Crystal Defects Perfect crystal - every atom of the same type in the correct equilibrium position (does not exist at T > 0 K) Real crystal - all crystals have some imperfections - defects most atoms are in ideal locations, a small number are out of place • Intrinsic defects - present for thermodynamic reasons • Extrinsic defects - not required by thermodynamics, can be controlled by purification or synthetic conditions • Chemical defects (foreign atom, mixed crystals, nonstoichiometry) • Geometrical defects (vacancy, interstitials, dislocations, boundaries, surface) Defects dominate the material properties: Mechanical, Chemical, Electrical, Diffusion Defects can be added intentionally 1 Crystal Perfect crystal Real crystal Classes of Crystal Defects Point defects (OD) places where an atom is missing or irregularly placed in the lattice structure - lattice vacancies, self-interstitial atoms, substitution impurity atoms, interstitial impurity atoms Linear defects (ID) groups of atoms in irregular positions - dislocations Planar defects (2D) interfaces between homogeneous regions of the material - grain boundaries, stacking faults, external surfaces Volume defects (3D) spaces of foreign matter - pores, inclusions Classes of Crystal Defects Point 99999999999 3999<£*1AP9999 9999g|*g9 099 99999*999999 99999999999 999999999999 99999999999 9999999AA999 9999 99W99 999999999999 99999999999 9999999999 99999999v 999JSE39999 99ttg*999999 99999V999999 Substitution Impurity Atom Defects Point defects - an atom is missing or is in an irregular position in the lattice • self interstitial atoms • interstitial impurity atoms • substitutional atoms • vacancies 5 Point Defects o vacancy o\o o v interstitial o o óh o o oo o/o substitutional oQo o 0|0 0 self interstitial Point defects - an atom is missing or is in an irregular position in the lattice • self interstitial atoms • interstitial impurity atoms • substitutional atoms • vacancies 6 Point Defects - Ionic Compounds interstitial imputity cation vacancy anion vacancy BA antisite defect substitution of a cation substitution of an anion AB antisite defect Types of Point Defects in Ionic Compounds Vacancy Interstitial Substitutional Frenkel Schottky Schottky: a pair of vacancies, missing cation/anion moved to the surface Frenkel: ions moved to interstitial positions, vacancies 8 Point Defects Schottky - equal numbers of vacancies at both A and B sites preserving charge balance, found in compounds where metal ions are able to assume multiple oxidation states Frenkel - found in open structures (wurtzite, sphalerite, etc) with low coordination numbers, open structure provides room for interstital sites to be occupied Schottky Frenkel Vacancies There are naturally occurring vacancies in all crystals Equilibrium defects - thermal oscillations of atoms at T > 0 K The number of vacancies increases as the temperature goes up The number of vacancies: Nv = TV exp • N = the total number of sites in a crystal • Nv = the number of vacancies • AHa = the activation energy for the formation of a vacancy • R = the gas constant Nv goes up exponentially with temperature T í ma V RT Crystal Energies AG = AH-TAS Point defects = equilibrium concentration, Enthalpy is positive, configurational entropy positive Minimum on free energy Extended defects = no equilibrium concentration, Enthalpy is HIGHLY positive, configurational entropy cannot outweight No minimum on free energy Metastable - dislocations, grain boundaries, surface Heating - minimize free energy: polycrystalline —> single crystal grain growth Grains with high dislocation density consumed Atoms move across grain boundary Typical point defects in crystals Alkali halides Schottky (cations and anions) Alkaline earth oxides Schottky (cations and anions) Silver halides Frenkel (cations) Alkaline earth fluorides Frenkel (anions) Typical activation energies for ion diffusion Na+ in NaCl ~ 0,7 eV CI- in NaCl ~leV Schottky pair ~ 2,3 eV (1 