Defects and Diffusion

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1 Defects and Diffusion Goals for the Unit Recognize various imperfections in crystals Point imperfections Impurities Line, surface and bulk imperfections Define various diffusion mechanisms Identify factors controlling diffusion processes

2 Defects in Materials Types of defects How are defects introduced Diffusion in materials Introduction to Defects All real structures are imperfect Real material properties are often dominated by the imperfections in the structure Some materials have little long-range structure at all (glasses, some polymers)

3 Types of defects Point defects Vacancies Interstitials Impurities Extensive chemical changes Solid solutions Not a defect in intentional alloying or doping Line defects (1-dimensional) Dislocations - in metals Types of Defects, cont. Interfacial defects (2-dimensional) Surfaces - both interior (pore walls) and exterior (surface of material) Interfaces -(grain boundaries) Bulk-Volume defects (3- dimensional) Cracks, foreign inclusions, other phases (including pores)

4 Point defects Vacancy An empty atomic site Interstitial An atom somewhere other than an atomic site Self-interstitial Impurity interstitial Substitutional impurity Some foreign species on an atomic site How are point defects introduced? Some types are thermally generated Direct result of thermal vibration of the atomic array The concentration of thermally-produced defects increases exponentially with increasing temperature

5 How are point defects introduced? Added solutes (impurities or dopants) How do they get uniformly distributed? Stoichiometry changes (cation/anion ratio changes) e.g., ZrO 2-δ, Fe 1-δ O How is uniform composition accomplished? Point Defects in Metals Self Interstitial Interstitial Impurity Vacancy Substitutional Impurity

6 Ceramic Point Defects Anion Vacancy Cation Vacancy Point Defects in Ceramics Substitutional Cation Impurity Interstitial Cation Impurity Substitutional Anion Impurity Anion impurity Interstitial (not shown)

7 Point Defects in Ceramics Schottky Defect (anion and cation vacancies ) Frenkel Defect (cation vacancy + cation interstitial) Anion Frenkel (anion vacancy+anion intersititial (not shown) Solid Solution All solids have some degree of impurities dissolved in them Unintentional - called impurities Intentional - called dopants or alloying additives Solute and solvent Solvent (present in greatest amount) Solute (present in minor concentration)

8 Hume-Rothery Rules Complete mutual solid solubility will occur between two metals if: Less than 15% difference in atomic radii Both have the same crystal structure in pure form Both have similar electronegativities Both have the same valence The more deviation, the less the solubility Can also be applied roughly in simple ceramics Solution of ~30 at% Cu dissolved in solid Ni(substitutional solid solution)

9 Disordered (normal) and ordered solid solution Cu 3 Au Interstitial solid solution

10 Solution of NiO in MgO (cations of same valence) Solution of Fe 2 O 3 in FeO (altervalent cation - vacancy charge compensation) cation vacancies

11 Point defects summary Linear defects Dislocations in Metals Linear (one dimensional) defect around which some of the atoms are misaligned

12 Types of Dislocations Edge Dislocation A portion of an extra plane of atoms Screw Dislocation Helical atomic displacement around a line extending through the crystal Mixed Dislocation Some edge, some screw nature Edge dislocation

13 Burgers vector Screw dislocation

14 Mixed dislocation Shear occurs by dislocation movement producing permanent (plastic) deformation by slip Slip plane Direction of dislocation movement

15 Examples of dislocations Mixed dislocation movement to cause slip Shearing Stresses Slip occurs along densely packed directions on densely packed planes (unlikely) (likely)

16 (Plane)[Direction] pairs designate slip systems (e.g., in ccp and hcp) Dislocation movement and ductility A large number of independent slip systems are required for good ductility in polycrystalline materials so grains can deform to accomodate their neighboring grains Common in many metal structures (esp. bcc and ccp) Dislocations are very complex in ceramic structures This and complications of like charged ions encountering each other during slip make dislocation movement almost impossible in ceramics Therefore ceramics are not ductile, they are brittle

17 Major slip systems in metal structures Impediments to easy dislocation movement Impurity atoms ( solute hardening ) Intersection with other dislocations (entanglement) ( work hardening ) Grain boundaries (dislocations pile up ) Small dispersed inclusions ( precipitation hardening ) All of these affect ductility and yield strength of a metal

18 Grain boundaries and other dislocations impede the movement of dislocations causing hardening 2-D Defects Twin boundaries Grain boundaries Surfaces

19 Twinning is common in some materials Small angle grain boundaries can be thought of as arrays of dislocations

20 Grain boundaries in a polycrystalline material Some details of surface structure

Other types of defects Bulk (Volume defects) Pores - common feature in parts made from powders Cracks Other phases (inclusions) Pores Diffusion in Materials Q.

