approximately 1168 atoms per unit cell arranged in a cluster substructure. Figure I.6 illustrates the model of the β-Al-Mg phase unit cell built according to the description given by Samson (1965). The phase β-Al-Mg features a structure with 23 crystallographic atom sites and 41 different coordination polyhedra (Samson, 1969). The coordination polyhedra comprise 672 icosahedra (ligancy 12), 252 Friauf polyhedra (ligancy 16), and 244 miscellaneous, more-or-less irregular polyhedra of ligancy 10–16.

Figure I.6: Unit cell of ß-Al-Mg according to the model of Samson (1965). The atomic radii of Al and Mg are rAl = 0.143 nm and rMg= 0.160 nm, respectively (Stöcker, 1994).

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I Complex Metallic Alloys The most important structural building blocks of the β-Al-Mg phase, are icosahedra and Friauf polyhedra. A wire model and perspective drawing of an icosahedron are shown in figures I.12 and I.14 (a), respectively, while the sphere and formal representations of the Friauf polyhedron are shown in figures I.7 (b) and (c). A Friauf polyhedron can be constructed on a basis of modifications performed on the cubic closest-packed arrangement of spheres of equal size. Removal of a tetrahedron of four contiguous spheres from the aggregate of sixteen spheres of such an arrangement results in the framework of 12 spheres located at the corners of a truncated tetrahedron shown in figure I.7 (a). In the β-phase structure, these atom sites are occupied by Al atoms (green spheres in figure I.7). The central cavity of the aggregate is occupied by a Mg atom (shown as red spheres in figure I.7). Other four Mg atoms arranged into a regular tetrahedron are located out of the centres of the four hexagonal faces of the truncated tetrahedron. Such a group of 17 atoms is called a Friauf polyhedron.

Figure I.7: Sphere models of: (a) the truncated tetrahedron, (b) the Friauf polyhedron and (d) the VF-polyhedron. consisting of 5 Friauf polyhedra (47 atoms). Formal representations of (c) the Friauf polyhedron and (e) the VF-polyhedron.

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I Complex Metallic Alloys The formal representation of the Friauf polyhedron shown in figure I.7 (c) does not indicate atoms located out of the centres of hexagons. This representation is accepted for the sake of better legibility in more complex aggregates in which each of these atoms is shared between two adjacent Friauf polyhedrons (as in the case of polyhedron shown in figure I.7 (e)) or between a Friauf polyhedron and other coordination shells. Five Friauf polyhedra arranged about an approximate fivefold rotational axis and sharing hexagonal faces form in the β phase an aggregate of 47 atoms, called VF-polyhedron by Samson (1965). A sphere model and a formal representation of the VF-polyhedron are shown in figures I.7 (d) and (e), respectively. Further stacking of VF-polyhedra (more details are given in Samson, 1969) leads to a large three-dimensional network of the cubic β-phase structure described by the idealized ordered model (1192 atoms per unit cell). However, a particular feature of the real β-phase structure is the inherent presence of disorder. This arises, in particular, from incompatibilities in the packing of the Friauf polyhedra, where certain vertices of adjacent polyhedra should be occupied by a large atom for one polyhedron and by a small atom for the other. It results in displacement disorder, substitutional disorder and fractional site occupation. Due to steric constraints 11 of the 23 different crystallographic sites have fractional occupations between 10 and 80%. To account for the disorder present, the idealized model of the β-phase structure requires some modifications of certain Friauf polyhedra into 15-atom complexes (described in details by Samson, 1969). This leads to the reduction of the total number of atoms from 1192 in the idealized, ordered model, down to 1168 atoms of the disordered model of the β-phase unit cell. Samson (1965) assumed that inherent disorder in β-Al-Mg is a result of the tendency to form the maximum number of icosahedral coordination shells. Accordingly, introduction of the disorder into the idealized model resulted in a gain of 48 icosahedra per unit of the structure.

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I.3 The κ-Al-Mn-Ni phase I.3.1 The к-phase in the Al-Mn-Ni system The existence of the hexagonal κ-phase in the Al-Mn-Ni alloy system (see figure I.8), was recently established by Balanetskyy et al. (2008). According to that work, κ-AlMnNi is formed incongruently at about 867 °C by the peritectic reaction

L+ φ + µ ↔ κ. The initial composition at this point is estimated as about Al80.0Mn18.5Ni1.5. Figure I.8 shows the projected compositional ranges of the equilibrium phases between 950 – 750 °C in the ternary Al-Mn-Ni alloy system. On this scheme, reprinted after Balanetskyy et al. (2008), the peritectic composition of the κ-phase is marked by a star.

Figure I.8: Scheme of the projected compositional ranges of the Al-rich Al-Mn-Ni phases between 950 and 750 °C. The peritectic composition of the κ-phase is marked by a star. TM stands for transition metals, X for vacancies in the Al- or TM sublattice. The stoichiometry of related binary intermetallics is shown by dotted lines. 17

I Complex Metallic Alloys According to the Balanetskyy et al. (2008) work, the extension and location of the κ-phase homogeneity region in the alloy system is strongly dependent on temperature. The κ-phase has practically the same Al content as µ-Al4Mn (i.e. about 80 at.%), however some Mn is replaced by Ni (figure I.8). An increase of the Ni-content in the κ-phase is accompanied by a small decrease of Al-content. Simultaneously, the melting temperature of the κ-phase decreases as well with increasing Ni content. At 850 °C, the κ -phase homogeneity region is relatively small, located along the line between the compositions Al80.0Mn18.8Ni1.2 and Al79.8Mn17.2Ni3.0. Further decrease of temperature from 850 to 750 °C is accompanied with an increase of the extension of the homogeneity region and a shift to a more Ni-rich composition. At 750 °C it is located along the line from Al80.0Mn17.7Ni2.3 to Al79.8Mn16.6Ni3.6. In spite of the very close composition range and close structural relationship between the κ-, and µ -phase, they are according to Balanetskyy et al. (2008) easily distinguishable in scanning electron microscope (SEM). Their diffraction patterns (figures I.9 and I.10) are also distinctly different. Patterns of selected area electron diffraction (SAED) and powder X-ray diffraction (PXRD) of the κ- and µ-phases from the Al-Mn-Ni system are shown (after Balanetskyy et al., 2008) in figures I.9 and I.10, respectively.

Figure I.9: Electron diffraction patterns of the κ-Al-Mn-Ni phase along [ 0 0 01 ] ( a ) , [ 211 0 ] ( b ) , and [ 110 0 ] ( c ) directions and of the µ-phase along [ 0 0 01 ] ( d ) , [ 211 0 ] ( e ) , and [ 11 0 0 ] ( f ) directions, respectively. (Balanetskyy et al., 2008)

18


Figure I.10: X-ray diffraction patterns (MoKα) of the κ-phase of composition Al80.3Mn17.5Ni2.2 (a), and of µ-Al4Mn phase of Al79.8Mn20.2 (b). Reflections labelled by a star belong to Si that was used as internal standard. (Balanetskyy et al., 2008)

19


I.3.2 The crystal structure of the к-phase According Balanetskyy et al. (2008), the κ-AlMnNi phase posses a hexagonal unit cell of space group P63 / m with lattice parameters of a = 1.7625(10) and c = 1.2516(10) nm (at the Al80.3Mn17.5Ni2.2 composition) and 227 atoms per unit cell. It is isostructural with the κ-phases in the Al-Cr-Ni (Li et al., 1997, Sato et al., 1997 and Marsh, 1998) and Al-Cr-Cu (Sugiyama et al., 2002 and Grushko et al., 2006) alloy systems (see Li et al., 1997, Kreiner and Franzen, 1995 and references therein for more details) and structurally closely related with the binary µ- and λ- phases of the Al-Mn system (Li et al., 1997, Sugiyama et al., 2002, Kreiner and Franzen, 1995, Uchida and Matsui, 2001 and Ranganathan et al. 2002). Since no structure refinement was performed yet for the κ-phase in Al-Mn-Ni system, the following description is based on a structure model given by Li et al., and Sato et al. in 1997 for κ-AlCrNi phase in which the Cr atoms are replaced by atoms of Mn. The structure of the κ-phase is based on icosahedral clusters grouped in more complex agglomerates, so-called I3-clusters and which is a building block typical for a great number of different intermetallic structures (Kreiner and Franzen, 1995). On the basis of such clusters one can construct complex “thick” PFP layers (where P stands for puckered and F for flat atomic layers), which are present in all mentioned above structurally related binary and ternary hexagonal phases. Figure I.11 shows a schematic representation of the κ-phase unit cell along the [211 0] direction. The structure consists of six layers stacking along the c-axis, two flat layers (F and f) and four puckered (P, P’, p and p’) arranged in a sequence PFP’pfp’. Layers above c = 0.5 (pfp’) can be generated by the operation of a twofold-screw symmetry (21 screw axis) along the c direction on layers below c = 0.5 (PFP’). The layer at c = 0.5 corresponds therefore to a pseudo-mirror plane with respect to its adjacent layers.

20


Figure I.11: Projection of the κ-AlMnNi phase unit cell along [ 2 1 1 0 ] direction ( rAl = 0.143 nm , rMn = 0.130 nm, rNi = 0.121 nm (Stöcke, 1994)). The unit cell features a structure of 2 flat and 4 puckered layers stacking along the c − axis . The model is based on a structure descriptions given by Li et al., and Sato et al. in 1997 for κ-AlCrNi phase in which the Cr atoms are replaced by atoms of Mn. The basic building blocks of the κ-phase structure are icosahedral clusters (Li et al., 1997 and Sato et al., 1997). Figure I.12 shows an icosahedron with a view direction along the twofold (a) and threefold axes (b). The corresponding schematic illustrations of the icosahedron in consecutive flat and puckered layers, formed by omitting bonds between the layers is shown in figure I.12 (c) (along twofold axis) and I.12 (d) (along threefold axis) respectively. The latter kind of representation is used subsequently in figure I.13 (b) for a description of the icosahedrons arrangements in consecutive layers. The atom positions in each layer are connected in solid lines forming nets, while the adjacent layers are indicated by the nets shown as broken lines.

21


Figure I.12: Icosahedral cluster in the κ-phase: wire models with a view direction along the twofold (a) and threefold (b) axes; schematic illustrations of the icosahedron in consecutive flat and puckered layers along the twofold (c) and threefold (d) axes, respectively.

Three separated icosahedra are formed in the κ-phase structure in the consecutive layers p’PF and another three in the layers FP’p. The two groups of the icosahedra are shown in figure I.13 (a). These are joined by sharing the triangular faces in the F-layer (marked in red in figure I.13 (b)). Another set of three icosahedral clusters, joined by sharing vertices and two face-to-face connected octahedra in the centre, is formed by atoms in the PFP’ layers. The icosahedrons of the latter aggregation, the so-called I3 cluster (Kreiner and Franzen, 1995) interpenetrate with the adjacent icosahedra of two previously described sets (figure I.13 (b)).

22


Figure I.13: Large atom clusters formed by icosahedra in the κ-phase structure: two groups of the icosahedra formed in the p’PF and FP’p layers (joined by sharing the triangular faces marked in red on (b)); three icosahedra sharing vertices form the I3 cluster in the PFP’ layer (c); p’PFP’p stacking sequence of a consecutive flat and puckered atom-layers forming clusters shown in (a) and (c). On (b) the atom positions in each layer are connected in solid lines forming nets, while the adjacent layers are indicated by the nets shown as broken lines. The group of atoms marked in blue corresponds to the perspective drawing in figure I.14 (b).

23

I Complex Metallic Alloys A transparent projection of the I3 cluster is shown schematically in figure I.13 (c), while a perspective drawing of the coordination icosahedra and three such clusters interpenetrating each other in the κ-phase structure along the c axis are shown in figure I.14 (a) and (b), respectively. A group of atoms shown in figure I.14 (b) is marked in blue in figure I.13 (b).

Figure I.14: A perspective drawing of a single icosahedral cluster (a) and of the three icosahedrons interpenetrating each other along the c axis (b). For clearness, two neighboring icosahedrons are shown in black, while the third one, interpenetrating two other is indicated in green on (b).

A whole aggregation of the nine interpenetrating icosahedra in p’PFP’p consecutive layers is referred to by Li et al. (1997) as ico-9 cluster. Another equivalent ico-9 cluster exists in the P’pfp’P layers and is related to the previous one by a 21 screw axis in the centre of the unit cell. Thus the ico-9 clusters, connected by two consecutive octahedral, propagates periodically along the c axis in the κ-phase structure.

24


I.4 Structure defects in CMAs Despite of the rather frequent occurrence of CMAs in binary and particularly in ternary and higher metallic alloy systems, the physical properties of these materials are largely unexplored. Although a large effort was spent in the recent decades on the crystallographic characterization of this class of materials, dedicated investigations focusing on their physical properties were carried out only in recently (e.g. Takeuchi and Mizutani, 1995, Klein et al., 2000, Feuerbacher et al., 2001, Belin-Ferré, 2002, Heggen et al., 2007, Smontara et al., 2007, Dolinšek et al., 2007, Jeglič et al., 2007 and Bauer et al. 2007). Among the results obtained, the plasticity of CMAs turned out to offer interesting properties and mechanisms. Due to the large lattice parameters encountered in these materials, the concepts that are used to describe the plastic deformation of simple crystalline materials are prone to failure (Feuerbacher et al., 2004). In the latter materials perfect dislocations are frequently observed to be carriers of crystal plasticity. The motion of these dislocations leaves the crystal structure entirely undistorted since the Burgers vectors correspond to translational invariant vectors of the lattice. Since the elastic line energy of a dislocation is proportional to the square of its Burgersvector length (Hirth and Lothe, 1992), the introduction of a perfect dislocation into a CMAs, featuring lattice constants of the order of several nanometers, would lead to very high elastic line energy, which might exceed physically reasonable values. Therefore, splitting of the perfect dislocations into partials, featuring Burgers vectors of only a fraction of a translational invariant lattice distance, becomes more favourable. In that case, however, dislocation motion is necessarily accompanied by the introduction of planar defects as it is observed for the β-AlMg and κ-Al-Mn-Ni phases in the present work (chapters VI and VII). Novel mechanisms to accommodate dislocations in large unit cell structures were found in some CMAs (Feuerbacher et al., 2004). For example in the C2 - Al - Pd - Mn phase (space group Fm3 , a = 1.552 nm, 248 atoms/unit cell) (Edler et al., 1998 and Sugiyama et al., 2000) a deformation mechanism involving splitting of a perfect [ 0 0 1] dislocation into two partials, combined with a local transformation to a structure with smaller lattice parameter, was observed (Feuerbacher et al., 2004). Figure I.15 shows a high-resolution TEM image of an example defect arrangement observed in the C2 - Al - Pd - Mn phase. The viewing direction

of the micrograph corresponds to the [110] crystallographic direction of the C2-phase. The defect arrangement observed, consists of a perfect [ 0 0 1] dislocation with a Burgers vector length of 1.55 nm. In order to lower the elastic line energy, it is split into two 1/2 [ 0 0 1] 25

I Complex Metallic Alloys partials with Burgers vector lengths of 0.78 nm. As a result, a stacking fault is formed between the partials. It is seen in figure I.15 in edge-on orientation. The Burgers vectors of the partials, however, still possess rather large lengths. To accommodate it in the structure a small portion of the material located in the compressive part of the partial dislocation strain field transforms to a body-centred structure possessing a slightly smaller lattice constant. The [110] crystallographic directions of the matrix and of the new structure are parallel to each other. A total arrangement appears in figure I.15 as a dumbbell-shaped object with two almost rectangular extremities. The terminating partial dislocations are marked by white arrows. These are located at the upper ends of the rectangular brighter-contrast areas consisting of the structure with the smaller lattice parameter. By this means, the dislocation with the large Burgers vector can be more easily accommodated into the face-centred structure of the

C 2 - Al - Pd - Mn phase.

Figure I.15: HRTEM image along the [ 1 1 0 ] direction of a defect arrangement in the C2-Al-Pd-Fe phase. The box marks the area of the material used for the identification of the transformed structure. (Feuerbacher et al., 2004)

26

I Complex Metallic Alloys The most exotic mechanism of plastic deformation of CMAs was observed in the ε 6 - Al - Pd - Mn phase. It involves a new type of structural defect called “metadislocation”

(Klein et al., 1999). Figure I.16 (a) shows a high-resolution TEM image of such a defect (in end-on orientation) taken along the [ 0 10] crystallographic direction (Feuerbacher et al., 2004). The dislocation core is situated at the position indicated by an arrow. Figure I.16 (b) shows a tiling representation of the situation from (a). The dislocation core is shown as a dark-grey polygon differing from the tiles used for the representation of the surrounding area (cf. chapter I.1). On the left of the core an area of the ε 6 - Al - Pd - Mn structure can be identified via the tiling of flattened hexagons. In the figure exterior the ε 28 - Al - Pd - Mn phase containing additional phason planes is present. The dislocation core is embedded into the structure by six “inserted” phason halfplanes.

Figure I.16: HRTEM image of a metadislocation in ε 6 - Al- Pd- Mn (a) and its schematic representation (b) along the [0 1 0] direction (Feuerbacher et al., 2004).

The Burgers vector of this metadislocation, analysed by means of contrast-extinction experiments (based on the “invisibility” criterion), is parallel to the [ 0 0 1] direction (pure edge dislocation) (Klein et al., 1999, Klein and Feuerbacher, 2003). Although, the dimensions of the area affected by the metadislocation obviously exceed the dimensions defined by the lattice constants of the surrounding structures (corresponding unit cells are marked on the figure I.4), the Burgers-vector length of a dislocation, on which the whole arrangement is based, corresponds to only a fraction of the lattice parameter. A Burgers-circuit analysis yields a Burgers-vector length of 0.183 nm. Despite the fact that the dislocation is a partial dislocation, the metadislocation as a whole (including the six attached phason halfplanes) can 27

I Complex Metallic Alloys glide in the ε28-Al-Pd-Mn phase without leaving behind the additional planar fault into the structure (Klein et al., 1999). Various [ 0 0 1] metadislocations occur in both ε6- and the ε28-phase. Their Burgers vectors differ by the moduli and a number of phason halfplanes associated with the respective partials. Up to date, defects featuring 2, 4, 6, 10, and 16 inserted phason halfplanes were observed, featuring Burgers vectors moduli of 0.480, 0.296, 0.183, 0.113, and 0.07 nm, respectively (Klein et al., 2000 and Heggen and Feuerbacher, 2006). The appearance of metadislocations in CMAs is an example of the interplay between the two different length scales defined by the cluster substructure on the one hand and the unit-cell dimension on the other hand. The partial-dislocation Burgers vector reflects the former length scale while the phason defects care about the constraints arising from the long-range lattice periodicity. Therefore, the plastic deformation properties of CMAs impressively demonstrate that the study of these materials can reveal interesting novel physical properties.

28

II Single-crystal growth In this chapter, an outline of the fundamentals of single-crystal growth, as well as the applied techniques, i.e. Bridgman and Czochralski and self-flux growth, are briefly discussed.

