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Crystal Defects Chapter 6 1 IDEAL vs. Reality 2 IDEAL Crystal 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: Crystal = Lattice + Motif (basis) 3 Real Crystal Deviations from this ideality. These deviations are known as crystal defects. 4 Is a lattice finite or infinite? Is a crystal finite or infinite? Free surface: a 2D defect 5 Vacancy: A point defect 6 Defects Dimensionality Examples Point 0 Vacancy Line 1 Dislocation Surface 2 Free surface, Grain boundary Stacking Fault 7 Point Defects Vacancy 8 Point Defects: vacancy A Guess 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. 9 Equilibrium? Equilibrium means Minimum Gibbs free energy G at constant T and P A crystal with vacancies has a lower free energy G than a perfect crystal What is the equilibrium concentration of vacancies? 10 Gibbs Free Energy G G=H–TS 1. Enthalpy H =E+PV 2. Entropy S =k ln W T Absolute temperature E internal energy P pressure V volume k Boltzmann constant 11 W number of microstates Vacancy increases H of the crystal due to energy required to break bonds D H = n D Hf 12 Vacancy increases S of the crystal due to configurational entropy 13 Configurational entropy due to vacancy Number of atoms: N Number of vacacies: n Total number of sites: N+n The number of microstates: W ( N n)! Cn n! N ! N n Increase in entropy S due to vacancies: ( N n)! n! N ! k[ln(N n)! ln n! ln N!] DS k ln W k ln 14 Stirlings Approximation ln N! N ln N N N ln N! N ln N N 1 0 1 10 15.10 13.03 100 363.74 360.51 100!=933262154439441526816992388562667004907159682643816214685\ 9296389521759999322991560894146397615651828625369792082\ 15 7223758251185210916864000000000000000000000000 DS k ln W k[ln(N n)! ln n! ln N!] ln N ! N ln N N DS k[(N n) ln(N n) n ln n N ln N ] DH n DH f 16 Change in G of a crystal due to vacancy DG DH DH n DH f G of a perfect crystal DG = DH TDS neq n TDS Fig. 6.4 DS k[(N n) ln(N n) n ln n N ln N ] 17 Equilibrium concentration of vacancy DS k[(N n) ln(N n) n ln n N ln N ] DH n DH f DG nDH f Tk[(N n) ln(N n) n ln n N ln N ] DG n 0 n neq DH f exp N kT neq 18 With neq<<N DH f exp N kT neq Al: Ni: DHf= 0.70 ev/vacancy DHf=1.74 ev/vacancy n/N 0K 300 K 900 K Al 0 1.45x1012 1.12x104 Ni 0 5.59x1030 1.78x10-10 19 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 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 Contribution of vacancy to thermal expansion V Nv DV N Dv V DN DV V Total expansion V=volume of crystal v= volume associated with one atom N=no. of sites (atoms+vacancy) Dv DN v N Lattice parameter increase vacancy 22 Experimental determination of n/N DV Dv DN V v N 3DL 3Da n L a N n DL Da 3 N a L Problem 6.2 Linear thermal expansion coefficient Lattice parameter as a function of temperature XRD 23 Point Defects Interstitial impurity vacancy Substitutional impurity 24 Defects in ionic solids Frenkel defect Cation vacancy + cation interstitial Schottky defect Cation vacancy + anion vacancy 25 Line Defects Dislocations 26 Missing half plane A Defect 27 An extra half plane… 28 …or a missing half plane What kind of defect is this? A line defect? Or a planar defect? 29 Extra half plane No extra plane! 30 Missing plane No missing plane!!! 31 An extra half plane… Edge Dislocation 32 …or a missing half plane This is a line defect called an 33 EDGE DISLOCATION 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 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 935 1 1 2 2 3 3 4 4 5 6 7 8 9 5 6 7 8 936 1 2 3 4 5 6 7 Burgers vector b 8 9 Slip plane slip no slip boundary = edge dislocation 1 2 3 4 5 6 7 8 937 Slip plane t b no slip dislocation slip Dislocation: slip/no slip boundary b: Burgers vector magnitude and direction of the slip t: unit vector tangent to the dislocation line 38 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 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 1 2 3 4 Burgers vector b 5 6 7 Line vector slip 8 9 Slip plane no slip t What is the relationship between the directions of b and t? b t 1 2 3 4 5 6 7 8 9 40 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 Screw Dislocation b || t Slip plane t unslipped slipped b 1 2 3 42 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 Positive Edge Dislocation Screw Dislocation Extra half plane above the slip plane Left-handed spiral ramp b parallel to t Negative Extra half plane below the slip plane Right-handed spiral ramp b antiparallel to t 44 Burgers vector Johannes Martinus BURGERS Burger’s vector Burgers vector 45 S F 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 9 2 8 3 A closed Burgers Circuit in an ideal crystal 7 6 5 4 4 5 6 7 3 8 2 1 9 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 46 The Burgers circuit fails to close !! F b 9 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 S 1 2 8 3 same Map the Burgers circuit on a real crystal 7 6 5 4 4 5 6 7 3 8 2 1 9 16 15 14 13 12 11 10 9 8 7 6 RHFS convention 5 4 3 2 1 47 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 Those who can, do. Those who can’t, teach. G.B Shaw, Man and Superman Happy Teacher’s Day b is a lattice translation b Surface defect If b is not a complete lattice translation then a surface defect will be created along with the line defect. 