Alloy Solidification Def. Partition coefficient k x s and x L are the mole fraction of solute in the solid and liquid respectively. T3T3 T1T1 T2T2 kx 0.

Alloy Solidification

Def. Partition coefficient k

L

s

x

xk

xs and xL are the mole fraction of

solute in the solid and liquid

respectively.

T3

T1

T2

kx0 x0x0 / k

We will consider 3 limiting cases of the solidification process. :

(a) Infinitely slow (equilibrium) solidification

(b) Solidification with no diffusion in the solid and perfect mixing in the liquid.

(c) Solidification with no diffusion in the solid and only diffusional mixing in

the liquid.

(a) Equilibrium solidification.

@ composition x0

0kxxs will be the composition of the 1st amount of solid to solidify.

* Note that k is constant for straight liquidus and solidus.

As the temperature is lowered more solid forms.

For slow enough cooling mixing in liquid and solid is perfect and xs and xL will

follow the solidus and liquidus lines respectively

At T3 the last liquid to freeze out has a composition xo

(b) No diffusion in solid, perfect mixing in liquid (i.e., by stirring).

Again the composition of the first quantity of solid solidifying in kx0.

Since kxo < xo this solid will not contain as much solute as the liquid and so

initially a quantity of solute x0 – kx0 is rejected back in to the liquid.

The slightly raises the solute content of the liquid, so the temperature of the

liquid must be decreased below T1 for further solidification to occur.

As this proceeds, the liquid become progressively richer in solute and

solidification requires progressively lower temperatures.

At any stage of the solidification process local equilibrium is assumed to exist at

the liquid / solid interface. The solid composition will not be homogeneous.

Note that the liquid can become much richer in solute than x0 / k and may even

reach the eutectic composition.

The average composition of the solid during the solidification process is always

lower that the equilibrium composition at the solid / liquid interface.

sx

T3

T1

T2

kx0 x0x0 / k

xSxL

Integration yields:

)1(0 )1( k

ss fkxx using 0@0 fkxxs

and)1(

0 k

LL fxx

These relations are known as non-equilibrium lever rule equations. Note that for

k < 1 & no solid state diffusion there will always be some eutectic in the last

drop of liquid to solidify.

The variation of xs along a solidified bar can be obtained by equating the solute

rejected into the liquid when a small amount of solid forms with the resulting

solute increase in the liquid. This gives:

LsssL dxfdfxx 1

Where fs is the volume fraction solidified.

(c) No diffusion in solid, Diffusional mixing in liquid

Composition profile when temperature is between T2 and T3.

Steady state solidification at T3.

Solid Liquid

x0

kx0

v

D / v

xo/k

Solid Liquid

x0

kx0

Composition profile @ T3 and below showing the final transient.

Steady state

x0

xeutectic

xmax

(1) Initial transient develops as first solid to solidify has composition x0k

and solute is “rejected” into liquid.

(2) Solute concentration in liquid builds up ahead of S / L interface.

Eventually a steady – state concentration of liquid at the S / L interface

is achieved. This is set by a balance between rejected solute

and diffusion of solute away from the S / L interface. i. e,

Solute rejected

)( sL CCvJ iLdx

dCD

,

sLiL

CCvdx

dCD

,

Note the similarity of this eqn. to the eqn. describing solidification of pure

metals.

In the case of pure metal, solidification is controlled by removal of latent heat.

For alloys, solidification is controlled by removal of excess solute.

The latent heat conduction is 104 times faster than solute diffusion. i.e., solute

diffusion is rate limiting.

The steady – state concentration gradient is formed by solving the diffusion eqn

:

vD

x

k

kxxL /

exp1

10

(D / v) is the characteristic width of the concentration profile.

When the S / L interface ~ (D / v) from the end of the bar, the solute

concentration rises rapidly to a final transient and eutectic formation.

Constitutional Supercooling-Dendrite solidification

How is a planar front stabilized during solidification ?

Solid

( D / v)

v Liquidx0 / k

xo

xL

T3

T1

T2

kx0 x0x0 / kTLT1

T3

Distance, S (composition)

L

i

dT

dS

compositionally super-cooled liquid

L

i

dT

dS

Critical gradient

e

i

dT

dS

e L

i i

dT dT

dS dS

The interface is in local equilibrium, i.e., the interface temperature is Tliquidus.

The rest of the liquid can follow any temperature gradient such as dTL/dS

If :

The liquid in front of the solidification front exists below its equilibrium

freezing temperature, i.e., it is super-cooled . Compositional or

“constitutional” supercooling. From the diagram

vD

TT

dS

dT

i

e

/31

e L

i i

dT dT

dS dS

When T1 and T3 are the liquidus and solidus temperatures for the bulk

composition x0. The condition for a stable planar interface is

1 3/

/L i

T TdT dS

D v

Large interface velocities and large ( T1 – T3 ) will find to de-stabilize the

interface. Normally the temperature gradients and growth velocities are not

individually controllable in an experiment and both are determined by the rate

of heat conduction in the solidifying alloy.