II.1 Fundamentals of single-crystal growth Successful development of the single-crystal growth procedures of two- or more-component phases requires thorough knowledge and understanding of the solidification behaviour of the material of interest. The phase diagram, which is a multidimensional map of the equilibrium states of an alloy system as a function of the composition and other outer conditions (temperature and pressure), is indispensable tool to tackle the problem. Figure II.1 shows a scheme of a fictitious binary phase diagram of the alloy system of the components A and B. On the vertical axis representing the temperature, the melting points of the pure components A and B are labelled as TA and TB, respectively.

Figure II.1: Scheme of a binary phase diagram. Incongruent solidification behaviour is illustrated: By cooling a melt with composition c3, growth of the solid phase β with composition c1 is initiated at T3. One discerns congruent and incongruent solidification behaviour. Congruent solidification takes place if the composition of the solidifying phase equals the composition of the melt. In the fictitious phase diagram shown in figure II.1 congruent solidification takes place if a melt of pure element A or B is cooled down below the melting temperature TA or TB, respectively. 29

II Single-crystal growth An example of a real phase that solidifies congruently is the β-Al-Mg (figure I.5). Incongruent solidification takes place in the case if the composition of the solidifying phase differs from that of the initial liquid. Lowering the temperature of this melt, as we would do in a solidification experiment, we force the system to follow the first encountered liquidus line downwards, i.e. towards lower temperatures. This means, the melt composition will vary along with the solidification process advancement. Simultaneously, the chemical composition of the phase solidifying from such a melt will change as well. An example of the incongruent solidification can be described on a basis of a melt of the chemical composition c1 in figure II.1. Despite the fact that this composition is located within the chemical stability range of the incongruently solidifying β phase, the latter cannot solidify from such a melt. Upon the temperature drop a melt with composition c1 first meets the liquidus line of the L+α field, i.e. it enters the area of the primary solidification of the α phase. In order to grow the β phase of final composition c1, a melt with the B component content higher then than c2 and lower than c4 has to be used. Within this compositional range the β phase is the primarily solidifying phase. By cooling a melt with an initial composition c3 (c2 < c3 < c4) as shown in figure II.1, the β phase of the composition c1 solidifies. Since the crystallizing phase contains a dominant quantity of the component A, the concentration of this element in the melt will decrease. The melt composition shifts, therefore, to higher concentrations of element B along the liquidus line of the L+β region. Simultaneously, the composition of the β phase shifts during solidification from c1 to higher concentration of element B along the solidus line, i.e. the phase boundary between the stability range of the β phase and the L+β phase region. The growth of pure β phase is possible until the melt composition reaches the eutectic point E. At this place, the solidifying β phase reaches composition c1+X, i.e. the limit of the stability range of β. Further temperature decrease would initiate the eutectic solidification of the β and γ phase. An example of the real incongruently solidifying complex metallic alloy is the κ phase from the Al-Mn-Ni system. The phase diagram of the three-component system (i.e. ternary alloy) is a three-dimensional object. In such a representation, the compositions of the alloy constituents are indicated on an equilateral triangle. Every corner of the compositional triangle represents 100% of each element. The temperature is indicated on the axis perpendicular to the plane of this triangle. Two-dimensional representations of the ternary system can be given in a form of cuts perpendicular (isothermal sections) or parallel (isopleths or vertical section) to the temperature axis. The interpretation of the ternary-alloy phase diagrams is more difficult than in the case of binary alloys, because of the additional degree of freedom arising from the presence of the third constituent. However, the general rules that apply are analogous. Detailed description one can find e.g. in Prince (1966). 30

II Single-crystal growth

II.2 Single-crystal growth techniques The crystal growth techniques described in this section involve a solidification process directly from a melt. The basic prerequisite enabling the application of this kind of growth to a given phase is that there must be a primary solidification area of the desired phase. In the case of complex metallic alloys the growth of large single crystals (exceeding 1 cm3) is a challenging task because of narrow stability ranges, other competing phases and/or incongruent solidification behaviour of these phases. From the variety of techniques available Bridgman, Czochralski and self-flux growth techniques are described below. These have already proven to be successfully applicable to the single-crystal growth of some complex metallic alloys (Feuerbacher et al., 2003) and, therefore, chosen for the grow attempts of the β - Al - Mg and κ - Al - Mn - Ni phases.

II.2.1 Bridgman technique A particularly useful approach to single-crystal grow is that commonly known as the Bridgman method. It was first used by Bridgman in 1925 (Laudise, 1970). Figure II.2 shows an example scheme of the setup for crystal growth according to the Bridgman technique. The melt of appropriate composition is kept in a vertical crucible, placed under an alumina envelope in order to protect the melt from contamination. The crucible is surrounded by a heater which generates an isothermal hot zone (dashed line in figure II.2). Solidification of the melt takes place starting at the lowermost part of the crucible. Crystallization process is initiated by slow movement of the melt downwards out of the hot zone, with the bottom of the crucible ahead. In figure II.2 the lower half of the crucible has left the hot zone and correspondingly, the melt in this part is solidified. Nucleation of as small as possible number of grains is attained by geometrically reducing the volume in which nucleation events take place. This can be realized by choosing a conical crucible, tapered into a sharp tip and assurance that nucleation starts in its narrowest part. The latter can be with an application of cold finger (water cooled rod) being in thermal contact with the top of crucible and providing a small defined temperature gradient.

31


Figure II.2: Schematic representation of the setup for single-crystal growth according to the Bridgman technique. Solidification starts at the tip of the crucible, when the crucible is slowly pulled downwards out of the hot zone of the heater. The dashed line represents the isotherm corresponding to the current temperature within the hot zone. The standard velocities used in this technique are of the order of 1 to 10 mm/h. The diameter of the obtained crystal depends on the diameter of the crucible used. A maximum volume for crystals of incongruently solidifying phases is limited by the size of the temperature window of the primary solidification area according to the phase diagram.

32


II.2.2 Czochralski technique Another well known method of single-crystal growth is the Czochralski technique. The solidification process is initiated in this approach at a seed crystal dipped into a melt. One discerns two kinds of seeding, namely homogeneous and heterogeneous. The former is of particular importance, since it allows for a deliberately oriented growth. If one uses an oriented single crystal as a seed, the newly solidified material preferentially continues the predetermined orientation of the seed (Feuerbacher et al., 2003). Figure II.3 schematically illustrates the set-up for the crystal growth according to the Czochralski technique.

Figure II.3: Schematic representation of the setup for single-crystal growth according to the Czochralski technique. The solidification process is initiated at a seed crystal dipped into a melt.

The melt of suitable composition is contained in an alumina crucible. The melt temperature is maintained at a value just above the solidification point. The seed-crystal is dipped carefully into the melt by lowering the pulling rod to which it is connected. After wetting, the seed is slowly lifted again in order to achieve a stable meniscus at the solid-liquid interface. Solidification of the melt occurs at this location due to heat dissipation mainly into the pulling rod. Crystal growth is performed by the continuous lifting of the pulling rod. A thermocouple is dipped into the melt for precise control of the process temperature. In order to increase the probability that the crystal grows in the form of one single grain, a ”thin neck”, i.e. a region in which the crystal has a diameter of less than 1 mm, is pulled in an 33

II Single-crystal growth early stage of the growth. This geometrical narrowing ensures that only one grain is selected for the further growth. The pulling rod and the crucible counter-rotate during the growth process in order to ensure a good homogenization of the melt. The diameter of the meniscus, where the solidification takes place, can be controlled by the melt temperature and/or the pulling speed. The principal rule is that an increase of the melt temperature (and/or pulling speed) leads to a decrease of crystal diameter and vice versa. The speeds of pulling in the Czochralski technique are usually chosen of the order of 1 to 10 mm/h. The second parameter that determines diameter of the crystal grown, i.e. the melt temperature, is restricted by the lower temperature limit of the primary solidification window. Nevertheless, due to the solidification mechanism employed in this technique, the resulting crystals are free of physical constraints imposed by the crucible (Laudise, 1970).

II.2.3 Self-flux growth technique The technically simplest single-crystal growth procedure is the so called self-flux growth technique. It is based on a slow cooling process of a melt of appropriate composition according to a well defined temperature program in a suitable crucible. The setup used for the crystal growth by means of this technique is schematically illustrated in figure II.4. The vertical tip-shaped crucible containing the melt is closed with a second, inverted crucible, placed on the top of the previous one. The latter is used for trapping the rest of the melt upon decanting. To avoid oxidation and evaporation during the growth process in which the melt is subjected to high temperatures for a very long-time, the crucibles are sealed in a protecting quartz tube filled with Argon (figure II.4). A total setup is placed in a furnace with a low temperature gradient and a high-precision temperature regulation.

34


Figure II.4: (a) Schematic representation of the setup for single-crystal growth according to the self-flux growth technique. Crystals grow freely into the isothermal melt. (b) After decantation – the crystal grown is separated from the residual melt.

The crystal growth is preceded by the melt homogenization. This step is carried out at a temperature well above the melting temperature of the alloy. Subsequently, a slow cooling process follows, during which the alloy is almost isothermally kept close to equilibrium. Very small cooling speeds between the liquidus and solidus temperature are necessary to keep the nucleation rate during the growth process as small as possible. Further reduction of the number of nucleation events can be achieved by reducing the volume in which these events take place. This is realized via application of the tip-shaped conical crucible in combination with a cold finger (water cooled platinum rod) positioned directly under the narrowest crucible part (tip). The crystal growth is terminated by a decantation process. The latter is performed by centrifugation or simply turning the quartz tube upside-down before reaching the solidification temperature of the next phase. The rest-melt is trapped in the second crucible, where it solidifies (figure II.4 (b)). Despite of the relative simplicity of the technique itself, the procedure described can lead to satisfying results if the starting composition of the melt and the temperature program are chosen in good accordance with the phase diagram of the alloy system. The crystals grown are of a very high structural quality, since they can grow freely into the isothermal melt. Their size however is limited by the crucible size, which itself is limited by the volume of homogeneous temperature field in the furnace. 35

III Crystal plasticity and plastic deformation In the following chapter an introduction to the fundamentals of crystal plasticity is given. The basic principles from the theory of thermally activated motion of dislocations are introduced and experimental routines for the determination of thermodynamic activation parameters are described. Finally, the procedures of deformation experiments performed in the frame of the present work are outlined.

III.1 Fundamentals of crystal plasticity Deformation experiments aim to determine the behaviour of the given material subjected to the external forces under specified environmental conditions. A stress σ acting on the sample in the compression∗ direction can be calculated according to σ =

F , where A stands for the A

sectional area of the material sample to which the external force F is applied. The strain, being the geometrical measure of deformation, is defined as ε =

∆l . ∆l and l0 are the length l0

variation and the initial length of the sample, respectively. The total strain consists of an elastic and a plastic part

ε = ε plast + ε el .

(III.1)

At the very beginning of deformation process (small strain values) the material straining features purely elastic character. That means, the material is able to return to its original state (size and/or shape) when the load is removed. Within this narrow deformation range stress and strain are proportional to each other. This can be described by Hooke’s law σ = E ⋅ ε el , where E is the material-specific Young’s modulus. The proportionality between stress and strain vanishes when the deformation reaches the transition between the elastic and plastic regime. A larger applied force may lead to a permanent change in the shape of the sample or even to its structural failure.

∗

In the simplified uniaxial deformation experiments one discerns tension or compression loading. The deformation experiments performed within the present work have been carried out in compression. However, the considerations described hold for tensile tests as well.

37

III Crystal plasticity and plastic deformation In many materials plastic deformation takes place by dislocation slip, where net planes of the material are displaced. Dislocations move along a slip plane, which usually corresponds to a low-index crystallographic plane. The geometry of the slip system (i.e. the dislocation slip plane and direction) determines the Schmid factor mS m S = cos λ cos Φ ,

(III.2)

where Φ and λ correspond to the angles between compression direction and slip-plane normal

n and between compression direction and slip direction, respectively. These angles are illustrated in a schematic deformation sample in figure III.1. The possible Schmid-factor values can range between 0 and 0.5.

Figure III.1: Schematic representation of a slip system and its orientation with respect to the compression direction. The Schmid factor is calculated from the angles between compression direction and slip-plane normal Φ and compression direction and slip direction λ, respectively. The motion of dislocation possessing a Burgers vector modulus b over a distance x contributes to the straining of a material. The corresponding macroscopic deformation of a sample is given by

ε plast = ms ρbx ,

(III.3)

where ρ is the dislocation density. The time derivative of (III.3) yields the Orowan equation (Evans and Rawlings, 1969)

ε& plast = m s ρ bv ,

(III.4)

where ε& plast is the plastic strain rate and v is the dislocation velocity. An important assumption for the validity of the Orowan equation is that the dislocation density is constant ( ρ& = 0 ). 38

III Crystal plasticity and plastic deformation r A driving force K of dislocations movement generated due to the stress σ applied at a r deformation sample acts perpendicular to the dislocation line direction l . This so called

Peach-Köhler force can be calculated according to r r r r K = (σ ⋅ b )× l .

(III.5)

The relation between the stress applied at the deformation sample σ and the resulting shear stress τ in the given slip system, depends on the relative orientation of the slip system and the deformation direction, which is taken into account via the Schmid factor

τ = mSσ .

(III.6)

Therefore, according to the equation (III.5) the driving force K of dislocations moving within the given slip system amounts to K =τ ⋅b⋅l .

(III.7)

Oppositional to the shear stress τ the internal stresses τi counteract the movement of a dislocation. These stresses originate from long-range stress fields of microstructural obstacles like point defects, precipitates or other dislocations. The effective stress τeff acting on a dislocation is therefore the shear stress τ reduced by τi (Seeger, 1958)

τ eff = τ − τ i ,

(III.8)

K = τ eff ⋅ b ⋅ l .

(III.9)

and hence

The total energy barrier corresponding to the encountered obstacle, which is to be overcome by the moving dislocation, is given by the Helmholtz free energy ∆F = ∆G + ∆W (figure III.2). ∆W denotes the work-term which corresponds to the part of the dislocation energy supplied by the effective stress τeff x2

∆W = ∫ τ eff lbdx .

(III.10)

x1

Integration of equation (III.10) with τeff(x) = const yields ∆W = τ eff lb∆x ,

(III.11)

where ∆x = x2-x1 is the distance moved by the dislocation to overcome the obstacle. If the effective stress τeff is larger than the friction stress τf at energetic obstacles, the dislocation will continuously move and overcome the obstacles. However, if the effective stress is smaller than that of the friction process, the dislocation cannot overcome the obstacle and remains halted in front of it. Figure III.2 illustrates schematically a situation of a dislocation (shown in red) blocked at position x1. 39

III Crystal plasticity and plastic deformation

Figure III.2: Schematic illustration of thermal activation. The dislocation (red) remains at the stable position x1 if the effective stress τeff is smaller than the friction stress τf of the obstacle. The energy barrier ∆G can be overcome by means of thermal fluctuations. Due to thermal fluctuations appearing at temperatures T > 0 K, there exists a non-zero probability to overcome the obstacle. The probability is given by (Vineyard, 1957)

P = exp

− ∆G , kT

(III.12)

where k is Boltzmann’s constant and T is the absolute temperature. The Gibbs free energy ∆G corresponds to the energy barrier which has to be thermally overcome and is given by (Gibbs, 1964) x2

∆G = ∫ (τ f − τ eff )lbdx ,

(III.13)

x1

where l is the length of the dislocation line at the obstacle. If the thermal activation is the rate-controlling process, the dislocation velocity v is v = v0 ∆xP ,

(III.14)

where v0 is the attempt frequency (Granato et al., 1964). In this case the probability of thermal activation (III.12) can be combined with the Orowan equation (III.4) (Schöck, 1965) as

ε& plast = ε&0 exp

− ∆G kT

(III.15)

with the pre-exponential factor ε&0 = ρb∆xv0 . If the energy barrier is overcome isothermally and at constant stress, the Gibbs free energy is a thermodynamic variable of state. The differential is d (∆G ) = −∆SdT − V * dτ eff with the definitions

40


∆S ≡ −

∂ (∆G ) ∂T τ eff

(III.16)

V* ≡ −

∂ (∆G ) . ∂τ eff

(III.17)

and

T

*

∆S is the activation entropy and V is called the activation volume. The latter can be calculated according to (Kocks et al., 1975) V * = lb∆x = b∆A

(III.18)

and can be interpreted as the material volume involved in thermally activated overcoming of an obstacle by a dislocation featuring the Burgers vector b. Figure III.3 illustrates the geometrical interpretation of the activation volume. The dislocation line is pinned at two points indicated by P and stays at a stable position in front of an obstacle (solid line). By means of thermal fluctuations the dislocation can overcome the obstacle (broken line) and pass the distance ∆x . ∆A corresponds to the area that is passed by a dislocation line during this event.

Figure III.3: Schematic description of the activation volume. The dislocation line is pinned at the positions labelled P. The solid and broken lines represent respectively the dislocation before and after overcoming the middle obstacle. The activation area ∆A is the area covered by the dislocation line during thermally-activated event. On the basis of equation III.11 the work term can be calculated from the activation volume, according to

∆W = τ eff V * .

(III.19)

41


III.2 Incremental tests The experimental determination of the thermodynamic activation parameters is performed by incremental tests during plastic deformation. They are calculated from the macroscopic parameters available from the deformation experiment, i.e. stress, strain, temperature, and time.

Stress-relaxation experiment The activation volume is a parameter of a great importance providing information about the nature of dislocation obstacles. It can be determined from the deformation experiment according to

V =

kT ∂ ln ε& plast mS ∂σ

(III.20) T

This equation contains the experimental parameters ε& plast , i.e. the strain rate of deformation and the applied stress σ (Evans and Rawlings, 1969). This experimental activation volume V is related to V* via V = V * (1 + ∂τ i ∂τ T ) (Evans and Rawlings, 1969, Hirth and Nix, 1969). V and V* are identical if τi is independent of τ. If dislocation motion is controlled by thermally activated overcoming of obstacles, the experimental activation volume approximately takes the same size as the obstacle volume (Krausz and Eyring, 1975). For plastic deformation controlled by diffusion processes, the experimental activation volume is expected to amount r3 r to about V ≈ b , where b is the average Burgers vector. In order to obtain the stress dependence of the strain rate, which according to equation III.20 is necessary for evaluation of the activation volume, one has to perform an additional stressrelaxation test in the course of the plastic deformation experiment. In the stress-relaxation experiment the deformation machine is suddenly stopped during plastic deformation. The control mode is switched from constant-strain-rate to constant-strain and the stress decrease is measured as a function of time. Since the total strain of the sample remains constant during the stress-relaxation test, the total strain rate is given by

ε& = ε& plast + ε&el = 0 and elastic strain transfers into plastic strain, i.e. ε&el = −ε& plast . Due to Hooke’s law, the elastic strain decreases proportionally to the stress ( ε el ∝ σ ) and hence, during the stress-relaxation experiment we have ε& plast ∝ −σ& . Taking into account the latter relation, the activation volume can be determined, since the equation (III.20) can be written as

V = 42

kT ∂ ln( −σ& ) mS ∂σ T

(III.21)

III Crystal plasticity and plastic deformation Another important parameter that can be calculated on a basis of stress-relaxation experiment is the stress exponent m (from the approach that ε& plast ∝ σ m ), frequently used to classify deformation processes (e.g. Poirier, 1985). According to Ilschner (1973):

m=

∂ ln ε& plast ∂ ln σ

.

