ppt slides - 3
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physical metallurgyTRANSCRIPT
Fundamentals of Solidification
Outline
• Introduction
• Homogeneous nucleation
• Heterogeneous nucleation
• Growth and microstructure
• Summary
Introduction
• There are two types of solidification
– Glass formation
• Physical properties such as viscosity change
smoothly across the solidifying region
– Phase transition
• Some physical properties change abruptly,
such as viscosity, heat capacity
Introduction
• Solidification by phase transition is
modelled as two stage
– Nucleation
• Homogeneous nucleation
• Heterogeneous nucleation
– Growth
Nucleation and Grain Growth
• Nucleation;– Homogeneous nucleation: very pure metal, substantial
undercooling (0.2Tm)– Heterogeneous nucleation: nucleation agents (5ºC
undercooling)• Grain growth
– Planar: pure metal– Dendritic: solid solution
• Grain size – depends on number of nuclei and cooling rate.
Crystal Nucleation and Growth
“Manufacturing Processes for Engineering Materials,” by Serope Kalpakjian
7
Nucleation Rate
• The rate at which nucleation of a new phase occurs
is critical to the prediction of phase transformation
behavior.
• We will contrast homogeneous nucleation
(extremely rare!) with heterogeneous nucleation
(typical) rates.
• Why study homogeneous nucleation? Useful
foundation and simplest to understand.
• Bottom line: the quickest transformation wins!
8
Thermodynamics of nucleation
• How should we understand nucleation?
• The crucial point is to understand it as a balance
between the free energy available from the driving force,
and the energy consumed in creating new interface
(between parent and product phases). Once the rate of
change of free energy becomes negative, then an
embryo can grow.
• Parallel to the Griffith analysis: once the rate of (free)
energy change becomes negative with crack length
increase, then the crack can grow without limit.
9
Nucleation paths• It is important to remember that the actual outcome is
always the process that leads most rapidly to the change for
which a (thermodynamic) driving force exists.
• Anisotropy in the interfacial energy forces growing grains to
adopt anisotropic shapes in order to minimize high energy
orientations of the interface.
• Anisotropy in growth rates has a similar effect.
• Heterogeneous nucleation on surfaces, pre-existing
interfaces (grain boundaries), dislocations etc. is very
important.
• Elastic energy plays a major role in constraining nucleation.
10
Homogeneous Nucleation
• Assume that the new, product phase appears as spherical
particles.
• Free energy released by transformation is proportional to
the volume.
• Free energy consumed by creation of interface is
proportional to the surface area of particle and the
interfacial energy, g.
• Net change in free energy per particle, ∆Gr:
∆Gr = -4π/3 r3 ∆GV + 4πr2 g.
• Differentiate to find the stationary point (at which the rate of
change of free energy turns negative).
11
Critical radius, free energy
• d(∆Gr) = 0 =
-4π/ r*2 ∆G* + 8πr*g.
• From this we find the critical
radius and critical free energy.
r* = 2g/∆GV
∆G* = 16πg3/3∆GV2
• Crucial difference from
solidification: the role of elastic
energy!
Not at ∆Gr=0!!!
12
Elastic energy
• Why does elastic energy play such an important role in solid state phase transformations?
• Volume changes on transformation of order a few % are typical. Elastic energy is symmetric: net (hydrostatic) tension or compression leads to an increase in elastic energy. This elastic energy cost for creation of a new phase, ∆GS (= Ee2/2), must be subtracted in proportion to the volume of new phase.
∆Gr = -4π/3 r3 (∆GV - ∆GS) + 4πr2 g.d(∆Gr) = 0 = -4π r2 ∆G* + 8πr*g. r* = 2g / (∆GV - ∆GS);∆G* = 16πg3 / 3(∆GV - ∆GS)2.
Homogeneous nucleation
rr
Homogeneous nucleation
• No preferred nucleation sites
– Spontaneous
– Random
• Those of preferred sites
– Boundary
– Surface
– Inclusion, …
Local free energy change
1. Liquid to solid 2. Interface
Single nucleus
Critical radius
0/ drGd
SL
SL
GGr
2*
2
3
3
16*
SL
SL
GGG
(GL-GS) vs. supercooling
Free energy density vs. temperature
liquid
solid
temperature
Free energy density
19
Homogeneous Nucleation: examples
• Two examples of homogeneous nucleation in the solid
state are known.
1) Cu with 1-3% Co can be heat treated to precipitate Co
homogeneously. We will examine this case in a homework
aimed at predicting TTT diagrams.
2) Ni superalloys will precipitate Ni3Al homogeneously at
small undercoolings because of the small lattice misfit and
small interfacial energy.
• Why only these cases? Small interfacial energy, and small
elastic energy difference.
• Everything else: heterogeneous!
20
Elastic Anisotropy
• Remember that most crystalline solids are elastically
anisotropic: this means that the shape of a new phase is
likely to be anisotropic.
• If either the parent or the product phase is more compliant
in a particular direction, larger dimensions parallel to this
direction will be favored over stiffer directions. This is
offset by the interfacial energy term which must increase
as the surface-to-volume ratio increases.
• Example: Guinier-Preston zones in the Al-Cu system,
which are platelets on {100}.
21
Interfacial Energy
• Even in solidification, the anisotropy of the interfacial
energy matters. The energy of the solid-liquid interface
varies depending on which crystallographic surface is
involved. {111} surfaces tend to have the lowest energy in
fcc metals.
• In solid state precipitation, the anisotropy of the interface
matters even more! This is because there are two
crystalline surfaces involved in the interface. If the crystal
structures are different (often the case) then low energy
interfaces require good atomic matching between the two
planes. Sometimes this results from combining close-
packed interfaces.
