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T7008T Phase Transformations 2010 John C. Ion
T7008T
Phase Transformations in Metals and Alloys
John Ion
Division of Engineering Materials
E-mail: [email protected]
Office: E316
Phone: 491249
T7008T Phase Transformations 2010 John C. Ion
Lecture 3
Crystal Interfaces and Microstructure
What are the most important interfaces in metallic systems?
Why are crystal interfaces and microstructure important in phase
transformations?
How do we achieve equilibrium in polycrystalline materials?
How do interfaces control kinetic transformations such as grain
growth?
What are interphase interfaces in solids?
How do we classify the different types of phase transformation?
Issues to address...
T7008T Phase Transformations 2010 John C. Ion
T7008T Phase Transformations 2010 John C. Ion
Count Alois von Beckh Widmanstätten
13 July 1753 – 10 June 1849
Austrian printer and scientist
Director of the Imperial Porcelain
works in Vienna
In 1808 Widmanstätten was flame
heating iron meteorites and
noticed special patterns…
http://www.facebook.com/pag
es/Count-Alois-von-Beckh-
Widmanstatten/14342953900
3005
The discovery was acknowledged by
Carl von Schreibers, director of the
Vienna Mineral and Zoology Cabinet,
who named the structure after
Widmanstätten
However, the discovery should be
assigned to the Englishman G.
Thomson, as four years earlier he was
using nitric acid to clean the rust off
meteorites, noticed the same patterns,
but published his findings in Italian (he
was living in Naples at the time)
T7008T Phase Transformations 2010 John C. Ion
André Guinier
1911 - 3 July 2000
French physicist, born in Nancy
George Dawson Preston
8 August 1896 – 22 June 1972
British physicist, born in Oundle
Simultaneously discovered
Guinier-Preston (GP) zones in
age hardening aluminium
copper alloys in 1938
T7008T Phase Transformations 2010 John C. Ion
4 October 1903 – 25 August 1992
Born in Birmingham, England
Renowned metallurgist and historian
of science
Studied metallurgy at the University
of Birmingham (BSc) and at the
Massachusetts Institute of
Technology (Sc.D)
Perhaps most famous for his work
on the Manhattan Project where he
was responsible for the production
of fissionable metals
Cyril Stanley Smith
c. 1952: Smith is holding a
small glass capsule full of
soap bubbles that he used to
illustrate how surface forces
control the growth of grains
in solid materials
T7008T Phase Transformations 2010 John C. Ion
Georg (Yuri Viktorovich) Wulff
Russian mineralogist
In 1878, Gibbs proposed that for
the equilibrium shape of a
crystal, the total surface Gibbs
free energy of formation should
be a minimum for a constant
volume of crystal
Wulff (1901):
“The length of a vector drawn
normal to a crystal face will be
proportional to its surface energy“
(the Gibbs-Wulff theorem)
In 1953 the American Conyers
Herring gave a proof of the
theorem and a method for
determining the equilibrium
shape of a crystal
Three types are important in metallic systems:
1. free surfaces of a crystal (solid/vapour interface)
2. grain boundaries (α/α interfaces)
3. interfaces between phases (interphase interfaces, α/γ)
All crystals possess the first type
The second type separates crystals with essentially the same
composition and crystal structure, but a different orientation in space
The third separates two different phases that may have different
crystal structures and/or compositions (and therefore includes
solid/liquid interfaces)
The majority of phase transformations in metels occur by the growth
of a new phase (β) from a few nucleation sites within the parent
phase (α)
Types of interface in metallic systems
T7008T Phase Transformations 2010 John C. Ion
Interfacial free energy
T7008T Phase Transformations 2010 John C. Ion
The free energy of a system containing an interface of area A and
free energy γ per unit area is:
𝐺 = 𝐺0 + 𝐴𝛾
where 𝐺0 is the free energy of the bulk system
Consider a wire frame suspending a liquid film with a movable bar:
If a force F moves a small
distance dA, the work done is FdA
The free energy of the system is
increased by dG:
𝑑𝐺 = 𝛾𝑑𝐴 + 𝐴𝑑𝛾 = 𝐹𝑑𝐴
∴ 𝐹 = 𝛾 + 𝐴𝑑𝛾
𝑑𝐴
Assuming 𝑑𝛾
𝑑𝐴= 0, 𝐹 = 𝛾
Solid/vapour interfaces
T7008T Phase Transformations 2010 John C. Ion
Assume that the structure of solids may be discussed in terms of
a hard sphere model
The atomic configurations on the three closes packed planes in
fcc crystals are:
Atoms in the layers nearest the surface are without some of their
neighbours
Atoms on a {111} surface, for example, are missing three of their
twelve nearest neighbours
This may be used to calculate the energy of a surface
Surface energy
T7008T Phase Transformations 2010 John C. Ion
If the bond strength of the metal is ε, then each bond may be
considered to lower the internal energy of each atom by 𝜀
2
Every surface atom with three ”broken bonds” has an excess internal
energy of 3𝜀
2 compared with atoms in the bulk
For a pure metal, ε may be estimated from the latent heat of
sublimation 𝐿𝑠 (the sum of the latent heats of fusion and vaporisation):
𝐿𝑠 = 12𝑁𝑎𝜀
2
for an fcc structure in which 12𝑁𝑎 broken bonds are formed
The energy of a {111} surface 𝐸𝑠𝑣 is therefore approximately
𝐸𝑠𝑣 = 0.25 𝐿𝑠/𝑁𝑎
T7008T Phase Transformations 2010 John C. Ion
Variation of surface energy with plane orientation
The energy E of different planes in a crystal varies systematically with
