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Imperfections in ordered structures
Point defects Line defects - dislocations Planar defects – stacking faults
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Point defects native – vacancies – interstitials
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Point defects impurities – functional position – substitutional
– unintentional position – substitutional – interstitial
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Interstitial positions in fcc lattice
in close-packed structure octahedral – ro ~ 41% of R tetrahedral – rt ~ 23% of R R – atomic radius of the atoms in hard-sphere model approximation 4 octahedral positions/cell 8 tetrahedral positions/cell
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Concetration of point defects
thermodynamic equilibrium – minimized Gibbs free energy G = H – TS enthalpy of defect formation Arrhenius plot quenching – non-equilibrium concentration of defects
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Ionic crystals Schottky defect – Frenkel defect – unoccupied anion
and cation sites Frenkel defect – atom displaced from its lattice position to an interstitial site
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Line defects – Dislocations
dislocation – central object in ductility of crystalline materials dislocations were introduced to explain the plasticity of crystalline solids theoretical estimate of the shear strength – σ ~ G/5 – G/30 the observed values are by 3 – 4 orders of magnitude lower applied shear stress – motion of dislocation within the slip plane
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Basic milestones Volterra (1907) – dislocations in elastic continuum
Taylor, Orowan, Polanyi (1934) – concept of dislocations in crystals Frenkel, Kontorova (1938) – string model Peierls, Nabarro (1940, 1947) – dislocation motion, barrier model Shockley (1953) – parcial dislocations in fcc lattice Hirsch (1956) – observation of dislocations by TEM Lang (1958) – imaging of dislocations by X-ray topography Ray, Cockayne (1969) – observation of partial dislocations by weak beam technique (TEM)
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Edge dislocation Burgers vector b – geometrical parameter
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finish-start/right-hand
Definition of the Burgers vector FS/RH convention finish-start/right-hand edge dislocation
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Definition of the Burgers vector
screw dislocation
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Basic axioms n ξ b The Burgers vector is conserved, it does not change
along the dislocation. For curved dislocation the character of the dislocation changes (edge vs. screw). ξ b n b and ξ define the slip plane
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Basic axioms A dislocation cannot end inside a perfect crystal.
ends at the free surface creates closed loop ends on an other dislocation Burgers vector of a perfect dislocation must equal to one of the lattice translation vectors.
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Energy of the dislocations
elasticity – stress and strain fields of dislocations – Volterra – 1907 ξ ‖ z E ~ b2 Burgers vector – always the shortest lattice vector
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Frenkel-Kontorova model
the motion of dislocation cannot be solved within the framework of theory of elasticity 1938 – first model based on atomic structure results – the existence of a maximum value for dislocation velocity – the limit is the sound velocity – increase of the dislocation energy with velocity – analogy with the theory of relativity Frank, van der Merwe (1949) – first theory of misfit dislocations based on Frenkel-Kontorova model
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Peierls stress Peierls-Nabarro model of dislocation continuum
atoms at the interface
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Peierls stress Peierls-Nabarro model of dislocation
Peierls stress – the force needed to move a dislocation within a plane of atoms w – dislocation width b – Burgers vector G – shear modulus ν – Poisson ratio w b glide plane
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interaction with point defects
Motion of dislocations conservative – glide – sklz non-conservative – climb – šplhanie interaction with point defects
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Intersection of dislocations
direction of motion emission of point defects edge segment
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Force acting on dislocations
Peach-Koehler formula
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Dislocation interaction
force – external – mechanical loading – internal – from other dislocations range – long range – between parallel dislocations – short range – between intersecting dislocations attraction Fx x repulsion
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Dislocation walls formation of stable arrays – dislocation walls
small angle grain boundaries
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Interaction with point defects
