Physical Metallurgy 14 th Lecture

1. Physical Metallurgy14 th Lecture MS&E 410 D.Ast dast@ccmr.cornell.edu 255 4140

2. Solidification

3. 1. Overview of Metal processing

4. Melt Cast Roll Draw Sheet Wires Powder Metallurgy starts with powders. Ni powder is made decomposing Nickel Carbonyl, other powders electro-chemically

5. Big Mill: Iron ore, coal, limestone Mini-Mill: Scrap and Electricity Notes Pig Iron is high Carbon Converter blows oxygen through melt, reducing C. Stocked billets reheated (US), German mills use hot billets (continuos processing)

6. Not Alloy Hot Rolled Sheets and Plates • · Not Alloy Hot Rolled Narrow Strip • · Not Alloy Hot Rolled Flat Products • · Cold Rolled Sheets • · Electrical Sheets (other than GOES) • · Metallic Coated Sheets • · Organic Coated Sheets • · Tin Mill Products • · Quarto Plates • · Wide Flats • · Non Alloy Merchants Bars and Light Sections • · Alloy Merchant Bars and Light Sections • · Rebars • · Stainless Bars and Light Shapes • · Stainless Wire Rod • · Stainless Steel Wire • · Fittings (<609.6 mm) • · Flanges (Other than stainless steel) • · Gas Pipes • · Hollow sections • Fittings (<609.6 mm) • · Flanges (Other than stainless steel) • · Gas Pipes • · Hollow sections Products

7. 2001 Data US production 36 b$, exported to Europe 0.9b$ EU production 66 b$, exported to US 7.5 b$ S.A. production 11 b$, exported to US 3.9 b$ China production 73 b$, exported to US 5.3 Europe (including Russia) ~ 3 times USA

8. Excursion into the Glory Days of US Steel The Bill Gates of his times …he too, started a foundation …his house, too was huge

9. Liquid Metals (Melts)

10. Liquid melts can be modeled as “solids with vacancies” or as stuffed full with dislocation loops that can climb with a velocity limited by the diffusion of vacancies to and from these hypothetical dislocation loops

11. Liquids as “solids with ~ 5% vacancies, or 1013 dislo/cm2 • maximum speed of solidification as climb of dislocation loops • vmax D/b  10-5 (cm2/s) /2.10-8  5 m/s • D diffusion coefficient in liquids (generally 10-5); b Burgers vector • Enthalpy of melting => breaking one of 12 (fcc) or 8 (but remember second nearest neighbors count in bcc !) => DHm  0.08 DHevep Predicts that if I cool a metal faster than several meters per second it will freeze into a metallic glass.

12. 3. Nucleation Can be homogeneous or heterogeneous

14. Nucleus of critical size Strain energy (liquid)  0, s  500 ergs/cm2 , DHm  1eV/atom, hence DG is zero at Tm and increases linearly with undercooling as DT/Tm . In Hg, at DT/Tm=0.2, the critical nucleus size is  200 atoms.

15. David Turnbull Became famous by studying homogeneous nucleation in Hg at GE and moved to Harvard (see Chalmers). I I put this paper on our the website Once upon a time you could do fundamental research at an industrial research lab.

16. DT/Tm switch Phase diagrams are equilibrium diagrams - do not contain kinetics Heterogeneous nucleation rate can be calculated if surface energy with nucleating foreign substance is known (see Turnbull paper

17. The Cu-Ni phase diagram. There is spinodal at low T. Amount of undercooling depends on impurities and cooling rate. Images show that the “first formed nuclei” are coated with an alloy containing less Ni - lowering Ts - as you would predict from the phase diagram

18. Metallographic image of Cu60Ni40 Note that under equilibrium you would have a single homogeneous solution. The phase diagram does not kinetics.

19. Heterogeneous Nucleation

20. 4. Crystal Growth Extremely well understood in Si. There is nothing we do not know..

22. Undercooling required • Counteracting heat of solidification => needs to be removed from interface • Removal of heat of solidification via • Radiation • Conductance • is the rate limiting step in single crystal growth of Si. G Low T High T Low Gv High Gv Driving force is DG which increases linearly in T below the melting point. Control over the thermal gradient is crucial.

23. Rate Theory Similar to diffusion derivation Forward jump => attachment Backward jump => detachment Gradient in free energy, not from concentration but from T Maximum Rate When not limited by T removal (splash cooling), upper limit is 10 ..20 m/sec for reasons discussed previously. Limit set by Diffusion coefficient divided by “atom size”

24. Attachment of adatoms Preferred sites are kinks in surface steps and screw dislocations. High index planes are difficult to grow - ask me why

25. Spiral, spirals everywhere..

26. The growth velocity depends on direction. High index plans tend to grow faster than low index planes because they are less closed packed. But they are also tighter stacked, so..

