STRUCTURE OF QUASICRYSTALS AND RELATED PHASES

Transcription

1 STRUCTURE OF QUASICRYSTALS AND RELATED PHASES ANANDH SUBRAMANIAM Guest Scientist (Alexander Von Humboldt Fellow) Electron Microscopy Group Max-Planck-Institut für Metallforschung STUTTGART Ph: (+49) (0711) , Fax: (+49) (0711) November 2004

2 OUTLINE OVERVIEW DEFINITION DISCUSSION PROJECTION FORMALISM CLUSTER BASED CONSTRUCTION Mg-Zn Zn-(Y, La) SYSTEMS Babuji

3 HYPERBOLIC EUCLIDEAN SPHERICAL SPACE nd + t UNIVERSE PARTICLES ENERGY STRONG WEAK ELECTROMAGNETIC GRAVITY FIELDS METAL SEMI-METAL SEMI-CONDUCTOR INSULATOR BAND STRUCTURE ATOMIC STATE / VISCOSITY NON-ATOMIC GAS SOLID LIQUID LIQUID CRYSTALS STRUCTURE AMORPHOUS QUASICRYSTALS RATIONAL APPROXIMANTS CRYSTALS SIZE NANO-QUASICRYSTALS NANOCRYSTALS

4 VARIOUS SPACES INVOLVED 1D, 2D, 3D 4D, 5D, 6D 7D,..., ND PHYSICAL SPACES QC HYPERSPACES GENERALIZED HYPERSPACES PARALLEL SPACE (E ) PERPENDICULAR SPACE (E ) REAL SPACE RECIPROCAL SPACE

5 QUASICRYSTALS (QC) ORDERED PERIODIC QC ARE ORDERED STRUCTURES WHICH ARE NOT PERIODIC CRYSTALS QC AMORPHOUS CRYSTALS (XAL) MODULATED STRUCTURES (MS) INCOMMENSURATELY MODULATED STRUCTURES (IMS) QC Can be thought of as IMS which cannot be constructed with a single unit cell but can be thought of as covering with a single prototile

6 SYMMETRY XAL QC t R C R CQ t translation inflation R C rotation 2, crystallographic 3, 4, 6 R CQ R C + 5, other 8, 10, 12 QC are characterized by inflationary symmetry and can have disallowed crystallographic symmetries DIMENSION OF QUASIPERIODICITY (QP) QP QP/P QP/P QC can have quasiperiodicity along 1,2 or 3 dimensions HIGHER DIMENSIONS QC can be thought of as crystals in higher dimensions (which are projected on to lower dimensions) QP XAL

7 THE FIBONACCI SEQUENCE Fibonacci Ratio 1/1 2/1 3/2 5/3 8/5 13/8 21/13 34/21... = ( 1+5)/2 WHERE IS THE ROOT OF THE EQUATION x 2 x 1 = 0 Convergence of Fibonacci Ratios Ratio n

8 B A B A Deflated sequence a Rational Approximants B A B B A B B A B A B B A B A B B A B B A B A B B A B B A b ba bab babba 1-D QC Penrose tiling Schematic diagram showing the structural analogue of the Fibonacci sequence leading to a 1-D QC

9 LIST OF QC.ppt

10 FOUND! THE MISSING PLATONIC SOLID [2] [1] [1] I.R. Fisher et al., Phil Mag B 77 (1998) 1601 [2] Rüdiger Appel, Mg-Zn-Ho

11 DISCUSSION

12 STRUCTURE OF QUASICRYSTALS QUASILATTICE APPROACH (Construction of a quasilattice followed by the decorationof the lattice by atoms) PROJECTION FORMALISM TILINGS AND COVERINGS CLUSTER BASED CONSTRUCTION (local symmetry and stagewise construction are given importance) TRIACONTAHEDRON (45 Atoms) MACKAY ICOSAHEDRON (55 Atoms) BERGMAN CLUSTER (105 Atoms)