eV/molecule = 96.49 kJ/mol) 12 Extrinsic Defects The addition of the dopant (an impurity) into a perfect crystal = point defects in the crystal NaCl heated in Na vapors Na is taken into the crystal and changes the compostion NaClH>Na7+xCl Na atoms occupy cation sites an equivalent number of unoccupied anion sites Na atoms ionize, Na+ ions occupy the cation sites, the electrons occupy the anion vacancies - F centers - color This solid is now a non-stoichiometric compound as the ratio of atoms is no longer the simple integer 13 Non-stoichiometric Compounds Non-stoichiometry can be caused by •introducing an impurity (doping) •the ability of an element to show multi-valent character. Vanadium oxide varies from VO0 79to V0729 other examples: TiOx, NixO, UOx and LixW03 Covalent compounds - held to together by very strong covalent bonds which are difficult to break, do not show a wide range of composition Ionic compounds - do not show a wide range because a large amount of energy is required to remove / add ions 14 Non-stoichiometric Compounds Non-stoichiometric ionic crystals a multi-valent element - changes in the number of ions can be compensated for by changes in the charge on the ions, therefore maintaining charge balance but changing the stoichiometry non-stoichiometric compounds have formulae with non-integer ratios and can exhibit a range of compositions. The electronic, optical, magnetic and mechanical properties of non-stoichiometric compounds can be controlled by varying their composition. 15 Non-stoichiometric Compounds Non-stoichiometric superconductor YBCO YBa2Cu3065 a multi-valent element = Cu YBa2Cu306 8_7 o 90 K superconductor YBa2Cu306 45_6 7 60 K superconductor YBa2Cu306 0_6 45 antiferromagnetic semiconductor critical 6,30 6,35 6,40 6,45 6,50 6,55 6,60 6,65 6,70 6,75 6,80 6,85 6,90 6,95 7,00 Oxygen content 16 Dislocations Line imperfections in a 3D lattice • Edge • Screw • Mixed $-§£>*&£> 17 Edge Dislocation Extra plane of atoms Burgers vector - Deformation direction - For edge dislocations it is perpendicular to the dislocation line dislocation extra net plane direction of slip ----------------------> core ä 18 Edge Dislocation Screw Dislocation A ramped step Burgers vector - Direction of the displacement of the atoms - For a screw dislocation it is parallel to the line of the dislocation Line vector Burgers vector,b 20 Screw Dislocation 21 Deformation When a shear force is applied to a material, the dislocations move Plastic deformation = the movement of dislocations (linear defects) The strength of the material depends on the force required to make the dislocation move, not the bonding energy Millions of dislocations in a material - result of plastic forming operations (rolling, extruding,...) Any defect in the regular lattice structure (point, planar defects, other dislocations) disrupts the motion of dislocation - makes slip or plastic deformation more difficult 22 Deformation Dislocation movement produces additional dislocations Dislocations collide - entangle - impede movement of other dislocations - the force needed to move the dislocation increases -the material is strengthened Applying a force to the material increases the number of dislocations Called "strain hardening" or "cold work" 23 Slip • When dislocations move slip occurs - Direction of movement - same as the Burgers vector • Slip is easiest on close packed planes • Slip is easiest in the close packed direction • Affects - Ductility - Material Strength 24 Schmidt's Law In order for a dislocation to move in its slip system, a shear force acting in the slip direction must be produced by the applied force. _,Ä ^ a = F/A Normal to slip plane Slip direction Tr = Fr / A - Resolved Shear Stress 25 Schmidt's Law Fr = F cos(X) A = A0/cos(4>) t = a cos((|)) cos(T) Where: t = Fr / A = resolved shear stress in the slip direction a = F/A0 = unidirectional stress applied to the cylinder 26 Surface and Grain Boundaries The atoms at the boundary