Ceramic Materials. Chapter 2: Crystal Chemistry

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1 Ceramic Materials Chapter 2: Crystal Chemistry F. Filser & L.J. Gauckler ETH-Zürich, Departement Materials HS

2 Order of the atoms in a solid type, strength and direction of the bonds determines the atom s spatial order in a solid. the strength of a bond is determined by the potential well. The order of the atoms in a solid determines its crystal structure. The crystal structure (spatial filling) is determined by (a) the stoichiometry (chemical composition), (b) the ratio of radii of the ions and (c) the type of the bond (its tendency towards covalent bonding). 2

3 Bond types in Ceramic Solids metallic ionic covalent no preferred spatial direction of the bond delocalized electrons and conduction bands -> charge neutrality of a bond is not required maximum coordination, densest packing attraction holds the solid together no prefered direction charge neutrality of the bond ionic bonds unfavorable charge neutrality of the bonds direction of the bond is very important, and prevails the principle of achieving max. packing density 3

4 Mechanism of Bond Formation: Ionic Bonding electron reception electron donation electron transfer ordering of the ions in a crystal structure Rocksalt crystal 5

5 Equilibrium of Attraction and Repulsion: Ionic Bonding Sum attracting repelling E repulsion r Ion s Distance E attraction r 0 = equilibrium distance Potential Sum of attracting and repelling potential bringing together a cat-ion and an an-ion 6

6 Equations for the Potentials: Ionic Bond E E E E E E net att rep att rep net z z e B n r r 2 2 z1 z2 e B n 4 0 r r 7

7 Equilibrium Distance and Energy of a Bond: Ionic Bond E 2 denet z1 z2 e n B 0 2 n 1 dr r r 4 0 r0 r0 bond 0 2 z1 z2 e r n 0 0 What can we do with the knowledge of E bond? 8

8 Lattice Energy: Ionic Bonding Example: Structure type AB (NaCl) NaCl lattice structure (Rocksalt) 9

9 Lattice Energy: Ionic Bonding Interaction of charges within the lattice structure (NaCl) (Equation is only valid for ions of equal charge) E sum 2 z1 z2 e r n E sum z z e 4 r n E sum 2 z1 z2 e B n 4 0 r0 r0 E Lattice N Av z z e 4 r n 10

10 Madelung Constant Structure type Stoichiometry Rocksalt NaCl AB Cesiumchloride CsCl AB Zinc blende ZnS AB Wurtzite ZnS AB Fluorite CaF 2 AB Rutile TiO 2 AB Cadmiumiodide CdI 2 AB Corundum Al 2 O 3 A 2 B We find big, non-neglible differences in a simple calculation of the Madelung constant like before vs its precise calculation!!! 11

11 Literature on the Calculation of the Madelung Constant References for Madelung's Constant: M. L. Glasser and I. J. Zucker, Lattice sums, Theoretical Chemistry: Advances and Perspectives, v. 5, ed. D. Henderson, Academic Press, D. Borwein, J. M. Borwein and K. F. Taylor, Convergence of lattice sums and Madelung's constant, J. Math. Phys 26 (1985) ; MR 86m: J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, Wiley, 1987; MR 99h: I. J. Zucker and M. M. Robertson, Exact values for some two-dimensional lattice sums, J. Phys. A: Math. Gen. 8 (1975) ; MR 54 #9515. K. F. Taylor, On Madelung's constant, J. Computat. Chem. 8 (1987) ; MR 88h: A. Hautot, A new method for the evaluation of slowly convergent series, J. Math. Phys 15 (1974) ; MR 53 #9575. R. E. Crandall, New representations for the Madelung constant, Experim. Math. 8 (1999)

12 Ionic Bonded Solids Ions are modeled as rigid and charged spheres. Ions possess an ionic radius which is a function of its atomic number and its valency. Coulomb attraction is effective along the direct connection line of the spheres centerpoints. No ion overlap because of the high repulsion forces at short inter-ionic distance. cations want to be surrounded by as much anions as possible closest packed spheres = highest atomic packing density for ionic bonding Cationen take a maximum distance from each other, anions will do the same. Ionic bonds are isotrop, i.e. they are non-directional. The ionic ratio (cations to anions) determines the spatial structure of the ions. 13