27. DT is degree of undercooling Spiral Growth Crystal growth rate is proportional to DT (which sets the arrival rate) At constant incorporation rate angular velocity increases linearly to center ? Haasen’s book says DT1

28. Many real systems tend to be diffuse - because it eliminates the difficulty of finding kinks and ledges. Si is an exception because it has a different coordination number in the solid and liquid In situ, high T, x-ray imaging shows that liquid droplets are occasionally caught behind the crystallization front. Since the density of liquid Si > solid Si, they freeze by sending out Si self interstitials

29. Small undercooling Small T gradient Protrusions into liquid encounter higher T, nucleation rate decreases 5. Heat Transfer Effects Stable x-stal front

31. Heat of solidification produces a local bump in T profile • Negative dT/dx in liquid => unstable • Any statistical protrusion of S/L interface into L will speed up Factors favoring Dendritic Growth High solidification rate => High heat of solidification Inefficient heat removal

32. Self heating forces dendrite to seek cooler areas via branches

34. Cobalt Samarium Copper Note: 3-D structure 2-D image !

35. Useful dendrites • Dentric Growth of solar silicon How it works The dendritic web process for growing long thin ribbon crystals of silicon and other semiconductors is described. Growth is initiated from a thin wirelike dendrite seed which is brought into contact with the melt surface. Initially, the seed grows laterally to form a button at the melt surface; when the seed is withdrawn, needlelike dendrites propagate from each end of the button into the melt, and the web portion of the crystal is formed by the solidification of the liquid film supported by the button and the bounding dendrites. Apparatus used for dendritic web growth, material characteristics, and the two distinctly different mechanisms involved in the growth of a single crystal are examined. The performance of solar cells fabricated from dendritic web material is indistinguishable from the performance of cells fabricated from Czochralski grown material. (Seidensticker 1977)

36. Self test Why does the snow flake have a dendrite structure ? And why are the branches offset … rather than mirror symmetric ?

37. 6 . Solidification of alloys Alloy with higher “Tm composition” precipitates first, followed by “lower Tm”compositions. At infinite times, diffusion will homogenize it

38. Notes • Coring => Kinetic effect • Casting => Spatially inhomogeneous cooling => coring • Two useful equations: • a) if convection mixing in liquid, but no diffusion in solid, concentration in solid, cs • (1) • b) If no convection in liquid, no diffusion in solid, concentration,c, of liquid in front of solid • (2) • f is the fraction of liquid solidified, k is partition coefficient

39. Notes The first equation is the Scheil equation and the basis for zone refining. k is the ratio of the slopes of solidus and liquidus k is determined by making linear fits

41. Quantitative dendrite formation due to supercooling • Negative dT/dx in liquid is rare - much rarer then dendrites • Higher concentration of impurities in front of growing . Crystal (snow plowing), lowering Ts • “ Dissolved impurity profile” leads to to dendrite growth . even if temperature profile in melt has a positive slope • Quantitative evaluation by taking equation 2 and a linear fit . to liquidus line (I.e. linear depression of Tm with concentration • Well known annoyance to crystal growers trying to grow . heavily doped Si single crystals

42. Temperature at which liquid freezes increases with distance (less “snow plowing” pile up) Constitutional supercooling Smallest T slope to suppress constitutional supercooling Temperature in liquid increases linearly with distance from interface. Critical T gradient depends on solidification velocity,v, see also eq. 2

43. Excursion into practical metallurgy: welding superalloys Higher T gradient=> dendritic Lower gradient => cellular

44. Growth Velocity of Dendrite as function of undercooling • Tip of growing dendrite must have radius of curvature of critical nucleus size, r* - otherwise it would either shrink or grow. As r* = 2g/Gv = 2g/aDT (a is a constant) dendrite radius proportional to 1/DT • Let tip - hypothetically - grow or shrink => consider heat balance Hence v a 1/(DT/r) aDT2

45. Notes • High undercooling => high growth velocity => high surface to volume required to shed heat => high snow plowing, I.e. segregation • Low growth velocity => reverse of above Cellular dendritic growth

46. Simple view • Dendrite at lower velocity sheds less heat • Less surface area needed - bulges will do • Hottest point • Alloying elements lowering melting point “core” towards bottom of cusps • Bottom of cups freeze last, incorporate Tm lowering impurities

47. The quantitative theory is rather difficult and concludes that the dendrite spacing is not an equilibrium value but history dependent (how growth velocity was changed). I put a copy of it on our Website - should you be interested.

48. Excursion into Low Cost Solar Silicon Growth • The segregation of impurity into cell walls has interesting consequences. • Consider 2-D growth (EFG growth of Solar Silicon) • The periodic trail of impurities triggers the formation of twins • The result is an array of twin boundaries parallel to the ribbon Liquid Silicon is pulled up by capillary forces between high purity graphite dies and pulled as a thin sheet from the top of the die. The method avoids the wasteful cutting of Si single crystals into wafers

49. Etched High resolution Raman High resolution TEM showed that the lattice distance of the {111} twin planes in EFG material is reduced by about 15%, as compared to the regular silicon–silicon bond length across the twin plane. This can be explained by the insertion of a single carbon layer on the (111) twin plane (Fig. 8) [10]. Ab-initio simulations of the total energy of the proposed lattice structure and the comparison with high resolution electron microscopy (HREM) images are in quantitative agreement with the model [26].

50. The twin boundaries are single layer insertion of Carbon atoms. The carbon atoms originate from the graphite die (the surface of which converts to SiC) Lower pulling speeds would reduce dendritic growth front but would worsen throughput => increase cost