Heterogeneous nucleation
! undercooling of a few K sufficient
N heterogeneous > N homogeneous due to reduced nucleus/melt surface
Solid
Liquid
Fig. 5-4. Sketch of
homogeneous and
heterogeneous
The
nucleation.
latter occurs at
Tm
existing surfaces.
"Gc
heterogeneous
f = f (cos #)
= f"Gc
homogeneous
(5.8)
cos #: wetting angle
Nucleation and Growth
Topic 4
M.S Darwish
MECH 636: Solidification Modelling
d in eq. (5.8) as function of
Fig. 5-5. (Left) Nucleation at a wall and wetting angle #. (Right) f define
wets the solid substrate.
cos # for a flat substrate. As expected nucleation is facilitated if the melt
Heterogeneous nucleation facilitated by:
- similar crystal structure (low misfitstrain)
- chemical affinity
Objectives
By the end of this lecture you should be able to:
Explain the term homogeneous as applied to nucleation events
Understand the concept of critical size and critical free energy
Differentiate between unstable cluster (embryos) and stable nuclei
Derive expressions for (r*,N, ...) in terms of Gv & T.
List typical heterogeneous nucleation sites for solidification
Understand the term wetting or contact angle,
Explain why the wetting angle is a measure of the efficiency of a
particular nucleation site
Write an expression relating critical volumes of heterogeneous and
homogeneous nuclei.
Introduction
During Solidification the atomic arrangement changes from a random or
short-range order to a long range order or crystal structure.
Nucleation occurs when a small nucleus begins to form in the liquid, the
nuclei then grows as atoms from the liquid are attached to it.
The crucial point is to understand it as a balance between the free
energy available from the driving force, and the energy consumed in
forming new interface. Once the rate of change of free energy
becomes negative, then an embryo can grow.
Energy Of Fusion
GV = GL GS = HV TS
HV = LV
Stable
solid
V
= hm
s
Stable
liquid
liquid
!G
solid
HV hmV
=
Tm =
S sS
V
hmV
GV = hm T
s
sTm
hmV
T V T
=
1 = hm
s Tm s Tm
LV T
=
Tm
GS
hmV
S =
sTm
GL
!T
Tm
Temperature
Homogeneous Nucleation
ASL = 4r 2
Liquid
Liquid
SL
Solid
VS =
4 3
r
3
G2 = G1 + !G
G1
G1 = (VS + VL )GVL
G2 = VS GVS + VL GVL + ASL SL
LV T
GV =
Tm
G = G2 G1
-ve
= VS (G G
S
V
L
V
)+ A
SL SL
= VS GV + ASL SL
+ve
4 3
G = r GV + 4r 2 SL
3
1. When r is smaller than some r* an increase
in r leads to an increase of G -> unstable
2. When r is larger than some r* an increase
in r leads to a decrease of G -> stable
Critical radius
Not at G=0!!!
Differentiate to find the stationary point (at which
the rate of change of free energy turns negative).
interfacial
energy ! r2
!G
d (G)
=0
dr
4( r
) G
+ 8r = 0
From this we find the critical radius and critical
free energy.
!G*
0
r*
!Gr
Volume free-energy
!r3"T
2 SL 2 SL Tm 1
r =
=
GV LV T
16
G =
3G
3
SL
2
V
3
2
16 SL
Tm
1
=
2
2
3LV (T )
4 3
G = r GV + 4r 2 SL
3
Cluster and Nuclei
if r<r* the system can lower its free
energy by dissolution of the solid
!G
Unstable solid particles with r<r* are
known as clusters or embryos
if r>r* the free energy of the system
decreases if the solid grows
Stable solid particles with r>r* are
referred to as nuclei
Since G = 0 when r = r* the critical
nuclei is effectively in (unstable)
equilibrium with the surrounding liquid
interfacial
energy ! r2
!G*
0
r*
!Gr
Volume free-energy
!r3"T
Effect of Undercooling
At r* the solid sphere is at equilibrium with its
surrounding thus the solid sphere and the liquid
have the same free energy
2 SL
GV =
r
interfacial
energy ! r2
!G
Stable
solid
Stable
liquid
r2" > r1"
liquid
!G
*
!G
solid
2" SL
r2#
How r* and G* decrease with undercooling T
r*
2" SL
r1#
GS
!Gr!
