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Heat Flow and Interface Stability
Elemental metals - solidification rate controlled by rate at which latent
of fusion can be conducted away from the solid/liquid interface
Heat conduction can be either into the solid OR liquid depending on
temperature gradient near the interface
Case I:
Solid growing into superheated liquid
Solid/liquid interface growing with a
velocity v
Temperature gradients in both solid and
liquid with interface at melting temp. Tm
Planar solid/liquid interface moving with
velocity v, heat flow balance
K STS' = K LTL' + vLv
K’s are the thermal conductivities
T’s are the thermal gradients
LV is the latent heat of fusion
The above equation holds for growth of
any planar interface even when heat is
conducted into the liquid (TL’ < 0)
Consider a small solid protrusion at the
solid/liquid interface growing into the
liquid
Locally at the protrusion - temp. gradient
into the liquid is increased, temp. gradient
in solid decreases
More heat flow from the liquid into the
protrusion, slower rate of heat
extraction
into the solid
Phase
Transformations
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Therefore, slower rate of growth of the
solid at the protrusion as compared with
the planar regions - solid protrusion is
removed with time
Interface protrusion is unstable
Case II:
Solid growing into undercooled or
supercooled liquid
Prostrusion at solid/liquid interface negative temperature gradient at the
interface becomes more negative steeper gradient in liquid
Faster rate of heat extraction into the
liquid near the protrusion
Solid protrusion grows faster
Protrusions at interface are stable
Planar solid/liquid interface growing into
a supercooled liquid is inherently
unstable
Case I type situation (heat conduction
through the solid) typically takes place
when the solidification takes place from
the mould walls - cooler than the molten
liquid
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Phase Transformations
Case II type situation (heat conduction into the liquid during
solidification) - takes place during the initial stages of solidification
for nucleation taking place on heterogeneous nucleation sites within
the melt (e.g. impurity particles) - substantial undercooling (or
supercooling) of the liquid might be required
Solid nucleus forming on an impurity particle will be growing into a
undercooled liquid - latent heat conducted into the liquid
Solid particle will develop many stable protrusions in different
directions during growth - arms will grow from the solid particle
Initially spherical nucleus
Interface becomes unstable
Starts developing primary arms
in many directions - usually
crystallographic (e.g. for cubic
metals along <001> directions
along the cube axes)
As primary arms grow and
elongate, the interfaces become
unstable and break up forming
secondary arms, then tertiary
arms and so on
Shape of this type of solid Dendrite (comes from the Greek
word for tree)
Dendritic growth morphology
In pure metals these dentrites
are referred to as thermal
dendrites
Phase Transformations
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Solidification of Single Phase Alloys
Consider a binary alloy with the phase diagram as shown below:
Liquidus and solidus are
considered straight lines
Define partition function as:
k=
XS
XL
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XS and XL are the equilibrium solute contents at the ends of the tie
line at a certain temperature
k - independent of temp.
Solidification of such an alloy is a rather complex process - depends
on a combination of temp. gradients, cooling rates, growth velocities
Simplified case:
Planar growth of liquid/solid interface
Unidirectional heat flow or heat extractions along a single direction
Special furnace in which heat flow occurs only along a single
direction
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Phase Transformations
Three limiting cases of solidification:
1.  Infinitely slow (equilibrium) solidification.
2.  Solidification with no diffusion in the solid but perfect mixing in
the liquid.
3.  Solidification with no diffusion in the solid and only diffusional
mixing in the liquid
Equilibrium solidification:
easiest case
Alloy of composition X0 starts
solidifying at temperature T1.
First solid to form has
composition, kX0.
Cooling further below T1,
more solid is formed.
Cooling rate is sufficiently
slow for diffusion to take place
in the solid phase allowing for
equilibrium homogeneous
solid composition to be
achieved
Relative amounts of solid and
liquid at any temp. - lever
rule
Since solidification is 1D,
solute conservation implies
that the shaded areas in the
adjacent figure are equal
Solidification is completed at
T3 and the last solidifying
liquid has composition X0/k
5
Phase Transformations
No Diffusion in Solid, Perfect Mixing in Liquid:
Often cooling rate is too rapid for substantial diffusion in the growing
solid phase - assume no diffusion in the solid phase
Liquid composition is kept homogeneous during cooling by
continuous stirring
1D heat extraction as previous case: Solidification starts at temp. T1
First solid to form has composition kX0 < X0, purer than the liquid
from which it forms
Solute rejected into the liquid - raises its composition above X0
Temp. of interface has to be reduced before further solidification
occurs - next layer of solid to form has a composition slightly richer in
solute than kX0.