of a grain or on the surface are not surrounded by other atoms - lower CN, weaker bonding Grains line up imperfectly where the grain boundaries meet Dislocations can not cross grain boundaries Tilt and Twist boundaries Low and High angle boundaries 27 Low Angle Tilt Boundary Low Angle Tilt Boundary = Array of Edge dislocations D = b sin# D = dislocation spacing b = Burgers vector 0 = misorientation angle 28 Low Angle Twist Boundary (a) (t>) Low Angle Twist Boundary = a Screw dislocation 29 Crystal Energies Point defects = equilibrium concentration, Enthalpy is positive, configurational entropy positive Minimum on free energy Extended defects = no equilibrium concentration, Enthalpy is HIGHLY positive, configurational entropy cannot outweight, no minimum on free energy Metastable - dislocations, grain boundaries, surface Heating - minimize free energy: polycrystalline —> single crystal grain growth Grains with high dislocation density consumed Atoms move across grain boundary 30 Effect of Grain Size on Strength • Material with a small grain = a dislocation moves to the boundary and stops - slip stops • Material with a large grain = the dislocation can travel farther • Small grain size = more strength Hall-Petch Equation Gy = G0 + Kd-1/2 Gy = yield strength (stress at which the material permanently deforms) d = average grain diameter g0 , K= constants 32 Control of the Slip Process • Strain hardening • Solid Solution strengthening • Grain Size strengthening 33 Amorphous Structures • Cooling a material off too fast - it does not have a chance to crystallize • Forms a glass • Easy to make a ceramic glass • Hard to make a metallic glass • There are no slip planes, grain boundaries in a glass 34 Amorphous Structures V ŕ Temperature 35 Diffusion Interdiffusion: atoms migrate from regions of large to lower concentration Initial state (diffusion couple) After elapsed time Concentration Profiles Concentration Profiles 36 Diffusion CaTi03-NdA103 diffusion couple fired at 1350 °C/ 6 h 37 Diffusion - Fick's First Law Fick's first law describes steady-state diffusion ^H dn de Adt diffusion flux J = diffusion flux [mol s-1 m-2] D = diffusion coefficient diffusivity [m2 s-1] dc/dx = concentration gradient [mol m-3 m-1] A = area [m2] Velocity of diffusion of particles (ions, atoms ...) in a solid mass transport and concentration gradient for a given point in a solid 38 Diffusion - Fick's First Law Typical diffusion coefficients for ions (atoms) in a solid at room temperature 10 13 cm2 s-1 In solid state ionic conductors (e.g. Ag-ions in a-Agl) the values are greater by orders of magnitude (» 10~6 cm2 s1) 39 Mechanisms of Diffusion Diffusion = the mechanism by which matter is transported into or through matter Diffusion at the atomic level is a step-wise migration of atoms from lattice site to lattice site Conditions for diffusion: • an adjacent empty site • atom possesses sufficient energy to break bonds with its neighbors and migrate to adjacent site (activation energy) The higher the temperature, the higher is the probability that an atom will have sufficient energy Diffusion rates increase with temperature 40 Mechanisms of Diffusion Along Defects = Vacancy (or Substitutional) mechanism - Point Defects - Line Defects Through Interstitial Spaces = Interstitial mechanism Along Grain Boundaries On the Surface o o •_____» O o o o o o o 1 o o •— o o o o o o o o o o o o o o o Interstitial Substitutional Vacancy Mechanisms of Diffusion Vacancies are holes in the matrix Vacancies are always moving An impurity can move into the vacancy Diffuse through the material Vacancy Mechanisms of Diffusion Atoms can move from one site to another if there is sufficient energy present for the atoms to overcome a local activation energy barrier and if there are vacancies present for the atoms to move into. The activation energy for diffusion is the sum of the energy required to form a vacancy and the energy to move the vacancy. oooo oooo