(III.22)

T

Therefore, from the stress-relaxation test we get m=

∂ ln( −σ& ) . ∂ ln σ T

(III.23)

Temperature changes The Gibbs free energy of activation cannot be determined directly from a deformation experiment. However, the activation enthalpy ∆H which is connected with the Gibbs free energy via (III.24)

∆H = ∆G + T∆S can be determined according to (Evans and Rawlings, 1969):

∆H = −kT 2

∂ ln ε& plast ∂σ

T

∂σ ∂T

.

(III.25)

ε plast

It can be determined due to the combination of the stress-relaxation experiment described above with temperature cycling. The latter is used to obtain the temperature dependence of the stress ∆σ/∆T. In order to perform temperature-change testing during a dynamic-compression test, the deformation is interrupted and the sample is unloaded. Then the temperature is changed by amount ∆T. After an equilibration time, the sample is reloaded at a new temperature and deformed with the initial strain rate. The resulting stress difference ∆σ can be determined in the stress-strain curve.

43


III.3 Experimental testing procedure All deformation experiments performed in the frame of present work involve the same sequence of the incremental tests. Figure III.4 illustrates an example of a stress-strain curve. The stress-relaxation tests correspond to vertical indentations within the deformation curve labelled as “R”. The range of the deformation performed within the temperature cycling is labelled as “TC”. The upper-yield point (UYP) and the lower-yield point (LYP) of the stressstrain curve are indicated. After reaching the lower-yield point in the stress-strain behaviour, a stress-relaxation test is performed for 60 or 120 s. The subsequent unloading of the sample is followed by a temperature increase of 10 °C. After an equilibration time of about 1200 s, the sample is reloaded and deformed by further 0.5 to 1.5% of plastic strain. Subsequently a second stress-relaxation test is performed. It is followed by a further unloading and temperature change back to the initial value. A third stress-relaxation test is performed after 0.5 to 1.5% of plastic strain and followed by further dynamic compression at the initial temperature and strain rate. The dashed line in figure III.4 denotes the interpolated course of the deformation curve.

Figure III.4: An exemplary stress-strain curve of a deformation experiment. The sequence of performed stress-relaxation tests “R” and temperature changes “TC” is illustrated. The upper- and the lower- yield points are indicated with “UYP” and “LYP”, respectively. The dashed line denotes the interpolated course of the deformation curve.

44

IV Characterization of structure defects by TEM IV.1 Imaging in TEM A frequently applied technique for imaging lattice defects in TEM is to set up two-beam conditions. These are achieved by orientation of the TEM specimen in such a way that only the direct beam and one diffracted beam are excited, i.e. the incident beam is diffracted at only one set of lattice planes. Using now an objective aperture one can blank the transmitted or the diffracted beam, permitting only one to form the image. Correspondingly, the bright-field or dark-field imaging conditions are achieved. The strain fields of dislocations cause a local bending of the atomic planes with respect to the surrounding perfect crystal. If the Bragg condition for the incident electron wave is fulfilled in the strain field of the dislocation but not in the surrounding area, the intensity of the directly transmitted beam at the position of the dislocation is reduced (and that of the diffracted beam is increased). Under these conditions the dislocation line appears as a dark line on a brighter background in the bright-field image, or analogously, as a bright line on a darker surrounding in the dark-field image (Hull and Bacon, 1984). By means of the analysis performed under two-beam imaging conditions, a quantitative characterization of dislocation strain fields can be performed. The dislocation contrast is r extinct if the reciprocal lattice vector g applied for the image formation is perpendicular to r r r the dislocation strain field u , i.e. the “invisibility” criterion g ⋅ u = 0 is fulfilled (Williams and Carter, 1996). If this strain field can be described directly by means of the dislocation r Burgers vector b , the dislocation is invisible if r r (IV.1) g ⋅b = 0. This can be used to determine the direction of the Burgers vector. The latter is possible if two linearly independent excitation vectors fulfilling the invisibility criterion (IV.1) are found. However, for pure edge dislocations, or dislocations with a mixed edge and screw character, a residual contrast is frequently observed even if condition (IV.1) is fulfilled. A full extinction of these dislocations contrast is often achieved only if the condition r r r (IV.2) g ⋅b ×l = 0 r is simultaneously fulfilled, taking also the line direction l of the dislocation into account (Edington, 1975).

45

IV Characterization of lattice defects by TEM A dislocation moving through the crystal may introduce planar defects into the crystal matrix. A stacking fault is a planar defect which according to its name is a local region in the crystal where the regular sequence has been interrupted. A constant displacement between the r crystal parts on both sides of the fault plane is characterized by the displacement vector R . An electron wave passing through the stacking fault is subjected to a phase shift induced by the displacement. A corresponding phase factor r r α = 2π g ⋅ R

(IV.3)

is therefore added into the transmitted and diffracted beams and causes a contrast formation in TEM (Edington, 1975). When the fault is inclined to the specimen surface, the image consists of a series of alternate bright and dark fringes running parallel to the intersection of the fault plane and the foil surface. The appearance of the stacking fault is determined by the sign of the phase shift. Gevers (1972) has shown that under bright-field imaging conditions the outer fringes of a stacking fault are dark if sin(α )〉0 and bright if sin(α )〉0 . These conditions can be used for determination of the displacement vector modulus. If no phase shift is present, no contrast appears and accordingly the stacking fault is invisible. That means, an extinction of a stacking fault contrast takes place if the scalar product of excitation vector and displacement vector equals zero or an integer: r r g ⋅ R = 0,±1,±2,... ,

(IV.4)

Deformation twins may be as well introduced into the material structure by plastic deformation. These kinds of defect are formed when a region of a crystal undergoes homogeneous shear producing the original crystal structure in a new orientation. In the simplest cases, this results in the atoms of the original crystal (”parent”) and those of the product crystal (“twin”) being mirror images of each other by reflection in a composition plane called twin boundary (Hull and Bacon, 1984). A high density of planar defects with common crystallographic orientation may be reflected in a reciprocal space in a form of streaks running perpendicularly to the thin dimension of defects. These arise in diffraction patterns as a result of modifications to the shape of reciprocal lattice points introduced either by the shape of crystal defects or the lattice strain associated with them (Edington, 1975). This feature can be used in qualitative defects analysis to ascertain the habit plane of planar defects.

46

IV Characterization of lattice defects by TEM

IV.2 Defect-density determination We consider our material as a solid body of the volume V in which discontinuities are located. Geometrically the discontinuities are regarded as a set of a finite number of straight lines (dislocations) or regular surfaces (planar defects). The defect density is a measure for their number in a volume. For dislocations, their density is defined as the entire dislocation length L in a given material volume V (e.g. Underwood, 1970)

LV =

L V

(IV.5)

with the unit cm/cm3 = cm-2. For internal surfaces (e.g. stacking faults, twin boundaries etc.), the most useful stereological parameter describing their total density is surface-area-to-volume ratio AV , which is defined as the entire area of the internal surface A in a given material volume V (e.g. Underwood, 1970) AV =

A , V

(IV.6)

with the unit cm2/cm3 = cm-1. However, evaluation of the stereological quantities in three-dimensional space is mostly inaccessible for direct measurements. Hence, determination of these parameters is usually performed on the basis of the related two-dimensional quantities, defined on section planes or projections (testing planes) of the considered microstructure, e.g. on TEM micrographs. The intersections of the structural elements featuring linear character with the testing planes appear as points. Number of these intersection points N counted within the area A defines their density within the testing plane (e.g. Underwood, 1970)

NA =

N . A

(IV.7)

Correspondingly, intersection traces of the structural elements featuring surface character appear as lines. A total length of the intersection lines L counted within the area A defines the density of these line-traces within the testing plane (e.g. Underwood, 1970) LA =

L . A

(IV.8)

Described two- and three-dimensional quantities are proportional to each other. However, the proportionality factor is dependent on the geometric orientation of the investigated defects

47

IV Characterization of lattice defects by TEM with respect to the testing planes on which N A or LA are determined. Therefore, an accurate knowledge on these orientation relationships is required in order to determine the correct density.

Dislocations Following the reasoning proposed by Schoeck (1961), the density of dislocations featuring line direction orientations within the element of the solid angle dΩ = sin ϕ dϕ dθ around the direction ϕ ,θ

may be defined as p( ϕ ,θ )dΩ , where ϕ and θ are the zenith and the

azimuthal angle of a spherical coordinate system, respectively. As shown in figure IV.1 the zenith angle ϕ is measured from the z axis of corresponding Cartesian system with 0 ≤ ϕ ≤ π . The azimuthal angle θ is measured from the y axis around the z axis, within the

testing plane (corresponding to the xy -plane) with 0 ≤ ϕ ≤ 2π . An example set of linear defects (simplified to the straight lines) embedded in an arbitrary material volume V and the corresponding geometrical relations are schematically shown in figure IV.1.

Figure IV.1: A schematic representation of an exemplar set of dislocations (red lines) with orientation defined by angles ϕ and θ , embedded in an arbitrary material volume V .

The contribution of the set of dislocations featuring line direction orientations within the range dΩ to the density of all intersection points ( N A ) measured on the testing surface is 48

IV Characterization of lattice defects by TEM p( ϕ ,θ ) cos ϕ dΩ . Therefore, the relation between the total density of the intersection points measured on the testing surface depends on the orientations and corresponding partial densities of dislocations existing in the material reads π 2

2π

(IV.9)

N A = ∫ ∫ p( ϕ ,θ ) cos ϕ sin ϕ dϕ dθ . ϕ =0 θ =0

On the other hand, the contribution of this dislocation set to the total length of dislocations within the arbitrary volume V can be defined as

l( ϕ ,θ ) dΩ = ∫ ∫ p( ϕ ,θ ) dΩ h( x' , y' ) dx' dy' ,

(IV.10)

X 'Y '

where h(x' , y' ) is the length of dislocations inside the volume V and the integration extends over the projection of this volume on the area A(X'Y' ) defined by the X' and Y' axes. The projection area A(X' Y' ) is chosen in such a way that its normal is parallel to the line direction of the considered dislocation set, defined by particular

ϕ

and

θ

angles. Then

∫ ∫ h( x' , y' ) dx' dy' = V and the total length L of dislocations (featuring all possible line X 'Y '

direction orientations) within the volume V is expressed by an equation π/ 2 2 π

L = ∫ l(ϕ,θ ) dΩ = V ∫ p(ϕ,θ )dΩ = V ∫ ∫ p( ϕ ,θ ) sin ϕ dϕ dθ . Ω

(IV.11)

ϕ =0 θ =0

Ω

Therefore finally we obtain an expression describing the density of dislocations

LV =

L π / 2 2π = ∫ ∫ p( ϕ ,θ ) sin ϕ dϕ dθ . V ϕ =0 θ =0

(IV.12)

Taking into account equations (IV.9) and (IV.12) one can derive the characteristic equation describing the real (sought) dislocations density ( LV ) as a function of their intersection points density ( N A ) measured on the testing plane

LV = f ( N A ) .

(IV.13)

This relation can be defined for every dislocation arrangement existing in the material. Although the description given above consider the linear defects simplified to straight lines, it applies also for random networks or for closed dislocation loops, since any linear object can be approximated by set of straight elements. These elements can take an arbitrary positions and orientations when the considered line as a whole does so.

49

IV Characterization of lattice defects by TEM Internal surfaces In the following, to keep the consistency of the reasoning, the description approach proposed by Schoeck (1961) for dislocations is extended for needs of the present work over the case of internal surfaces. As shown in figure IV.2 the spherical coordinate system is again assigned in such a way, that the zenith angle ϕ is measured from the z axis of the Cartesian system. The azimuthal angle θ is measured from the y axis around the z axis, within the testing plane. In this case we simplify our description to the set of parallel planes. For the simplicity, only one, exemplar set of such planar defects (red planes) is presented in figure IV.2. Nevertheless, also the following considerations are not restricted only to plane figures, since any surface can be approximated by plane elements.

Figure IV.2: A schematic representation of an exemplar set of internal surfaces (red planes) with the habit plane normal orientation defined by angles ϕ and θ , embedded in an arbitrary material volume V .

The orientation of the considered plane surface embedded in the material is defined by an orientation of the plane normal n, the direction of which is ϕ ,θ . The density of planar defects featuring the normal orientations within the element of the solid angle dΩ = sin ϕ dϕ dθ

around the direction

ϕ ,θ

may be defined as

p( ϕ ,θ )dΩ . The

contribution of this set of defects to the total density of the intersection lines ( L A ) measured 50

IV Characterization of lattice defects by TEM π  within the area of the cut surface is p( ϕ ,θ ) cos − ϕ  dΩ = p( ϕ ,θ ) sin ϕ dΩ . The total 2 

density of the intersection traces measured on the testing plane can be therefore described as π 2

2π

L A = ∫ ∫ p( ϕ ,θ ) sin 2 ϕ dϕ dθ .

(IV.14)

ϕ =0 θ =0

On the other hand, the contribution of the considered set of defects to the total area of the planar defects embedded within the arbitrary material volume V is given as

a( ϕ ,θ ) dΩ = ∫ ∫ p( ϕ ,θ ) dΩ h( x' , y' ) dx' dy' , X 'Y '

(IV.15)

where h( x' , y' ) is the height of the internal surfaces enclosed by the material volume V . The integration extends over the projection of this volume on the area A( X ' ,Y ' ) defined by the

X ' and Y ' axes. This projection area A( X ' ,Y ' ) is chosen in such a way that its normal is parallel to the direction along which the height h( x' , y' ) of the given parallel planar defects set is defined. Therefore, we have again: ∫ ∫ h( x' , y' ) dx' dy' = V and the total area A of internal X 'Y '

surfaces (taking into account all possible orientations) embedded within the volume V , is expressed by π 2

2π

A = ∫ a( ϕ ,θ ) dΩ = V ∫ p( ϕ ,θ ) dΩ = V ∫ ∫ p( ϕ ,θ ) sin ϕ dϕ dθ . Ω

Ω

(IV.16)

ϕ =0 θ =0

Subsequently, from (IV.16) we get an expression describing the density planar defects surface π

AV =

2π 2 A = ∫ ∫ p( ϕ ,θ ) sin ϕ dϕ dθ , V ϕ =0 θ =0

(IV.17)

analogous to the expression (IV.12) describing the dislocation density. On the basis of equations (IV.14) and (IV.17) one can derive the relation describing the real (sought) density of internal surfaces ( AV ) as a function of their intersection traces ( L A ) density measured on an arbitrary testing plane AV = f ( L A ) .

(IV.18)

51

Experimental part

53

V Single-crystal growth of CMAs In this chapter the single-crystal growth of the investigated phases β - Al - Mg and κ - Al - Mn - Ni is described. Bridgman, Czochralski and self-flux growth techniques were

chosen for grow attempts of both materials. Basic descriptions of these techniques as well as fundamentals of single-crystal growth itself are given in chapter II.

V.1 Experimental procedures Generally, constituents with a purity of 99.999% for Al, 99.98% for Mg and 99.99% for Mn and Ni were used for the preparation of alloys. Only for Czochralski growth, Al with a purity of 99.9999% was used. Preliminary alloys of desired composition were produced using a levitation induction melting furnace (LIMF) in a water-cooled copper crucible under protective Ar atmosphere. Repeated heating to temperatures well above the melting point was performed to ensure a high homogeneity of the alloy. Such prepared master alloys were subsequently used in the crystal-growth experiments. In several attempts, the growth parameters were varied in order to find the optimum growth conditions and the corresponding appropriate growth techniques. The parameters of the growth processes for all experiments are listed in table V.1 and V.2 for the β - Al - Mg and κ - Al - Mn - Ni phases, respectively. Characterization of the crystals grown was carried out by means of light microscopy (LM), scanning electron microscopy and transmission electron microscopy (TEM). The electron microscopes used for materials characterisation were JEOL instruments: 840A and 7400F (SEM) and 4000FX and 2000FX (TEM). Thin foils for TEM investigations were prepared according to the procedures described in chapter V.2.1 and VI.2.1. Alloy compositions were determined by energy-dispersive X-ray analysis (EDX). For a number of selected samples, the chemical composition was measured by inductively coupled plasma optical emission spectroscopy. These analyses were used for calibrating the EDX measurements. Differential thermal analysis (DTA) was performed in Ar atmosphere with heating and cooling rates of 10 K/min. For DTA, samples of typically 20-50 mg were placed in alumina crucibles and covered by alumina powder to protect them against oxidation and evaporation. Phase identification in the samples produced was performed by means of powder X-ray diffraction (PXRD) carried out on a STOE diffractometer in transmission mode with a position sensitive detector. We used Cu-Kα1 and Mo-Kα1 radiation for β - Al - Mg and κ - Al - Mn - Ni , respectively. The presence or absence of grain boundaries, i.e. the single-crystal nature of the crystals produced, was checked by means of X-ray Laue diffraction operated in backreflection geometry. 55

V Single-crystal growth of CMAs

V.2 Growth of β-Al-Mg single crystals On a basis of initial experiments performed to probe the Al-Mg phase diagram in the vicinity of the β phase (Lipińska, 2005 and Lipińska-Chwałek et al., 2007), it was concluded that solidification of a melt of composition Al61.5Mg38.5 should lead to homogeneous single-βphase material. Accordingly, this composition was used for the single-crystal growth experiments. The primarily important temperature parameter is the melting point of the β phase, deduced to average 447 °C. Table V.1 summarizes the parameters of the growth experiments performed for the β - Al - Mg phase.

Table V.1 Process parameters for the single-crystal growths performed. BM, CZ, and FG denote the Bridgman-, Czochralski-, and flux-growth technique, respectively.

Sample name

Crucible type

Atmosphere [mbar]

Growth velocity

Grain size*

Porosity

[cm3]

[%]

[mm/h] β-BM1

Graphite

260

20

0.01

0.75

β-BM2

Graphite

260

5

0.01

1.3

β-BM3

Graphite

260

1

0.04

1.6

β-BM4

Graphite

260

1

0.25

0.85

β-BM5

Alumina

260

5

0.2

0.35

β-BM6

Alumina

260

2

0.12

0.35

β-CZ1

Alumina

400

15

3.6

1

β-CZ2

Alumina

400

15

3.2

0.6

β-CZ3

Alumina

400

15

4.2

0.7

β-FG1

Alumina, flat bottom

800

-

17

0.8

β-FG2

Alumina, tip shaped

800

-

10

0.67

β-FG3

Alumina, 800 17 0.45 flat bottom * Bridgman crystals: average grain size. Czochralski-crystals: complete single crystal. Flux-grown crystals: largest grain.

56


V.2.1 Bridgman growth Six single-crystal growth experiments of the β phase were performed by means of the Bridgman technique. The relevant growth parameters are listed in table V.1. The melt was kept in a tapered tip-shaped crucible and placed under a protective atmosphere in a vertical tube furnace. Solidification is induced by slow removal of the crucible from the hot zone of the furnace. For further details see chapter II.2.1. Two crucible materials, graphite and alumina, and varying growth speeds ranging from 20 mm/h down to 1 mm/h were used. The furnace temperature was kept constant at 520 °C in all experiments. The main conclusion that is to be drawn is that, at least with the growth-parameter ranges accessible for the equipment employed, the Bridgman technique is not the best choice for single-crystal growth of β - Al - Mg . In none of the approaches it was possible to produce single-crystalline samples.