13 Covalent Bonded Solids The direction of the bonds are the main factor. The atomic orbitals mainly determine the direction of the bonds. Highest atomic packing density is sacrificed for the direction of the bonds. A less dense packing for covalent bonded solids in comparison to ionic bonded solids. 14

14 Metallic Bonded Solids Free electrons in metals, i.e. valency electrons can t be allocated to one atom and they move freely within the solid body. No limitation because of charge neutrality. No limitations because of stoichiometry Ions are modeled as spheres, the bond is non-directional. All ions and electrons possess the same attraction. In pure metals all ions are of the same size, therefore ions pack as dense as possible, i.e. want to achieve highest packing density Simlarily for alloys and intermetallic phases. However, in some cases the different radius of the atoms prevents it from achieving the closest packing density. 15

15 Ionic Radius Each neutral atom possess a radius which is determined by its outer electron orbit (Elektronenschale). Atomic radius decreases within a period of the periodic system (horizontal from left to right). Ionisation of an atom (electron donation or reception) changes its radius. If valence electrons are donated (cation), then the remaining electrons will be stronger attracted to the nucleus and the ionic radius decreases Example: neutral charged Na atom: r Na = 1.86 Å, BUT r Na+ = 0.98 Å. If more than one electron is donated, then the ionic radius decreases. If valence electrons are received (anion), then the ionic radius increases Example: neutral charged Cl atom, r Cl = 1.07 Å, BUT r Cl- = 1.81Å. If more than one electron is received then the ionic radius increases. 16

16 Packing Density in Ionic Bonded Solids stable stable instable 17

17 Ionic Radius and Coordination Number in Ionic Bonded Solids r Cation /R Anion Coordination number CN Name linear Geometry triangular planar tetrahedral octahedral cubic coordinated 18

18 Packing Density in Covalent Bonded Compounds The coordination number is determined by: number of valence electrons in each atom number of valence electrons which participate in the bonding hybridisation of the orbitals (sp, sp2, sp3) Atoms of Group IV A to VII A show number of bonding NB assuming single bonds: NB = 8 - NV NB = number of bondings per atom NV = number of valence electrons for that atom 19

Packing Density in Covalent Bonded Solids: Carbon Diamond: sp 3 hybride: single bonds (s), hence the coordination number is 4 Graphite: sp 2 hybride: single bonds (s), planar,

19 Packing Density in Covalent Bonded Solids: Carbon Diamond: sp 3 hybride: single bonds (s), hence the coordination number is 4 Graphite: sp 2 hybride: single bonds (s), planar, coordination number is 3 the free unpaired electron per C atom is responsible for a weak bonding between the platelike layers of the graphite. more ceramic examples: SiC, Si 3 N 4, AlN 20

20 Packing Density for Metals pure metals: r/r=1 hence coordination number 12 => closest packed, highest packing density fcc 21

21 fcc / cpp - metals bcc - metals C B A C B A 22

Coordination number 8 Packing density 68 vol-% Simple

22 Other Packing Density than the most Dense Packing in Metalic Bonded Solids Body-Centered Cubic (BCC) Coordination number 8 Packing density 68 vol-% Simple Cubic (SC) = Coordination number 6 Packing density 52 vol-% 23

23 Why should metals have also other packing densities than fcc? 24

24 Why should metals have also other packing densities than fcc? E Melting temperature: Melting temperature: Gruppe IA ( o C) Gruppe IIA ( o C) Li (181) Be (1290) Na (98) Mg (650) K (63) Ca (839) Rb (39) Sr (769) Cs (29) Ba (729) E Thermal Expansion Coefficient: Thermal Expansion Coefficient: Gruppe IA ( o C) Gruppe IIA ( o C) (x 10-6 cm/cm) (x 10-6 cm/cm) Na (70) Be (12) K (83) Mg (25) 25

25 Why should metals also have other packing densities than fcc? Group VB and VIB, transistion metal (V, Cr, Nb, Mo, Ta, W, Fe) These elements possess partial filled d orbitals in their base state. The d-electrons are split-up in either bonding or antibonding orbitals. This split favors the bcc structure over a closest packed structure (hcp, fcc) 26

26 Atomic and Ionic Radii From this section you should learn: the concept of atomic radii The concept of ionic radii and how they change with: the atomic number in the periodic system the coordination number the oxidation state / oxidation number for coordination numbers of CN 6 and 8, respectively 27

