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Chap. 2
THERMODYNAMICS OF
SOLIDIFICATION
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Chap. 2
2.1 EQUILIBRIUM
Equilibrium is attained when the Gibbs free energy (a
function of P, T, and composition) is at a minimum.
 G 
 G 
 G 
 dn i ....  0
dG ( P, T , ni ...)  
 dT  
 dP  
 T  P ,ni
 P  T ,ni
 ni  P ,T
 G 

 i  
 ni  T , P ,n ...
j
dG   i dni   j dn j  ...  0
Number of moles of component i.
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Chap. 2
2.1 EQUILIBRIUM
For a multiphase system, a condition for equilibrium is that the chemical
potential of each component must be the same in all phases.

i  i

Although equilibrium conditions do not actually exist in real systems,
under the assumption of local thermodynamic equilibrium, the liquid and
solid composition of metallic alloys can be determined using equilibrium
phase diagrams.
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Chap. 2
2.2 THE UNDERCOOLING REQUIREMENT
The driving force of any phase transformation
including solidification is the change in free energy.
Thermodynamics demonstrates that in a system without
outside intervention, the free energy can only decrease.
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Chap. 2
2.2 THE UNDERCOOLING REQUIREMENT
The entropy is a measure of the amount of disorder in the
arrangement of atoms in a phase.
The liquid possesses a larger
degree of disorder than the
solid. Thus, the entropy of the
liquid is higher than the entropy
of the solid.
Characteristics
Solid
Thermal vibration from their equilibrium position at lattice points
Liquid
Structural disorder ( disappeared long-range order)
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Chap. 2
2.2 THE UNDERCOOLING REQUIREMENT
A certain amount of heat, the heat of fusion, is required to melt a specific
material. Since the heat of fusion is the energy required to disorganize a
mole of atoms, and the melting temperature is a measure of the atomic
bond strength, there is a direct correlation between the two.
Let us start our analysis of solidification by introducing
a number of simplifying assumptions.
A) pure metal
B) constant pressure
C) flat solid/liquid (S/L) interface
( the radius of curvature of the interface is r=∞ )
D) no thermal gradient in the liquid
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Chap. 2
2.2 THE UNDERCOOLING REQUIREMENT
L  S  0
OR
G L  GS  0
G  H  TS
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Chap. 2
2.2 THE UNDERCOOLING REQUIREMENT
Gv  GL  GS  ( H L  H S )  Te (S L  S S )  0
H f  Te S f
GV  H f  T
Gv
T 
S f
S f 
H f
Te
H f
Te
Te  T
 H f
 S f T
Te
Thermodynamics does not allow further
clarification of the nature of undercooling.
Kinetics considerations must be introduced
to further understand this phenomenon.
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Chap. 2
2.3 KINETIC UNDERCOOLING
When a S/L interface moves, the net transfer of atoms at the interface
results from the difference between two atomic processes.
atoms in solid -> atoms in liquid
:
melting
atoms in liquid -> atoms in solid
:
solidification
Rate of melting (S->L)
 GM
 dn 
   p M nS v S exp  
 dt  M
 k BT



Rate of solidification (L->S)
 GS
 dn 
   p S n L v L exp  
 dt  S
 k BT



p : probability
n : number of atoms per unit area
v : the vibration frequency
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Chap. 2
2.3 KINETIC UNDERCOOLING
p M , S  f M , S  AM , S
f : the probability that an atom of sufficient energy is moving toward the interface
A : the probability that an atom kicked back by an elastic collision
dn dt M
 dn dt S
V  Rate  of  solidifica tion  Rate  of  melting  VC  VC exp  G RTi 
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Chap. 2
2.3 KINETIC UNDERCOOLING
Fv  Gr  GT  Gc  FP
The four right hand terms are the change in free energy, because of
curvature, temperature, composition and pressure variation, respectively.
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Chap. 2
2.4 CURVATURE UNDERCOOLING
Liquid-solid interface is not planar especially in the case at the
beginning of solidification. (different from the assumption C)
If solidification begins at a point in the liquid, a spherical
particle is assumed to grow in the liquid, and an additional
free energy associated with the additional interface, different
than ΔGV must be considered. This additional energy, results
from the formation of a new interface, and is a function of
the curvature of the interface.
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Chap. 2
2.4 CURVATURE UNDERCOOLING


1
K 


l
r
r
A
1
1
K 


v
r1
r2
(2D)
(3D)
sphere (r1  r2 ) : K  2 / r
cylinder (r1  , r2  r ) : K  1 / r
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Chap. 2
2.4 CURVATURE UNDERCOOLING
Assuming that the radius of the spherical particle is r,
when the particle increases by dr, the work resulting
from the formation of a new surface, d ( 4r 2  )
dr
must be equal to that resulting from the decrease of
the free volumetric energy, i.e., d  4 3

 r G v 
dr  3

2
Gv  
r
Gv  K
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Chap. 2
2.4 CURVATURE UNDERCOOLING
S f Tr  K
Tr  Te  T 
r
e

S f
K  K
Tr : curvature undercooling
Ter
: equil. temperature for a sphere of radius r

: Gibbs-Thomson coefficient
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Chap. 2
2.5 THERMAL UNDERCOOLING
Let us now relax assumption (D), and allow a
thermal gradient to exist in the liquid.
TT  Te  T
*
T* is the interface temperature.
GT  S f (Te  T )
*
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Chap. 2
THE END.
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