Employing of the alumina crucibles yielded polycrystalline material with grain sizes ranging from 4 × 10-5 to 0.07 cm3. With the graphite crucibles the grain sizes between 0.12 and 0.35 cm3 (table 1) were obtained, i.e. also far below the desired material volume of 1 cm3. Due to sticking of the solidified alloy to the walls of the graphite crucible, severe problems with removal of the ingots from the crucibles were encountered. However, SEM/EDX characterizations combined with PXRD confirmed that all Bridgman samples contained pure β-phase material. The Bridgman ingots, therefore, could be used for the preparation of homogeneous (but unoriented) initial seeds for Czochralski growth.

V.2.2 Czochralski growth Three successful single-crystal growth experiments of the β phase were performed by means of the Czochralski technique (table V.1). Homogeneous pulling was carried out at 15 mm/h with a seed rotating at 25 turns per minute. Details of the setup are given in chapter II.2.2. For growths β-CZ1 and β-CZ2 (see table 1), a polycrystalline seed cut from the Bridgman ingot β-BM5 was used. The crystals produced are displayed in figures V.1 (a) and (b). The thin neck grown during the first stage of the pulling-process is shown on the right-hand side. The volumes of the crystals were 3.6 and 3.2 cm3 for β-CZ1 and β-CZ2, respectively. X-ray Laue analysis performed at several spots on the surface of the crystals showed that they were single crystalline. No evidence for the presence of grain boundaries was found. However, since the seed employed was polycrystalline, the orientation of the long axis of the crystals could not be selected deliberately. Growth β-CZ3 was carried out using a homogeneous, single-crystalline [1 0 0] - oriented seed. The latter was cut from a single crystal produced by means of the flux-growth technique 57

V Single-crystal growth of CMAs (described below). The crystal grown is displayed in figure V.1 (c). The surface shows clear faceting, in particular at the bottom part. The volume of the crystal is 4.2 cm3. X-ray Laue analysis performed at several spots on the surface of the crystal demonstrated that it is single crystalline. The long axis of the crystal is, according to the seed orientation, parallel to a [1 0 0] lattice direction.

Figure V.1: Outer appearance of crystals β-CZ1 (a), β-CZ2 (b) and β-CZ3 (c) grown by means of the Czochralski technique. SEM/EDX and PXRD investigations confirmed that all the crystals grown by means of the Czochralski technique are single phased β-Al-Mg and the composition corresponds to the nominal composition of the alloy prepared. Figure V.2 shows an example of PXRD pattern of the β-Al-Mg phase grown by means of the Czochralski technique (β-CZ2).

Figure V.2: X-ray diffraction pattern of the β-Al-Mg phase grown by means of the Czochralski technique. The strongest peaks are labelled with the {hkl} values of β-Al3Mg2. 58

V Single-crystal growth of CMAs Samples taken from the top and bottom of each crystals, i.e. corresponding to early and late stages of the growth, showed identical composition within the precision of EDX measurements. Single-crystalline material used for plastic deformation experiments (see chapter VI) was grown by means of the Czochralski technique (β-CZ2 crystal). The composition of the crystal used in these experiments was determined to average 38.6 at.% Mg and 61.4 at.% Al. Figure V.3 shows SAED patterns of β-Al-Mg grown by means of the Czochralski technique. The absence of satellite spots and diffuse scattering reflects the high quality of the crystal. The porosity of the crystals was determined to be of the order of 1% and below (see table V.1).

Figure V.3: Electron diffraction pattern of the ß-Al-Mg phase along the [ 0 0 1 ] (a), [ 0 1 1 ] (b) and [ 1 1 1 ] (c) crystallographic directions.

V.2.3 Self-flux growth Three successful single-crystal growth experiments of the β phase were performed by means of the self-flux growth technique (table V.1). The two-element alumina crucible containing Al61.5Mg38.5 melt was sealed in a quartz ampoule under an argon atmosphere of 800 mbar and a dedicated temperature programme was run: First, the material was homogenized at 600 °C for 1 h in order to achieve good homogenization of the melt. Then the temperature was lowered at 10 K/h to 540 °C and subsequently to 300 °C at 1 K/h. During the latter cooling stage, the solidification temperature was crossed and the melt started to crystallize. When the temperature of 300 °C was reached, the ampoule was taken from the furnace. For more details of the setup used see chapter II.2.3. The crystals produced are shown in figure V.4. The surfaces of the crystals appear dirty due to oxidation and residuals of the removed crucibles. In spite of the latter, the presence of grain boundaries can be seen clearly. Crystals β-FG1 and β-FG3, shown in figure V.4 (a) and (c), 59

V Single-crystal growth of CMAs respectively, were grown using a flat-bottom crucible. β-FG1 contains one grain boundary with a (1 1 1) habit plane, which divides the crystal in two parts of 16.8 and 0.9 cm3 in volume. β-FG3 has a very similar outer appearance. It also has one large grain, comparable in volume to the large grain in β-FG1, but the small part is subdivided by some additional grain boundaries. All grain boundaries are parallel to (1 1 1) crystallographic planes. X-ray Laue analysis performed at several spots on the surface of the larger part of crystals β-FG1 and β - FG3 confirmed their single-crystallinity. β-FG2 (figure V.4 (b)) in an attempt to reduce

the initial nucleation volume, was grown in a tip-shaped crucible which was expected to lead to a smaller number of parallel (1 1 1) grain boundaries. In contrast to these expectations, however, β-FG2 displayed a considerably higher number of grain boundaries. SEM/EDX and PXRD investigations confirmed that the flux-grown crystals are single phased β-Al-Mg. The composition measurements range between 38 and 39 at.% Mg. TEM investigations performed on the β-FG1 and β-FG2 crystals confirmed their high quality. The porosity of the crystals was determined as 0.45 - 0.8%.

Figure V.4: Outer appearance of crystals β-FG1 (a), β-FG2 (b), and β-FG3 (c) grown by means of the self-flux growth technique.

60


V.3 Growth of к-Al-Mn-Ni single crystals According to the studies of Balanetskyy et al. (2007) on the solidification behaviour of the Al-rich region of ternary Al-Mn-Ni alloy system, it could be concluded that solidification of a melt of composition Al91Mn6.5Ni2.5 should lead to homogeneous single-κ-phase material. Accordingly, this composition was used for the single-crystal growth experiments. The primarily important temperature parameter is the peritectic reaction point of the κ-phase formation, deduced to average 867 °C (c.f. of Balanetskyy et al., 2008 and section I.3.1). Table V.2 summarizes the parameters of the growth experiments performed for the κ - Al - Mn - Ni phase.

Table V.2 Process parameters for the single-crystal growths performed for κ- Al- Mn- Ni . BM, CZ, and FG denote the Bridgman-, Czochralski-, and flux-growth technique, respectively.

Sample name

Crucible type

Atmosphere

Growth velocity

Grain size*

[mbar]

[mm/h]

[cm3]

κ-BM1

Alumina

1 ·10-6

1

and [ 1 0 0 ] directions are observed and indicated by black arrows. The inset in the lower right corner of each micrograph indicates the corresponding crystallographic directions.

81

VI Plasticity of the β-Al-Mg phase

Figure VI.7: Bright-field TEM images and corresponding SAED patterns of β -Al-Mg samples deformed at 250 °C up to the UYP (a), the LYP (b) and ε = 6 % (c) and (d). The plane normal of the particular images lies close to the [ 0 1 1 ] directions. In the right lower corner of each micrograph main crystallographic directions of the β -Al-Mg fcc structure are indicated. Two families of planar defects featuring { 1 1 1 } and ( 1 0 0 ) habit planes are indicated in (a) by red and green arrows, respectively.

82

VI Plasticity of the β-Al-Mg phase The amount of planar defects observed increases with strain. For the sample deformed up to

ε = 6%, the effect of a very high number of {1 1 1} planar defects can be clearly observed also in the corresponding [ 0 1 1] diffraction pattern. Streaking of the diffraction spots in directions perpendicular to the habit planes of the planar defects is visible in the diffraction patterns of the strongly deformed sample ((c) and (d)). Clear evidence for the fact that the streaks originate from the presence of planar defects is found when comparing the micrographs shown in (c) and (d). In (c), a very high density of planar defects occurs in both (1 1 1) and (1 1 1) planes and, correspondingly, streaking is observed along directions perpendicular to both plane families. In the micrograph (d), a high number of planar defects occurs only within (1 1 1) planes. Correspondingly, the streaking is observed only in one direction, i.e. in [1 1 1] . It is worth to mention, that no signs of such an effect could be found for less-deformed samples. In TEM investigations performed, only defects with habit-plane normal vectors parallel to and [ 1 0 0 ] are found. In the following dislocations and planar defects featuring

{1 1 1} and (1 0 0 ) habit planes are referred to as {1 1 1} - and (1 0 0 ) defects, respectively.

(100)-dislocations In the following, an analysis of the invisibility conditions for (100)-dislocations is given. Figure VI.8 shows a bright-field TEM micrograph of the β - Al - Mg sample deformed up to the UYP. The same sample area with a specimen normal close to the [100] direction is shown under various imaging conditions (a-c) (see insets). The black arrows mark fixed position in the sample. The applied two-beam conditions correspond to the reciprocal vectors r r r g = ( 3 11 3) (a) , g = (0 8 8) (b) , and g = ( 0 8 8) (c) . Several (100)-dislocations extend within the specimen plane and correspondingly long dislocation-line segments are visible in figure VI.8. One bent (100)-dislocation line is visible on the left-hand side of the micrographs, indicating hereby, that (100) − dislocations are actually dislocation loops elongated in [ 0 1 1 ] direction. Depending on the applied two-beam conditions,

the

contrast

of

particular

dislocation

segments

is

extinct.

In

(a)

(100) − dislocations are fully visible in the bright-field contrast, while in (b) and (c), the contrast of some segments is extinct. Positions of segments invisible in (b) or (c) are indicated by black arrows. In (b) and (c) those dislocation segments are invisible which possess line directions parallel to the reciprocal vectors of the applied imaging conditions.

83


Figure VI.8: Bright-field TEM images of the β phase, close to the [100] zone axis. The r r corresponding two-beam conditions are g = ( 3 11 3) ( a ) , g = (0 8 8) ( b ) , and r g = ( 0 8 8 ) ( c ) . The contrast of (100)-dislocation segments featuring a line direction parallel to the applied reciprocal vector (insets in the upper-right corners of the micrographs) is extinct in the respective images.

Considering equation (IV.1), the observed cases of defects invisibility reveal that the Burgers vector of (100)-dislocations is parallel to [1 0 0] . Nevertheless, only specific segments of the dislocation are invisible in the respective micrographs. Despite the fact that all reciprocal vectors applied in (a-d) fulfil equation (IV.1), the loop segments featuring the line direction not parallel to the operating reciprocal vector show a strong blurry residual contrast. 84

VI Plasticity of the β-Al-Mg phase This is due to the pure edge character of the (100)-dislocations (i.e. the Burgers vector is perpendicular to the line direction of all occurring loop segments). As described in chapter IV.1, the contrast of pure edge dislocations can be completely extinct only if condition (IV.2) is fulfilled. In this case, additionally to conditions (IV.1), the r r reciprocal vector g has to be parallel to the dislocation line direction l . Figure VI.8 reveals r r that only dislocation segments with l parallel to g (cf. insets) are completely invisible. Dislocation segments which remain visible in figure VI.8, do not simultaneously fulfil condition (IV.2) and, hence, cause residual contrast. Since the Burgers vector of (100)-dislocation is parallel to the normal vector of the dislocation habit plane, it can be concluded that c-axis dislocations move by a pure climb mechanism. A technique commonly used in TEM for the determination of the length of a dislocation Burgers vector is convergent-beam electron diffraction (CBED) (e.g. Tanaka et al., 1988). However, in CMA crystals featuring large lattice parameters, the density of Kikuchi lines is too high for unambiguous determination of the number of splitting nodes in the strain field of the dislocation (Feuerbacher et al., 2004). Hence, determination of the Burgers-vector length by means of CBED was found practically impossible for β - Al - Mg . Some additional information on the modulus of the present defects might be obtained, however, by employing TEM imaging of higher resolution. Figure VI.9 shows a TEM micrograph of the β phase deformed up to the LYP. The image normal is parallel to the [0 11 ] zone axis (cf. inset in the right lower corner of the micrograph). The inset in the upper left corner of the micrograph indicates the crystallographic directions according to the β − Al − Mg structure. A stacking fault caused by the motion of a (100)-dislocation crosses the micrograph diagonally. The normal vector of its habit plane is parallel to [1 0 0 ] . Green rectangles indicate unit cells of β - Al - Mg on both sides of the stacking fault. The displacement vectors of particular parts of stacking fault are reflected in the overlap of the rectangles.

85


Figure VI.9: HRTEM micrograph of the β -Al-Mg phase along the [ 0 1 1 ] zone axis. A stacking fault with a habit-plane normal parallel to [1 0 0] is terminated by two partial dislocations (indicated with red arrows). Some unit cells of the β -Al-Mg phase are indicated with green rectangles on both sides of the stacking fault in two places, i.e. between two partials and after the second partial (very right part of the micrograph). The r corresponding displacement vectors R are reflected in the overlap of the green rectangles which can be seen in the magnified insets indicated with blue frames.

The planar defect is terminated by two partials, which are indicated in the image by red arrows. The extended planar defect follows on the right hand side of the second partial, while the distance between two terminating partials amounts to about 86 nm. One can observe an indication of (100)-dislocation splitting as well in figure VI.8 (a). The place of such splitting is indicated with two red arrows in the dislocation loop located on the left-hand side of the micrograph. The splitting distance that can be roughly evaluated from figure VI.8 amounts between 50 and 100 nm. These values fit very well with the distance evaluated from figure VI.9. The observed (100)-dislocations are therefore considered as partials generating planar defects upon movement. The displacement vector of the extended stacking fault is concluded 86

VI Plasticity of the β-Al-Mg phase to correspond to the Burgers vector of this dislocation. According to the defects-invisibility analysis described above, the Burgers vector of (100)-dislocations has only a component along the [1 0 0 ] direction. The length of the displacement vector, hence, can roughly be estimated from figure VI.9. For the part of planar defect included between two partials it r amounts to approximately | R |= 0.35 nm which corresponds, within the measurement accuracy, to a removal of an eighth part of the β - Al - Mg unit cell along the [100] axis. For the staking fault part following the second partial (visible at the very right side of the image) the length of the displacement vector is doubled. This allows to draw a conclusion that the r Burgers vectors of both partials are of the same size (equal to | R |= 0.35 nm ), and the resulting Burgers vector of (100)-dislocation, consisting of two partials, amount to the quarter of the β - Al - Mg unit cell along the [100]. This value is further discussed in section VI.3.

{111} - dislocations In the following, an analysis of the invisibility conditions for {111}-dislocations is given. Figure VI.10 shows TEM micrographs of the sample deformed up to the UYP. A TEM specimen was cut with its surface normal vector close to the [111] zone axis. Bright-field r image is obtained by using two-beam conditions corresponding to g = (8 8 0) (a) and r g = (0 8 8) (b). The (111)-dislocation located in the centre of the micrograph is indicated by white and black arrows at several, characteristic image places. Since the dislocation line lies within the plane of the TEM sample, long dislocation segments are visible as dark lines. In (a) the (111)-dislocation is visible with full diffraction contrast, while in (b) the contrast of r particular dislocation segments is extinct. For the reciprocal vector g = (0 8 8) , governing the bright-field image formation in figure VI.10 (b), an invisibility of dislocation segments featuring specific line direction orientation is observed. Dislocation segments with line direction parallel (indicated by white arrows in both micrographs) and perpendicular (indicated by black arrows) to the applied reciprocal vector are invisible. On the other hand, a dislocation segment that possesses an intermediate line direction (not parallel and not r perpendicular to the operating g = (0 8 8) ) features residual contrast. This segment can be seen in figure VI.10 (b), on the right-hand side of the most upper white arrow. Due to complete invisibility of several dislocation segments that takes place for two-beam conditions applied in figure VI.10 (b), it is concluded that the Burgers-vector direction is r perpendicular to the reciprocal vector g = (0 8 8) (cf. equation (IV.1)), i.e. it lies within the (0 1 1) plane. Nevertheless, only specific segments of the dislocation are invisible. In the

87

VI Plasticity of the β-Al-Mg phase following, it is argued that the Burgers vector lies within this plane along the [2 1 1 ] direction.

Figure VI.10: Bright-field TEM images of the β phase, close to the [111] zone axis. r r The corresponding two-beam conditions are g = ( 8 8 0 ) (a) and g = ( 0 8 8 ) (b). The contrast of (111)-dislocation segments featuring a line direction parallel to the applied reciprocal vector is extinct in the respective images.

A Burgers vector parallel to [2 1 1 ] direction is in good agreement with the contrastextinction experiments shown in figure VI.10 (b). Vertical and horizontal dislocation r segments are indicated by white and black arrows and possess, in case of b being parallel to [2 1 1 ] direction, pure edge and pure screw character, respectively. The contrast of these segments is completely extinct in figure VI.10 (b) since the invisibility criteria (equation (IV.1) and (IV.2)) for both, edge and screw dislocations are fulfilled for the respective segments. Also high resolution TEM observations performed on planar defects occurring in {1 1 1 } planes indicate that glide of dislocations featuring Burgers vectors parallel to direction is involved in these planar defects formation.

88

< 211 >

VI Plasticity of the β-Al-Mg phase Figure VI.11 shows a TEM micrograph of the β phase deformed at 250 ºC up to ε = 6 % . The image normal is parallel to the [ 0 1 1 ] direction. Green rectangles indicate corresponding unit-cell projections. The area decorated with a big red rectangle is shown in figure VI.12 at higher magnification. Several planar defects of various thicknesses expand diagonally across the area. These are not conventional planar defects, involving displacement at a sharp interface, but modified slabs of nanometre-thick microtwins. The mirror planes of these defects possess normal vectors parallel to [ 1 1 1 ] crystallographic direction, while the displacement vectors corresponding to the general material shear introduced via formation of the microtwins are parallel to [ 2 1 1] direction. Detailed description of these defects is given referring to the example shown in figure VI.12.

Figure VI.11: TEM image of the β -Al-Mg phase along the [0 1 1] zone axis. Microtwins with habit-plane normal parallel to [ 1 1 1] direction are visible as long diagonal objects of finite thickness. Unit-cell projections of the β -Al-Mg phase are indicated as green rectangles within the area corresponding to undistorted material.

A microtwin with [1 1 1] mirror-plane normal diagonally crosses the micrograph area. Green and red rectangles (or their parts), indicate some β - Al - Mg unit cell projections of the undistorted and twinned material, respectively. The lower inset on the right-hand side of figure VI.12 denotes crystallographic directions of the undistorted β - Al - Mg . The microtwin introduces a definite shift of the β - Al - Mg crystal parts located on both sides of the micro 89

VI Plasticity of the β-Al-Mg phase twin. The thickness of the microtwin observed amounts to 4/3 of the unit cell along [111] direction. The displacement vector corresponding to the relative shift of the crystal parts located on both defect sides is indicated in the micrograph by a red arrow. It amounts to r approximately | R |= 2.8 nm which corresponds, within the accuracy of the measurement, to 2/3 of the β - Al - Mg unit cell along the [ 2 1 1] direction.