dislocation climb high (non-equilibrium) vacancy concentration mechanical stress
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Interaction with point defects
climb force Fcl Fcl L h chemical potential = Gibbs free energy/particle
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Growth of dislocation loops
non-equilibrium point defect concentration growth of dislocation loops dislocation loop climb force acting on dislocation tension in dislocation line
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Consequences of the Peierls barrier
slip planes – lattice planes with largest interplanar distances Peierls relief – determines the direction of dislocation lines direction of motion
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Peierls relief metallic bond – low σPN covalent bond – high σPN
vybočenia metallic bond – low σPN covalent bond – high σPN strongly localized objects – kinks - vybočenia
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Selection rules Burgers vectors – shortest lattice translation vectors
vybočenia Burgers vectors – shortest lattice translation vectors dislocation orientation – along the Peierls relief – directions with the smallest indices slip planes – lattice planes with the largest interplanar distances direction of slip is given by the orientation of the Burgers vector slip system – combination of the slip planes and the slip directions plasticity of polycrystalline materials requires five independent slip systems
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Dislocations in fcc lattice
vybočenia shortest lattice vectors b vectors – slip planes – 12 slip systems – 5 independent
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Dislocations in bcc lattice
vybočenia Dislocations in bcc lattice shortest lattice vectors plane b vectors – similar reticular density in different lattice planes no preferred slip plane
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Dislocations in bcc lattice
vybočenia Dislocations in bcc lattice plane
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Dislocations in bcc lattice
vybočenia Dislocations in bcc lattice plane
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Dislocations in hcp lattice
vybočenia Dislocations in hcp lattice b vectors – 2 basal slip – plane 3 1
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Dislocations in hcp lattice
vybočenia Dislocations in hcp lattice prismatic slip bc vectors – ba vectors – 2 ba+ bc 3 1
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Dislocations in hcp lattice
vybočenia Dislocations in hcp lattice pyramidal slip I. ba+c vectors – ba vectors – 2 3 1
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Dislocations in diamond lattice
vybočenia Dislocations in diamond lattice stacking of {111} planes plane (111) three positions in fcc lattice – ABCABCABC
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glide set dislocations are formed
vybočenia Dislocations in diamond lattice [112] projection of Si lattice B A [111] C B d111 A shuffle set glide glide set dislocations are formed in diamond lattice
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Dislocation motion in covalent crystals
vybočenia Dislocation motion in covalent crystals additional parameters – energy of kink formation and kink migration secondary Peierls barrier introduced for kink migration dislocation velocity both energies ~ 1 eV strong dependence of dislocation velocity on T !
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Stacking faults – vrstevné chyby
vybočenia Stacking faults – vrstevné chyby ABCABCABCABC fcc ABABABABABAB hcp one plane missing – intrinsic stacking fault ABCABABCABC one excess plane – extrinsic stacking fault ABCABACABCAB
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Stacking faults and partial dislocations
vybočenia Stacking faults and partial dislocations SF – terminate at the free surface of crystal – bounded by partial dislocation type of partial dislocation reveals the process leading to the creation of SF A C B A C B vacancy condensation – intrinsic SF condensation of interstitials – extrinsic SF SF is bounded by a Frank parcial dislocation –
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vybočenia Stacking faults and partial dislocations AB AC ABCABCABCABC CABCABCA ABCACABCABCA plane B is missing
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Shockley partial dislocations
vybočenia Shockley partial dislocations 30° mixed dislocation 90° edge dislocation energy of SF ~ 50 mJ/m2 weak beam imaging perfect 60° dislocation – splitted into two Shockley partial bounding an intrinsic SF in the dislocation core
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Microtwinning ABCABCABCABC ABCABABCABCA ABCABACABCAB ABCABACBCABC
vybočenia Microtwinning ABCABCABCABC glide of one Shockley partial dislocation – formation of intrinsic SF ABCABABCABCA ABCABACABCAB ABCABACBCABC ABCABACBABCA repetition of the process – formation of a microtwin formation by plastic deformation or at the process of crystal growth random distribution – polytypism – ZnS, SiC microtwin
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