# TOPIC 2. STRUCTURE OF MATERIALS III

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1 Universidad Carlos III de Madrid MATERIALS SCIENCE AND ENGINEERING TOPIC 2. STRUCTURE OF MATERIALS III Topic 2.3: Crystalline defects. Solid solutions. 1

2 PERFECT AND IMPERFECT CRYSTALS Perfect crystal: All the atoms are in the ideal lattice positions (only true at 0 K). Real crystal: atoms vibrate - There positions that are not occupied (vacancies) - atoms displaced from ideal positions - There are defects that modify the properties 2

3 PERFECT AND IMPERFECT CRYSTALS Thermodynamics Justify of defects, as ΔG crystal Creation of a defect ΔH > 0 and ΔS > 0 Configuration Entropy S = k B ln W K B = Boltzmann constant = JK -1 W=way of distributing n defects in N possible positions arbitrarily In 1 mol of C + there are N A possibilities to create 1 vacancy W When T [defects] Vacancy or defect: ΔH<TΔS ΔG= ΔH-TΔS<0 3

4 DEFECTS TYPES (according to dimensions) 1. Point : Vacancies Interstitials Substitutional Schottky Frenkel Order-Disorder 2. Linear: dislocations: Edge Screw Mixed 3. Complex: clusters 4. Planar or Extended External Surfaces Grain Boundaries Twin Boundaries Stacking Faults 4

5 POINT DEFECTS: VACANCIES AND INTERSTITIALS Vacancy: When an atom is missing from a position where it should be. Interstitial: An atom from the crystal occupies the place of an interstitial (self-interstitial or interstitialcy) n v = nº vacancies per cm 3 N= nº of lattice points per cm 3 n v /N = fraction of vacancies ΔH v = Formation energy (J.mol -1 ) vacancies concentration at equilibrium They are produced : during solidification (impurities, alloys) by particle bombardment with E during plastic deformation (processing) when T thermal vibrations n v 5

6 POINT DEFECTS: SCHOTTKY AND FRENKEL DEFECTS IN IONIC CRYSTALS Schottky Defect: Vacancy in an ionic crystal. In order to maintain neutrality Anionic and cationic vacancies appear Frenkel Defect: migration of an ion from its normal position to an interstitial position (combination vacancy-interstitial) 6

7 POINT DEFECTS: SCHOTTKY AND FRENKEL Nº of Schottky defects: n s Nº of Frenkel defects : n F ΔH S = E for Schottky defect creation ΔH F = E for Frenkel defect creation N= nº of lattice positions N i = nº of interstitial positions T= Temperature (K) In a crystal ΔH S ΔH F the defect with the lowest ΔH will form nº defects increases with temperature 7

8 POINT DEFECTS: SCHOTTKY AND FRENKEL Compound ΔH (ev) ΔH (10-19 J) Schottky defects MgO CaO LiF LiCl NaCl KCl Frenkel defects ZrO CaF SrF AgCl AgBr NaCl (T f =801 ºC) ΔH s = 3.69x10-19 J T=300K n s = 2.64x10 4 vacancies/mol T=1000K n s = 9.38x10 17 vacancies/mol MgO ΔH s = 10.57x10-19 J T=300K n s = 2.12x10-32 vacancies/mol T=1000K n s = 1.39x10 7 vacancies/mol ΔH s (MgO) > ΔH s (NaCl) n s low, more difficult to create vacancies. 8

9 POINT DEFECTS: ORDER-DISORDER IN SOLID SOLUTIONS Order-disorder phenomena (in substitutional solid solutions): Atoms of one sub-lattice occupy positions corresponding to the other and vice versa (metallic alloys). Solids with elements that have similar electronegativity β' 450 ºC Cu/Zn β (bcc) T>390 ºC Disordered T<390 ºC Ordered Zn Cu or vice versa Brass β (ordered) Brass β : Cu centre Zn corners average gold-copper atom copper atom gold atom Cu-Au Alloys Brass β: Cu and Zn arbitrarily distributed in BCC 9

10 LINEAR DEFECTS : DISLOCATIONS DISLOCATIONS They are defects that produce distortion in the lattice situated around a line. Characteristics: they can be displaced in the interior of a crystal by applying relatively low forces and can produce a complete displacement over crystalline planes. Explain: - E theoretical ( Young modulus) > E experimental - Plastic deformation in metals (workability, ductility) Formation: - during solidification - through plastic or permanent deformation of crystal - through concentration of vacancies - through atomic disarrangements in solid solutions Types: - edge dislocation, or Taylor dislocation - screw dislocation, or Burgers dislocation - mixed dislocation 10