27 Atomic Radii The periodic system of the elements: 28

28 Ionic Radius = Bond Length Ionic radius can t be measured isolated, but it can be derived from the bond length in elements and compounds. (see - Shannon, Acta Cryst. (1976) A32 751) Oxygen ion is assumed to: r 0 = 1.26 Å 29

Different Ionic Radii: Ions can be approximated as

element (metalic radius) covalent radius = d/2 in

29 Different Ionic Radii: Ions can be approximated as rigid spheres. Element or Compound Pure Ionic bonding Elements or Compounds, ( Alloys ) Metals atomic radius = d/2 in the element (metalic radius) covalent radius = d/2 in simple bonding (s) Nonmmetals atomic radius = d/2 in the element covalent radius = d/2 in simple bonding (s) 30

30 Radius of Anions & Cations in Periods and Groups of the Periodic System dec dec dec inc inc increasing radius increasing radius decreasing radius 31

31 Ionic Radius The ionic radius (in pm) of iso-charged ions grew with increasing nucleus charge (atomic number) 32

32 Determination of the Ionic Radius electron density electron density map of NaCl crystals 33

33 Ionic Radius References: Krug et al. Zeit. Phys Chem. Frankfurt 4 36 (1955) Krebs, Fundamentals of Inorganic Crystal Chemistry, (1968) 34

34 Coordination Number CN Bonding CN Length (Å) C-O Si-O Si-O Ge-O Ge-O Sn IV -O Pb IV -O Pb II -O The ionic radius of an element increases with increasing CN. 35

35 Variation of the Ionic Radius with its CN 36

36 Valency Bonding CN Length (Å) C-O Si-O Si-O Ge-O Ge-O Sn IV -O Pb IV -O Pb II -O The ionic radius of an element increases with increasing CN. The ionic radius decreases with increasing valency. 37

37 Main Group in the Periodic System Bonding CN Length (Å) C-O Si-O Si-O Ge-O Ge-O Sn IV -O Pb IV -O Pb II -O The ionic radius of an element increases with increasing CN. The ionic radius decreases with increasing valency. The ionic radius increases within a main group of periodic system from top to down (increasing atomic number) Anions are often larger than cations. 38

38 Interstices in Crystalline Solids 39

39 Tetrahedral Hole (Interstice) Spatial space of the tetrahedral interstice. 40

40 Octahedral Hole (Interstice) R X R A cross section of the octahedral interstice 41

coordination (CN = 6) R= radius of the large ions r = radius

41 Rules for the Ionic Radii Ratio (Coordination Number CN =6) Calculation of the ionic radii ratio in case of octahedral coordination (CN = 6) R= radius of the large ions r = radius of the small ions R R r cos R R r ( 2 1) R r R r

42 Rules for the Ionic Radii Ratio (Coordination Number CN = 8) For coordination number CN = 8 : unit cell length a = 2R ions are in touch along the room diagonal of the cell: a 3 = 2(R+r) division: 3 = (R+r)/R multiplication: 3R = R+r then: R( 3-1) = r r/r = 3-1 =

43 Radius ratio: Limits of a Coordination Configuration If r/r < 0.414, then the cation is too small and wiggles in the octahedral hole If r/r > 0.414, then the anions will be moved away from each other If r/r << or >> 0.414, then the coordination number changes Coordination smallest r/r ratio linear, 2 - triangular planar, tetrahedral, octahedral, cubic, closest packing, This simple rules works very often, however not in all the cases! 44

44 Additional means for the classification of crystals 1) Maps for crystal structures (structure maps) e.g. for A x B y O z compounds: draw a diagram showing radius of A ions vs radius of B ions and mark the limits for the different structures (properties) in that map! 2) Mooser-Pearson Graphs Focus on the amount of the convalent bond. Draw a graph showing the difference of electronegativity of the elements versus the principal quantum number and determine the limits for the different structures. 3) Structure - Property - Maps e.g. for Perovskites ABO 3 draw a graph using the polarisation of A ion vs the polarization of B ion and mark the limits of a property of the compound. 45

45 A 2 BO 4 Compounds 46

46 Mooser-Pearson Graph AB Compounds 47

47 ABO 3 Structure Property - Map Perovskiktes: ABO 3 LaCoO 3 LaMnO 3 T T Doshi, R., et al. (1999): Development of Solid Oxide Fuel Cells that operate at 500 C Journal of the Electrochemical Society: 146(4):1273 Kamata, K.,Nakamura, T., Sata, T.(1974): On the State of d- electrons in perovskite-tape compounds ABO 3, Bulletin of Tokyo Ceramics: Institute of Technology Crystal Chemistry, 120:73 Chap 2 48

