solidification son dagıt(28.04.2013)
TRANSCRIPT
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Mid-term exam -1 : %20
Mid-term exam -2 : %20
Final exam : %60
%100
SOLIDIFICATION
Prof. Dr. . Aydn ATASOY
Department of Metallurgical and
Materials Engineering
Technical University of stanbul
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141
GIBBS FREE ENERGY
For reactions at constant temperature andpressure
the relative stability of a system is determined by its Gibbs free energy:
H : enthalpy (heat content) of the system
T : absolute temperature (Kelvin)
E : internal energy of the system (kinetic energy from atomic vibrations +
potential energy from bonds between the atoms)
p : pressure
v : volume
S : entropy (measure of the randomness of the system; degree of disorder)
T S : mixing energy
p v : For condensed systems (i.e. solids and liquids) p v is relatively small
T = constant
p = constantSTHG vpEH
STEG EH
Liquid - Solid Phase Transformation 141-0
Liquid Solid For T = T , G = GT
L s
,
G
T
GGL
Temperature
G = G GLs
G = E
T S
= 0
T TFor
G = E TT
Undercooling
T = (T T) 0
E 0 G 0
Solidification Nucleation + Growth
v
v v v
m
m
m
vm
S = ETm
vv
v v
v
s m
T
T
T = T TTT
Solid grains
TimeTemperature
0
141-1
Cooling Curve and Nucleation
m nm
n
Liquid
Liquid + First nuclei
n
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141b
r
A Spherical solidparticle called embryo forms homogeneously
from the liquid
Liquid atoms
Embryo/liquid interfacial
area :
A= 4 r2
Volume of a spherical
embryo of radius r :
34 r3
V=
DEFINITION OF THE VOLUME FREE ENERGY CHANGE
Solid (s)
Liquid ()
: volume of l iquid phase : volume of solid phase
Free energy of the system : Free energy of the system:
Total free energy change of the system after the solidification of a solid phase
Volume free
energy change =
free energy change
per unit volume
: free energy per unit
volume of liquid phase
: free energy per unit
volume of solid phase
141aa
: Solid-liquid interfacial area
: Solid-liquid interfacial free energy
Liquid ()
G v
Gsv)vv( s
v
v s
ss
s
vsvs2AGvGvvG )(
GvAGvG)vv(GGG vsssvsvs12
A s
s
GGG vsvv
ssvsssvsvs AGvA)GG(v
)v(
GvGv1
Tn
1 2
Homogeneous NUCLEATION in Liquids 141
Liquid Solid For T =T G = E T Sv = 0r* = critical size For T T G =(E Tn) / T
undercooling
G
G
r*r (radius of particle)
(4/3) rGV
4 rssurfacefree energychange
volume free energy change
Total free energy change
a r r* embryor r* stable nucleus (GT / r) = 0
0
For r = r*
T m
m
m
*
mv
v v
vTTT nmn
: Latent heat of solidification0LE mv
(GT)
s
2v
3T r4Gr)3/4(G
TE
T2r
nv
ms*
TE3
)T16(G
2n
2v
2m
3s*
a0
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141 aaa
of radius r*Stable embryo( nucleus)
of radius (r* + r)1 atom
from the liquid ][ ] [
+
G 0T
Unstable embryo+[ ]
141a
The number of critical nuclei per unit volume of liquid , n*,
at any temperature is given by
n* = N exp(G*k T
)
N : the number of atoms per unit volume of liquid
(The number of sites available for nuclei formation in the liquid)
L
L
Two-dimensional representation of an
instantaneouspicture of the liquid
structure. Many close-packed
crystal-like embryos(coloured) are
present
B
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141eHomogeneous Nucleation Rate in Liquids
N
= N L exp(G*
k T
GD
k T) exp( )
NL
:Number ofavailable nucleation sites ( here; each atom)
in the unit volume of the liquid.(The number of atoms per
unit volume of the liquid.)
: Thermal vibration frequency of the atoms in the liquidG
D
: Activation free energy for the diffusion of atoms in the
liquid
The addition ofone more atom to each of the critical-sized embryos
will convert them into stable nuclei :
hom.
hom. B B
0 K Tm
n*
N
n*
N
T
T4 3 2 1
TTT
T
T : For small undercoolings G* is largebut the rate of diffusion is rapid
and hence number of critical nuclei (n*)
is small. (Large grains)
1
T3 : For large undercoolings G* is small
but the rate of diffusion is very slow
and hence the nucleation rate is again
low.
T
4
: At intermediate undercoolings, diffusion
rate is fairly rapid and G* is not too largeand there is a maximum in the nucleation
rate. (Small grains)
T :
2
For very large undercoolings,
the diffusion rate and G* areextremely small and hence no embryo
can reach the crystal unit cell size (r* a )(Amorphous solid)
141d
0
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100 m
Hyper-eutectic Al-wt%15Si alloy
Primary dendrites of Al
Primary dendrites of Si
Fine eutectic
microstructure
grows onprimary
dendrites of Al
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141m
= N exp( ) exp( )
Heterogeneous Nucleation Rate in Liquids
het. s
D
D
G*k T
Gk T
N :Number of sites available for heterogeneous nuclei
formation per unit volume of the liquids
: Thermal vibration frequency of the atoms in the liquidG : Activation free energy for the diffusion of atoms in the
liquid
Nhet.