Figure VI.12: HRTEM image of the β -Al-Mg phase along the [ 0 1 1 ] zone axis. A microtwin with a mirror-plane normal parallel to [ 1 1 1] diagonally crosses the micrograph area. Unit-cell projections of the β -Al-Mg phase are indicated with green and blue rectangles within undistorted and twinned material, respectively.

Analysis of other microtwins in these terms revealed that the thickness of the microtwins always corresponds to a multiple of 1/3 of the unit cell along [ 1 1 1 ] crystallographic direction. The corresponding displacement vector amounts to 1/6 of the unit cell along [ 2 1 1] direction. These values are further discussed in section VI.3.

VI.2.3 Defect density In the following an evaluation of the density of linear (dislocations) and planar (stacking faults and microtwins) defects in the β - Al - Mg phase is given. The density of dislocations is calculated separately for such moving on {111}, {100} planes. Analogously, density of planar 90

VI Plasticity of the β-Al-Mg phase defects is determined separately for defects formed by movement of particular dislocation types. Additionally, within defects (both, of linear and planar type) featuring the {100} habit planes, these with a habit-plane normal being parallel to the deformation direction (i.e. defects in (100) planes) are distinguished from defects with the habit-plane normal being perpendicular to the deformation direction (i.e. defects lying in (010) and (001) planes). The (010) and (001) planes are equivalent with respect to the deformation direction, which is reflected in their similar frequency of occurrence. Therefore, only average values of densities evaluated for defects (both, of linear and planar type) featuring these two habit planes are given in table VI.1. Since microtwins with {111} habit planes, upon observation conditions applied for density determination (magnification and edge-on orientation), exhibit the appearance similar to planar defects (i.e. their thickness is not resolved), the evaluated values correspond only to the average area occupied by these defects in the volume. If referred as the area corresponding to the microtwin mirror planes, the calculated values have to be multiplied by 2 (two mirror planes, each on every side of the microtwin, separating the twinned material from the undistorted structure). Density of dislocations is determined from the TEM specimens cut with surface normal parallel to [001] and [010] directions (which correspond also to the viewing directions). Dislocation densities can be then calculated from the amount of dislocations segments N encountered in an observed area A, taking different orientations of these segments into account. By considering an isotropic dislocation distribution in one habit-plane, the relation resulting from comparison of equations (IV.9) and (IV.12) can be simplified to (Schöck, 1961; c.f. equation [10] therein)

LV =

π

NA N π = 2 sin γ 2 A ⋅ sin γ

,

(VI.1)

where γ is an angle that the dislocations habit-plane normal encloses with the viewing direction (i.e. [001] or [010], corresponding to z-axis in figure IV.1). In particular cases, the density measurements of {100}-dislocations were performed on defects in {100} planes with the normal vectors in orientation perpendicular with respect to the normal of TEM specimen surface. Therefore, for both observation directions ([001] and [010]), γ amounts to 90 ° for {100}-dislocations, while for all {111}-dislocations the observation geometry corresponds to γ of about ~54.73 °. Densities of planar defects produced by movement of dislocations on {111} and (100) planes, are determined from TEM specimens cut with surface normal parallel to the < 011 > directions being perpendicular to the compression direction (100). For stacking faults lying within (010) and (001) planes, their density is determined from the TEM specimens with the surface normal parallel to the [100] direction. Hence, the thickness of the specimen 91

VI Plasticity of the β-Al-Mg phase corresponds in each case, to the height of the analysed planar defect (in the edge-on orientation). The corresponding defect density can be, therefore, calculated directly from the length of the defects traces L measured within an observed area A. For planar defects embedded within a specified family of planes, featuring an orientation which is defined by an angle ϕ , their normal encloses with the viewing direction (c.f. figure IV.1), the relation resulting from comparison of equations (IV.14) and (IV.17) takes the form:

AV =

LA . sin ϕ

(VI.2)

In the present case, where determination of the planar defects density is performed on defects being each time in edge-on orientation with respect to the viewing direction, γ amounts to 90°. The relation (VI.2), therefore, simplifies to:

AV = LA =

L . A

(VI.3)

The investigated TEM specimens were prepared from the samples featuring various deformation states. Systematic compression experiments were carried out on the β-Al-Mg phase at 250 °C up to different strain stages (UYP, LYP and ε = 6%, see above), and reference investigations were carried out on undeformed material which was heat treated in the same way as the deformed sample. This allowed performing an analysis of the deformation process from the point of view of the microstructure evolution. The densities of particular defects determined according to the equations given above are summarized in table VI.1. These densities are mean values obtained from measurements performed on a number of randomly selected TEM-specimen areas observed under magnification conditions, depending on the defects visibility. For undeformed and deformed up to the UYP samples, about 100 TEMspecimen areas were analysed at a magnification of 20k. In the material subjected to higherstrain compression experiments, distinction of individual dislocations was precluded due to extremely high defect densities and corresponding overlap of their strain fields. Therefore, measurements of dislocation densities were performed only on reference samples and samples deformed to the UYP and less (cf. table VI.1). Analysis of the planar defects, however, could be performed further if dark-field TEM images at higher magnifications were employed. Correspondingly, for material deformed up to the LYP and ε = 6%, about 30 TEM-specimen areas were analysed at a magnification of 50k. The evaluated densities of the dislocation types differ by about one order of magnitude, and both increase during deformation by about two orders of magnitude.

92


Table VI.1: Defect densities in undeformed and deformed (250 °C) β − Al − Mg . Densities are given separately for dislocations ( LV ) and planar defects ( AV ) featuring {111}*, (100) and (010) or** (001) habit planes.

Type of

PLANAR DEFECTS

DISLOCATIONS

defects

Defects density ( LV or AV )

Defects habit

undeformed

plane

sample***

{111}*

1·107 cm-2

(100) (010) or**

1.5 ·108 cm-2

UYP

LYP

ε = 6%

deformed

deformed

deformed

sample

sample

sample

1 ·109 cm-2

-

-

5 ·108 cm-2

-

-

1 ·108 cm-2

-

-

5 ·104 cm-1

3 ·105 cm-1

1.5 ·106 cm-1

1 ·104 cm-1

1 ·105 cm-1

1 ·105 cm-1

1 ·103 cm-1

1 ·103 cm-1

1·103 cm-1

(001) {111}*

2 ·104 cm-1

(100) (010)

1·103 cm-1

or** (001)

* Summarized density of all defects existing in four {111} planes, equivalent in the fcc structure in respect to the applied deformation direction [100] ** Density of defects embedded to only one of two {010} planes, equivalent in the fcc structure in respect to the applied deformation direction [100]; an average value of defects density evaluated for these two perpendicular families of planes is given. *** In undeformed samples all three {010} planes are crystallographically equivalent; an average value of defects density evaluated for these three perpendicular families of planes is given.

93


VI.3 Discussion VI.3.1 Macroscopic deformation behaviour Uniaxial compression tests were performed on single-crystalline β - Al - Mg at constant strain rate of 10-4 s-1. The investigations of the macroscopic deformation behaviour were performed by Roitsch et al. (2007) at temperatures between 200 and 375 °C, corresponding to the homologous temperature TH∗ between 0.60 and 0.90. These experiments revealed the brittle-to-ductile transition of β - Al - Mg to reside between 200 and 225 °C (0.65 to 0.69 of TH) at the employed strain rate. A maximum flow stress was accomplished at 225 °C and it

amounted to 780 MPa. The presence of the pronounced yield-point effects was observed on stress-strain curves at all investigated temperatures. The microstructural investigations of β - Al - Mg revealed an increase of the structural defect density in the course of the plastic deformation process. A difference of dislocation density between undeformed material and samples deformed up to the upper yield point amounts to about three times for (100) dislocations and to about two orders of magnitude for {111} defects. An increase of dislocation density is accompanied by an increase of density of the planar defects generated by movement of the particular kind of dislocations. It is concluded, therefore, that the deformation process in the β - Al - Mg phase is based on dislocation motion. The presence of yield-point effects after incremental tests is very often associated with dislocation recovery process. It involves a decrease of the mobile dislocation density during annealing in the unloaded state or during stress relaxation (Hull and Bacon, 1984). Therefore, some small yield-point effects observed after incremental tests in β - Al - Mg indicate that material recovery takes partially place already at the time scale corresponding to these tests. Thermodynamic activation parameters of the deformation process were determined by incremental tests (see chapter III.2). The activation enthalpy ∆H (figure VI.3) is larger than the work term ∆W by about a factor of 4, which indicates that the deformation process is thermally activated. Moreover, the activation enthalpy shows linear temperature dependence in the investigated temperature range. According to Gibbs (1969), this indicates that no change of deformation process takes place in the investigated temperature range.

∗

The homologous temperature TH = T / Tm is the absolute value of the apparent temperature T scaled by the

absolute melting temperature Tm, which for β - Al - Mg amounts to Tm = 724 K (Murray, 1988).

94


No values of the diffusion enthalpy are reported in the literature for β - Al - Mg . In order to check whether the rate of dislocation movement is a diffusion controlled process, the diffusion parameters of the pure constituent elements can be used for comparison. These amount to 1.28 eV (Messer et al., 1974) and 1.40 eV (Shewmon, 1956) for pure Al and pure Mg, respectively. Values for other Al-Mg alloys range between 1.2 and 1.33 eV (Stoebe et al., 1965). Assuming that the diffusion enthalpy of β - Al - Mg compares to these values, it would differ from the calculated activation enthalpy by a factor of about 2. Therefore, it is concluded, that the rate-controlling deformation mechanism is not given by a diffusion process. The activation volume of β - Al - Mg evaluated by Roitsch (2008) reaches the same order of magnitude as those found for other CMA phases (cf. chapter VII). It amounts to about 0.5 nm3 at a moderate stress of 300 MPa (c.f. Figure VI.2). Scaled by the atomic volume Va (i.e. the volume of the unit cell VCell = 22.43 nm3 divided by 1168 atoms per cell) it amounts to V/Va = 26 . This value is of the same order of magnitude as the number of atoms in clusters

present in the material, i.e. icosahedra, Friauf polyhedra or VF-polyhedra (Samson 1965 and 1969). Feuerbacher et al. (2001) have shown, on the example of ξ'-Al - Pd - Mn that the large activation volumes in CMAs may result from the thermally activated overcoming of structurebuilding atom clusters. Therefore, possible obstacles in β - Al - Mg , which may cause the large values of the activation volume can be provided by the cluster substructure present in the material. Accordingly, the friction between dislocations and the atom clusters may be supposed to act as rate-controlling process of dislocation motion in β - Al - Mg . A Schmid factor of 0.47 was employed in the calculations of activation parameters of β - Al - Mg . This value corresponds to the maximum Schmid factor available for the glide of

< 211 > -dislocations on {1 1 1} planes (see section VI.2). Although, dislocation climb was observed in β - Al - Mg as well, quantitative analysis performed indicates that this mechanism plays only a marginal role in the material deformation. From the results of the defect density measurements performed within present work (see Table VI.1) it can be realised that the amount of planar defects produced by dislocations climbing in (1 0 0) planes remains constant after approaching the LYP of the deformation process. Since the incremental tests are performed within later deformation stages, the climb mechanism cannot contribute considerably to analysed activation processes. It is therefore admissible, to consider the glide mechanism with {1 1 1} < 211 > glide system as the primary process of plastic deformation in β - Al - Mg , and to make an allowance for this mechanism when determining the Schmid

factor for thermodynamic activation analysis. 95


VI.3.2 Microstructural analysis The studies of the microstructural mechanisms contributing to the plastic deformation processes of β - Al - Mg were performed by means of TEM. The microstructural analyses were performed on samples deformed plastically in uniaxial compression tests, carried out at 250 °C and the constant strain rate of 10-4 s-1 up to the UYP, LYP, and ε = 6 % . Comparative investigations were performed on undeformed reference sample, which was heat treated in the same way as the deformation sample strained up to ε = 6 % . Already the initial observations of plastically deformed samples performed by means of LM show that single-crystalline β - Al - Mg exhibits inhomogeneous deformation behaviour. The diagonal shear traces

appearing upon deformation on the sample surface (figure VI.5) indicate the existence of shear planes with {111} crystallographic orientation. These were also revealed upon chemical preparation of TEM samples taken from plastically deformed material (figure VI.6). The subsequent TEM investigations revealed an existence of linear and planar defects in a density increasing with strain (figure VI.7). This observation indicates that the deformation process of β - Al - Mg is mediated by dislocation motion. Since dislocations generate stacking faults

during movement, they are considered as partial dislocations. The appearance of partials in β - Al - Mg phase is in accordance with the considerations on the elastic line energy of the

linear present in CMA. Since this energy is proportional to the square of the Burgers-vector length (Hirth and Lothe, 1992), the presence of partial dislocations, featuring Burgers-vector lengths corresponding to a fraction of the lattice periodicity only, is energetically more reasonable for materials featuring giant unit cells than occurrence of perfect dislocations. Two different mechanisms of dislocation movement were revealed in β - Al - Mg to contribute to its deformation process. Dislocation motion takes place on {1 1 1} and on (1 0 0) planes. Correspondingly, stacking faults with normal vectors parallel to < 1 1 1 > and [1 0 0] directions are observed. r

The Burger vectors b of particular dislocation types were determined by means of contrastextinction experiments. The analysis performed on (100) dislocations, shown in figure VI.8, revealed the Burgers vector of these defects to be parallel to the [100] direction. It is, therefore, parallel to the normal vector of the dislocation habit-plane and perpendicular to the dislocation line direction. Correspondingly it could be concluded that (100)-plane dislocations are of pure edge character and move by means of a pure climb mechanism. The occurrence of dislocation climb is frequently observed mechanism of the plastic deformation behaviour of the CMA phases (c.f. chapter VII). The Burgers-vector length of (100)-dislocations in β - Al - Mg is estimated from the micrograph shown in figure VI.9. An extended stacking fault

96

VI Plasticity of the β-Al-Mg phase caused by the motion of the (100)-dislocation consisting of two partials is visible. The r displacement vector R of the resulting stacking fault amounts about 1/4 of unit cell along the [1 0 0] crystallographic direction. As visible in figures VI.8 and VI.9, (100)-dislocations split into two partials with parallel line directions. The distance between the partials is of the order of 50 to 100 nm. These partials are concluded to feature the Burgers vector 1 8 [1 0 0] with the r modulus b = 0.35 nm, which is a physically acceptable length with regard to the corresponding elastic line energy. The splitting of 1 4 [1 0 0] -dislocation into two 1 8 [1 0 0] − partials obviously takes place in order to reduce the size of the occurring Burgers vector. The energy of the stacking fault produced between two resulting 1 8 [1 0 0] dislocations is obviously high, which leads to a short distance between partials. On the basis of invisibility conditions found for {1 1 1} -dislocations, it was proposed that Burgers vectors of these defects are parallel to the directions lying within the plane of dislocation movement. Accordingly, the mechanism of dislocation movement corresponds to a pure dislocation glide on {1 1 1} planes. Indeed, the occurrence of the glide mechanism within the (111) glide system is highly plausible under applied deformation geometry. In the uniaxial compression experiment performed with deformation direction parallel to the [1 0 0] direction the (111)glide system contributes very efficiently to the deformation process, since a high Schmid factor of mS = 0.47 is present. A detailed analysis of {1 1 1} -dislocations character was performed on a basis of HRTEM observations of stacking faults associated with linear defects. An exemplar HRTEM micrograph of such a defect in edge-on orientation is shown in figure VI.12. It reveals that the {1 1 1} -planar-defects are not conventional stacking faults, involving displacement at a sharp interface, but structurally modified material slabs of several-nanometre thickness. The structure within these slabs can be matched to β - Al - Mg unit cells of modified orientation. The observed planar defects are therefore considered as microtwins (c.f. figure VI.12). Mirror planes of these defects possess normal vectors parallel to < 1 1 1 > directions, while the displacement vectors corresponding to the general material shear introduced via formation of the microtwins are parallel to directions lying within defects habit planes. The thickness of the observed defects corresponds to a multiple of the 1/3 of the β - Al - Mg unit cell along < 1 1 1 > crystallographic directions. The respective displacement vector amounts to 1/6 of the unit cell along directions per single defect thickness (i.e. per 1/3 of the β - Al - Mg unit cell along < 1 1 1 > direction). Hence, it is concluded that the microtwins are

created by dislocations with Burgers vectors 1/6 < 2 1 1 > moving on {1 1 1} planes. Varying 97

VI Plasticity of the β-Al-Mg phase thicknesses of the microtwins occur by different numbers of dislocations, which have moved on adjacent layers and can hence only occur in multiples of 1/3 of the unit cell along [ 1 1 1 ] direction. The dislocation directly corresponds to a Shockley partial, which is very striking feature of the β - Al - Mg phase. Although, very large lattice parameters are present, the occurring carriers of the material deformation resemble defects, which are typical for the simple fcc materials (e.g. Ray and Cockayne, 1971, Lee et al., 2001). If calculated in respect to the lattice parameter of the β-Al-Mg phase (a = 2.82 nm) , the resulting Burgers vector r a length of such defect amounts to about | b |= = 0.575 nm . With regard to the elastic line 2 6 energy of dislocation, this value is rather high and splitting to smaller partials is highly plausible. The shear mechanism observed in present work resembles the shear transformations observed by Heggen et al. (2008) in other CMA phase, Al13Co4. There, movement of a novel type of metadislocation introduces a structurally modified slab of m- Al13Co4 phase within the o-Al13Co4 phase matrix. However, the mechanism observed in β - Al - Mg involves only an alteration of the structure orientation within the slabs affected by linear defects moving in {1 1 1} planes. The underlying mechanisms of these defects formation are not understood yet, because single dislocations terminating stacking faults could not be found for analysis in HRTEM in the given experiment conditions.

98

VII Plasticity of the к-Al-Mn phase In this chapter deformation experiments performed on the κ - Al - Mn - Ni phase are described. Uniaxial compression tests at constant strain rate with compression direction parallel to the c-axis of the hexagonal structure were carried out. Thermodynamic activation parameters of the deformation process are evaluated and discussed below. A comparative microstructural analysis of the annealed and plastically deformed material is performed by means of TEM. The underlying microstructural deformation mechanisms are determined and discussed with respect to the macroscopic deformation behaviour.