13 HIGHER DIMENSIONS ARE NEAT E2 GAPS S2 E3 REGULAR PENTAGONS SPACE FILLING

14 PROJECTION METHOD QC considered a crystal in higher dimension Additional basis vectors needed to index the diffraction pattern 2D 1D E Window E e 2 e 1 Irrational QC Slope = Tan () Rational RA (XAL)

15 KINDS OF STRUCTURES OBTAINED BY PROJECTION FORMALISM Strip Projection Plane Irrational Rational Irrational Tiles Irrational dimensions Arrangement Quasiperiodic Quasicrystal (QC) Tiles Rational dimensions Arrangement Quasiperiodic Quasiperiodic Superlattice (QPSL) Rational Tiles Irrational dimensions Arrangement Periodic QC Approximant Tiles Rational dimensions Arrangement periodic QPSL Approximant

16 Real Space Reciprocal Space 1. Rational Lengths L and S arranged periodically 2. Rational lengths L and S arranged in a Fibonacci chain 3. Irrational length L and S arranged periodically 4. Irrational length L and S arranged in a Fibonacci chain Periodic Periodic with satellites Peak positions periodic Intensities aperiodic Aperiodic Diffraction properties of various distributions of scatterers.

17 Progressive lowering of dimension starting with an N-fold symmetry in ND space N-fold symmetry Hypercubic Lattice viewed along [ ] N 1s N-D Quasiperiodic tiling 2D RA Approximants Sequence of numbers Sequence of a s and b s Polynomial Equation Convergence of sequence Length of a /length of b Root of Polynomial Eq. 1D Repeating Sequence 0D Rational Number

18 ND-0D.ppt

19 GENERALIZED PROJECTION METHOD [A] a 1, a 2, a 3,..., a N : a set of vectors in E [B] b 1, b 2, b 3,..., b N : a set of vectors in E W : the acceptance region or window in E {n 1, n 2, n 3,..., n N } are a set of integers in N dimensional space such that n 1 a 1 + n 2 a 2 + n 3 a n N a N is accepted as a point in E if and only if: n 1 b 1 + n 2 b 2 + n 3 b n N b N W Linear deformations of E do not affect the pattern produced in E, i.e. if E is m dimensional and T is a non singular m m matrix, then: n 1 (Tb 1 ) + n 2 (Tb 2 ) n N (Tb N ) TW, if and only if n 1 b 1 + n 2 b n N b N W The pattern in E will have a period n 1 a 1 + n 2 a 2 + n 3 a n N a N for any {n 1, n 2, n 3,..., n N } such that n 1 b 1 + n 2 b 2 + n 3 b n N b N = 0

20 2D AND 3D QUASILATTICS AND THEIR APPROXIMANTS (QC & RA)

21 RATIONAL APPROXIMANTS TO THE PENROSE TILING WITH ORTHOGONAL BASIS VECTORS {1/1 1/1} RA to the Penrose tiling Lattice with rectangular unit cell ABCD Fourier transform of the lattice a set of 10-fold spots are marked with circles.

22 RA to the Penrose tiling {1/1 2/1} {3/2 1/1} { 2/1}

23 RATIONAL APPROXIMANTS WITH APPROXIMATIONS ALONG BASIS VECTORS 72 APART {1/1 1/1}e RA to the Penrose tiling Lattice with rectangular unit cell ABCD and parallelogram cell EFGH Fourier transform of the lattice with remnant of the 10-fold symmetry marked by circles.

24 ICOSAHEDRAL QUASILATTICE 5-fold [1 0] 3-fold [2+1 0] 2-fold [+1 1]

25 E V 1 V 2 V 3 B [p/q,, ] p q q q q q {1/1 } P PENTAGONAL QUASILATTICE

26 {1/1 } T TRIGONAL QUASILATTICE B' = E V 1 V V 2

27 Three dimensional covering with triacontahedra Lord, E. A., Ranganathan, S., and Kulkarni, U. D., Current Science, 78 (2000) 64