22
Nucleation rate• 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.
• Population (concentration) of critical embryos, C*, is given by a
Boltzmann factor, where C0 is the number of atoms per unit
volume:
C* = C0 exp -(∆G*/kT)
• The rate at which a critical embryo is formed, f, depends on the
migration of atoms, i.e. diffusion, which is again given by a
Boltzmann factor, where ∆Gbulk is the activation energy for (bulk)
diffusion (∆Gm in P&E), and w is of the same order as the atomic
jump frequency:
f = w exp -(∆Gbulk/kT).
23
Nucleation rates, contd.
• Based on this approach,
we can now understand
the extremely strong
dependence of nucleation
rate on under cooling.
• Note that the net effect of
elastic energy is to offset
(decrease) the equilibrium
transformation
temperature.
24
Effect of undercooling
• The effect of undercooling on the nucleation rate is drastic,
because of the non-linear relation between the two quantities.
• By incorporating the previous expression into the nucleation rate
we obtain the following:
C* = C0 exp -(∆G*/kT) =
• Finally the nucleation
rate is the product
of C* and f:
N = f C*
C* C0 exp 163
3 GV GS 2kT
C0 exp16 3
3HT
Te
GS
2
kT
25
Nucleation Rate
• The combined equations are as follows.
• The nucleation rate is the product of
C* and f. Note that the product of w and C0 is a large
number because w is of the order of the atomic vibration
frequency, and C0 is the number of atoms per unit volume.
N f C * expGbulk
kT
C0 exp
163
3HT
Te
GS
2
kT
Heterogeneous nucleation
• Nucleation site
– Mold walls
– Inclusion
– Interface
– Surface
– Impurity
27
Heterogeneous Nucleation
• Heterogeneous nucleation must occur on some substrate:
grain boundaries
triple junctions
dislocations
(existing) second phase particles
• Consider a grain boundary: why is it effective? Answer:
by forming on a grain boundary, an embryo can offset its
“cost” in interfacial energy by eliminating some grain
boundary area.
Liquid
Inclusion
Nucleus IL
NL
IN
R
r
h
a
Heterogeneous nucleation
29
Grain boundary nucleation
• The semi-angle, q = cos-1gaa/2gab
• As for solidification, the radius of the spherical caps
depends only on the interfacial energy, so:
r* = 2gab/(∆GV-∆GS)
but a shape factor modifies the critical free energy:
∆G* = 16πgab3/3(∆GV-∆GS)2 S(q)
= 16πgab3/3(∆GV-∆GS)2 0.5(2 + cosq)(1 - cosq)2
Grainboundary in alpha
30
Other heterogeneous sites
• Other sites for
heterogeneous nucleation
have been listed.
• For the same contact angle,
grain corners (quadruple
points) are more effective
than grain edges (triple
lines), which are more
effective than grain
boundaries.
31
Heterogeneous nucleation rate
• The rate of heterogeneous nucleation, Nhet, is described by a
very similar equation as previously described for homogeneous
nucleation, Nhomo. The critical difference is in the critical free
energy, ∆G*, and the density of sites, C1.
• Homogeneous:
Nhomo = exp -(∆Gbulk/kT) C0 exp -(∆Ghomo*/kT)
• Heterogeneous:
Nhet = exp -(∆Gbulk/kT) C1 exp -(∆Ghet*/kT)
• For grain boundary nucleation, for example, the ratio of site
densities, C1/C0 = /D, where D is the grain size, and is the
boundary thickness.
Local free energy change
SLLSbeforeafter AGGVGGG
SLSL rGGrG 23 43
4
Spherical nucleus:
Thermodynamic barriers
Heterogeneous nucleation barrier
Homogeneous nucleation barrier
System free energy
• Ideal solution: Particle of different sizes
• ni particles with each contains i atoms
• n particles with each contains 1 atom
STGnG ic
ii
ii nn
nn
nn
nnkS lnln
Number of nuclei
• At equilibrium
0/ ic nG
i
i
nn
n
kT
Gln
inn
kT
Gnni exp
kT
Gnni
*exp*
when
Number of nuclei
Boltzmann formula:
Critical nuclei:
Inoculating agents
• Small interface energy
– Similar crystal structure
– Similar lattice distance
– Same physical properties
– Same chemical properties
Casting refinement
• Adding inoculating agents
– Overheat might melt the agents
• Surface refinement
– Coat agents on mold walls
• Pattern induced solidification
Growth and microstructure
T. F. Brower and M.C. Flemings, Trans. AIME, 239, 1620 (1967)
H.B. Dong and P.D. Lee, Acta Mater. 53 (2005) 659
Growth and microstructure
Outer chilled zones
Outer chilled zones
Outer chilled zones
Outer chilled zones
Pure metals: Formation of shell because temperature gradient is the key factor in grain growth.
Outer chilled zones
re-melted?
Pouring temperature
survived?
Microstructure of ingot
• Chilled zone
– Fine equiaxed grains.
– Pure substance: Continuous shell.
– Solution: Particles
– Particles flushed away from wall into the
central
• Re-melted
• Survived – nucleus
Intermediate columnar zone
Columnar grains grows
The grain is overtaken by neighbors.
Intermediate columnar zone
Growth and overtaken
Intermediate columnar zone
Columnar growth blocked
Central equiaxed zone
• Equiaxed grain
– Nucleation:
• Supercooling
• Falling particles
• Dendrite fragments– Elevated pouring
temperature:
• Larger equiaxed
grains
• More columnar zone
– Anisotropic properties
• Magnetic materials
• Turbo blade.
• More equiaxed zone
– Isotropic properties
– Less segregation
Structure and properties