NTNU

1

Solidification, Lecture 3

1

Interface stabilityConstitutional undercoolingPlanar / cellular / dendritic growth front

Cells and dendritesGrowth of dendritesPrimary & secondary arm spacing

 

NTNU

2

Growth

Controlling phenomenon Importance Driving force

Diffusion of heat Pure metals ΔTt

Diffusion of solute Alloys ΔTc

Curvature Nucleation ΔTr

DendritesEutectics

Interface kinteics Facetted ΔTk crystals

NTNU

3

Morphologies of the s/l front

Increasing growth rate

Causes instability of s/l front - more branching

planar cellular dendritic

NTNU

4

Solute redistribution

C0

T0

C0l

s

T

C

• 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

C0/k

C0k

NTNU

5

Solute boundary layer

Thickness, dependson diffusion, Dl andgrowth velocity, V

V2>V1

Ref. 1

Reproduced from:W. Kurz & D. J. Fisher:Fundamentals of SolidificationTrans Tech Publications, 1998

€

≈2Dl /V

€

Cl =C0 + ΔC0 exp(−VzDl

)

NTNU

6

Steady state growth

• Fully developed solute bondary layer

• Rejected solute from solid balanced by diffusion in liquid

Cl

Gc

Concentration gradient in liquidat interface, Gc

€

Gc = (dCldz

)z=0 = −VDl

ΔC0

NTNU

7

Constitutional undercooling

Ref. 1

Local variations inliquid concentration, Cl

causes local variationsin liquidus temperature, Tl

mGc

Temperature gradient: GLiquidus temperature gradient: mGc

Undercooling: φ=mGc-G

G

Reproduced from:W. Kurz & D. J. Fisher:Fundamentals of SolidificationTrans Tech Publications, 1998

NTNU

8

Constitutional undercooling• Undercooling if G<mGc

Constitutional undercooling in all ”normal” casting operations

• Example: Al-0.1%SiΔT0=4 KD=3x10-9 m2/sG=2x104 K/mV>1.5x10-5 m/sV needs to be less than 15 μm/s or G needs to increase to avoid constitutional undercooling

€

GV

<ΔT0

Dl

NTNU

9

Stability of planar front

Breakdown of planar front with constitutional undercooling

Reproduced from:W. Kurz & D. J. Fisher:Fundamentals of SolidificationTrans Tech Publications, 1998

NTNU

10

Morphological development of the s/l front

Increasing const. undercooling

planar cellular dendritic

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

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

NTNU

13

Growth temperatures of cells and dendrites

Reproduced from:W. Kurz & D. J. Fisher:

Fundamentals of Solidification

Trans Tech Publications, 1998

NTNU

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

NTNU

15

Columnar dendrite growth

Al-30%Cu

NTNU

16

Equiaxed dendritic growth

Al-30%Cu

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

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

NTNU

19

Secondary dendrite arm coarsening

Al-20%Cu

Secondary arm

spacing, λ2

,increases during growth

NTNU

20

Secondary dendrite arm spacing

dtdTK

ft 31

2 fAt

Reproduced from:W. Kurz & D. J. Fisher:

Fundamentals of Solidification

Trans Tech Publications, 1998

NTNU

21

Dendrite growth, summary

Reproduced from:W. Kurz & D. J. Fisher:

Fundamentals of Solidification

Trans Tech Publications, 1998

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 = 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

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, λ1, and secondary arms (branches) with spacing λ2.

• Dendrites coarsen as they grow increasing λ2 with local solidification time.