28 (a) Bergman cluster (b) Mackay double icosahedron Important clusters underlying the structure of quasicrystals and their approximants. (a) Bergman, G., Waugh, J. L. T., and Pauling, L., Acta Cryst., 10 (1957) 2454 (b) Ranganathan, S., and Chattopadhyay, K., Annu. Rev. Mater. Sci., 21 (1991) 437 = 1

29 The structure of the Al 3 Mn decagonal phase Hiraga, K., Kaneko, M., Matsuo, Y., and Hashimoto, S., Phil. Mag. B67 (1993) 193 = 2

30 (a) Cluster of three dodecahedra (b) Four vertex-connected icosahedra Arrangement of sub-units in complex hexagonal phases (a) Singh, A., Abe, E., and Tsai, A. P., Phil. Mag. Lett., 77 (1998) 95 (b) Kreiner, G., and Franzen, H. F., J. Alloys and Compounds, 221 (1995) 15 = 3

31 IQC ( = 1) DQC ( = 2) Mackay Approximant Taylor Approximant Little Approximant Robinson Approximant IQC ( = 1) HQC ( = 3) Key: shows a twinning operation Relation between IQ C and its approxim ants with D Q C, its approxim ants and H Q C via the twinning operation

32 A quadrant of the stereogram of the decagonal phase with indices derived by the twinned icosahedron model

33 Stereogram of the Taylor phase obtained by twinning of the Mackay approximant to the icosahedral phase

34 Quadrant of the stereogram corresponding to I3 cluster

35 = 1 = 2 Icosahedral Quasicrystal = 3 Decagonal Quasicrystal Digonal Quasicrystal = 1 Pentagonal Quasicrystal Cubic R.A.S. Mackay Bergman Trigonal Quasicrystal Hexagonal Quasicrystal Hexagonal R.A.S. Orthorhombic R.A.S. Taylor Little Robinson Monoclinic R.A.S. = 108 o R.A.S. Orthorhombic R.A.S Monoclinic R.A.S. = 90 o Trigonal R.A.S. Orthorhombic R.A.S. Monoclinic R.A.S. 120 o Unification scheme based on the twinning of the icosahedral cluster

36 EXPERIMENTAL Mg-Zn Zn-(Y, La) SYSTEMS

37 METASTABLE PHASES IN Mg-BASED ALLOYS QUASICRYSTALS RATIONAL APPROXIMANTS & RELATED STRUCTURES METALLIC GLASSES NANOCRYSTALS & NANOQUASICRYSTALS

38 MILESTONES IN Mg-BASED QUASICRYSTAL RESEARCH Mg-Zn-Al First Mg-Based QC (Icosahedral) P. Ramachandrarao, G.V.S. Sastry 1985 Mg-Zn-Al-Cu Quaternary System N.K. Mukhopadhyay, G.N. Subbanna, S. Ranganathan, K. Chattopadhyay 1986 Mg-Zn-Ga Stable QC W. Ohashi, F. Spaepen 1987 Mg-Zn-RE Icosahedral QC Z. Luo, S. Zhang, Y. Tang, D. Zhao 1993 Mg-Al Cubic QC P. Donnadieu, A. Redjaimia 1995 Mg-Zn-RE Decagonal QC T.J. Sato, E. Abe, A.P. Tsai 1997 Mg-Zn-RE QC without underlying atomic clusters E. Abe, T.J. Sato, A.P. Tsai 1999

39 IMPORTANT PHASES IN THE Mg-Zn-RE SYSTEMS Composition e/a Phase, Symmetry Comments Mg 3 Zn 6 RE 2.1 Icosahedral, Fm53 a R = RE = Y, Gd, Tb, Dy, Ho, Er dia (0.352, 0.360) Mg 40 Zn 58 RE Decagonal, 10/mmm RE = Y, Dy, Ho, Er, Tm, Lu Mg 24 Zn 65 RE 10 (S) 2.1 Hexagonal superlattice, P6 3 /mmc a = 1.46 nm, c = 0.86 nm Mg 24 Zn 65 RE 10 (M) 2.1 Hexagonal superlattice, P6 3 /mmc a = 2.35 nm, c = 0.86 nm Mg 24 Zn 65 Y 10 (L) 2.1 Hexagonal superlattice, P6 3 /mmc a = 3.29 nm, c = 0.86 nm Mg 12 ZnY 2.07? Mg 3 Zn 3 Y cf16, Fm3m dia < RE = Y, Sm, Gd Related to IQC RE = Sm, Gd Related to IQC RE = Sm Related to IQC a S : a M : a L = 3 : 5 : 7 Mg 7 Zn 3 2 oi142, Immm 1/1 RA to IQC Mg 4 Zn 7 2 mc110, B2/m Related to DQC MgZn 2 2 hp12, P6 3 /mmc Related to S, L & M phases