21 Other types of defects Bulk (Volume defects) Pores - common feature in parts made from powders Cracks Other phases (inclusions) Pores Diffusion in Materials Q. How do changes in microstructure and chemical composition actually occur? A. Atoms must be able to move around (this is called diffusion ) Diffusion occurs in solids, liquids and gases Redistribution of non-uniform chemical species is called impurity diffusion or interdiffusion Random atomic movement can also occur in chemically uniform materials (called self diffusion )

22 Diffusion is driven by nonuniformity Concentration Profile Diffusion Diffusion is necessary for: Redistribution of chemical species Physical changes in microstructure Densification of powder compacts Deformation at high temperature (creep) Formation of solid state reaction products One kind of conductivity in ceramics (ionic)

23 Atoms in a perfect crystal would not move around because there would be no places for them to move to (all sites would be occupied)--all would be locked in place Point defects must be present in a crystal to permit atomic movement (diffusion) In a way, atomic diffusion is actually the movement of defects Diffusion Mechanisms- Vacancy Diffusion Only adjacent atoms can move into a vacancy Vacancy moves in opposite direction of atomic motion Rate depends on concentration of vacancies

24 Diffusion Mechanisms- Interstitial Diffusion Interstitial atom can move into any adjacent empty interstitial position (usually smaller atoms) Rate depends on concentration of interstitial atoms (Usually faster than vacancy diffusion) Interdiffusion forming a solid solution

Diffusion occurs by random jumps After many random jumps by an atom, it s displacment can be calculated by the theory of random walks Quantitative Description of Diffusion The rate of diffusion is

25 Diffusion occurs by random jumps After many random jumps by an atom, it s displacment can be calculated by the theory of random walks Quantitative Description of Diffusion The rate of diffusion is characterized by describing atomic fluxes at particular locations in the material Critical quantities J = atomic flux (atoms/cm 2 -s) (dc/dx) = concentration gradient (atoms/cm 4 ) D = diffusion coefficient (cm 2 /s)

26 Illustration of critical quantities Interrelating the quantities Fick s first law: J = -D dc dx (negative sign indicates that the direction of diffusion flux is down the concentration gradient from high to low concentration) For steady state diffusion (local flux doesn t change with time), Fick s First Law can be solved directly

27 Non-steady state diffusion The diffusion flux at a particular point varies with time (There is a net accumulation or depletion of the diffusing species at a given location) i.e., local concentration of diffusing species changes with time as diffusion proceeds This is the most common situation Non-steady state diffusion Fick s Second Law governs 2 c t = D c 2 x Many solutions exist for particular geometries (initial and boundary conditions)

28 A Non-Steady State Situation Surface concentrati on held constant at c s Conentration and gradient change at given location with time Factors that Influence Diffusion Diffusing Species Magnitude of diffusion coefficient, D - indicates the rate at which atoms diffuse Both diffusing species and host material influence the coefficient

29 Factors influencing diffusion, cont. For example: For the host species of iron: Self diffusion at 500 C (Fe atoms moving in Fe) D = 1.1 x m 2 /s (vacancy diffusion) Carbon impurity diffusion at 500 C (C moving in Fe): D = 2.3 x m 2 /s (interstitial diffusion) This shows the contrast between rates of vacancy and interstitial diffusion Factors that Influence Diffusion Temperature Very strong effect on the diffusion coefficient: D= D o exp Q d RT (Arrhenius Equation) D o = T independent preexponential Q d = the activation energy for diffusion (J /mol, or ev/ atom) R = the gas constant, 8.31 J/ mol- K or x 10-5 ev/ atom T = absolute temperature, (K) A large activation energy results in a small D ln D = ln D o Q d R 1 T Plot lnd vs 1/T - get straight line (to measure activation energy and D o )

30 Temperature dependence of diffusion coefficient (activation energy) Carbon in α-fe Other Diffusion Paths (Besides through volume of the crystal) Atomic migration often occurs more rapidly along so-called short circuiting paths Dislocations Grain boundaries External surfaces However, there is usually small total area for this to occur - so not always important

31 Volume, grain boundary and surface diffusion Ag in Ag Diffusion and Materials Processing Properties and microstructure of materials are altered through diffusion Heat treatment is used to cause these modifications to occur in a reasonable time frame (accelerating effect of higher temp.) This is one of our most valuable tools for modifying materials

32 Summary Recognize various imperfections in crystals Point imperfections Impurities Line imperfections (dislocations) Bulk imperfections Define various diffusion mechanisms Identify factors controlling diffusion processes

Imperfections, Defects and Diffusion

Imperfections, Defects and Diffusion Lattice Defects Week5 Material Sciences and Engineering MatE271 1 Goals for the Unit I. Recognize various imperfections in crystals (Chapter 4) - Point imperfections