1. NTNU 1 Solidification, Lecture 3 1 Interface stability Constitutional undercooling Planar / cellular / dendritic growth front Cells and dendrites Growth of dendrites Primary & secondary arm spacing

2. NTNU 2 Growth Controlling phenomenonImportanceDriving force Diffusion of heatPure metalsΔT t Diffusion of soluteAlloysΔT c CurvatureNucleationΔT r Dendrites Eutectics Interface kinteicsFacettedΔT k crystals

3. NTNU 3 Morphologies of the s/l front Increasing growth rate Causes instability of s/l front - more branching planar cellular dendritic

4. NTNU 4 Solute redistribution C0C0 T0T0 C0C0 l s T C Lower solubility of alloying elements in s than in l k=C s /C l <1 m= dT l /dC<0 Enrichment of solute in liquid during solidification C 0 /k C0kC0k

5. NTNU 5 Solute boundary layer Thickness,  depends on diffusion, D l and growth velocity, V V 2 >V 1 Ref. 1 Reproduced from:W. Kurz & D. J. Fisher: Fundamentals of Solidification Trans Tech Publications, 1998

6. NTNU 6 Steady state growth Fully developed solute bondary layer Rejected solute from solid balanced by diffusion in liquid ClCl GcGc Concentration gradient in liquid at interface, G c

7. NTNU 7 Constitutional undercooling Ref. 1 Local variations in liquid concentration, C l causes local variations in liquidus temperature, T l mG c Temperature gradient: G Liquidus temperature gradient: mG c Undercooling: φ=mG c -G G Reproduced from:W. Kurz & D. J. Fisher: Fundamentals of Solidification Trans Tech Publications, 1998

8. NTNU 8 Constitutional undercooling Undercooling if G<mG c Constitutional undercooling in all ”normal” casting operations Example: Al-0.1%Si ΔT 0 =4 K D=3x10-9 m 2 /s G=2x10 4 K/m V>1.5x10 -5 m/s V needs to be less than 15 μm/s or G needs to increase to avoid constitutional undercooling

9. NTNU 9 Stability of planar front Breakdown of planar front with constitutional undercooling Reproduced from:W. Kurz & D. J. Fisher: Fundamentals of Solidification Trans Tech Publications, 1998

10. NTNU 10 Morphological development of the s/l front Increasing const. undercooling planar cellular dendritic

11. NTNU 11 Cellular growth Cells grow at low constitutional undercooling No side branching Direction antiparallell to heat flow Accumulation of solute between cells Adjustment of cell spacing by stopping or division of cells Reproduced from:W. Kurz & D. J. Fisher: Fundamentals of Solidification Trans Tech Publications, 1998

12. NTNU 12 Transformation from cells to dendrites Dendrites form at higher const. undercooling Side branches Growth in preferred crystallographic directions Reproduced from:W. Kurz & D. J. Fisher: Fundamentals of Solidification Trans Tech Publications, 1998

13. NTNU 13 Growth temperatures of cells and dendrites Reproduced from:W. Kurz & D. J. Fisher: Fundamentals of Solidification Trans Tech Publications, 1998

14. NTNU 14 Dendrites Primary arms, λ 1 Secondary arms, λ 2 Distinct angles between arms (90o for cubic) Reproduced from:W. Kurz & D. J. Fisher: Fundamentals of Solidification Trans Tech Publications, 1998

15. NTNU 15 Columnar dendrite growth Al-30%Cu

16. NTNU 16 Equiaxed dendritic growth Al-30%Cu

17. NTNU 17 Solute boundary layer in dendritic growth Al-30%Cu Yellow-red: low C Green-blue: high C Faster growth and sharper dendrite tips when thin boundary layer

18. NTNU 18 Solute rejection from dendrite Growth at low undercooling Radial solute diffusion Growth determined by diffusion and curvature Supersaturation, , (undercooling) determines growth rate & tip radius Reproduced from:W. Kurz & D. J. Fisher: Fundamentals of Solidification Trans Tech Publications, 1998

19. NTNU 19 Secondary dendrite arm coarsening Al-20%Cu Secondary arm spacing, λ 2,increases during growth

20. NTNU 20 Secondary dendrite arm spacing Reproduced from:W. Kurz & D. J. Fisher: Fundamentals of Solidification Trans Tech Publications, 1998

21. NTNU 21 Dendrite growth, summary Reproduced from:W. Kurz & D. J. Fisher: Fundamentals of Solidification Trans Tech Publications, 1998

22. NTNU 22 Summary/ Conclusions Solute in an alloy will redistribute during solidification. In eutectic systems (k<1), alloying elements will enrich in the liquid. With limited diffusion, solute will pile up at the s/l interface and form a boundary layer. Width of the boundary layer is inversely proportional to growth rate At steady state the boundary layer is fully developed. Growth of a solid with constant composition = C 0 The liquid boundary layer causes local variations of liquidus temperature ahead of the s/l interface. If the liquidus temperature gradient, mG c is larger than the actual temperature gradient, G, the liquid will be constitutionally undercooled. Constitutional undercooling occurs in most casting operations of alloys Constitutional undercooling leads to breakdown of a planar growth front

23. NTNU 23 Summary/ Conclusions Cells form at low constitutional undercooling, just after breakdown of planar front. Cells have no side branches and grow independent of crystallographic orientation, antiparallell to heat flow. Cells grow at temperatures far below liquidus. Dendrites grow at high constitutional undercooling. They grow just below liquidus in preferred crystallographic directions. Solute diffuses radially at the dendrite tip. Growth undercooling and growth morphology is determined by curvature and diffusion. Dendrites are characterized by a primary arms (trunk) with a spacing, λ 1, and secondary arms (branches) with spacing λ 2. Dendrites coarsen as they grow increasing λ 2 with local solidification time.