VII.1 Macroscopic deformation behaviour VII.1.1 Experimental details The Al-Mn-Ni phase single crystals subjected to deformation experiments were grown by means of the self-flux-growth technique as discussed in chapter II.2.3. The material was characterized as described in chapter V.3. The crystal was oriented by Laue X-ray diffraction in back-reflection geometry. Rectangular samples of about 1 x 1 x 3 mm3 in size for uniaxial deformation experiments were cut from the crystal by spark erosion. The long axis of the samples, i.e. the compression direction, corresponds to the [0 0 0 1] direction (c-axis) of the hexagonal structure. The normal vectors of side faces correspond to [2 11 0 ] and [11 0 0] directions of the κ - Al - Mn - Ni structure, respectively. All surfaces were carefully ground and polished in order to prevent crack formation due to surface roughness. Special attention was paid to obtain flat and plan-parallel end faces in order to prevent inhomogeneous stress fields in the sample. The deformation experiments were carried out as uniaxial compression tests in a modified Zwick Z050 testing system under closed-loop control. The experiments were performed in air at temperatures between 680 °C and 740 °C, i.e. within the temperature range of confirmed existence of the pure κ - Al - Mn - Ni phase (c.f. chapter V.3). A constant strain rate of 10-5 s-1 was applied. Additional incremental tests, i.e. stress-relaxation tests and temperature changes, were performed as described in chapter III.2. After deformation, the samples were rapidly unloaded and quenched on a cold metal plate in order to preserve their microstructure.

99

VII Plasticity of the κ-Al-Mn-Ni phase

VII.1.2 Results The stress-strain behaviour of the κ - Al - Mn - Ni phase between 680 and 740 °C is shown in figure VII.1. Vertical dips in the curves are due to incremental tests. A stress-relaxation experiment and a part of deformation experiment carried out at temperature increased by 10 °C are exemplarily labelled “R” and “TC”, respectively, in the stress-strain curve at 680 °C. Dashed lines in figure VII.1 indicate interpolated courses of the stress-strain curves. The deformation experiments were terminated at total strains between 4 and 8%.

Figure VII.1: Stress-strain curves of κ-Al-Mn-Ni at temperatures between 680 and 740 ° C at a strain rate of 10-5 s-1. A stress relaxation test and a temperature change are exemplarily labelled “R” and “TC”.

All curves in figure VII.1 show a pronounced yield-point effect, i.e. a strong overshoot of the stress at the onset of plastic deformation. Reloading of the sample does not induce additional yield-point effects. In general, after stress-relaxations as well as after unloading for temperature changes the stress smoothly approaches an almost constant flow stress without exhibiting an overshoot. At the highest deformation temperature work hardening was observed. It started at a strain of about ε = 3.5 % , after reloading the sample subsequent to 100

VII Plasticity of the κ-Al-Mn-Ni phase incremental tests. The experiment at 740 °C was performed up to higher value of total strain than other experiments in order to observe the further material behaviour after the hardening was noticed. A constant tendency of the stress increasing was observed and deformation was terminated at 8% of total strain. Figure VII.2 shows the stress at the UYP (triangles) and LYP (circles) as a function of temperature. The stresses at these points decrease continuously with increasing temperature. The magnitude of the yield drop, i.e. the difference between upper and lower yield point, amounts to a constant value of about 110 MPa for all temperatures which corresponds to 29 up to 63% of the respective lower yield stress. The dashed lines in figure VII.2 serve as a guide to the eye.

Figure VII.2: Stress at the UYP (triangles) and LYP (circles) of κ-Al-Mn-Ni as function of temperature. Dashed lines are a guide to the eye. In the course of the deformation process horizontal and vertical lines appear on the sample surface. Figure VII.3 shows the LM image of a sample deformed in air at 720 °C . After the deformation experiment its surface was completely covered by such lines. Figure VII.3 (b) is an enlargement of an area marked by a green rectangle in (a). Fine steps or cracks oriented perpendicular and parallel to the compression direction (parallel to the long dimension of the deformation sample) are observed on the sample surface. The majority of lines, however, feature the orientation perpendicular to the direction of the force applied in deformation experiment. Samples deformed at other temperatures exhibit a similar surface appearance.

101


Figure VII.3: Deformation sample (a) and magnified detail of the surface region (b) after deformation performed in air at 720 °C up to ε = 4%. Figure VII.4 shows the results of a stress relaxation test carried out at 720 °C. The test was terminated after 2 minutes. In (a) the stress is plotted as function of time. The natural logarithm of the slope ln(−σ& ) is plotted as function of stress in (b). A linear fit of the latter is indicated by a solid line. Its slope delivers, according to equation (III.21), the experimental activation volume V. A Schmid factor of ms = 1 was implied in this calculation since the dominating microstructural mechanism of the plastic deformation in this material is dislocation climb (see discussion in section VII.3).

Figure VII.4: Stress-relaxation test on κ-Al-Mn-Ni at 720 °C. The stress plotted as function of time is shown in (a). The natural logarithm of the slope ln( −σ&) plotted as function of stress is shown in (b). The solid line in (b) is a linear fit.

102

VII Plasticity of the κ-Al-Mn-Ni phase The stress dependence of the activation volume is shown in figure VII.5. The dashed curve is a fit of the determined experimental activation volumes at different stresses. The hyperbolic function is used to fit the available data since equation describing activation volume posses a hyperbolic form. Additionally, a hyperbolic dependence of these parameters was observed for other CMAs. In present case a function V = 85 / σ could be assigned to fit the data.

Figure VII.5: Activation volume V of κ-Al-Mn-Ni evaluated from stress-relaxation experiments as a function of stress σ. The dashed curve follows the hyperbolic function V = 85 / σ . The stress exponent m is calculated according to equation (III.22) and its temperature dependence is shown in figure VII.6. The stress exponent is nearly constant in the investigated temperature range. The dashed horizontal line corresponds to the average value of m = 5.3.

103


Figure VII.6: Stress exponent m of κ-Al-Mn-Ni evaluated from stress-relaxation experiments as a function of temperature. The dashed line indicates the average value of m = 5. 3 . The activation enthalpy ∆H, calculated according to equation (III.25), is shown figure VII.7 (triangles). The determined values range between 5.9 and 7.9 eV. The solid line shows a linear fit of the ∆H data. The work term, corresponding to the part of the energy which is supplied by the applied stress, is calculated according to equation (III.19) by neglecting existence of the internal stresses, i.e. ∆W ≈ τV . It is shown in figure VII.7 as circles. The work term is constant in the observed temperature range and amounts to about 0.5 eV. The same, the work term is more than ten times smaller than the activation enthalpy. The dashed line shows a linear fit of ∆W data.

104


Figure VII.7: Activation enthalpy ∆H (triangles) and work term ∆W (circles) of κ-Al-Mn-Ni as a function of temperature. The solid and dashed lines are linear fits of the activation enthalpy and the work term, respectively.

105


VII.2 Microstructural analysis VII.2.1 Experimental details The microstructural investigations on the Al - Mn - Ni phase were carried out by means of TEM at 200 kV (JEOL 4000FX). These studies were performed on the sample material deformed at 720 °C. For comparative investigations, undeformed reference sample that was heat treated in the same way as the deformed one, was employed. Fundamentals of the microstructural analysis by TEM are described in chapter IV. The sample material was cut into slices of about 0.5 mm thickness by means of a high precision wire saw. Samples were cut with plane normal parallel to the [0 0 0 1] , [2 11 0 ] and

[11 0 0] directions of the κ - Al - Mn - Ni structure∗, i.e. parallel and perpendicular to the compression direction. Further sample preparation was performed by standard procedures including subsequent grinding, dimpling and polishing with an alumina suspension. The final thinning of the TEM samples was performed by means of chemical etching, since the standard preparation procedures involving an argon-ion milling were found to cause beam damage of the material. Due to this damage the samples with very rough surface and uneven thickness were obtained. For chemical TEM-sample preparation an etchant with composition O66N17S17 (Vol.%) was used, involving the following constituents: (O) orthophosphoric acid (H3PO4, 84%), (N) nitric acid (HNO3, not smoking), and (S) sulphuric acid (H2SO4, 68%). Powder samples crushed by means of a mortar were investigated as well.

VII.2.2 Results Figure VII.8 shows a bright-field TEM micrograph of a self-flux grown Al - Mn - Ni sample deformed at T = 720 °C along [0 0 0 1] , up to ε = 4% . The TEM specimen was prepared parallel to the compression direction and the plane normal of figure VII.8 lies close to the [0 0 0 1] zone axis. The inset on the left-hand side of the image indicates the directions according to the hexagonal structure of the κ - Al - Mn - Ni phase. The orientation of the TEM specimen with respect to the geometry of the deformation sample is illustrated by the

∗

The indexing applied in the microstructural analysis described in this chapter corresponds to the hexagonal structure of the κ - Al - Mn - Ni phase, which was confirmed to exist in a pure form in the sample material at the temperature rage above 630 °C (c.f. chapter V.3). Accordingly, the observed structural defects introduced into the material during deformation are related to the κ-phase structure, since these were produced at the temperature range of κ - Al - Mn - Ni existence, i.e. as a result of κ-phase plastic deformation.

106

VII Plasticity of the κ-Al-Mn-Ni phase inset on the right-hand side of the image. The applied two-beam condition corresponds to the r reciprocal vector g = ( 7 1 8 0) (inset in the upper left corner).

Figure VII.8: Bright-field TEM image of deformed Al-Mn-Ni phase. The TEMspecimen possesses a surface normal parallel to [ 0 0 0 1 ] direction. The black and the white arrows indicate two different sets of dislocations. The scheme on the right-hand side of the micrograph illustrates the specimen orientation with respect to the compression direction of the deformation sample. The applied two-beam condition corresponds to r g = ( 7 1 8 0) (inset in the upper left corner).

Two different types of dislocations are visible in figure VII.8. One of them possesses a line direction parallel to the compression direction [0 0 0 1] , i.e. parallel to the normal of the TEMspecimen plane. Correspondingly, only short projections of the line segments of this dislocation type (marked in the image with white arrows) are visible in the micrograph. These are referred in the following as c-axis dislocations. The second type of dislocations possesses line directions which lie within the TEM-specimen plane, i.e. are perpendicular to [0 0 0 1] compression direction. The long segments of this dislocation type feature blurry contrast in figure VII.8. These are marked in the image with black arrows and referred as basal-plane dislocations. In the following, line directions, habit planes and Burgers vectors of both types of dislocations are analyzed separately. 107


Basal-plane dislocations Several segments of basal-plane dislocations visible in figure VII.8 as long dark lines (black arrows) are parts of loops extending in the (0 0 0 1) plane (basal plane of the hexagonal structure). Since the TEM sample was prepared with a surface normal parallel to the [0 0 0 1] direction, long segments of basal-plane dislocations are lying within the TEM-specimen plane. The dislocation loops feature facetted shape and several straight segments of the dislocation lines can be observed in figure VII.8. The orientations of these line segments correspond to specific crystallographic directions of hexagonal structure of κ - Al - Mn - Ni . They are aligned along < 2 11 0 > or < 11 0 0 > directions, which correspond to the long and short diagonal axes of the hexagonal unit cell. The straight loop segments possess an equal distribution along these directions. The occurrence of preferential directions is not observed.

Figure VII.9: Bright-field TEM micrographs of the Al - Mn - Ni phase in the orientation close to the [ 0 0 0 1 ] direction. Applied two-beam conditions correspond to r r r g = ( 3 5 8 0 ) ( a ) , g = ( 7 1 8 0 ) ( b ) and g = ( 7 7 0 0 ) ( c ) . The particular segments of the basal–plane dislocation, featuring the line direction parallel to the applied reciprocal vector are invisible in the respective images.

In the following, an analysis of invisibility conditions for the basal-plane dislocations is given. Figure VII.9 shows a TEM micrograph of a basal-plane dislocation loop segment under various imaging conditions (a-c). The applied two-beam conditions correspond to r r r g = ( 3 5 8 0) (a) , g = ( 7 1 8 0) (b) and g = ( 7 7 0 0) (c) . Considering equation (IV.1), the observed extinctions of defects contrast reveal that the Burgers vector of c-axis dislocations is 108

VII Plasticity of the κ-Al-Mn-Ni phase parallel to [0 0 0 1] direction. However, as in the case of c-axis dislocations, only specific r dislocation-loop segments, featuring the line direction l parallel to the employed reciprocal r vector g (cf. insets) are invisible in the respective micrographs. This is due to the pure edge character of basal-plane dislocations, the contrast of which can be completely extinct only if additionally to equation (IV.1) the condition (IV.2) is fulfilled as well (cf. chapter IV.1), i.e. if r r the reciprocal vector g is parallel to the dislocation line direction l . Dislocation segments which remain visible in figure VII.9 do not fulfil condition (IV.2) and their residual contrast can be observed. Accordingly, the Burgers vector of basal-plane dislocations is parallel to the normal vector of the dislocation habit plane. Therefore, it can be concluded that basal-plane dislocations move by means of a pure climb mechanism.

c-axis dislocations The dislocations featuring line direction parallel to [0 0 0 1] crystallographic direction (caxis) are shown in figure VII.10 in nearly end-on orientation. The plane normal of this micrograph is almost parallel to the [0 0 0 1] direction. The diffraction pattern corresponding to this zone axis is shown in the lower right corner of the figure. Between the dislocations, the stacking faults are present (in nearly edge-on orientation). Hence, it is concluded that the caxis dislocations are partial dislocations. The habit planes of these dislocations are arranged according to specific lattice directions reflected in the orientation of planar defects. Their r normal vectors n lie within the ( 0 0 0 1) plane and are parallel to < 11 0 0 > directions which correspond to the short diagonal axes of the hexagonal unit cell. Hence, three different orientations of these habit planes are observed. The black arrow indicates a position where the c-axis dislocation mowing on plane (11 0 0) split into other c-axis dislocations, which move within two different habit planes, corresponding to

(1 0 1 0 ) and ( 0 1 1 0 ) planes,

respectively.

109


Figure VII.10: Bright-field TEM image of c-axis dislocations in nearly end-on orientation. Stacking faults featuring the habit planes with normal vectors parallel to < 1 1 0 0 > crystallographic directions are visible between the individual partials in nearly edge-on orientation. The TEM-specimen possesses a surface normal close to [ 0 0 0 1 ] direction. The SAED pattern along this zone axis is shown in the lower right r corner of the image. The applied two-beam condition corresponds to g = ( 2 7 5 0) (inset

Several c-axis dislocations are shown in a TEM micrograph in figure VII.11. Here the specimen was cut parallel to the [11 0 0] direction, i.e. perpendicular to the compression direction. The inset in the right-hand side of the micrograph illustrates the TEM-specimen orientation with respect to the compression direction. The c-axis is indicated in the image by a long black arrow. Since the c-axis-dislocation lines lie within the plane of the TEM sample, long segments of these defects are visible as dark diagonal lines (white arrows). One bent caxis dislocation line is visible in the micrograph centre, indicating hereby that c-axis dislocations are actually dislocation loops elongated in [0 0 0 1] direction. The habit plane of these dislocations corresponds to the (1 1 0 0 ) plane, which does not coincide exactly with the image plane. Three dislocation lines leave the plane of the sample in the lower-right corner of figure VII.11, leading to a vanishing of the contrast of these partial dislocations.

110


Figure VII.11: Bright-field TEM image of c-axis dislocations in a specimen prepared perpendicular to the compression direction (cf. inset on the right-hand side). The

[ 0 0 0 1] direction (long black arrow) corresponds to the line direction of the partials. The image normal is close to the [ 1 1 0 0 ] direction. The applied two-beam condition r corresponds to g = ( 3 5 8 0 ) (inset in the upper left corner). Segments of basal-plane and c-axis dislocations are indicated by short red and white arrows, respectively.

Additionally, a stacking fault in edge-on orientation (indicated with bold black arrow) is visible as a thin diagonal line perpendicular to the [0 0 0 1] direction. This is terminated by the basal-plane dislocation (located far from the place where the image was taken, therefore not visible in figure VII.11). Hence, it is concluded that the basal-plane dislocations are partial dislocations. Due to their orientation with normal vector parallel to [0 0 0 1] , the stacking faults are not observable in the TEM specimens prepared with surface normal parallel to [0 0 0 1] as e.g. in figures VII.8 and VII.10. Short segment of another basal-plane dislocation is visible in figure VII.11 and indicated with a red arrow.

111

VII Plasticity of the κ-Al-Mn-Ni phase In the following, an analysis of the invisibility conditions for c-axis dislocations is given. Figure VII.12 shows TEM micrographs of elongated dislocation loop from the centre of figure VII.11 under various imaging conditions (a-d). The applied two-beam conditions correspond r r r r to g = (3 5 8 2) (a) , g = ( 3 5 8 4 ) (b) , g = (0 0 0 4) (c) and g = ( 3 5 8 0 ) (d) . Depending on the applied two-beam conditions, the contrast of particular loop segments featuring the line direction parallel or almost parallel∗ to the employed reciprocal vector is extinct.

Figure VII.12: Bright-field TEM micrographs close to the [ 5 3 2 0 ] zone axis. Applied r r r two-beam conditions correspond to g = ( 3 5 8 2) (a) , g = ( 3 5 8 4 ) (b) , g = ( 0 0 0 4 ) (c) r and g = ( 3 5 8 0 ) (d) . The contrast of dislocation segments featuring the line direction parallel to the applied reciprocal vector is extinct in the respective images.

∗

r Besides the g = ( 0 0 0 4 ) the operating reciprocal vectors can be only “almost” parallel to the dislocation line

directions, since they do not lie within the dislocation-habit plane (1 1 0 0 ) ; they originate from planes possessing the [ 5 3 2 0 ] zone axis, which is tilted from the habit-plane normal by about 8.2° towards

[ 2 1 1 0].

112

VII Plasticity of the κ-Al-Mn-Ni phase Considering equation (IV.1), the contrast extinctions observed reveal that the Burgers vector of c-axis dislocations is parallel to [ 5 3 2 0 ] direction. Nevertheless, only specific segments of the dislocation are invisible in the respective micrographs. Despite the fact that all reciprocal vectors applied in (a-d) fulfil equation (IV.1), the loop segments featuring the line direction unparallel to the operating reciprocal vector show a strong blurry residual contrast. This is due to the pure edge character of c-axis dislocations (i.e. the Burgers vector is perpendicular to the line direction of all loop segments). As described in chapter IV.1, the contrast of pure edge dislocations can be completely extinct only if the applied reciprocal r vector g , fulfilling general invisibility criterion (IV.1) is parallel to the dislocation line r direction l (i.e. condition (IV.2) is also fulfilled). Figure VII.12 reveals that only dislocation r r segments with l parallel or almost parallel to the applied g (cf. insets) are invisible. In the case of dislocation segments, which remain visible in figure VII.12, their line direction r deviates strongly from the orientation parallel to the operating g , which causes a strong residual contrast of these segments. Since the Burgers vector of c-axis dislocations is parallel to [ 5 3 2 0 ] direction, it possesses components in direction parallel and perpendicular to the normal vector of the dislocation habit plane. Movement of c-axis dislocations is therefore concluded to consist of mixture of glide and climb mechanism. The climb component is however considerably larger than that of the glide mechanism. Figure VII.13 shows a TEM micrograph of a specimen oriented along the [0 0 0 1] zone axis comprising a stacking fault caused by the motion of a < 5 3 2 0 > -c-axis dislocation. The stacking fault is indicated by black arrow. The normal vector of its habit plane is parallel to [11 0 0] direction. Green hexagons indicate unit cells of κ - Al - Mn - Ni on both sides of the r stacking fault. The displacement vector R of the stacking fault is shown in the micrograph as a red arrow.

