Chapter 5 Imperfections in Solids
Chapter 5 2D Defects and Introduction to Diffusion
Imperfections in Solids Issues to Address... What types of defects arise in solids? Can the number and type of defects be varied and controlled? How do defects affect material properties? Are defects undesirable? How do point defects in ceramics differ from those in metals? In ceramics, how are impurities accommodated in the lattice and how do they affect properties?
Imperfections in Solids There is no such thing as a perfect crystal. What are these imperfections? Why are they important? Many of the important properties of materials are due to the presence of imperfections.
Types of Imperfections Vacancy atoms Interstitial atoms Substitutional atoms Extra of missing electron 0D Point defects Dislocations Grain Boundaries Surface Bloch walls 1D Line defects 2D Area defects Voids, Cracks 3D Bulk defects
Types of Point Defects (1) Vacancies (2) Solid Solution - Substitutional Solid Solution - Interstitial Solid Solution small size atoms such as C, O, N - Complete Solid Solution Ni(FCC, 0.125nm) and Cu(FCC, 0.128nm) Size Difference is 2.3% Hume-Rothery Rule (3) Electrons or Holes in Semiconductors
Point Defects Vacancies: -vacant atomic sites in a structure. Vacancy distortion of planes Self-Interstitials: -"extra" atoms positioned between atomic sites. distortion of planes selfinterstitial
Point Defects Self-interstitial Vacancy Interstitial impurity atom Substitutional impurity atom
Equil. Concentration : Point defects Equilibrium concentration varies with temperature! No. of defects No. of potential defect sites. Each lattice site is a potential vacancy site N D N exp Q D kt Activation energy Temperature Boltzmann's constant (1.38 x 10-23 J/atom K) (8.62 x 10-5 ev/atom K)
Measuring Activation Energy We can get Q from an experiment. Measure this... N D Q exp D N kt Replot it... ND N exponential dependence! ND ln N 1 slope -QD/k T defect concentration 1/T
Estimating Vavancy Conc. Find the equil. # of vacancies in 1m of Cu at 1000C. Given: = 8.4 g/cm 3 ACu = 63.5g/mol QV = 0.9eV/atom NA = 6.02 x 10 23 atoms/mole 0.9eV/atom N D N exp Q D = 2.7 x?10 kt -4 For 1m 3, N = x Answer: 3 1273K 8.62 x 10-5 ev/atom-k NA x 1m 3 = 8.0 x 10 28 sites ACu ND = 2.7 x?10-4 x?8.0 x 10 28 sites = 2.2x 10 25 vacancies
Observing equil. Vacancy Conc. Low energy electron microscope view of a (110) surface of NiAl. Increasing T causes surface island of atoms to grow. Why? The equil. vacancy conc. increases via atom motion from the crystal to the surface, where they join the island. Click on image to animate Island grows/shrinks to maintain equil. vancancy conc. in the bulk.
Point Defects / Ionic Materials_1 + + - + - + - - + + + - + - - + + - + - + - + = + - - + = + - + - + - + ++ - + - - + + - Schottky defect Frenkel defect Charge Neutrality Required -
Point Defects / Ionic Materials_2 A. Stoichiometric defect a) Schottky defects + - Density affected Ionic crystal CN is high b) Frenkel defects voids, interstices Ionic Silver halide (AgCl) Ag + is small Cf. AgBr : both (a) & (b)
Point Defects / Ionic Materials_3 Impurities must also satisfy charge balance Ex: NaCl Na + Cl - Substitutional cation impurity 2 Ca cation vacancy Na Na 2 initial geometry Ca Substitutional anion impurity impurity 2 Ca resulting geometry anion vacancy 2 O initial geometry Cl O 2 Cl impurity resulting geometry
Point Defects / Ionic Materials_4 B. Non- stoichiometric defects a) Metal excess alkali halides b) Metal deficiency d-block elements FeS, FeO : Fe 2+ Fe 3+ missing c) Color centers F center (F=Farbe) For the anionic atom, vacancy in a crystal filled by one or more electrons d) Impurities [CaF 2 ] NaCl + SrCl 2 AgCl + CdCl 2
Point Defects / Alloys Two outcomes if impurity (B) added to host (A): Solid solution of B in A (i.e., random dist. of point defects) OR Substitutional alloy (e.g., Cu in Ni) Interstitial alloy (e.g., C in Fe) Solid solution of B in A plus particles of a new phase (usually for a larger amount of B) Second phase particle --different composition --often different structure.
