Solidification, Lecture 3

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

2. Growth Controlling phenomenon Importance Driving force Diffusion of heat Pure metals ΔTt Diffusion of solute Alloys ΔTc Curvature Nucleation ΔTr Dendrites Eutectics Interface kinteics Facetted ΔTk crystals

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

4. Solute redistribution • Lower solubility • of alloying elements • in s than in l • k=Cs/Cl<1 • m= dTl/dC<0 • Enrichment of solute in • liquid during solidification T l C0 C0k T0 C0/k C0 s C

5. Solute boundary layer Thickness,  depends on diffusion, Dl and growth velocity, V V2>V1 Reproduced from:W. Kurz & D. J. Fisher: Fundamentals of Solidification Trans Tech Publications, 1998 Ref. 1

6. Steady state growth • Fully developed solute bondary layer • Rejected solute from solid balanced by diffusion in liquid Concentration gradient in liquid at interface, Gc Cl Gc

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

8. Constitutional undercooling Undercooling if G<mGc Constitutional undercooling in all ”normal” casting operations Example: Al-0.1%Si ΔT0=4 K D=3x10-9 m2/s G=2x104 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. 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. Morphological development of the s/l front planar cellular dendritic Increasing const. undercooling

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

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

14. Dendrites • Primary arms, λ1 • Secondary arms, λ2 • Distinct angles • between arms • (90o for cubic) Reproduced from:W. Kurz & D. J. Fisher: Fundamentals of Solidification Trans Tech Publications, 1998

15. Columnar dendrite growth Al-30%Cu

16. Equiaxed dendritic growth Al-30%Cu

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. 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. Secondary dendrite arm coarsening Al-20%Cu Secondary arm spacing, λ2,increases during growth

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

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

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 = C0 • The liquid boundary layer causes local variations of liquidus temperature ahead of the s/l interface. If the liquidus temperature gradient, mGc 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. 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, λ1, and secondary arms (branches) with spacing λ2. • Dendrites coarsen as they grow increasing λ2 with local solidification time.