113


Figure VII.13: TEM micrograph of the Al-Mn-Ni phase along the [0 0 0 1 ] direction. A r stacking fault is indicated by black arrow. The displacement vector R of the planar defect is indicated with a red arrow. Some unit cells corresponding to the κ-Al-Mn-Ni phase are drawn in green.

From the described above analysis of defects-invisibility conditions, it is known that the Burgers vector of c-axis dislocations is parallel to the [ 5 3 2 0 ] direction. Accordingly, the length of the displacement vector of a stacking fault produced by movement of c-axis dislocations can roughly be estimated from figure VII.13. Thickness of the inserted material slab corresponds to about 2/5 of the hexagonal unit cell of κ - Al - Mn - Ni along the b-axis. Accordingly, the displacement vector of the observed planar defect amounts, within the r measurement accuracy, to approximately | R |= 1.2 nm along [ 5 3 2 0 ] direction. This value is further discussed in section VII.3.

114


VII.2.3 Defect density In the following, the evaluation of density of planar (stacking faults) and linear (dislocations) defects in Al - Mn - Ni phase is given. The density of dislocations is calculated separately for basal-plane and c-axis defects. Analogously, density of stacking faults is determinate separately for defects formed by movement of particular dislocation types. Density of c-axis dislocations and of stacking faults produced by movement of these dislocations within {1 1 0 0} planes, are determined from TEM specimens cut with surface normal parallel to the [0 0 0 1] direction. The relation that results from comparison of equations (IV.9) and (IV.12), when describing the case of linear defects featuring the line direction with one defined crystallographic orientation is reduced to the following form

LV =

NA N = , cos ϕ A cos ϕ

(VII.1)

where ϕ is an angle between the specific dislocation line direction and the observation direction (for present TEM observations it is normal of the TEM sample surface) i.e. the z axis from figure IV.1. In the particular case of measurements performed on c-axis dislocations within the TEM sample with surface normal parallel to the [0 0 0 1] direction, the surface normal of the specimen is parallel to the line direction of these dislocations (cosϕ = 1) . Therefore the thickness of the specimen corresponds to the length of the observed dislocation line and the dislocation density can be calculated directly from the number of dislocations N in an observed area A

LV =

N . A

(VII.2)

The situation is analogous for the measurements performed on planar defects produced by caxis dislocations. When TEM sample with a surface normal parallel to the [0 0 0 1] direction is employed, the specimen thickness corresponds to the height of the observed planar defect (in edge on orientation). The density of stacking faults can be, therefore, calculated directly from the measured length of the stacking faults traces L within in an observed area A according to the equation (VI.3) as it was already made for measurements performed in the β - Al - Mg phase (cf. chapter VI).

For basal-plane dislocations and stacking faults generated in ( 0 0 0 1) planes by movement of these dislocations, the respective defects densities are determined from TEM specimens prepared with surface normal parallel to the [1 1 0 0] direction. Density of basal-plane 115

VII Plasticity of the κ-Al-Mn-Ni phase dislocations can be then calculated from the equation (VI.1), considering an isotropic distribution of linear defects in a plane featuring its normal vector perpendicular to the observation direction (i.e. direction [ 1 1 0 0] in present case). Analogously, the normal vector of planar defects generated by basal-plane dislocations is perpendicular to the applied observation direction. TEM specimen thickness corresponds, therefore, to the height of the observed planar defects. The density of stacking faults can be again calculated according to the equation (VI.3). The investigated TEM specimens were prepared from a sample deformed at 720 °C and 10-5 s-1 up to total strain of 4%. Reference investigations were carried out on undeformed material which was heat in the same way as the deformed sample. The densities of particular defects were determined according to equations referred above and are gathered in table VII.1. These densities are mean values obtained from investigations performed on a number of randomly selected TEM-specimen areas (200 for undeformed and 40 for deformed sample) observed under magnification of 25k times. The number of areas investigated for undeformed sample was increased in order to obtain better statistics, since an obviously low defects density was observed. In undeformed material, the evaluated densities of the respective dislocation or stacking fault types differ by about one order of magnitude. Upon deformation, the amount of c-axis dislocations increases by about one order of magnitude and by about two orders of magnitude increases density of basal-plane dislocations. In the case of planar defects, their amount rises by about two orders of magnitude for both defect types.

Table VII.1: Densities of dislocations ( LV ) and stacking faults ( AV ) in undeformed and deformed (720 °C, ε = 4 %) Al-Mn-Ni phase. Densities are given separately for basalplane and c-axis dislocations and for planar defects in {1 1 0 0} and ( 0 0 0 1 ) planes.

Type of defects

Defects density ( LV or AV ) undeformed sample deformed sample

116

c-axis dislocations

1 · 107 cm-2

3 · 108 cm-2

basal-plane dislocations

3 · 106 cm-2

3 · 108 cm-2

{1 1 0 0} stacking faults

7 · 102 cm-1

4 · 104 cm-1

( 0 0 0 1) stacking faults

75 cm-1

2.5 · 103 cm-1


VII.3 Discussion VII.3.1 Macroscopic deformation behaviour Uniaxial compression experiments at constant strain rate of 10-5 s-1 were successfully performed in the temperature range of 680 to 740 °C. This corresponds to the homologous temperature range TH between 0.84 and 0.89 (Tm = 1140 K, according to Balanetskyy et al., 2007). One of the most prominent features of the macroscopic deformation behaviour of κ - Al - Mn - Ni is the strong yield-point effect present in the material stress-strain curves,

which are shown in figure VII.1. Within the investigated temperature range, the yield drop following the stress maximum amounts up to 40% of the latter (at TH = 0.84). This value is much larger than usually encountered for CMAs. For instance, in the Al13Co4 phase, a maximum yield drop found amounts to about 28% (Heggen et al., 2007). In β-Al-Mg this value reaches 20% (cf. chapter VI), while almost no yield-point effect (below 3%) was observed in ξ’-Al-Pd-Mn (Feuerbacher et al., 2001). Only one CMA material, the µ phase from the binary Al-Mn alloy system, was reported until now to exhibit the yield drop of higher value. For this binary alloy the yield drop approaching to about 50% (at TH = 0.86) of a stress value corresponding to the UYP of the material was reported (Roitsch et al., 2007). In the microstructural investigations two sets of dislocations were identified as carriers of plastic deformation in the

κ - Al - Mn - Ni phase. In the sample deformed at 720°C

( ε total = 4 % ) the dislocation density is higher than in the undeformed reference sample by more than one order of magnitude for c-axis defects and by about two orders of magnitude for basal-plane dislocations. A comparison of dislocation densities of deformed and undeformed material confirms that the deformation mechanism in κ - Al - Mn - Ni is based on dislocation motion. According to the model elaborated by Johnston and Gilman (1959), there exists a strong correlation between the flow stress of the material and dislocation density in the course of its plastic deformation. The authors have demonstrated that the rate of dislocation multiplication depends strongly on the applied stress and enhanced dislocation multiplication can take place by crossing a stress threshold. Therefore, the appearance of a yield-drop effect at the onset of plastic deformation can be referred to a strong increase in the number of mobile dislocations contributing to the deformation process. The large magnitude of these yield-drop effects implies that the stress dependence of the dislocation-multiplication rate in the κ - Al - Mn - Ni phase is very high.

117

VII Plasticity of the κ-Al-Mn-Ni phase The presence of yield-point effects after incremental tests is usually associated with dislocation recovery process i.e. the decrease of the mobile dislocation density during unloading at elevated temperatures (Hull and Bacon, 1984). The observed absence of yieldpoint effects in the stress-strain curves of κ - Al - Mn - Ni after performed incremental tests indicates that recovery takes place at low rates in this material. Thermodynamic activation parameters of the deformation process were evaluated due to incremental tests. The activation volume of κ - Al - Mn - Ni (figure VI.5) was calculated employing a Schmid factor value of ms = 1, i.e. assuming climb mechanism as a dominating mode of dislocation motion. Although no Schmid factor can be defined for dislocation climb, a value of ms = 1 is an assumption widely accepted in the literature for dislocation climb occurring under normal stresses (e.g. Nandy and Banerjee, 2000, Mitra et al., 2004, Malaplate et al., 2005), since the climbing dislocations resolve the full applied stress.

The concluded deformation mechanism based on dislocation climb is in good agreement with the values of the stress exponent of

m = 5.3 (figure VI.6) observed for

the κ - Al - Mn - Ni phase. According to Kassner and Pérez-Prado (2000) the “five-power-law creep”, i. e. the creep deformation with m = 5, is a signature of a dislocation climb mechanism. The activation volume V of κ - Al - Mn - Ni scaled by the atomic volume Va (i.e. the average volume per atom), amounts to V/Va = 19 at a stress of 300 MPa. This value is of the same order of magnitude as the number of atoms in present clusters, i.e. icosahedra or I3-clusters. These clusters may act as obstacles for dislocation motion and might therefore increase the values of activation volume observed. Accordingly, the rate controlling process of dislocation motion in κ - Al - Mn - Ni might be provided by the interaction of dislocations with these clusters. Friction between dislocations and the cluster substructure is also assumed as a ratecontrolling factor of dislocation motion in β - Al - Mg (c.f. section VI.3.1). High values were also found for the activation enthalpy ∆H of κ - Al - Mn - Ni (figure VI.7). The activation enthalpy ∆H is about fifteen times larger than the work term ∆W, which indicates that the deformation process is thermally activated. In the investigated temperature range an activation enthalpy amounts from about 6 to 8 eV. These magnitudes, however, exceed the physically reasonable values of an activation enthalpy since processes involving such high energy values would run at very low rates. Consequently, any effects would not be observable on usual laboratory time scales. Additionally, by extrapolation of the activation enthalpy to lower temperatures in figure VII.7, a value of ∆H = 0 is reached at about 700 K. According to Gibbs (1969), however, the activation enthalpy should be proportional to temperature and extrapolation of its temperature behaviour down to low temperatures should lead to ∆H = 0 at T = 0 K, if no change of deformation process takes place in the considered 118

VII Plasticity of the κ-Al-Mn-Ni phase temperature range. Therefore, the present values of the activation enthalpy of κ - Al - Mn - Ni are to be considered unreliable and only can be used for very rough estimates. For the concluded climb mechanism of dislocation movement, the activation enthalpy ∆H is generally assumed to correspond to the enthalpy of lattice self-diffusion. This assumption originates from the fact that diffusion processes are typically the rate-controlling factor for dislocation climb. No values of the diffusion enthalpy for κ - Al - Mn - Ni are reported in the literature. The value for self-diffusion in pure Al amounts to 1.28 eV (Messer et al., 1974), while the diffusion enthalpies for Mn and Ni in Al range between 2.2 and 2.4 eV (Beke et al., 1987) and between 1.76 and 3.5 eV (Berkowitz et al., 1954, Hoshino et al., 1988, Hancock, 1971, Hancock and McDonnell, 1971 and Lutze-Birk and Jacobi, 1975), respectively. Assuming that the diffusion enthalpy for κ - Al - Mn - Ni compares to these values, the activation enthalpy evaluated in present work reveals a significant difference to the diffusion enthalpy. Very similar deformation behaviour was reported recently by Roitsch (2008) for µ - Al - Mn , which is a CMA phase closely related to κ - Al - Mn - Ni . There exist several

apparent similarities between both materials. First of all, the main chemical constituents of the κ and µ phase are Al and Mn and the proportion of the atomic content of these elements is

close to 4:1 in both compounds. The hexagonal lattice, icosahedral atoms arrangements in the local structure and the deformation mechanism mediated by dislocation climb are also a common feature of κ - Al - Mn - Ni and µ - Al - Mn . At this point it is indispensable to discuss the numerous similarities, which are found in the macroscopic deformation behaviour of these materials as well. A striking resemblance between the magnitude of the yield drops in the stress-strain curves of κ - Al - Mn - Ni and µ - Al - Mn was already mentioned at the very beginning of present discussion. The further similarities of these materials are reflected in values of the calculated activation parameters. The stress dependence of the activation volume evaluated for µ - Al - Mn was fit with the hyperbolic function V = 85.5/σ (Roitsch, 2008), which is very

close to the fit function employed in present work for κ - Al - Mn - Ni ( V = 85/σ ). The resulting values of the activation volume scaled by the atomic volume of the particular phase are also very close (according to Roitsch (2008) V/Va = 20 for µ - Al - Mn at σ = 300 MPa ). The stress exponent found by Roitsch (2008) for µ - Al - Mn amounts to approximately m = 5.5 , which is very close to m = 5.3 determined here for κ - Al - Mn - Ni . Finally, comparably high values of the activation enthalpy (from 5 to 9 eV at the temperature range between 700 and 850°C) and a deviation from their proportional temperature dependence, as found in present work for κ - Al - Mn - Ni (see above), are also reported by Roitsch (2008) for µ - Al - Mn . For the latter phase, the author interprets this phenomenon by a continuous

119

VII Plasticity of the κ-Al-Mn-Ni phase transition between two deformation-controlling processes at different temperatures. For the concluded dislocation climb mechanism, diffusion processes at adequate rates are necessary. At higher temperatures, however, the lattice diffusion is most probably only a subordinated factor and dislocation-cluster friction might be a rate-controlling mechanism. It is suggested in the interpretation of the activation volume (see discussion above), that atom clusters act as obstacles against dislocation movement, and that the rate-controlling mechanisms for dislocation motion are provided by thermally activated overcoming of these obstacles in the investigated temperature range. Following the interpretation of Roitsch (2008), the discrepancy between the values of diffusion- and activation enthalpy can be explained if the friction between dislocations and the cluster substructure determines the magnitude of the activation enthalpy in the investigated temperature range, rather than lattice diffusion. At lower temperatures, however, where the (extrapolated) activation enthalpy approximates the assumed diffusion enthalpy, diffusion might become a factor that strongly controls deformation processes based on dislocation climb mechanisms. Taking into account the structural similarities and the agreement of the experimental results between κ - Al - Mn - Ni and µ - Al - Mn , it can be can assumed that these conclusions hold for κ - Al - Mn - Ni as well.

VII.3.2 Microstructural analysis The microstructural analysis of the processes contributing to the plastic deformation of κ - Al - Mn - Ni was performed by means of TEM. Nevertheless, already the initial

observations of plastically deformed samples, which were performed by means of LM, show that single-crystalline material of the Al - Mn - Ni phase exhibits inhomogeneous deformation behaviour. The horizontal and vertical lines appearing upon deformation on the sample surface (figure VII.3) indicate that the deformation processes are localized within the specific crystallographic planes. Accordingly, two different types of dislocations, i.e. c-axis and basalplane defects, were identified upon subsequent TEM investigations, to move on {1 1 0 0} and ( 0 0 0 1) crystallographic planes, respectively. The analysis of invisibility conditions for basal-plane dislocations is shown in figure VI.9. It revealed the Burgers vector of these defects to be parallel to the [ 0 0 0 1] direction. It is, therefore, parallel to the normal vector of dislocation habit-plane and perpendicular to dislocation line direction. Correspondingly, it could be concluded that basal-plane dislocations possess pure edge character and move by means of a pure climb mechanism. The Burgersvector length of basal-plane dislocations in κ - Al - Mn - Ni was difficult to be estimated on a basis of performed experiments. As discussed in chapter VI, a technique commonly used in 120

VII Plasticity of the κ-Al-Mn-Ni phase TEM for determination of the length of dislocation Burgers vector is convergent-beam electron diffraction (CBED) (e.g. Tanaka et al., 1988). However, as in the case of β − Al − Mg , the density of Kikuchi lines of the Al - Mn - Ni phase is too high for

unambiguous determination of the number of splitting nodes in the strain field of the dislocation (Feuerbacher et al., 2004). Therefore, in this material determination of the Burgers-vector length by means of CBED was found practically impossible. Unfortunately, also no high quality high resolution TEM images along crystallographic directions perpendicular to the c-axis of this material could be obtained up to now. Nevertheless, the formation of planar defects in ( 0 0 0 1) planes due to the movement of basal-plane dislocations is observed (e.g. figure VI.12). The basal-plane dislocations are therefore considered as partial dislocations, i.e. dislocations featuring the Burgers-vector lengths r corresponding to only a fraction of the lattice periodicity along c-axis, i.e. | b | < 1.24 nm . The appearance of partial dislocations in the Al - Mn - Ni phase is in accordance with the considerations on the elastic line energy of the linear defects occurring in CMAs. The presence of partial dislocations featuring Burgers-vector lengths of only a fraction of the lattice periodicity in materials featuring giant unit cells is energetically more reasonable than occurrence of perfect dislocations. The c-axis dislocations are dislocation loops elongated in [ 0 0 0 1] direction moving within {1 1 0 0} planes of the κ - Al - Mn - Ni structure. From contrast-extinction experiments in r TEM it was found that their Burgers vectors b lie within the ( 0 0 0 1) plane and are parallel to the < 5 3 2 0 > directions. It can be concluded, therefore, that c-axis dislocations possess pure edge character and move by means of a mixture of climb and glide. The displacement-vector length of the stacking faults generated by movement of c-axis dislocation is estimated from the micrograph shown in figure VII.13. The thickness of the stacking fault observed amounts to about 2/5 of the hexagonal unit cell of κ - Al - Mn - Ni along b-axis. The corresponding displacement vector of this defect amounts, within the r measurement accuracy, to approximately | R |= 1.2 nm . This corresponds to about 2/5 of the hexagonal unit cell along the [ 5 3 2 0 ] direction, i.e. the determined direction of the c-axisdislocation Burgers vector. If considered, however, as a magnitude of the Burgers vector of a single dislocation, this length exceeds the physically acceptable value with regard to the elastic line energy of dislocation. In consequence, it is concluded that the planar defect was formed by motion of several smaller partials moving on adjacent layers, rather than of one single c-axis dislocation. To conclude however on the possible modulus of the Burgers vector, more HRTEM investigations are needed to observe a single partial terminating a planar defect. 121

VII Plasticity of the κ-Al-Mn-Ni phase Nevertheless, the observed size of the displacement vector of planar defects occurring in

{1 1 0 0} planes reflects some characteristic distances in the hexagonal unit cell of κ - Al - Mn - Ni along the < 5 3 2 0 > direction. It can be described in terms of a [0 0 0 1]

projection of the crystal structure. Figure VII.14 shows a schematic representation of the κ - Al - Mn - Ni structure along the [0 0 0 1] crystallographic direction according to the

structural model adapted in present work (cf. chapter I.3.2). The displacement vector of r | R |= 1.2 nm corresponding to a {1 1 0 0} -stacking fault, as observed in figure VII.13, is indicated in this figure by red arrows.

Figure VII.14: Projection of the atoms stacking in the κ-Al-Mn-Ni phase along [ 0 0 0 1 ] direction based on a model given by Li et al., and Sato et al. in 1997 for κ-Al-Cr-Ni phase in which the Cr atoms are replaced by atoms of Mn. Green, red and blue spheres represents Al, Mn and Ni atoms, respectively. The hexagonal unit cell of κ-Al-Mn-Ni is drawn in black. The red arrows indicate the length of the vector corresponding to 2/5 of the hexagonal unit cell along [ 5 3 2 0 ] direction.