50 Elastic strain field associated with an edge dislocation N+1 planes Compression Above the slip plane Tension Below the slip plane N planes 51 Line energy of a dislocation Elastic energy per unit length of a dislocation line 1 2 E b 2 Shear modulus of the crystal b Length of the Burgers vector Unit: J m1 52 Energy of a dislocation line is proportional to b2. Thus dislocations with short b are preferred. b is a lattice translation b is the shortest lattice translation 53 b is the shortest lattice translation SC 100 BCC 1 111 2 FCC 1 110 2 DC 1 110 2 NaCl 1 110 2 CsCl 100 54 A dislocation line cannot end abruptly inside a crystal Slip plane Slip plane b no slip dislocation slip Dislocation: slip/no slip boundary slip no slip 55 A dislocation line cannot end abruptly inside a crystal C Q D B P A 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 A dislocation line cannot end abruptly inside a crystal C Q Q D B P P A A Dislocation Line AB Dislocation Line APQ It can end on a free surface 57 Dislocation can end on a grain boundary Grain Boundary Grain 1 Grain 2 58 The line vector t is always tangent to the dislocation line A dislocation loop t b b t t b slip t No slip b The Burgers vector b is constant along a 59 dislocation line Can a loop be entirely edge? Prismatic dislocation loop Example 6.2 b b Cylindrical slip plane (surface) 60 Dislocation node b2 t Node t t b1 b3 b2 b1 b1 + b2 + b3 = 0 b3 61 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 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 Dislocation Motion Glide (for edge, screw or mixed) Cross-slip (for screw only) Climb (or edge only) 64 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 Glide of an Edge Dislocation 66 Glide of an Edge Dislocation crss crss is crss critical resolved shear stress on the slip plane in the direction of b. 67 Glide of an Edge Dislocation crss crss is crss critical resolved shear stress on the slip plane in the direction of b. 68 Glide of an Edge Dislocation crss crss is crss critical resolved shear stress on the slip plane in the direction of b. 69 Glide of an Edge Dislocation crss crss is crss critical resolved shear stress on the slip plane in the direction of b. 70 Glide of an Edge Dislocation A surface step of magnitude b is created if a dislocation sweeps over the entire slip plane crss Surface step, not a dislocation crss 71 slip b Dislocation motion no slip t Shear stress is in a direction perpendicular to the GLIDE motion of screw dislocation 72 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 Cross-slip of a screw dislocation Slip plane 1 b 1 2 3 Change in slip plane of a screw dislocation is called cross-slip 74 Climb of an edge dislocation The motion of an edge dislocation from its slip plane to an adjacent parallel slip plane is called CLIMB Slip plane 2 1 ? glide 3 glide climb 4 2 Obstacle Slip plane 1 75 Atomistic mechanism of climb 76 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 Dislocations in a real crystal can form complex networks From Callister 78 A nice diagram showing a variety of crystal defects http://www.tf.uni-kiel.de/matwis/amat/def_en/index.html 79 Surface Defects 80 Surface Defects External Free surface Internal Grain boundary Stacking fault Same phase Twin boundary Interphase boundary Different phases 81 External surface: Free surface Area A Broken bonds If bond are broken over an area A then two free surfaces of a total area 2A is created Area A 82 External surface: Free surface nA=no. of surface atoms per unit area nB=no. of broken bonds per surface atom =bond energy per atom 1 n A nB 2 Surface energy per unit area Area A Broken bonds Area A If bond are broken over an area A then two free surfaces 83 of a total area 2A is created What is the shape of a naturally grown salt crystal? Why? 84 Surface energy is anisotropic Surface energy depends on the orientation, i.e., the Miller indices of the free surafce nA, nB are different for different surfaces Example 6.5 & Problem 6.16 85 Internal surface: grain boundary Grain Boundary Grain 1 Grain 2 A grain boundary is a boundary between two regions of identical crystal structure but 86 different orientation Optical Microscopy, Experiment 5 Photomicrograph an iron chromium alloy. 100X. Callister, Fig. 4.12 87 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 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 Tilt boundary Edge dislocation model of a small angle tilt boundary Grain 1 C C B A Grain 2 b sin 2h 2 Or approximately b B 2 2h b tan h Eqn. 6.7 90 A Stacking fault C B A C B A C B A FCC Stacking fault A C B A B A C B A FCC HCP 91 Twin Plane C B A C B A C B A C B A Twin plane C A B C A B C B A C B A 92 Edge Dislocation 432 atoms 55 x 38 x 15 cm3 93 Screw Dislocation 525 atoms 45 x 20 x 15 cm3 94 Screw Dislocation (another view) 95 A dislocation cannot end abruptly inside a crystal Burgers vector of a dislocation is constant 96 B A L P Q C D Front face: an edge dislocation enters 720 atoms 45 x 39 x 30 cm3 97 G F R S E H Back face: the edge dislocation does not come out !! 98 F G Screw dislocation B A S b N M b L P E D R Q H Edge dislocation C Schematic of the Dislocation Model 99 A low-angle Symmetric Tilt Boundary 477 atoms 55 x 30 x 8 cm3 100 R. Prasad Dislocation Models for Classroom Demonstrations Conference on Perspectives in Physical Metallurgy and Materials Science Indian Institute of Science, Bangalore 2001 101 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 A Prismatic Dislocation Loop 685 atoms 38 x 38 x 12 cm3 103 Slip plane Prismatic Dislocation loop 104 d c a b A Prismatic Dislocation Loop Top View 105 Welcome to Cochranes Manufacturers and suppliers of quality, affordable educational equipment, toys and Kites since 1962. Our reputation is based on innovation, quality and value. Please select a zone to continue... Science & Education | Kites | Birdscaring | Promotions & IncentivesAll material copyright Cochranes of Oxford Ltd. Tel: +44 (0)1993 832868. Fax: +44 (0)1993 832578 . Email [email protected] 106 Resources The following resources are available: Crystal Dislocation Models for Teaching Three-dimensional models for dislocation studies in crystal structures … Format: PDF | Category: Teaching resources Click here to open 107