48 Ionic Radius: Summary The ionic radius of an element increases with increasing coordination number and decreasing valency. The ionic radius increases with increasing atomic number within the main groups in the periodic system The ionic radius ratio can be calculated and in lots of cases to predict the ionic coordination with its help. 49

49 Principles of coordination I Coordinationscheme polyhedron examples cubic close packing ccp (Cu, Ne, etc) hexagonal close packing hcp (Mg, He, etc) 50

50 Principles of coordination II 51

51 Propensity towards Tetrahedric Coordination Many compounds show a tetrahedric coordination despite their ionic radius ratio (r c /r A ). For example, many compounds with ionic radius ratio of r c /r A =0.414 crystalize in a tetrahedric coordination like zinc blende or wurtzite. This is due if the covalent character of the bonds is pronounced, for example, if : 1) cations with a high polarizing ability, i.e. Cu 2+, Al 3+, Zn 2+, Hg 2+ are combined with anions which are readily to polarize, i.e. I -, S 2-, Se 2-. and: 2) atoms are used which are likely to become sp 3 -hybridized orbitals, i.e. Si, C and Ge. 52

52 Ionic Bonded Solids radius ratio r C /r X stoichiometry AX ZnS NaCl CsCl AX 2 SiO 2 TiO 2 CaF 2 A 2 X 3 Corundum ( -Al 2 O 3 ) ABX 3 CaCO 3 and Perovskites A 3 X 4 AB 2 X 4 Spinels 53

53 Zinc blende Wurtzite Rocksalt Material Science I AX: Compilation of possible Structures Compound ZnS NaCl CsCl r Cation /R Anion Coordination Number

54 AX: Zinc blende Representative ZnS, -SiC, GaAs 0.40 Ionic Radius Ratio ZnS Cation Zn 2+ Coordination Number C.-Polyhedron Ionic Radius [Å] 4 Tetrahedron 0.74 Anion S 2- Coordination number C.-Polyhedron Ionic Radius [Å] 4 Tetrahedron

55 AX: Zinc blende (ZnS) Zinc blende 56

56 AX: Wurtzite Representative Ionic Radius Ratio ZnS ZnS, AlN, BeO, ZnO 0.40 Cation Zn + Coordination Number C.-Polyhedron Ionic Radius [Å] 4 Tetrahedron 0.74 Anion S - Coordination Number C.-Polyhedron Ionic Radius [Å] 4 Tetraedron

57 AB: Wurtzite Wurtzit 58

58 AX: NaCl Representative Ionic Radius Ratio NaCl NaCl, CaO, MgO, FeO 0.54 Cation Na + Coordination Number C.-Polyhedron Ionic Radius [Å] 6 Octahedron 0.97 Anion Cl - Koordinationszahl C.-Polyhedron Ionic Radius [Å] 6 Octahedron

59 AX: NaCl Structure of Rocksalt 60

60 Quartz Type Rutile Type Fluorite Type Material Science I AX 2 : Compilation of Structures Compound SiO 2 TiO 2 CaF 2 r Cation /R Anion Coordination Number

61 AX 2 : Quartz Representative Ionic Radius Ratio SiO 2 SiO Cation Si 4+ Coordination Number C.-Polyhedron Ionic Radius [Å] 4 Tetrahedron 0.42 Anion O 2- Coordination Number C.-Polyhedron Ionic Radius [Å] 2 linear Coord

62 AX 2 : Quartz high cristobalite 63

63 AX 2 : Rutile Representative Ionic Radius Ratio TiO 2 TiO 2, PbO 2, GeO Cation Ti 4+ Coordination Number C.-Polyhedron Ionic Radius [Å] 6 Octahedron 0.68 Anion O 2- Coordination Number C.-Polyhedron Ionic Radius [Å] 3 planar 3-coord

64 AX 2 : Rutile Structure of Rutile 65

65 AX 2 : Fluorite Representative Ionic Radius Ratio CaF 2 CaF 2, ZrO 2, CeO Cation Ca 2+ Coordination number C.-Polyhedron Ionic Radius [Å] 8 Cube 0.99 Anion F - Coordination Number C.-Polyhedron Ionic Radius [Å] 4 Tetrahedron

