Solidification son dagıt(28.04.2013)

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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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Solidification son dagıt(28.04.2013)