Alloy Solidification
Def. Partition coefficient k
k 
T1
T2
T3
xs
xL
xs and xL are the mole fraction of
solute in the solid and liquid
respectively.
kx0 x0
x0 / 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
xs  kx0will 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.
The average composition of the solid during the solidification process is always
lower that the equilibrium composition at the solid / liquid interface.
xS xL
T1
T2
T3
xs
kx0 x0
x0 / k
Note that the liquid can become much richer in solute than x0 / k and may even
reach the eutectic composition.
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:
xL  xs  df s
 1  f s  dxL
Where fs is the volume fraction solidified.
Integration yields:
xs  kx0 (1  f s ) ( k 1)
and
xL  x0 f L
using
xs  kx0 @ f  0
( k 1)
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.
(c) No diffusion in solid, Diffusional mixing in liquid
Solid
Liquid
x0
kx0
Composition profile when temperature is between T2 and T3.
Solid
xo/k
v
Liquid
x0
kx0
D/v
Steady state solidification at T3.
xeutectic
Steady state
xmax
x0
Composition profile @ T3 and below showing the final transient.
(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
J  v(CL  Cs )
 dC 
 D

 dx  L ,i
 dC 
 D
  vCL  Cs 
 dx  L ,i
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 :


1 k
x 
xL  x0 1 
exp 


k
 D / v   

(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
x0 / k
v
Liquid
T1
T2
T3
xL
xo
 dTe 


dS

i
T1
T3
( D / v)
Critical gradient
TL
 dTL 


 dS i
Distance, S (composition)
kx0 x0
x0 / k
 dTe   dTL 

 

dS
dS

i 
i
compositionally
super-cooled liquid
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 :
 dTe   dTL 

 

dS
dS

i 
i
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
T1  T3
 dTe 

 
D / v 
 dS i
When T1 and T3 are the liquidus and solidus temperatures for the bulk
composition x0. The condition for a stable planar interface is
 dTL / dS i 
T1  T3
 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.