BB
=0r (excellent wetting)
G* = 0
0180(partial wetting)
G* G*het. hom.
= 180(no wetting)
G* = G*het. hom.
141k
0
N
T
het.
hom.
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T
Water
V
WaterWaterMet
allicsubstrate
(c
hill)
a b
Heater
Cooler
Heatinsulator
z
T IT E
V
T
0(t)V f
TqLiquid
Liquid
0
0
TssT fs
Water
G
G
CC E1
)t(TI f
0
x
0(t)T
Gq
f
0.onstCGz
TG
0z
q
0C .onstVV pot .ConstTI
Vpot
Vcon
Tft Tct
)t(V f
Liquid
cru.
Vcru
Basic Techniques of Directional Solidification
Bridgman technique Directional casting technique
: System coordinates
: Coordinate system moving with the solid-liquid interfaceyx
V : Rate of interface movement
Tq : Temperature imposed by the temperature gradient (G) arising fromthe heat flow occurring in the casting.
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b
TT kk TT kk
5nm
2)Tk(/ IB 10)Tk(/ IB 12)Tk/(L IBf
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141
At equilibrium the free energy reaches an absolute minimum
and the entropy reaches the maximum value that the system can exhibit.
Free energy change for the reaction:
T= constant
Gl
E
Gs
E
S S
G = G G =s l
E = E E =S = S S =
All spontaneous reactions occur when the system can lower its free energy
l s
Free energy change
Activation energy
Entropy change
In an isolated system:
For condensed systems (i.e. solids and liquids) v is relatively small
H H H = H H = Entalpy change
EvpEH
l
l
l
s
s
s
s
s
s
l
l
l
0STESTHGGGs
l
141t
REACTION RATE (How fast does a reaction occur?)
1 2 3
An atom is vibrating about the position 1
To lower the free energy the atom
must move into position 3 (free energy
change : G)But it must first overcome the energybarrier which is called as
activation free energy barrier
Activation free energy to overcome the energy barrier can be supplied to
the atom as thermal energy in the form of atomic vibrations.
Ga
Ga
Arrangement of atoms
a
2
3
Stable
solid
Unstable
Gs
Gl
G*
Gibbsfreeenergy
1 0)GG(G*
a
l
T > 0
0GGG s l
TTI
GMeta-stable
Liquid Solid
G
0
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141u
Statistical Thermodynamics
Probability that atom has sufficient thermal energy to overcome
the barrier of G :
R= molar gas constant = 8.314 J/(mol K) = 1.98 cal/(mol
K)
k = Boltzmanns constant =
N = number of AvogadroA
Here, the particle is an atom!
Joule106.1eV118
)Kparticle/(J1038.1N/R23
A
)Kparticle/(cal103.324
mol/particles1002.623
TkIB
GaexpP
a
141
: vibration frequency of an atom in a system (number ofattemptsper unit time to overcome the energy barrier;typically
The rate at which the atoms overcome the barrier (number of
successfuljumps per unit time):
s10113
Tkr
IB
a
)hkl()s(
Gexpf
l
Tkr
IB
a
)hkl()(s
GGexpf
l
s
f)hkl(
: Crystallographic factor; the ratio of the number of
growth sites occupied by the liquid atoms to the number
of sites available in the solid at interface.
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TkTkrr
IBIB
a
)hkl(s
Gexp1
Gexpa)(aV f
lsl
a
D
Tk
Gexp
2I
IB
a
I
TkTkIB
DIB
I
)hkl(
Gexp1V
Gexp1
a
DV f
a
Df
a
D
a
DV
0
I)hkl(
0
II
D
Continuous growth
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1.5
TkN
G
ABs
C
f )hkl(
5
3
25.11
a
0
2
1
0.5
0.5
10
0.20 0.4 0.6 0.8 1.0
0
= 10 = 5
= 3
= 1b
f )hkl(
)TkN(/GABsC
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a-4
7
c
)0110(
)0001(
141
(0001)
c
(0001)
a2
1
a-1
b
brC y0V
b-1 b-2
123
4
5
6
a-2
a-3
b
b
141
141
219
219
(111)
kiz dzlemleri
BA
A211 A121
A112
B211
(111)A
(111)B
B
)111(
211
121
112
211
121
112
Twin planes
Repeatable growth defects in faceted crystals : Screw dislocation with a spiral ramp,
twinning with re-entrant corner, and twist boundary with steps. Depending upon
the type of defect present, the faceted crystal can exhibit various morphologies:needles in the case of screw dislocations, or plates in the case of twinnings (Si in Al-Si)
or twist boundaries (graphite in cast iron).
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