122

VII Plasticity of the κ-Al-Mn-Ni phase The area of the model projection restricted by red arrows and black lines parallel to [ 1 1 2 0 ] direction corresponds to the material volume that, in terms of a Volterra-construction (e.g. Hirth and Lothe, 1992), can be inserted upon deformation via movement of c-axis dislocations producing a planar {1 1 0 0} defect. It can be observed in figure VII.14 that red arrows connect the atomic positions of very similar, icosahedral configurations of their surrounding. Although in the applied deformation geometry, there is no shear stress motivating a dislocation glide in {1 1 0 0} planes (ms = 0), a driving force for occurrence of a < 1 1 2 0 > glide component in the c-axis dislocation movement can originate in the structural characteristics of the

κ - Al - Mn - Ni phase. If only pure climb mechanism in {1 1 0 0} planes, introducing a material slab of a thickness observed in figure VII.13 took place, more next-neighbour discrepancies would have to be introduced within the inserted material slab boundary than in the case when additional slip of dislocation in {1 1 0 0} < 1 1 2 0 > occurs. A comparison of both example situations is shown in figure VII.15. The projections represent our material with the stacking fault in (1 1 0 0) plane of the thickness reported above. Part (a) of the drawing in figure VII.15 shows the situation resulted in a pure climb mechanism taking place in (1 1 0 0) plane, while part (b) shows the atom ordering of the material in which additionally relative a shift of the crystal parts on both sides of the defect occurs in (1 1 0 0) [ 1 1 2 0 ] system. The corresponding displacement vectors are indicated with red arrows on both image parts. The slab of material placed between horizontal lines connecting heads and ends of red arrows in figure VII.14 is inserted in figure VII.15 into material featuring a perfect κ-phase atoms ordering. The insertion position corresponds to the horizontal line connecting the heads of arrows in figure VII.14. This means that the introduced material slab follows the atoms ordering of the matrix material at the upper defect border and the discrepancies in the next-neighbour atom arrangements will occur at its lower border, which is indicated in figure VII.15 with dashed line. The orange rings above the dashed line indicate the “perfect” positions of atomic columns, which would exist if the undistorted material stacking was continued upwards following the perfect structure of the crystal part placed below the stacking fault. It can be observed that more atom positions existing within planar defect overlap with the “perfect” locations in (b) than in (a). This means that the introduction of a small relative shift in [ 1 1 2 0 ] direction of the crystal parts located on both sides of the planar defect reduces the amount of the next-neighbour discrepancies which arise from the introduction of the material slab by the climb component of c-axis dislocation movement. Therefore, the movement of < 5 3 2 0 > -c-axis dislocations can be concluded to result in the introduction of energetically favourable planar defects. 123


Figure VII.15: Schematic projections of the atom positions in the κ-Al-Mn-Ni phase along [ 0 0 0 1 ] direction including a stacking fault in (1 1 0 0) plane with a defect thickness corresponding to about 2/5 of the hexagonal unit cell of κ-Al-Mn-Ni along the baxis. Atomic arrangements on the border of planar defect (above dashed line), in the situation if (a) only pure climb mechanism would take place in (1 1 0 0) plane and (b) mixture of climb and glide mechanism resulting in a displacement vector corresponding to 2/5 of the hexagonal unit cell along [ 5 3 2 0 ] direction appears, are enhanced.

The two climb mechanisms observed in κ - Al - Mn - Ni , i.e. climb of basal-plane dislocations and climb component of c-axis dislocations movement are complementary and can effectively interact by exchanging vacancies or interstitial atoms via diffusion process. In such case only diffusion distances between the involved dislocations sets need to be passed rather than the distances between dislocations and sample surface. Climb of basal-plane dislocations (primary mechanism) corresponds to the deformation of the sample by removing atomic ( 0 0 0 1) planes (positive climb). Absorption of vacancies necessarily takes place during this process. Climb of c-axis dislocations in {1 1 0 0} planes of the κ - Al - Mn - Ni phase corresponds, on the other hand, to negative climb (causes broadening of a sample) and, hence, acts as source of vacancies for the primary climb mechanism.

124

VII Plasticity of the κ-Al-Mn-Ni phase The same deformation mechanisms are observed in the structurally related to the κ phase, hexagonal µ-Al-Mn and in cubic phase Mg32(Al,Zn)49 as well (Roitsch, 2008). This fact is quite remarkable, since for widely investigated family of structurally simple materials, only a few materials are reported in the literature, which show a plastic-deformation behaviour primarily mediated by this mode of dislocation motion (e.g. Le Hazif et al., 1968, Edelin and Poirier, 1973). The climb processes in all three CMA phases, κ-Al-Mn-Ni, µ-Al-Mn and Mg32(Al,Zn)49, take place on (0 0 1) planes ( (0 0 0 1) in the hexagonal structures). In the applied deformation geometry, the climbing dislocations contribute efficiently to the deformation process by removing atomic layers with normal vector parallel to the compression direction. Dislocation climb is, however, a non-conservative process of dislocation motion, i.e. it needs to be accompanied by atom transport via lattice diffusion. The positive climb mechanism corresponding to removal of atomic layers in (0 0 1) or (0 0 0 1) planes acts, therefore, as sinks for vacancies. Since the formation rate of thermal vacancies is usually too low to compensate the vacancy consumption taking place, a strong decrease of the vacancies concentration arises in the dislocation neighbourhood. The resulting vacancyconcentration gradient generates considerable chemical stress in the material (Le Hazif et al., 1968). Accordingly, if no vacancy source is present, the occurring chemical stress counteracts the driving force of the positive climb processes and therefore hampers the deformation process taking place. Therefore, in the CMA materials in which dislocation climb provides the primary mechanism of the material deformation, i.e. in κ-Al-Mn-Ni, µ-Al-Mn and Mg32(Al,Zn)49, the secondary climb mechanisms driven by the chemical stress are present. The latter cannot contribute directly to the straining of the deformed material in the compression experiments, since no mechanical force, that could drive motion of such dislocations, occurs in the applied deformation geometry. They correspond, however, to negative climb mechanism that causes the sample broadening. By inserting the atomic layers into sample material the secondary climb mechanisms act as efficient sources of vacancies for the primary climb mechanisms. Such, complementary climb mechanisms (Feuerbacher, 2008) in κ-Al-Mn-Ni, µ-Al-Mn and Mg32(Al,Zn)49 ensure short diffusion distances for vacancies transport. In the following, a rough estimation of plausible diffusion distances which have to be covered during κ - Al - Mn - Ni deformation is given. The diffusion distance x can be calculated according to the Einstein-Smoluchowski equation (e.g. Tilley, 2004): x = 6 Dt ,

(VII.1)

where D is the diffusion coefficient and t is the time of diffusion process. Since no values for diffusion rates in κ - Al - Mn - Ni are available in the literature, the diffusion coefficient of Al self-diffusion calculated for the potential temperature of 1000 K, 125

VII Plasticity of the κ-Al-Mn-Ni phase i.e. D ≈ 6 ⋅ 10 −6 mm 2 s −1 (Stöcker, 1994), is adopted here. A plastic deformation of a sample by an amount of strain corresponding to the length change below a single c-lattice parameter of κ - Al - Mn - Ni , i.e. 0 < ∆L < c = 1.24 nm can be assumed as the length change caused by

climb of one basal-plane dislocation (It was concluded above that the basal plane dislocations possess a Burgers vector length corresponding to a fraction of a κ - Al - Mn - Ni lattice r periodicity along c-axis, i.e. | b | < c = 1.24 nm ; therefore, a length change caused by climb of one basal-plane dislocation will be smaller than ∆L = 1.24 nm ). Taking into account a strain rate of 10-5 s-1 applied in the deformation experiments, the deformation time up to about 0.06 s is necessary to approach the maximum value of the assumed strain. If this time value is assumed as effective diffusion time t in our material, an average diffusion distance within which the vacancies or interstitial atoms can be transported to provide the formation of the planar defect of a given thickness amounts to about 10-3 mm. This value is of the same order of magnitude as average distance between the two interacting sets of dislocations in undeformed κ - Al - Mn - Ni and even an order of magnitude higher than distance between these defects in the sample deformed at 720°C (according to the experimental dislocationdensities measurements performed). On the other hand, the estimated diffusion range is about three orders of magnitude smaller than the average distance between the dislocation and the sample surfaces. The possibility of the vacancies transport from the sample surface and of interstitial atoms in opposite direction is therefore excluded as the process providing the desired material exchange. This testifies the necessity of diffusion interactions between basalplane and c-axis dislocations which have to take place in order to provide an effective deformation mechanism of κ - Al - Mn - Ni based on dislocation climb.

126

Summary In the present work, the single-crystal growth routes and the plasticity of selected complex metallic alloys were investigated. Macroscopic as well microstructural examinations on two materials, face-centred cubic β-Al-Mg and hexagonal κ-Al-Mn-Ni were carried out. High-quality single crystals of the two phases of interest were successfully produced. The single-crystal growth route of both materials was developed by employing three selected growth methods, Bridgman, Czochralski and self-flux growth technique. In the case of the Samson phase (β-Al-Mg), the crystallographically oriented single crystals of a size exceeding 3cm3 were gown by means of the Czochralski technique, while the application of the self-flux growth technique resulted with unoriented single crystals, the size of which exceeds 17 cm3. Within the growth experiments performed for the κ-Al-Mn-Ni phase, the largest single crystals (grain size ~0.2 cm3) of the Al-Mn-Ni phase of a composition corresponding to the stability range of κ - Al - Mn - Ni were produced by means of the self-flux growth technique. The homogeneous κ - Al - Mn - Ni phase was confirmed by means of the in-situ TEM to exist in the single-crystalline material grown at the temperature range above 630 °C. The determination of the deformation behaviour of both phases was performed at elevated temperatures (between 200 and 375 °C for β - Al - Mg and between 680 and 740 °C for κ - Al - Mn - Ni ) on the produced single crystals. Using single-crystalline material ensures the determination of the intrinsic material properties without effects of secondary phases or grain boundaries. Uniaxial compression tests were performed at constant strain rates of 10-4 s-1 ( β - Al - Mg ) and 10-5 s-1 ( κ - Al - Mn - Ni ) along [10 0] and [0 0 01] directions, respectively. Stress-strain curves of the κ - Al - Mn - Ni phase were recorded for the first time. Both materials were found to feature ductile behaviour at the elevated temperatures. Thermodynamic activation parameters of the deformation processes at different temperatures were determined by means of incremental tests. Activation volumes of both materials indicate that atom clusters, present in the crystal structures, form primary obstacles against dislocation motion. It is argued that friction between dislocations and cluster substructure provides the rate-controlling mechanism of the deformation processes in the investigated temperature range. The activation enthalpy indicates that plastic deformation is thermally activated in both materials. Macroscopic deformation experiments were complemented by detailed microstructural investigations performed by means of TEM. Comparison analyses of defects density in deformed and undeformed samples reveal that deformation in both phases is mediated by dislocation motion. The underlying deformation mechanisms were successfully determined.

127

Summary All dislocations involved are partial dislocations and, accordingly, upon movement introduce stacking faults into the crystal structure. In β - Al - Mg the deformation process is efficiently mediated by dislocation glide mechanism taking place on {111} planes. Planar defects introduced into the material via motion of these dislocations feature a microtwin character and a nanometre-size thickness. In the low strain range of the deformation process, additionally dislocation climb mechanism takes place on planes featuring the normal vector parallel to the compression direction ( (1 0 0) planes). The latter mechanism plays only a subordinated role in the β-phase deformation and at the experimental conditions applied it is blocked when the strain value approaches the level corresponding to the LYP of the material stress-strain behaviour. The microstructural deformation mechanism of the κ - Al - Mn - Ni is based on the pure climb of prismatic dislocation loops taking place on planes with normal vector parallel to the compression direction ( (0 0 01) planes). This climb mechanism contributes efficiently to the deformation process by removal of atomic layers and, therefore, acts as sink for vacancies. It is accompanied by diffusion of vacancies towards the dislocation core, which produces a vacancy-concentration gradient within the material. This gradient causes the chemical stress which counteracts the driving force of the (0 0 01) -climbing process. A secondary climb process acts as a source for vacancies and is driven by the chemical stress caused by the primary (0 0 01) -climb. The two climb mechanisms are complementary and interact with each other by exchanging the vacancies via diffusion. The secondary climb mechanism takes place on planes featuring the normal vector perpendicular to the compression direction along < 1 1 0 0 > directions of the hexagonal κ-phase structure. This climb mechanism is

accompanied by a small glide component of the involved dislocations. The latter mechanism acts as a factor reducing the spatial incompatibilities on the boundary of the planar faults produced via dislocation climb in the {1 1 0 0 } planes. Several similarities are found in the microstructural deformation processes of the β - Al - Mg and κ - Al - Mn - Ni phases in comparison with other CMAs. In particular, the

partial character of dislocation segments and the friction between dislocations and the cluster substructure are favoured concepts in the deformation process of CMAs. The occurrence of a climb mechanism as a primary mode of dislocation movement, as observed on the example of the κ - Al - Mn - Ni phase, is frequently found as a basic feature of CMA phases, as well. Complementary climb systems, which interact via diffusion and affect each other by means of a chemical stress, were previously observed in the other as-far investigated CMAs. The mechanism encountered in the β - Al - Mg phase, based on the movement of partial

128

Summary dislocations which introduce the planar defects being altered-matrix-structure slabs of a definite thickness, reflects the mechanism observed recently in another CMA. The macroscopic as well as the microstructural features of investigated materials reflect the interplay between the two different length scales present in the CMA phases, which are defined by the cluster substructure on the one hand and the unit-cell dimension on the other hand. The values of the evaluated activation enthalpy and partial-dislocation Burgers vectors present in CMAs reflect the former length scale, while the crystallographic orientation of the habit planes for dislocation motion cares about the constraints arising from the long-range lattice periodicity.

129

Conclusions The growth routes for the single-crystalline material of the β - Al - Mg

and

κ - Al - Mn - Ni were developed by employing three selected growth methods, Bridgman,

Czochralski and self-flux growth technique

High quality single-crystalline material of the β - Al - Mg phase was successfully grown by means of the Czochralski method (oriented single crystals of the size exceeding 3 cm 3 ) and the self-flux growth technique (very large but not oriented single crystals of the volume approaching 17 cm 3 )

For the κ - Al - Mn - Ni phase, the largest single crystals of a size of about 0.2 cm 3 were produced by means of the self-flux growth technique. The complementary in-situ TEM investigations of the produced material confirmed the existence of a homogeneous κ - Al - Mn - Ni phase in a single-crystalline form at the temperature range above 630 °C .

The macroscopic deformation behaviour of both phases was determined on the produced single

crystals

by

the

uniaxial-compression

tests

at

elevated

temperatures.

Thermodynamic activation parameters of the deformation processes at different temperatures were determined by means of incremental tests.

Systematic uniaxial compression experiments on the single-crystalline β - Al - Mg samples were performed along [10 0] crystallographic direction at the temperature range 200 − 375 °C and constant strain rate of 10 − 4 s −1 . Additionally, set of compression experiments was performed at 250 °C up to the three deformation stages (UYP, LYP and ε = 6% ).

Deformation experiments of the κ - Al - Mn - Ni single crystals were carried out along [0 0 01] direction at the temperature range 680 - 740 °C and constant strain rate of 10 − 5 s −1 .

Both materials feature ductile behaviour at the investigated temperature range. Plastic deformation processes of the β - Al - Mg and κ - Al - Mn - Ni phases are identified as thermally activated. Evaluated values of the activation volume indicate that atom clusters form primary obstacles for dislocation motion in both materials. Friction between dislocations and cluster substructure is proposed as the rate-controlling mechanism of the deformation processes taking place in the investigated temperature range.

Deformation mechanisms of both materials were successfully determined via microstructural investigations performed by means of TEM. The plasticity of the 131

Conclusions investigated phases was revealed to be mediated in each case by movement of partial dislocations. Accordingly, stacking faults are present in deformed sample material.

Deformation process of the β - Al - Mg single crystals in the applied deformation geometry are mediated by dislocations glide taking place in the in the {1 1 1} < 1 1 2 > glide system. A subordinated mechanism of dislocation movement in β - Al - Mg was identified as a pure climb occurring in the (1 0 0)[1 0 0] system.

Deformation process of the κ - Al - Mn - Ni single crystals in the applied deformation geometry is based on the pure climb of prismatic dislocation loops in the (0 0 0 1)[0 0 0 1] system. A complementary mechanism of dislocation movement takes place on planes featuring the normal vector perpendicular to the compression, in the

{1 1 0 0}[ 5 3 2 0 ] system of the hexagonal κ-phase structure. The macroscopic as well as the microstructural features of investigated materials reflect the interplay between the two different length scales present in the CMA phases. The values of the evaluated activation enthalpy and partial-dislocation Burgers vectors reflect the length scale defined by the cluster substructure, while the crystallographic orientations of the habit planes of dislocation motion care about the constraints arising from the long-range lattice periodicity of CMAs.

132

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List of symbols and abbreviations ε ε el

strain elastic strain

ε plast

plastic strain

ε& ρ ρ&

strain rate density of dislocations rate of the dislocation-density increase

θ λ σ σ& τ τeff τi ϕ

azimuthal angle angle between compression direction and slip direction stress stress rate shear stress effective stress internal stresses zenith angle

Φ

angle between compression direction and slip-plane normal n

A ∆A

sectional area of the material sample to which the area passed by a dislocation line during overcoming an obstacle

AV

planar-defects density; surface-area-to-volume ratio

A(X' Y' )

projection area defined by the X' and Y' axes

a, b, c b r b bcc c1, c2,… E F ∆F F, f fcc r g

lattice parameters modulus of the Burgers vector Burgers vector. body centered cubic chemical composition of an alloy in the phase diagram Young’s modulus external force Helmholtz free energy flat layers of atoms stacking in the κ-phase along c axis face centered cubic reciprocal lattice vector

∆H h( x' , y' )

Gibbs free energy activation enthalpy height of the internal surfaces or length of straight dislocations in the material

k

volume Boltzmann’s constant, = 8617343(15) x 10-5 eV·K-1

∆G

139

List of symbols and abbreviations r K r l L L l l0

line direction of dislocation liquid phase in the phase diagram entire length of dislocations in the material length (of dislocation or deformation sample) initial length of the deformation sample

LA

density of the intersection lines within the specified area

LV

dislocations density; length-to-volume ratio

m mS n N

stress exponent Schmid factor slip-plane normal number of intersection points density of the intersection points on specified surface area

NA

driving force of dislocations movement

P, P’, p, p’ r R r R ∆S T TC TEM r u x v V* V v0 ∆W

puckered layers of atoms stacking in the κ-phase along c axis atomic radius relaxation test

BM CMA CZ DTA EDX FG HRTEM LIMF LM LYP PXRD SAED SEM

Bridgman technique complex metallic alloy Czochralski technique differential thermal analysis energy-dispersive X-ray analysis flux growth technique high resolution transmission electron microscope levitation induction melting furnace light microscopy lower-yield point and powder X-ray diffraction selected area electron diffraction scanning electron microscope

140

displacement vector of planar defect activation entropy absolute temperature temperature change transmission electron microscope dislocation strain field distance dislocation velocity activation volume material volume or experimental activation volume attempt frequency work term

List of symbols and abbreviations TEM UYP

transmission electron microscope and the upper-yield point

141