Complete Solid Solution_1 Conditions for substitutional solid solution (S.S.) W. Hume Rothery rule 1. r (atomic radius) < 15% 2. Proximity in periodic table i.e., similar electronegativities 3. Same crystal structure for pure metals 4. Valency All else being equal, a metal will have a greater tendency to dissolve a metal of higher valency than one of lower valency
Complete Solid Solution_2 Application of Hume Rothery rules Solid Solutions 1. Would you predict more Al or Ag to dissolve in Zn? 2. More Zn or Al in Cu? Element Atomic Crystal Electro- Valence Radius Structure nega- (nm) tivity Cu 0.1278 FCC 1.9 +2 C 0.071 H 0.046 O 0.060 Ag 0.1445 FCC 1.9 +1 Al 0.1431 FCC 1.5 +3 Co 0.1253 HCP 1.8 +2 Cr 0.1249 BCC 1.6 +3 Fe 0.1241 BCC 1.8 +2 Ni 0.1246 FCC 1.8 +2 Pd 0.1376 FCC 2.2 +2 Zn 0.1332 HCP 1.6 +2
Composition / Conversion_1 Specification of composition m1 weight percent C1 x 100 m m m 1 = mass of component 1 1 2 atom percent C ' 1 n m1 n m1 n m2 x 100 n m1 = number of moles of component 1
Composition / Conversion_2 Definition: Amount of impurity (B) and host (A) in the system. Two descriptions: Weight % CB = mass of B total mass x 100 Atom % Conversion between wt % and at% in an A-B alloy: # atoms of B C'B = total # atoms x 100 C'BAB CB/AB CB = x 100 C'B = C'AAA + C'BAB CA/AA + CB/AB Basis for conversion: mass of B = moles of B x AB mass of A = moles of A x AA atomic weight of B atomic weight of A
Semiconductors n-type Si p-type Si Adapted from http://www.redarc.com.au/
1D Defects Dislocations: are line defects, cause slip between crystal plane when they move, produce permanent (plastic) deformation. Schematic of a Zinc (HCP): before deformation after tensile elongation slip steps
Type of Dislocation Linear Defects (Dislocations) Are one-dimensional defects around which atoms are misaligned Edge dislocation: extra half-plane of atoms inserted in a crystal structure b to dislocation line Screw dislocation: spiral planar ramp resulting from shear deformation b to dislocation line Burger s vector, b: measure of lattice distortion
Edge Dislocation Edge of Dislocation
Screw of Dislocation Screw Dislocation Dislocation line Burgers vector b (a) b (b)
Edge, Screw, and Mixed Dislocations Mixed Edge Screw
Observation of Dislocation Dislocations are visible in electron micrographs
2D Defects / Surface - Unsaturated bonds of the surface atoms - Represented by surface energy (energy/unit area) - Surface energy is dependent on the crystallographic orientation - Densest atomic packing plane is the lowest surface energy plane - Surface energy reduced by adsorption of impurity atoms such as O, H, C
2D Defects / Grain Boundaries_1 Boundary between crystals with different crystallographic orientations (1) Small angle grain boundary (low angle G.B.) a) tilt boundary - edge dislocation pile up b) twist boundary - screw dislocation pile up misorientation θ < 10 (2) High angle grain boundary a) crystal growth from liquid b) grain boundary is an amorphous like area between crystals for a few atomic diameters (3) grain boundary energy is lower than surface energy but higher than lattice energy more susceptible to corrosion, chemical etching deformation
2D Defects / Grain Boundaris_2 Solidification- result of casting of molten material 2 steps Nuclei form Nuclei grow to form crystals grain structure Start with a molten material all liquid nuclei crystals growing grain structure liquid Crystals grow until they meet each other
2D Defects / Grain Boundaris_3 Grain Boundaries regions between crystals transition from lattice of one region to that of the other slightly disordered low density in grain boundaries high mobility high diffusivity high chemical reactivity
Grain Boundary Solidification / Solidification Grains can be - equiaxed (roughly same size in all directions) - columnar (elongated grains) ~ 8 cm heat flow Columnar in area with less undercooling Shell of equiaxed grains due to rapid cooling (greater T) near wall Grain Refiner - added to make smaller, more uniform, equiaxed grains.
