2. Crystal Defects Chapter 6 1

3. 2 IDEAL vs. Reality

4. 3 An ideal crystal can be described in terms a three-dimensionally periodic arrangement of points called lattice and an atom or group of atoms associated with each lattice point called motif: IDEAL Crystal Crystal = Lattice + Motif (basis)

5. Deviations from this ideality. These deviations are known as crystal defects. 4 Real Crystal

6. Is a lattice finite or infinite? Is a crystal finite or infinite? Free surface: a 2D defect 5

7. Vacancy: A point defect 6

8. DefectsDimensionalityExamples Point0Vacancy Line1Dislocation Surface2Free surface, Grain boundary Stacking Fault 7

9. Point Defects Vacancy 8

10. There may be some vacant sites in a crystal Surprising Fact There must be a certain fraction of vacant sites in a crystal in equilibrium. A Guess Point Defects: vacancy 9

11. What is the equilibrium concentration of vacancies? Equilibrium? A crystal with vacancies has a lower free energy G than a perfect crystal Equilibrium means Minimum Gibbs free energy G at constant T and P 10

12. 1. Enthalpy H 2. Entropy S G = H – T S Gibbs Free Energy G =E+PV =k ln W T Absolute temperature E internal energy P pressure V volume k Boltzmann constant W number of microstates 11

13. Vacancy increases H of the crystal due to energy required to break bonds  H = n  f 12

14. Vacancy increases S of the crystal due to configurational entropy 13

15. Number of atoms:N Increase in entropy S due to vacancies: Number of vacacies: n Total number of sites: N+n The number of microstates: Configurational entropy due to vacancy 14

16. Stirlings Approximation Nln N!N ln N  N 1 0  1 10 15.10 13.03 100363.74 360.51 100!=933262154439441526816992388562667004907159682643816214685\ 9296389521759999322991560894146397615651828625369792082\ 7223758251185210916864000000000000000000000000 15

17. 16

18.  G =  H  T  S n eq G of a perfect crystal Change in G of a crystal due to vacancy n GG HH TSTS Fig. 6.4 17

19. With n eq <<N Equilibrium concentration of vacancy 18

20. Al:  f =0.70ev/vacancy Ni:  H f =1.74 ev/vacancy n/Nn/N 0 K 300 K 900 K Al 01.45x10  12 1.12x10  4 Ni 05.59x10  30 1.78x10 -10 19

21. Contribution of vacancy to thermal expansion Increase in vacancy concentration increases the volume of a crystal A vacancy adds a volume equal to the volume associated with an atom to the volume of the crystal 20

22. Contribution of vacancy to thermal expansion Thus vacancy makes a small contribution to the thermal expansion of a crystal Thermal expansion = lattice parameter expansion + Increase in volume due to vacancy 21

23. Contribution of vacancy to thermal expansion V=volume of crystal v= volume associated with one atom N=no. of sites (atoms+vacancy) Total expansion Lattice parameter increase vacancy 22

24. Experimental determination of n/N Linear thermal expansion coefficient Lattice parameter as a function of temperature XRD Problem 6.2 23

25. vacancy Interstitial impurity Substitutional impurity Point Defects 24

26. Frenkel defect Schottky defect Defects in ionic solids Cation vacancy + cation interstitial Cation vacancy + anion vacancy 25

27. Line Defects Dislocations 26

28. Missing half plane  A Defect 27

29. An extra half plane… …or a missing half plane 28

30. What kind of defect is this? A line defect? Or a planar defect? 29

31. Extra half plane No extra plane! 30

32. Missing plane No missing plane!!! 31

33. An extra half plane… …or a missing half plane Edge Dislocation 32

34. If a plane ends abruptly inside a crystal we have a defect. The whole of abruptly ending plane is not a defect Only the edge of the plane can be considered as a defect This is a line defect called an EDGE DISLOCATION 33

35. Callister FIGURE 4.3 The atom positions around an edge dislocation; extra half-plane of atoms shown in perspective. (Adapted from A. G. Guy, Essentials of Materials Science, McGraw-Hill Book Company, New York, 1976, p. 153.) 34

36. 123456789 123456789 35

37. 123456789 123456789 36

38. 123456789 123456789 slip no slip boundary = edge dislocation Slip plane b Burgers vector 37

39. Slip plane slipno slip dislocation b t Dislocation: slip/no slip boundary b: Burgers vector magnitude and direction of the slip t: unit vector tangent to the dislocation line 38

40. Dislocation Line: A dislocation line is the boundary between slip and no slip regions of a crystal Burgers vector: The magnitude and the direction of the slip is represented by a vector b called the Burgers vector, Line vector A unit vector t tangent to the dislocation line is called a tangent vector or the line vector. 39

41. 123456789 123456789 slip no slip Slip plane b Burgers vector t Line vector Two ways to describe an EDGE DISLOCATION 1. Bottom edge of an extra half plane 2. Boundary between slip and no-slip regions of a slip plane What is the relationship between the directions of b and t? b  t 40