66 AX 2 : Fluorite Structure of Fluorite 67

67 A 2 X 3 : Corundum Representative Ionic Radius Ratio Al 2 O 3 Al 2 O 3, Fe 2 O 3, Cr 2 O 3, B 2 O Cation Al 3+ Coordination number C.-Polyhedron Ionic Radius [Å] 6 Octahedron 0.51 Anion O 2- Coordination number C.-Polyhedron Ionic Radius [Å] 4 Octahedron with 2 empty vertices

68 A 2 X 3 : Corundum C B A C B A Corundum - the large circles represent the oxygen ions- and - the small circles represent the alumium ions. 69

69 ABX 3 : Calcite Representative Ionic Radius Ratio CaCO 3 CaCO 3 Ca 2+ : [CO 3 ] 2- = 0.36 Cation Ca 2+ Coordination number C.-Polyhedron Ionic Radius [Å] 6 Octahedron 0.99 Anion [CO 3 ] 2- Coordination number C.-Polyhedron Ionic Radius [Å] a 6 Octahedron 2.72 a The bond length C-O in CO 3 -complex is 1.36 Å. 70

70 ABX 3 : Calcite Ca Calcite 71

71 ABX 3 : Calcite Ca Calcite 72

72 ABX 3 : Calcite planar CO 3 2- Calcite O C 73

73 ABX 3 : Calcite Ca O C Calcite 74

74 ABX 3 : Perovskite Representative Ionic Radius Ratio CaTiO 3 CaTiO 3, BaTiO 3 Ca 2+ :O 2- =0.75; Ti 4+ :O 2- =0.52 Cation Ca 2+ Coordination Number C.-Polyhedron Ionic Radius [Å] 12 cuboctahedron 0.99 Cation Ti 4+ Coordination number C.-Polyhedron Ionic Radius [Å] 6 Octahedron 0.68 Anion O 2- Coordination number C.-Polyhedron Ionic Radius [Å] 4 Planare 4-Coordination

75 ABX 3 : Perovskite Perovskite 76

76 ABX 3 : Perovskite - TiO 6 octahedron - CaO 12 cuboctahedron (Ca 2+ and O 2- possesses a cubic close packing) mainly ferroelectrika, superconductors, etc. Perowskite 77

77 A 2 BX 4 : Spinel Representative Ionic Radius Ratio MgFe 2 O 4 Fe 2 MgO 4, Mg 2 TiO 4 Mg 2+ :O 2- =0.50; Fe 3+ :O 2- =0.48 Cation Mg 2+ Coordination Number C.-Polyhedron Ionic Radius [Å] 4 Tetrahedron 0.66 Cation Fe 3+ Coordination Number C.-Polyhedron Ionic Radius [Å] 6 Octahedron 0.64 Anion O 2- Coordination Number K.-Polyhedron Ionic Radius [Å] 6 Octahedron

78 A 2 BO 4 Spinel 79

79 A 2 BO 4 Spinel 80

80 A 2 BO 4 Spinel 81

81 A 2 BO 4 Spinel 82

82 A 2 BO 4 Spinel 83

83 A 2 BO 4 Spinel 84

84 A 2 BO 4 Spinel 85

85 B 2 AX 4 : Spinel Example: MgO x Fe 2 O 3 = MgFe 2 O 4 O 2- Fe 3+ Mg 2+ Spinel 86

86 Summary Factors determining the crystal structure: Three major factors influence the crystal structure in solid compounds: 1.) the stoichiometry, 2.) the ratio of cation radius and anion radius and 3.) the propensity for the convalent bond type (sp 3 - hybridized bonding). 87

87 Additional Slides 88

88 What types of bonds do you know? Primary bonds: metallic, ionic or covalent bonds. Secondary bonds: Van-der-Waals, hydrogen or coordinative bonds. Primary bonds are generally much stronger then secondary bonds. 89

Lecture 04 Structure of Ceramics 1 Ref: Barsoum, Fundamentals of Ceramics, Ch03, McGraw-Hill, 2000.

MME 467 Ceramics for Advanced Applications Lecture 04 Structure of Ceramics 1 Ref: Barsoum, Fundamentals of Ceramics, Ch03, McGraw-Hill, 2000. Prof. A. K. M. Bazlur Rashid Department of MME, BUET, Dhaka