Theoretical Strength Point Defects Linear Defects Planar Defects Volume Defects Microscopy Please read Chapter 4Please read Chapter 4 ME 330 Engineering Materials Lecture 6 Imperfections in Solids

How Strong Are Crystalline Materials? We have learned... –All metals solidify as crystals –Studied symmetries, bond potentials and strengths Today, we will compute theoretical strength from this background Demonstrate why theoretical strength is purely theory –Actual strength is 2-3 orders of magnitude lower –Defects!!

Look back at the potential function for two atoms. Derivative is related to strength. Approximate strength curve with a sinusoid: –  o = theoretical cohesive strength For small atomic displacements, Displacement from equilibrium (r o ) is taken as r Elastic modulus: Equating: For most materials, a ~ r 0 Theoretical Cohesive Strength r a Strength oo roro Energy  (r)

How Close is Theory? Theory states: Way too high for common materials ( x too high) Look at “whiskers” –Small, “defect-free” fibers –Agreement is a little better. From: Hertzberg, p.76.

Failure in Tension vs. Shear Tensile Shear Previous analysis –Energy necessary to tear planes of atoms apart from each other –Tensile strength of primary bonds What about energy needed to cause slipping in shear? –Theoretical yield strength

Theoretical Shear Strength Frenkel Analysis Look at deformation along “slip planes” instead of 2 atoms –Close packed planes from last lecture –Determine shear stress to cause displacement along these planes –Assume entire plane translates simultaneously All bonds must rupture in a simultaneous manner Cause permanent deformation, not failure  yield strength  = max  = max Q x x Strength Energy(  ) P R Q 0 b  = 0  = min b x PQR a  = 0  = min R b

Approximate as sinusoid, periodic in b, For small angles, Assume elastic strains, For small shear strains, where a is the distance between slip planes (a  b) Frenkel Analysis Equating shear stresses Remember Mohr’s circle (uniaxial tension) x Strength 0 b   yy mm Still too high!

Modulus vs. Strength

Comparisons of Theoretical Strength Experimental strengths ~100x lower than theory Some experiments show discrepancy to be ~1000x Whiskers and fibers can come close, but not exact. Need an explanation for lower strength  defects! M. F. Ashby, Materials Selection in Mechanical Design, 1999, pg 424

Defects in Solids To this point we have assumed perfect order in crystals –Defects always exist in real materials –Sometimes we add “defects” - alloying Classifications of defects –Usually referring to geometry or dimension of defect –Point : 1-2 atomic positions ( m)- e.g. vacancies, interstitials –Line: 1-Dimensional (10 -9 to m)- e.g. dislocations –Interfacial: 2-Dimensional (10 -8 – m) - e.g. grain boundaries –Volume: 3-Dimensional (10 -4 – m) - e.g. pores, cracks

Point Defects Vacancy Interstitial impurity Self-interstitial Substitutional impurity Vacancy –Vacant lattice point –Most common point defect –Presence increases entropy –Important for diffusion –Equilibrium number of vacancies, N v, follow a Boltzman distribution : N is total # of atomic sites Q v is activation energy required to form a vacancy (  1 eV/atom) T is absolute temperature k is Boltzman’s constant(  1x10 -4 eV/atom*K)

Point Defects Vacancy Interstitial impurity Self-interstitial Substitutional impurity Self-interstitial –Interstitial: Naturally occurring spaces between atoms in crystal Interstitial size depends on crystal structure Largest interstitial sites are usually ~60% atomic radii –Atom of crystal crowded into interstitial site –Induce large lattice distortions –Much lower defect population

Point Defects Impurities always present Impurities can form solid solutions –No new structure formed –Remember solvent & solute definitions –Substitutional - replace host atoms –Interstitial- settle in interstitial sites –Will form substitutional s.s. if Atomic radii are close (  15%) Crystal structures match Electronegativities similar Valences are the same or close –Will form interstitial s.s. if Interstitial is much smaller Maximum concentration ~10% Otherwise, can form intermetallic compound (2 nd phase) How will these effect theoretical strength? Vacancy Interstitial impurity Self-interstitial Substitutional impurity

Alloying Elements We often add alloying elements to form solid solutions –Carbon in iron forms interstitial solid solution (most important system) –Copper and Nickel form substitutional solid solution Need to specify the concentration of an alloy –Composition: Relative amounts of constituents –Can be given in weight percent or atom percent –Conversions between composition schemes presented in book –Maximum concentration of C in Fe is ~2% –Will study in more detail later in semester

Linear Defects - Dislocations Most insightful argument for why theoretical strength is too high Introduce extra half plane of atoms Bond breakage restricted to vicinity of extra plane (dislocation line, ) Bonds break consecutively instead of simultaneously Upper portion translated relative to lower –Same result as before with less energy Responsible for plastic deformation Interactions cause strain hardening From: Callister b

Dislocations Types Edge, screw, mixed Burger’s vector ( ) –Indicates magnitude and direction of motion –Constructing Burger’s circuit –Edge dislocation –Screw dislocation Mixed –Both edge and screw behavior –Burgers vector constant despite changing type     From: Callister

Dislocations - microscopy Dislocation pileup in 18Cr-8Ni stainless steel thin foil Macrograph of slip character in different grains Frank-Reed source in Silicon crystal From: Hertzberg, p.71,78, 87. Introduced by –Solidification –Plastic deformation –Thermal processing

Interfacial Defects - Planar 2-Dimensional in extent External surfaces –Do not bond to maximum nearest neighbors  high surface energy Grain Boundaries –Boundary separating two grains (crystals) –Atoms bonded less regularly along these boundaries –Larger grains have lower total interfacial energy –High angle (more energy) vs. low angle boundaries Angle of misalignment Low-angle boundary High-angle boundary Angle of misalignment From: Callister

Interfacial Defects (continued) AAA BBB CCC BBB CCC AAA BBB CCC BBB AAA AAA BBB CCC AAA BBB Correct stacking sequence Twin boundary Stacking fault From: Hertzberg, p.76. Twin boundary –Mirror lattice symmetry –Annealing twins in FCC –Mechanical twins in BCC, HCP Stacking Faults –In FCC materials only –Interruption in ABCABC sequence Phase boundaries Domain walls

Volume Defects InclusionVoids/Pores Most common are second phases (e.g. salt in water) Cracks represent macroscopic volumetric defects

Buckyballs and Nanotubes Buckyballs –Single molecule, C60 (60 carbon atoms) –20 hexagons, 12 pentagons –7 angstroms in diameter Carbon Nanotubes –Single molecule, millions of atoms –~1nm diameter, up to 2 mm in length –Buckyball on each end –Strongest known material –Other amazing properties

New Concepts & Terms Planar defects –External boundaries –Grain boundaries Angle of misalignment High and low angle boundary –Twin boundary –Stacking fault Volume defect –Void/pore –Inclusion *Know concept, not definition Theoretical Strength of a material –Frenkel analysis Point defects –Vacancies –Self-interstitials –Interstitial impurity –Substitutional impurity –Solid solution –Solvent & solute Linear Defects –Edge dislocation –Screw dislocation –Mixed dislocation –Burger’s vector –Dislocation line

Next Lecture... Dislocations Strengthening Mechanisms Please read chapter 7Please read chapter 7