40 (a) (b) SEM micrograph of as-cast Mg 51 Zn 41 Y 8 alloy showing (a) Eutectic Microstructure (b) Four-fold dendrite

41 5-FOLD DEVELOPING INTO 6-FOLD 5-FOLD TO 6-FOLD6 SEM micrograph of as-cast Mg 51 Zn 41 Y 8 alloy showing distorted 5-fold dendrite growing into hexagonal shape Initial stages of growth

42 BFI [111] [110] [113] As-cast Mg 37 Zn 38 Y 25 alloy showing the formation of a cubic phase (a = 7.07 Å):

43 [112] [111] [011] SAD patterns from a BCC phase (a = 10.7 Å) in as-cast Mg 4 Zn 94 Y 2 alloy showing important zones

44 High-resolution micrograph SAD pattern BFI As-cast Mg 37 Zn 38 Y 25 alloy showing a 18 R modulated phase

45 [1 0] [1 1 1] [0 0 1] [ ] SAD patterns from as-cast Mg 23 Zn 68 Y 9 showing the formation of FCI QC

46 Uniform deformation along the arrow of the [0 0 1] 2-fold pattern from IQC giving rise to a pattern similar to the [ ] pattern

47 BFI SAD TEM micrograph of as-cast Mg 4 Zn 94 Y 2 alloy showing the formation of nanocrystalline Mg 3 Zn 6 Y phase BFI SAD Mg 4 Zn 94 Y 2 as-cast alloy heat treated at 350 o C for 20 hrs (corresponding to the MgZn 5.51 phase)

48 BFI from as-cast Mg 46 Zn 46 La 8 alloy showing patterns from APBs

49 BFI [001] [113] [111] Melt-spun Mg 50 Zn 45 Y 5 alloy showing the formation of a cubic phase (a = 6.63 Å)

50 Comparison of the [001] two-fold of the FCI QC (a) with the two-fold from other phase in the MgZnY (b), (c) and MgZnLa (d) systems

51 CONCLUSIONS A variety of Quasiperiodic and Rational Approximant structures can be realized using the Strip Projection Method, which serves to unify these structures using higher dimensions Structures with diverse kinds of symmetries can be generated using the Twinned Icosahedron Model, which further can be used to construct a unified framework based on the orientations of the icosahedron and the lowering of symmetry The Mg-Zn-RE systems serves a new model system for the study of quasicrystals and related phases Study of quasicrystals is fun ACKNOWLEDGEMENTS Dr. Eric A Lord Prof. S. Ranganathan Dr. K. Ramakrishnan Dr. Sandip Bysakh Dr. Steffen Weber

52 APPLICATIONS OF QUASICRYSTALS WEAR RESISTANT COATING (Al-Cu Cu-Fe-(Cr)) NON-STICK COATING (Al-Cu Cu-Fe) THERMAL BARRIER COATING (Al-Co Co-Fe-Cr) HIGH THERMOPOWER (Al-Pd Pd-Mn) IN POLYMER MATRIX COMPOSITES (Al-Cu Cu-Fe) SELECTIVE SOLAR ABSORBERS (Al-Cu Cu-Fe-(Cr)) HYDROGEN STORAGE (Ti-Zr Zr-Ni)

53 PENROSE TILING Inflated tiling

54 DIFFRACTION PATTERN 5-fold SAD pattern from as-cast Mg 23 Zn 68 Y 9 alloy