42. In general, there can be any angle between the Burgers vector b (magnitude and the direction of slip) and the line vector t (unit vector tangent to the dislocation line) b  t  Edge dislocation b  t  Screw dislocation b  t  b  t  Mixed dislocation 41

43. Screw Dislocation Line b t b || t    Screw Dislocation Slip plane slipped unslipped 42

44. If b || t Then parallel planes  to the dislocation line lose their distinct identity and become one continuous spiral ramp Hence the name SCREW DISLOCATION 43

45. Edge Dislocation Screw Dislocation PositiveNegative Extra half plane above the slip plane Extra half plane below the slip plane Left-handed spiral ramp Right-handed spiral ramp b parallel to tb antiparallel to t 44

46. Burgers vector Johannes Martinus BURGERS Burgers vectorBurger’s vector 45

47. 1 2 7 6 5 4 3 8 9 18234567 9 10111213 1 23 456 7 8 9 18 23 4 56 7 9 10 11 1213 A closed Burgers Circuit in an ideal crystal S F 14 1516 141516 46

48. 1 2 7 6 5 4 3 8 9 182345679101112 131415 1 23 456 7 9 1 2 3 4 5 6 8 7 91011 12 13 14 15 8 16 S b RHFS convention F  Map the same Burgers circuit on a real crystal The Burgers circuit fails to close !! 47

49. A circuit which is closed in a perfect crystal fails to close in an imperfect crystal if its surface is pierced through a dislocation line Such a circuit is called a Burgers circuit The closure failure of the Burgers circuit is an indication of a presence of a dislocation piercing through the surface of the circuit and the Finish to Start vector is the Burgers vector of the dislocation line. 48

50. Those who can, do. Those who can’t, teach. G.B Shaw, Man and Superman Happy Teacher’s Day

51. b is a lattice translation b If b is not a complete lattice translation then a surface defect will be created along with the line defect. Surface defect 50

52. N+1 planes N planes Compression Above the slip plane Tension Below the slip plane  Elastic strain field associated with an edge dislocation 51

53. Line energy of a dislocation Elastic energy per unit length of a dislocation line  Shear modulus of the crystal bLength of the Burgers vector Unit: J m  1 52

54. Energy of a dislocation line is proportional to b 2. b is a lattice translation Thus dislocations with short b are preferred. b is the shortest lattice translation 53

55. b is the shortest lattice translation FCC DC NaCl SC BCC CsCl 54

56. A dislocation line cannot end abruptly inside a crystal Slip plane slip no slip slipno slip dislocation b Dislocation: slip/no slip boundary Slip plane 55

57. A B A dislocation line cannot end abruptly inside a crystal C D Q P Extra half plane ABCD Bottom edge AB of the extra half plane is the edge dislocation line What will happen if we remove the part PBCQ of the extra half plane?? 56

58. A P Q A dislocation line cannot end abruptly inside a crystal It can end on a free surface A B C D Q P Dislocation Line ABDislocation Line APQ 57

59. Grain 1 Grain 2 Grain Boundary Dislocation can end on a grain boundary 58

60. A dislocation loop b b b b t t t t No slip slip   The line vector t is always tangent to the dislocation line The Burgers vector b is constant along a dislocation line 59

61. b Cylindrical slip plane (surface) Prismatic dislocation loop Can a loop be entirely edge? b Example 6.2 60

62. t t t b2b2 b3b3 b1b1 Node Dislocation node b 1 + b 2 + b 3 = 0 b1b1 b2b2 b3b3 61

63. A dislocation line cannot end abruptly inside a crystal It can end on Free surfaces Grain boundaries On other dislocations at a point called a node On itself forming a loop 62

64. Slip plane The plane containing both b and t is called the slip plane of a dislocation line. An edge or a mixed dislocation has a unique slip plane A screw dislocation does not have a unique slip plane. Any plane passing through a screw dislocation is a possible slip plane 63

65. Dislocation Motion Glide (for edge, screw or mixed) Cross-slip (for screw only) Climb (or edge only) 64

66. Dislocation Motion: Glide Glide is a motion of a dislocation in its own slip plane. All kinds of dislocations, edge, screw and mixed can glide. 65

67. Glide of an Edge Dislocation   66

68. Glide of an Edge Dislocation  crss  crss is critical resolved shear stress on the slip plane in the direction of b. 67

69. Glide of an Edge Dislocation  crss  crss is critical resolved shear stress on the slip plane in the direction of b. 68

70. Glide of an Edge Dislocation  crss  crss is critical resolved shear stress on the slip plane in the direction of b. 69

71. Glide of an Edge Dislocation  crss  crss is critical resolved shear stress on the slip plane in the direction of b. 70

72. Glide of an Edge Dislocation  crss Surface step, not a dislocation A surface step of magnitude b is created if a dislocation sweeps over the entire slip plane 71

