Post on 18-Mar-2020
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
Homogeneous Nucleation
Materials Science and EngineeringPhase Transformations in Metals
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
Materials Science and EngineeringPhase Transformations in Metals
Homogeneous Nucleation
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
Materials Science and EngineeringPhase Transformations in Metals
Homogeneous Nucleation
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.
<
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Materials Science and EngineeringPhase Transformations in Metals
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
Materials Science and EngineeringPhase Transformations in Metals
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
Materials Science and EngineeringPhase Transformations in Metals
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
Materials Science and EngineeringPhase Transformations in Metals
Heterogeneous Nucleation
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
Materials Science and EngineeringPhase Transformations in Metals
Heterogeneous Nucleation
)(*
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
Materials Science and EngineeringPhase Transformations in Metals
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
Materials Science and EngineeringPhase Transformations in Metals
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
Materials Science and EngineeringPhase Transformations in Metals
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
Heterogeneous Nucleation相變態
- 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
Heterogeneous Nucleation相變態
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
Heterogeneous Nucleation相變態
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
Heterogeneous Nucleation相變態
- 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.