Imperfections in ordered structures Point defects Line defects

Imperfections in ordered structures • Point defects • Line defects – dislocations • Planar defects – stacking faults

Point defects native – vacancies – interstitials

Point defects impurities – functional – unintentional position – substitutional – interstitial

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

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

Ionic crystals Schottky defect – unoccupied anion and cation sites Frenkel defect – atom displaced from its lattice position to an interstitial site

Line defects – Dislocations dislocation – central object in study of 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

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)

Edge dislocation Burgers vector b – geometrical parameter

Definition of the Burgers vector FS/RH convention finish-start/right-hand edge dislocation

Definition of the Burgers vector screw dislocation

Basic axioms The Burgers vector is conserved, it does not change along the dislocation. For curved dislocation the character of the dislocation changes (edge vs. screw). n ξ b b and ξ define the slip plane

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.

Energy of the dislocations elasticity – stress and strain fields of dislocations – Volterra – 1907 ξ ‖ z E ~ b 2 Burgers vector – always the shortest lattice vector

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 theory of relativity Frank, van der Merwe (1949) – first theory of misfit dislocations based on Frenkel-Kontorova model

Peierls stress Peierls-Nabarro model of dislocation continuum atoms at the interface continuum

Peierls stress Peierls-Nabarro model of dislocation Peierls stress – the force needed to move a dislocation within a plane of atoms w glide plane b w – dislocation width b – Burgers vector G – shear modulus ν – Poisson ratio

Motion of dislocations conservative – glide – sklz non-conservative – climb – šplhanie interaction with point defects

Intersection of dislocations direction of motion emission of point defects edge segment

Force acting on dislocations Peach-Koehler formula

Dislocation interaction force range – external – mechanical loading – internal – from other dislocations – long range – between parallel dislocations – short range – between intersecting dislocations attraction Fx repulsion x

Dislocation walls formation of stable arrays – dislocation walls small angle grain boundaries

Interaction with point defects dislocation climb mechanical stress high (non-equilibrium) vacancy concentration

Interaction with point defects climb force Fcl h L chemical potential = Gibbs free energy/particle

Growth of dislocation loops non-equilibrium point defect concentration growth of dislocation loops dislocation loop climb force acting on dislocation tension in dislocation line

Consequences of the Peierls barrier slip planes – lattice planes with largest interplanar distances Peierls relief – determines the direction of dislocation lines direction of motion

Peierls relief metallic bond – low σPN covalent bond – high σPN vybočenia strongly localized objects – kinks - vybočenia

Selection rules Burgers vectors – shortest lattice translation vectors dislocation orientation – along the Peierls relief – directions with the lowest indices slip planes – lattice planes with the largest interplanar distances vybočenia 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

Dislocations in fcc lattice shortest lattice vectors b vectors – slip planes – vybočenia 12 slip systems – 5 independent

Dislocations in bcc lattice shortest lattice vectors b vectors – vybočenia similar reticular density in different lattice planes no preferred slip plane

Dislocations in bcc lattice vybočenia plane

Dislocations in bcc lattice vybočenia plane

Dislocations in hcp lattice b vectors – vybočenia 2 basal slip – plane 3 1

Dislocations in hcp lattice ba vectors – vybočenia 2 bc vectors – ba+ bc 3 prismatic slip 1

Dislocations in hcp lattice ba vectors – vybočenia 2 ba+c vectors – 3 pyramidal slip I. 1

Dislocations in diamond lattice stacking of {111} planes plane (111) vybočenia three positions in fcc lattice – ABCABCABC

Dislocations in diamond lattice [112] projection of Si lattice B A [111] C vybočenia B d 111 A shuffle set glide set dislocations are formed in diamond lattice

Dislocation motion in covalent crystals vybočenia additional parameters – energy of kink formation and kink migration secondary Peierls barrier introduced for kink migration dislocation velocity both energies ~ 1 e. V strong dependence of dislocation velocity on T !

Stacking faults – vrstevné chyby ABABAB ABCABC hcp fcc ABCABC vybočenia ABCABACABCAB one plane missing – intrinsic stacking fault one excess plane – extrinsic stacking fault

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 A C vybočenia C B B A A vacancy condensation – intrinsic SF condensation of interstitials – extrinsic SF SF is bounded by a Frank parcial dislocation –

Stacking faults and partial dislocations AB AC vybočenia ABCABCABCACABCABCA plane B is missing

Shockley partial dislocations vybočenia energy of SF ~ 50 m. J/m 2 30° mixed dislocation 90° edge dislocation weak beam imaging perfect 60° dislocation – splitted into two Shockley partials bounding an intrinsic SF in the dislocation core

Microtwinning ABCABCA glide of one Shockley partial dislocation – formation of intrinsic SF repetition of the process – formation of a microtwin ABCABACABCABACBCABC ABCABACBABCA microtwin vybočenia formation by plastic deformation or at the process of crystal growth random distribution – polytypism – Zn. S, Si. C