!T
GL
Volume free-energy
!r3"T
!T, C
500
300
Tm
Temperature
Nuclei are stable
in this region
4( r
Embryos form in this
region and may redissolve
100
5x10-7
10-6
Critical radius of particle,
r* (cm)
1.5x10-6
) G
+ 8r = 0
Variation of r* and rmax with T
Although we now know the critical values for an embryo
to become a nucleus, we do not know the rate at which
nuclei will appear in a real system.
To estimate the nucleation rate we need to know the
population density of embryos of the critical size and the
rate at which such embryos are formed.
The population (concentration) of critical embryos is
given by
n r = n oe
r*
Gr
kT
rmax
k is the Boltzmann factor, no is the total number of atoms in the system
Gr is the excess of free energy associated with the cluster
!TN
!T
Homogeneous Nucleation Rate
taking a G equal to G*, then the concentration of
clusters to reach the critical size can be written as:
C = Coe
Ghom
kT
Nhom
clusters/m 3
!TN
The addition of one more atom to each of these
clusters would convert them into stable nuclei
If this happens with a frequency fo,
N hom = f oCoe
Ghom
kT
N hom = f oCoe
nuclei /m
r*
rmax
( T ) 2
nuclei /m
!T
3
16 SL
Tm2
A=
3L2V kT
!TN
The effect of undercooling on the nucleation rate is drastic, because of the non-linear
relation between the two quantities as is shown in the plot
!T
Heterogeneous Nucleation
16
G =
3G
3
SL
2
V
3
2
16 SL
Tm
1
=
2
2
3LV (T )
it is clear that for nucleation to be facilitated the interfacial energy term should be reduced
Liquid
" SL
Solid
"
Nucleating agent
" SM
!
!
ML = SM + SL cos
ML SM )
(
cos =
SL
" ML
Heterogeneous Nucleation
+ AML ) ML
G1 = (VS + VL )GVL + ( AML
ML + ASL SL + ASM SM
G2 = VS GVS + VL GVL + AML
Liquid
Liquid
" SL
Solid
" ML
" SM
Nucleating agent
Nucleating agent
"
" SM
G = G2 G1 = VS GV + ASL SL + ASM SM AML ML
Ghet
4 3
= r Gv + 4 SL S ( )
3
Ghom
2 + cos )(1 cos )
(
S ( ) =
4
" ML
<1
Critical r and G
2 SL
r =
GV
!G*
#
"Ghet
3
16
SL
G =
S ( )
2
3GV
!G
#
"Ghom
#
"Ghet
Critical value
for nucleation
#
"Ghom
r*
!
!G!
r
!Gr
!T
N
Nhet
Nhom
= 10 S ( ) = 104
= 30 S ( ) = 0.02
!T
model does not work for = 0
Heterogeneous Nucleation
Rate
n = n1e
Mould walls not flat
Ghet
kT
number of atoms in contact with
nucleating agent surface
N het = f1C1e
Ghet
kT
Critical radius
for solid
nuclei /m 3
number of atoms in contact with
nucleating agent surface per unit
volume
Exercise show that
1
G = V Gv
2
Nucleation in cracks occur with very little
undercooling
for cracks to be effective the crack
opening should be large enough to allow
the solid to grow out without the radius
of the solid/liquid interface decreasing
below r*
Nucleation of Melting
While nucleation during solidification requires
some undercooling, melting invariably occurs
at the equilibrium temperature even at
relatively high rates of heating.
this is due to the relative free energies of the
solid/vapour, solid/liquid and liquid/vapour
interfaces.
