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Physical metallurgy

Nucleation and growth kinetics

(異質成核推導補充)

2

Homogeneous Nucleation

Materials Science and EngineeringPhase Transformations in Metals

FIGURE 12.1 Schematic diagram showing the nucleation

of a spherical solid particle in a liquid.

Gibbs free energy (G)

- Gibbs free energy is a function of the internal energy of the system (enthalpy, H) and a

measurement of the randomness or disorder of the atoms or molecules (entropy, S).

- A transformation will occur spontaneously only when the change in free energy G has a

negative value.

Case of homogeneous nucleation

- For the sake of simplicity, let us first consider the solidification of a pure material, and that

nuclei of the solid phase form in the interior of the liquid as atoms cluster together so as to

form a packing arrangement similar to that found in the solid phase.

- There are two contributions to the total free energy change that accompany a solidification

transformation. The first is the free energy difference

between the solid and liquid phases, or the volume

free energy, Gv. The second energy contribution

results from the formation of the solid–liquid phase

boundary during the solidification transformation, or

the surface energy, .

3



FIGURE 12. 2 (a) Schematic curves for volume free energy and surface free energy contributions

to the total free energy change attending the formation of a spherical embryo/nucleus during

Solidification, (b) Schematic plot of free energy versus embryo/nucleus radius, on which is

shown the critical free energy change (G*) and the critical nucleus radius (r*).

23 43

4rGrAGVG vv

084)( 2

rGr

dr

GdvWhen r = r*

vGr

2*

2

323

)(3

16)

2(4)

2(

3

4*

vv

v

v GGG

GG

V: volume of spherical nucleus,

Gv: volume free energy,

A: surface area of spherical nucleus,

: surface free energy,

r: radius of spherical nucleus,

r*: critical radius of spherical nucleus,

G*: critical free energy, activation energy

4



The volume free energy change Gv is the driving force for the solidification

transformation, and its magnitude is a function of temperature. At the equilibrium

solidification temperature Tm, the value of Gv is zero, and with diminishing

temperature its value becomes increasingly more negative. It can be shown that

Gv is a function of temperature as

Hf: latent heat of fusion,

Tm: equilibrium melting temperature (K),

T: real solidification temperature (K),

m

mf

vT

TTHG

)(

TTH

T

T

TTHGr

mf

m

m

mfv

12

)(

22*

22

23

2

3

2

3

)(

1

3

16

)(3

16

)(3

16*

TTH

T

T

TTHGG

mf

m

m

mfv

5



FIGURE 12.3 Schematic free energy-

versus-embryo/nucleus radius curves for

two different temperatures. The critical

free energy change (G*) and critical

nucleus radius (r*) are Indicated for each

temperature.

TTH

Tr

mf

m 12*

22

23

)(

1

3

16*

TTH

TG

mf

m

Both the critical radius r* and the activation free energy G* decrease as

temperature T decreases. With a lowering of temperature at temperatures

below the equilibrium solidification temperature (Tm), nucleation occurs

more readily.

<

6


Nucleation rate

The number of stable nuclei n* (having radii greater than r*) is a function of

temperature as

Where K1 is related to the total number of nuclei of the solid phase

Another temperature-dependent step influences nucleation: the clusting of atoms by

short-range diffusion during the formation of nuclei. The diffusion effect is related to

the frequency at which atoms from the liquid attach themselves to the solid nucleus, d.

Where Qd is a temperature-independent parameter-the activation energy for diffusion,

and K2 is a temperature-dependenct constant. A decrease of temperature results in a

reduction in d.

The nucleation rate is simply proportional to the product of n* and d, that is

7


FIGURE 12.4 For Solidification: (a) number of stable nuclei v.s. temperature, (b) frequency of

atomic attachment v.s. temperature and (c) nucleation rate v.s. temperature.

8

Heterogeneous Nucleation


FIGURE 12.5 Heterogeneous nucleation of

a solid from a liquid. The solid–surface (SI),

solid–liquid (SL), and liquid–surface (IL)

interfacial energies are represented by

vectors. The wetting angle () is also shown.

In practical supercoolings are often on the order of only several degrees Celsius.The reason

for this is that the activation energy for nucleation is lowered when nuclei form on preexisting

surfaces or interfaces, since the surface free energy is reduced. In other words, it is easier for

nucleation to occur at surfaces and interfaces than at other sites. Again, this type of nucleation

is termed heterogeneous.

Let us consider the nucleation, on a flat surface, of a solid particle from a liquid phase. It is

assumed that both the liquid and solid phases “wet” this flat surface, that is, both of these

phases spread out and cover the surface.

cosSLSIIL

9



VS: volume of heterogeneous nucleus,

Gv: volume free energy,

ASL: area of solid-liquid interface,

ASI: area of solid-surface interface,

SL: surface free energy of solid-liquid interface,

SI: surface free energy of solid-surface interface,

r: radius of spherical nucleus,

r*: critical radius of spherical nucleus,

Ghet*: critical free energy, activation energy

rr

r sin r sin

ILSISISISLSLvShet AAAGVG

v

SL

Gr

2*

)()(3

162

3

*

SG

Gv

SLhet

2)sin( rASI

)cos1(2 2 rASL

)coscos32(3

33

r

VS

cosSLSIIL

)(SGGhet

SLv rGr

G 2

3

43

4

4

)cos1)(cos2()(

2

S

)(*

hom

* SGGhet

10



)(*

hom

* SGGhet

4

coscos32)(

3

S

FIGURE 12.6 Schematic free energy-versus-

embryo/nucleus radius plot on which is presented

curves for both homogeneous and heterogeneous

nucleation. Critical free energies and the critical

radius are also shown.