As process continues, liquid gets progressively richer in solute and
solidification takes place at progressively lower temperatures
At any temperature local equilibrium is maintained at the interface solid and liquid compositions given by end points of tie line
But since no diffusion in the solid is permitted the layers of solid
forming retain their original compositions
Therefore, mean composition of the solid is always lower than the
interface solid composition (given by the phase diagram) - shown by
the dashed line below:
6
Phase Transformations
Relative amounts of solid and liquid given by level rule applied to XS
and XL.
Liquid composition can become much richer than X0/k during
solidification and even reach the eutectic composition XE.
Soldification can finish by the final liquid undergoing eutectic
solidification and forming a two phase (α + β) structure
Average composition after complete solidification is still X0 but the
profile looks very different as compared to equilibrium solidification
7
Phase Transformations
Variation of XS along solidified bar obtained by equating solute
rejected by small (infinitesimal) layer of solid formed to solute
increase in the liquid
( X L − X S ) dfS = ( 1− fS ) dX L
X S = kX 0 ( 1− fS ) (
k−1
X L = X 0 f L( )
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k−1)
fS is the volume fraction of solid
Integrating using the boundary
condition XS = kX0 when fS = 0
Above equations are known as the non-equilibrium lever rule or Scheil
equations
8
Phase Transformations
Cellular and Dendritic Solidification
Diffusion of solute away from growing solid into liquid during alloy
solidification - analogous to conduction of latent heat of fusion into
liquid during pure metal solidification
Complication - solute gradient ahead of solid/liquid interface variation in equilibrium solidification temperature (liquidus line)
Actual liquid
temperature gradient
can follow any line,
e.g., TL
At steady-state, at the
interface,
TL = Te = T3
Temp. gradient TL lies
below the critical Te
value - so liquid is
supercooled below its
equilibrium
solidification temp. constitutional
supercooling (arising
from compositional or
constitutional effects)
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Phase Transformations
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Stable protrusions can form on a planar interface during alloy
soldification provided a region of constitutional supercooling exists
ahead of the growing solid
For the TL shown, the temp. of the tip of the protrusion will be higher
than that of the surrounding interface - unlike pure metal
solidification all the interface need not be isothermal in alloy
solidification
As long as tip temp. is < local liquidus temp., protrusion can still grow
If temp. in liquid (TL) > critical temp. gradient (Te), then the tip temp.
will exceed the local liquidus temp. and the protrusion will melt back
Under steady-state growth, the critical temp. gradient can be
calculated as follows:
TL' >
(T1 − T3 )
( D / v)
Where T1 and T3 are the liquidus and solidus temps. for comp. X0
Regrouping, condition for no constitutional supercooling is,
TL' ( T1 − T3 )
>
v
D
T1-T3 is the equilibrium freezing range of the alloy
Planar front solidification is most difficult in case of alloys with a
large equilibrium freezing range and high rates of solidification
Usually, alloys never solidify with a planar solidification front - only
under very specially controlled conditions
Temp. gradients and growth rates are not individually controllable
but depend in a complex manner on the rate at which heat is
conducted away from solidifying alloy
10
Phase Transformations
Process of breakdown of an initially planar solidification front can be
shown as below:
Formation of first protrusion - lateral rejection of solute - solute pileup at the base of protrusion - lowers equilibrium solidification temp.
locally - causes formation of recesses - triggers the formation of other
protrusions
Protrusions develop into long arms, or cells, parallel to the direction
of heat flow - cellular solidification
Lateral solute rejection concentrates at the cell walls - last liquid to
solidify at lowest temperature
Tips of the cells grow into the hottest liquid - therefore have
minimum solute concentration
Liquid between the cells - enriched in solute may even reach the
eutectic composition and then the cell walls may have a second phase
Cells have the same orientation as their neighbors and can join
together to form single grain
11
Phase Transformations
Cellular solidification is stable only for certain range of temp.
gradients - if gradient is too small, the cells or primary arms start
developing secondary arms, tertiary arms .etc. -
dendritic solidification
Reason from cells to dendrites is not fully understood
Probably associated with the creation of constitutional supercooling
within the liquid entrapped between the walls of growing cells - causes
interface instabilities in the lateral direction (perpendicular to the
growth direction of primary cellular arms)
For unidirectional solidification to continue, there should be minimal
temp. gradient that develops in the lateral direction - perpendicular
to growth direction
Therefore, typically the cell arm spacing is such that this lateral
gradient is reduced as much as possible - experimentally it is
observed that with faster cooling rates the spacing decreases consistent since faster cooling rates would allow less time for solute
diffusion into the liquid between the cellular arms - reducing the
spacing reduces the probability of developing constitutional
supercooling
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Phase Transformations