11 EDGE DISLOCATIONS (or TAYLOR) Geometric modification of the lattice: equivalent to an extra plane of atoms Energetic modification (they store energy) Direction of motion Dislocation line.- point A Slipping plane.- line of points William D. Callister, Jr., Materials Science and Engineering : An Introduction, John Wiley & Sons, Inc. 11

12 EDGE DISLOCATIONS (or TAYLOR) * Dislocation characterization: Burgers Vector (b) Burgers Vector: magnitude and direction of the lattice distortion associated with the dislocation Magnitude: distance 1-5 Direction: 1-5 or 5-1 b to dislocation line b to the direction movement 12

13 SCREW DISLOCATIONS (or BURGER) Locally curves some atom lines. The effect is as if a shear stress is applied to produce a distortion Direction of motion We can move from the inferior to the superior plane through the dislocation line. William D. Callister, Jr., Materials Science and Engineering : An Introduction, John Wiley & Sons, Inc. 13

14 SCREW DISLOCATIONS (or BURGER) * Burgers Vectors in screw dislocations: The magnitude and distance from 1-5 define b b to dislocation line SS b to the direction of movement We can trace a screw (spiral ramp) around the dislocation. dislocation line

15 MIXED DISLOCATIONS If there are many dislocations they are mixed ( MIXED dislocations) Screw Edge Burgers Vector = b The Burgers vector will be the same at all points along its line even though the dislocation might change direction and nature within a crystal. William D. Callister, Jr., Materials Science and Engineering : An Introduction, John Wiley & Sons, Inc. 15

16 DISLOCATIONS MOVEMENT Dislocation movement (edge and screw) when a shear stress is applied Slip direction Slip plane slip direction: the direction in which the dislocation moves slip plane: the plane in which the dislocation moves The combination of the slip direction and the slip plane is the slip system 16

17 PLASTIC DEFORMATION : SLIPPING OF DISLOCATIONS Dislocations movement parallel to the directions of maximum packing Lower distance between them + easy sliping Compact Direction E used to move a dislocation= E b 2 For metallic materials, the burgers vector for a dislocation will be the magnitude equal to the interatomic spacing. b = 2R E b 4R 2 NON Compact Direction E b 8R 2 A.R. West. Solid State Chemistry and its applications. Wiley.Chichester,

18 PLANAR DEFECTS: GRAIN BOUNDARIES They separate crystals with different orientation. Formation: during solidification: crystals are formed when nuclei grow simultaneously. Grain boundary They limit dislocations movement in the material 18

19 PLANAR DEFECTS : GRAIN BOUNDARIES Boundary width: 2-5 interatomic distances Very energetic zones (atomic packing < less than in interior ): - Solid state reactions - Atomic diffusion Materials properties = f (grain size) When grain size Mechanical resistance Dislocation movement is limited when there are many grain boundaries. William D. Callister, Jr., Materials Science and Engineering : An Introduction, John Wiley & Sons, Inc. 19

20 PLANAR DEFECTS: EXTERNAL SURFACES EXTERNAL SURFACE It is the end of the crystal or grain structure Coordination numbers at the surface < at the interior of the crystal E surface > E interior 20

21 PLANAR DEFECTS : STACKING FAULTS STACKING FAULTS In fcc and hcp structures hcp fcc perfect lattice :...ABABAB......ABCABCABC... Stacking fault:...ababcab......abcababc... 21

22 PLANAR DEFECTS : TWIN BOUNDARIES Twin boundary: Boundary that separates two parts of a crystal with a small difference in the orientation (misorientation) of the planes (crystal twinning). Highly symmetrical interface, often with one crystal the mirror image of the other Formation: Deformation process or during thermal treatments *Produce mechanical resistance make difficult dislocation gliding Twin boundary 22

23 PLANAR DEFECTS: TWIN BOUNDARIES Brass (70 Cu / 30 Zn) 23

24 SOLID SOLUTION 24

25 SOLID SOLUTION A) Substitutional solid solution B) Interstitial solid solution T>T f during cooling 70g Cu (fcc) + 30 g Ni (fcc) in liquid state new structure with a intermediate between Cu and Ni Small atoms located in octahedral sites or tetrahedral sites Ni Cu 25

26 SOLID SOLUTION : HUME-ROTHERY RULES Hume-Rothery rules for the formation of substitutional solid solutions : 1) The same crystal structure 2) Atoms or ions with similar radius For ceramics difference < 30% For metals difference < 15% For differences > 15% limited solubility (<1%) 5) They must have similar electronegativities 6) Valance: In ceramics: They must have the same valance. If all conditions are fulfilled we do not necessarily have total solubility -If one or more conditions are not fulfilled partial solubility 26

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