2D Defects / Twin Boundaries mirror lattice symmetry (FCC) A B C A C B A C A A B C A C B A B C A B C A C B C A B A B C A C A B C A Twin
3D Bulk Defects Inclusion Cavities : during casting Cracks
Optical Microscopy_1 Useful up to 2000X magnification. Polishing removes surface features (e.g., scratches) Etching changes reflectance, depending on crystal orientation. microscope close-packed planes micrograph of Brass (Cu and Zn) 0.75mm
Optical Microscopy_2 Grain boundaries... are imperfections, are more susceptible to etching, may be revealed as dark lines, change direction in a polycrystal. ASTM grain size number microscope polished surface surface groove grain boundary N = 2 n-1 no. grains/in 2 at 100x magnification Fe-Cr alloy
Summary Point, Line, and Area defects arise in solids. The number and type of defects can be varied and controlled (e.g., T controls vacancy conc.) Defects affect material properties (e.g., grain boundaries control crystal slip). Defects may be desirable or undesirable (e.g., dislocations may be good or bad, depending on whether plastic deformation is desirable or not.)
Chapter 6 Diffusion
Reading and Thinking Solid Solution Equilibrium Driving Force Activation Energy Steady State Diffusion Equation
Diffusion in Solids Issues to Address... How does diffusion occur? Why is it an important part of processing? How can the rate of diffusion be predicted for some simple cases? How does diffusion depend on structure and temperature?
Diffusion Demo Glass tube filled with water. At time t = 0, add some drops of ink to one end of the tube. Measure the diffusion distance, x, over some time. Compare the results with theory. t o x (mm) t 1 t 2 t 3 x o x 1 x 2 x 3 time (s)
What is Diffusion? Diffusion : Net flow of materials to one direction due to concentration (electrochemical potential) gradient. (1) Interdiffusion (2) Self-diffusion (3) Vacancy diffusion ( Substitutional diffusion ) (4) Interstitial diffusion --- small atoms
Inter-Diffusion Inter-diffusion : In an alloy, atoms tend to migrate from regions of high conc. to regions of low conc. Initially After some time
Self-Diffusion Self-diffusion : In an elemental solid, atoms also migrate. Label some atoms After some time C A D B C A B D
Diffusion Mechanisms / Alloy Vacancy Diffusion: atoms exchange with vacancies applies to substitutional impurities atoms rate depends on: --number of vacancies --activation energy to exchange. increasing elapsed time
Diffusion Mechanisms Interstitial diffusion smaller atoms can diffuse between atoms More rapid than vacancy diffusion
Atomistic Diffusion Diffusion Mechanisms Model Random Walk (Drunken Walk) Origin Step length λ t
Diffusion How do we quantify the amount or rate of diffusion? moles (or mass) diffusing mol kg J Flux = or 2 2 ( surface area)( time) cm s m s Measured empirically Make thin film (membrane) of known surface area Impose concentration gradient Measure how fast atoms or molecules diffuse through the membrane J = M At = l A dm dt M = mass diffused time J slope
Modeling Diffusion : Flux Flux: J 1 A dm dt Directional Quantity y Jy kg or atoms m 2 s m 2 s Jx Jz x z Flux can be measured for: --vacancies --host (A) atoms --impurity (B) atoms x-direction Unit area A through which atoms move.
Concentration Proflies & Flux Concentration Profile, C(x): [kg/m 3 ] Cu flux Ni flux Concentration of Cu [kg/m 3 ] Concentration of Ni [kg/m 3 ] Fick's First Law: flux in x-dir. [kg/m 2 -s] Position, x J x D dc dx The steeper the concentration profile, the greater the flux! Diffusion coefficient [m 2 /s] concentration gradient [kg/m 4 ]
Fick s 1 st Law C C λ unit area : 1 J J J D L R R L diff = 1 = vcλ 6 1 = v(c + δc)λ 6 1 dc = v(c + λ )λ 6 dx 1 2 dc = - vλ = - D 6 dx 1 6 vλ x 2 = x = 6Dt λ vt x dc dx t