It is always found that
SL + LV < SV
Therefore the wetting angle
=0
and no superheating is required for nucleation of the liquid
Growth of a Pure Solid
Solid
Solid
Liquid
Liquid
T
Tm
In a pure metal solidification is
controlled by the rate at which
the latent heat of solidification
can be conducted away from
the solid/liquid interface.
Tm
Solid
Liquid
Solid
Heat
Solid
Liquid
Heat
Solid
Liquid
Liquid
Tm
Tm
dTS
dTL
kS
= kL
+ vLV
dx
dx
Solid
Liquid
Solid
Liquid
Development of Thermal
Dendrites
dTS
dTL
kS
= kL
+ vLV
dx
dx
dTS
0
dx
dTL Tc
dx
r
Tm
!Tr
!To
TS
!Tc
TL,far
Solid
Liquid
dTL 1
k L Tc
v k L
dx LV
LV r
2Tm
2Tm
Tr =
r =
LV r
LV Tr
kL 1 r
v
1
LV r
r
r = 2r
Alloy Solidification
Limited Diffusion in Solid and Liquid
Solid
cL(x)
c1
k=cS/cL
T
cS
T1
T2
cL
Liquid
T3
co
DL/v
c
A
kco
co cmax co/k
ceut
critical
gradient
T
T1(co)
TL(x)
TL (cL)
T3(c1)
constitutional
undercooling
Solid
Liquid
0
xL
Constitutional undercooling and solidification morphology
Constitutional undercooling and solidification morphology
Fig. 5-9. How constitutional undercooling
affects
solidification morphology.
Crucial parameters:
Local solidification rate: if low, solute has time to diffuse
Fig. 5-9. How conaway from interface into bulk liquid
stitutional under! planar growth
cooling
affects
solidification mor grad T: > critical value ! no constitutional undercooling
phology.
(cf. Fig. 5-9).
Steep grad T + low SL interface velocity
(e.g. growth of Si single crystals)
Planar growth
" cell growth
Crucial parameters:
#
"
! planar growth
dendrite growth
#
no undercooling
Local
solidificationmoderate
rate: undercooling
if low, solutestrong
hasundercooling
time to diffuse
5-7
away from interface into bulk liquid
Summary
By considering the balance between the release of free energy by
transformation and the cost of creating new interface, the critical free
energy for nucleation and the critical size of the nucleus can be derived.
The exponential dependence of nucleation rate on undercooling means
that, in effect, no nucleation will be observed until a minimum
undercooling is achieved.
The undercooling required for nucleation is increased by volume changes
on transformation, but decreased by the availability of heterogeneous
nucleation sites.
Units
Consider the units of the various quantities that we have examined.
For driving force, the units are either Joules/mole (Gm) or Joules/m3 (Gv);
dimensions = energy/mole, energy/volume.
For interfacial energy, the units are Joules/m2; dimensions = energy/area.
For nucleation rate, the units are number/m3/s; dimensions are number/volume/
time.
For critical free energy, the units are Joules; dimensions are energy. What is less
obvious is how to scale the energy against thermal energy. When one calculates
a value for G*, the values turn out to be of the order of 10-19J, or 1eV. This is
reasonable because we are calculating the energy associated with an individual
cluster or embryonic nucleus, I.e. energies at the scale of atoms. Therefore the
appropriate thermal energy is kT (not RT).
For the activation energy (enthalpy) of diffusion, in the equation for nucleation
rate, the units depend on the source of the information. If the activation energy
for diffusion is specified in Joules/mole, then the appropriate thermal energy is
RT, for example.
For critical radius, the units are m (or nm, to choose a more practical unit);
dimensions = length.
Metal
Freezing Temp. (C)
Latent Heat of fusion (J/
cm3)
Surface enrgy (J/cm2)
Typical undercooling for
Hv
Ga
30
488
56 10-7
76
Bi
271
543
54 10-7
90
Pb
327
237
33 10-7
80
Ag
962
965
126 10-7
250
Cu
1085
1628
177 10-7
236
Ni
1453
2756
255 10-7
480
Fe
1538
1737
204 10-7
420
N2O
40