73.  slip no slip Dislocation motion Shear stress is in a direction perpendicular to the GLIDE motion of screw dislocation t b 72

74. Glide Motion and the Shear Stress For both edge and screw dislocations the glide motion is perpendicular to the dislocation line The shear stress causing the motion is in the direction of motion for edge but perpendicular to it for screw dislocation However, for edge and screw dislocations the shear stress is in the direction of b as this is the direction in which atoms move 73

75. 1 2 3 b Cross-slip of a screw dislocation Change in slip plane of a screw dislocation is called cross-slip Slip plane 1 Slip plane 2 74

76. Climb of an edge dislocation The motion of an edge dislocation from its slip plane to an adjacent parallel slip plane is called CLIMB    Obstacle climb glide  Slip plane 1 Slip plane 2 1 2 3 4 ? 75

77.   Atomistic mechanism of climb 76

78. Climb of an edge dislocation Climb up Climb down Half plane shrinks Half plane stretches Atoms move away from the edge to nearby vacancies Atoms move toward the edge from nearby lattice sites Vacancy concentration goes down Vacancy concentration goes up 77

79. From Callister Dislocations in a real crystal can form complex networks 78

80. http://www.tf.uni-kiel.de/matwis/amat/def_en/index.html A nice diagram showing a variety of crystal defects 79

81. Surface Defects 80

82. Surface Defects ExternalInternal Free surfaceGrain boundary Stacking fault Twin boundary Interphase boundary Same phase Different phases 81

83. External surface: Free surface If bond are broken over an area A then two free surfaces of a total area 2A is created Area A Broken bonds 82

84. External surface: Free surface If bond are broken over an area A then two free surfaces of a total area 2A is created Area A Broken bonds n A =no. of surface atoms per unit area n B =no. of broken bonds per surface atom  =bond energy per atom Surface energy per unit area 83

85. What is the shape of a naturally grown salt crystal? Why? 84

86. Surface energy is anisotropic Surface energy depends on the orientation, i.e., the Miller indices of the free surafce n A, n B are different for different surfaces Example 6.5 & Problem 6.16 85

87. Grain 1 Grain 2 Grain Boundary Internal surface: grain boundary A grain boundary is a boundary between two regions of identical crystal structure but different orientation 86

88. Photomicrograph an iron chromium alloy. 100X. Callister, Fig. 4.12 Optical Microscopy, Experiment 5 87

89. Grain Boundary: low and high angle One grain orientation can be obtained by rotation of another grain across the grain boundary about an axis through an angle If the angle of rotation is high, it is called a high angle grain boundary If the angle of rotation is low it is called a low angle grain boundary 88

90. Grain Boundary: tilt and twist One grain orientation can be obtained by rotation of another grain about an axis through an angle If the axis of rotation lies in the boundary plane it is called a tilt boundary If the angle of rotation is perpendicular to the boundary plane it is called a twist boundary 89

91. Edge dislocation model of a small angle tilt boundary     Grain 1 Grain 2 Tilt boundary A B C 2h2h b A B C Eqn. 6.7 Or approximately 90

92. Stacking fault CBACBACBACBACBACBA ACBABACBAACBABACBA FCC HCP 91

93. Twin Plane CBACBACBACBACBACBACBACBA CABCABCBACBACABCABCBACBA Twin plane 92

94.  Edge Dislocation 432 atoms 55 x 38 x 15 cm 3 93

95. Screw Dislocation 525 atoms 45 x 20 x 15 cm 3 94

96. Screw Dislocation(another view) 95

97.  A dislocation cannot end abruptly inside a crystal  Burgers vector of a dislocation is constant 96

98. A B C D P Q L 720 atoms 45 x 39 x 30 cm 3 Front face: an edge dislocation enters 97

99. E F G H R S Back face: the edge dislocation does not come out !! 98

100. Schematic of the Dislocation Model Edge dislocation b b G F A B R S M N D H E C P Q L Screw dislocation 99

101.     A low-angle Symmetric Tilt Boundary 477 atoms 55 x 30 x 8 cm 3 100

102. R. Prasad Dislocation Models for Classroom Demonstrations Conference on Perspectives in Physical Metallurgy and Materials Science Indian Institute of Science, Bangalore 2001 101

103. MODELS OF DISLOCATIONS FOR CLASSROOM*** R. Prasad Journal of Materials Education Vol. 25 (4-6): 113 - 118 (2003) International Council of Materials Education Paper is available on Web if you Google “Dislocaton Models” 102

104.   A Prismatic Dislocation Loop 685 atoms 38 x 38 x 12 cm 3 103

105. Slip plane Prismatic Dislocation loop 104

106. a b c d A Prismatic Dislocation Loop Top View 105

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108. Resources The following resources are available: Crystal Dislocation Models for Teaching Three-dimensional models for dislocation studies in crystal structures … Format: PDF | Category: Teaching resourcesClick here to openClick here to open 107