oooo OOOO OOOO OQr+m oo^ o oroo ooo# oo#o oooo oooo ---------------------------:----------:------------------:--------------------------------► increasing elapsed time 43 Interstitial Mechanisms of Diffusion ■ ■■■■■■■■ • There are holes between the atoms in the matrix ■ ■■■■■■■I * If the atoms are small enough, they can diffuse through the interstitial holes ■*■*■*■*■ • Fast diffusion Interstitial Atoms • An atom must be small to fit into the interstitial voids • H and He can diffuse rapidly through metals by moving through the interstitial voids • Interstitial atoms like hydrogen, helium, carbon, nitrogen, etc. must squeeze through openings between interstitial sites to diffuse around in a crystal • The activation energy for diffusion is the energy required for these atoms to squeeze through the small openings between the host lattice atoms. • Interstitial C is used to strengthen Fe = steel, it distorts the matrix • The ratio of r/R is 0.57 - needs an octahedral hole • Octahedral and tetrahedral holes in both FCC and BCC - however the holes in BCC are not regular polyhedra • The solubility of C in FCC-Fe is much higher than in BCC-Fe 45 Interstitial Atoms 46 Interstitial Atoms 47 Interstitial Atoms 48 Activation Energy • All the diffusion mechanisms require a certain minimum energy to occur • The activation energy • The higher the activation energy, the harder it is for diffusion to occur • The highest energy is for volume diffusion - Vacancy - Interstitial • Grain Boundary diffusion requires less energy • Surface Diffusion requires the least 49 Activation Energy Initial state Intermediate state Final state Energy barrier for diffusion 50 Energy Barrier for Diffusion o0oo o o o ooo oo o I distance 51 Diffusion Rate / D = Dn exp Q \ V RT J Diffusion coefficients show an exponential temperature dependence (Arrhenius type) D = the diffusivity, which is proportional to the diffusion rate D^ = D for T^ oo Q = the activation energy R = the gas constant T = the absolute temperature D is a function of temperature Thus the flux (J) is also a function of temperature High activation energy corresponds to low diffusion rates The logarithmic representation of D verus 1/T is linear, the slope corresponds to the activation energy and the intercept to D^ 52 Diffusion Coefficients C in Fe Ag in Ag ■40 -10 _____100 300 800 °C 53 Diffusion coefficients show an exponential temperature dependence (Arrhenius type): 0 = 0^ exp(-Q/kT) (D^: D for T^ oo, Q: activation energy of diffusion, k: Boltzmann-faktor) The logarithmic representation of D verus 1/T is linear, the slope corresponds to the activation energy and the intercept to D^. 10 cm!/s - 10"6 | KT1» D It)"11 10-« w -10 100 300 eoo°c '—i------1—r TT i m m C in Fe MF cm V s Mrs 10-* t I B'' 500 3,B 3,0 2,2 U-10"3 K"' -«— 1/r 10-a IQ"' 10-» xr11 — Ľsur ■f idche 7Si 1000 K n ace Korngrenzen grain boundaries Ag in Ag ag m ag Volumen/ / 20 18 16 % 12-10'* K B volume 54 Diffusion Velocity of diffusion of particles (ions, atoms ...) in a solid - mass transport and concentration gradient for a given point in a solidj fa [molcm-2s-1] (const. T) Jji flow of diffusion (mol s1 cm2); D^ diffusion coefficient (cm2 s1) Sc/ ôx: concentration gradient (mol cm3 cm1) (i.e. change of concentration along a line in the solid!) Knowledge of D allows an estimation of the average diffusion length for the migrating particles: = 2Dt (: average square of diffusion area; t: time) 55 Diffusivity of Fe and Carbon in Iron Diffusion Coefficient 1 e-21 - -1-------1--------1--------r- -i--------1--------1--------r 0.0006 0.0008 0.001 0.0012 Reciprocal Temperature 0.0014 ' ' 0.0016 56 Diffusion FASTER for: • open crystal structures • lower melting T materials • materials w/secondary bonding • smaller diffusing atoms • lower density materials Diffusion Diffusion SLOWER for: • close-packed structures • higher melting T materials • materials w/covalent bonding • larger diffusing atoms • higher density materials