FIGURE 12.7 Nucleation rate versus

temperature for both homogeneous and

heterogeneous nucleation. Degree of

supercooling (T) for each is also shown.

22

23

)(

1

3

16*

TTH

TG

mf

m

)exp()*

exp(kT

Q

kT

GN d

11


Heterogeneous versus Homogeneous Nucleation

Critical radius

The critical radius for heterogeneous nucleation is the same as for homogeneous.

Activation energy

The activation energy barrier for heterogeneous nucleation is smaller than the

homogeneous barrier.

Supercooling

A much smaller degree of supercooling is required

for heterogeneous nucleation.

12

Growth


kT

QCG exp

The growth step in a phase transformation begins once an embryo has exceeded the critical

size, r*, and becomes a stable nucleus. The growth process will cease in any region where

particles of the new phase meet, since here the transformation will have reached completion.

Particle growth occurs by long-range atomic diffusion, which normally involves several steps-

for example, diffusion through the parent phase, across a phase boundary, and then into the

nucleus. Consequently, the growth rate is determined by the rate of diffusion, and its

temperature dependence is the same as for the diffusion coefficient,

Q: activation energy, independent of temperature; C: constant, independent of temperature.

FIGURE 12.8 Schematic plot showing curves for

nucleation rate (N), growth rate (G), and overall

transformation versus temperature.

At some temperature, the overall transformation rate

is equal to some product of N and G. The third curve

for the total rate represents this combined effect.

The general shape of this curve is the same as for

the nucleation rate, in that it has a peak or maximum

that has been shifted upward relative to the curve.

13


Growth

FIGURE 12.9 Schematic plots of (a) transformation rate versus temperature, and (b) logarithm

time [to some degree (e.g., 0.5 fraction) of transformation] versus temperature. The curves in

both (a) and (b) are generated from the same set of data—i.e., for horizontal axes, the time

[scaled logarithmically in the (b) plot] is just the reciprocal of the rate from plot (a).

As we shall see below, the rate of transformation and the time required for the transformation to

proceed to some degree of completion are inversely proportional to one another.

First, the size of the product phase particles will depend on transformation temperature.

Secondly, when a material is cooled very rapidly through the temperature range encompassed

by the transformation rate curve to a relatively low temperature where the rate is extremely low,

it is possible to produce nonequilibrium phase structures.

14

SLsin

SLcos

Heterogeneous Nucleation相變態

SL

SMML

SLSMML

cos

cos

(4.14)

Driving force and critical size of heterogeneous nucleation

- Balance among tensions

Consider a solid embryo forming in contact

with a perfectly flat mold wall as depicted

in Fig. 4.7. Assuming SL is isotropic,

the total interfacial energy of the system

is minimized if the embryo has the shape

of a spherical cap.

SL: solid/liquid interfacial tension, ML: mold/liquid interfacial tension,

SM: solid/mold interfacial tension, : a wetting angle.

Note that the vertical component of SL remains unbalanced. Given time, this force would pull

the mold surface upwards until the surface tension forces balance in all direction.

Fig. 4.7 Heterogeneous nucleation of spherical

cap on a flat mold wall.

15


- Driving force

The formation of such an embryo will be associated

with an excess free energy

VS: volume of the spherical cap, ASL: areas of solid/liquid interface,

ASM: areas of solid/mold interface, SL: solid/liquid interfacial tension,

ML: mold/liquid interfacial tension, SM: solid/mold interfacial tension.

MLSMSMSMSLSLVShet AAAGVG (4.15)

33

333

33

0

3

23

0

2

0

3

0

2

0

2

0

222

2

0

2

0

2

0

2

0

coscos323

coscos3

1cos1

3

2

coscos3

1sin

3

2

coscos13

1sin

3

1

cossin3

1sin

sinsin

cos12cos2sin2sin

rV

rrV

rdrV

rddrV

rrrddrdrV

rrA

rrdrrddrA

S

S

S

S

rS

SM

SL

r

rsin

16


MLSMSMSLSLVShet

MLSMSMSMSLSLVShet

AAGVG

AAAGVG

(4.15)

cos

cos

cos

SMSLSLVShet

SLSMSLSLVShet

SLSMML

AAGVG

AAGVG

(4.14)

cosSMSL AA

32

22

22

222

coscos32

coscos1cos22

cossincos22

cossincos12

r

r

r

rr

hetG

SLV

SLV

SMSLSLVS

rGr

rGr

AAGV

233

3233

43

4

4

coscos32

coscos32coscos323

cos

17


Note that except for factor S(), the expression for heterogeneous nucleation is the same

as that obtained for homogeneous nucleation.

S() has a numerical value 1 dependent only on .

(4.16)

SrGr

G SLVhet

2

3

43

4

4

cos1cos2

4

coscos3223

S (4.17)

0 20 40 60 80 100 120 140 160 180

()

0

0.2

0.4

0.6

0.8

1

S(

)

18


- Critical nucleus size

dGhet/dr = 0 at r = r*

SG

GG

G

SrGr

G

SL

V

SLV

V

SL

het

SLVhet

2

3

*

23

24

3

24

43

4

0*8*4

0*8*4

043

4

2

2

*

23

SLV

SLV

rr

SLV

rGr

SrGr

SrGrdr

d

V

SL

Gr

2* (4.18)

SG

GV

SLhet

2

3*

3

16(4.19)

Fig. 4.8 The excess free energy of

solid clusters for homogeneous and

heterogeneous nucleation. Note r*

is independent of the nucleation site.

physical metallurgy - web.nchu.edu.twweb.nchu.edu.tw/~samsongnchu/chap15補充導證.pdf ·...

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