the orientation of the plane θ, taking a minimum corresponding to the
orientation of a close packed plane
T7008T Phase Transformations 2010 John C. Ion
Wulff construction
Possible section
through the plane
energy plot of an fcc
crystal
Length OA
represents the free
energy of a surface
plane whose normal
lies in the direction
OA
OB = 𝛾 001
OC = 𝛾 111
Boundaries in single-phase solids
T7008T Phase Transformations 2010 John C. Ion
The grains in a single-phase polycrystalline specimen are generally
in many different orientations and many different types of grain
boundary are therefore possible
The lattices of any two grains may be made to coincide by rotating
one of them about a single axis
Pure tilt boundary Pure twist boundary
T7008T Phase Transformations 2010 John C. Ion
Grains
Metallographic specimens are two dimensional sections of a three
dimensional structure
Two grains meet in a plane (a grain boundary)
Three grains meet in a line (a grain edge)
Four grains meet at a point (a grain corner)
Low and high angle grain boundaries
T7008T Phase Transformations 2010 John C. Ion
Lower density of atoms means:
high mobility
high diffusivity
high chemical reactivity
Soap bubble analogy: several grains of varying misorientation
T7008T Phase Transformations 2010 John C. Ion
row of
dislocations
(low angle)
disordered structure
(high angle)
T7008T Phase Transformations 2010 John C. Ion
Twins
The stacking sequence across a coherent twin boundary in the
fcc lattice is:
ABCABACBA
The twin plane is a plane of mirror symmetry (the crystals on
either side of it are twins)
The nearest neighbour packing is correct through the twin
plane; only the second nearest neighbours lie in the wrong sites
Twin boundaries
T7008T Phase Transformations 2010 John C. Ion
coherent e.g. {111} plane
in FCC
incoherent
grain boundary
energy γ as a
function of grain
boundary
misorientation φ
Measured boundary free energies for twin crystals
T7008T Phase Transformations 2010 John C. Ion
Crystal Coherent
(mJ m-2)
Incoherent
(mJ m-2)
Grain boundary
(mJ m-2)
Cu 21 498 623
Ag 8 126 377
Fe-Cr-Ni 19 209 835
tilt parallel to <100>
Al
tilt parallel to <110>
Al
Equilibrium in polycrystalline materials (I)
T7008T Phase Transformations 2010 John C. Ion
incoherent
annealing
twin boundary
coherent
annealing
twin boundary
Annealed (recrystallized) austenitic stainless steel
high angle
grain
boundary
low angle
grain
boundary
How do different grain boundary energies affect the microstructure of
a polycrystalline material?
T7008T Phase Transformations 2010 John C. Ion
Turbine blades in jet
engines may:
• be polycrystalline
• have a columnar
grain structure
• be a single crystal
Single crystal and polycrystalline materials
Polycrystalline blades are formed using a ceramic mould
Columnar grain structured blades are created using directional
solidification techniques and have grains parallel to the major stress
axes
Single-crystal superalloys are formed as a single crystal using a
modified version of the directional solidification technique, so there
are no grain boundaries in the material
T7008T Phase Transformations 2010 John C. Ion
Equilibrium in polycrystalline materials (II)
Consider the factors that control grain shapes in a recrystallised
polycrystal
Why do grain boundaries exist at all in annealed materials?
Boundaries are all high energy regions that increase the free energy of
a polycrystal relative to a single crystal
Therefore a polycrystalline material is never a true equilibrium structure
Grain boundaries in a polycrystal can adjust themselves during
annealing to produce a metastable equilibrium at the grain boundary
intersections
The conditions for equilibrium at a grain boundary junction may be
obtained by considering the forces that each boundary exerts on the
junction
T7008T Phase Transformations 2010 John C. Ion
Equilibrium in polycrystalline materials (III)
If the boundary energy is independent of orientation, there will be no
torque forces acting since the energy is a minimum in that orientation
The grain boundary then behaves like a soap film
For metastable equilibrium the boundary tensions must balance: 𝛾23
sin 𝜃1+
𝛾13
sin 𝜃2=
𝛾12
sin 𝜃3
T7008T Phase Transformations 2010 John C. Ion
Thermally activated migration of grain boundaries
A cylindrical
boundary is acted
on by a force 𝛾
𝑟
Tension forces balance in
three dimensions if the
boundary is planar or if it is
curved with equal radii in
opposite directions
In real metals there are always grain boundaries with net curvature
in one direction
Consequently a random grain structure is inherently unstable:
boundaries will tend to migrate towards ther centre of curvature
T7008T Phase Transformations 2010 John C. Ion
Two dimensional grain boundary configurations
Arrows indicate directions of boundary migration during grain growth
T7008T Phase Transformations 2010 John C. Ion
Grain growth in a soap solution (C.S. Smith)
Numbers are time in minutes
The higher pressure on the concave side of the films induces air
molecules in the smaller cells to diffuse through the film into the
larger cells, so that the smaller cells eventually dissolve
T7008T Phase Transformations 2010 John C. Ion
Grain growth in a polycrystalline metal
The effect of the pressure difference caused by a curved
boundary is to create a difference in free energy ∆𝐺 or chemical
potential ∆𝜇
In a pure metal ∆𝐺 = ∆𝜇:
∆𝐺 =2𝛾𝑉𝑚𝑟
= ∆𝜇
If atom Ⓒ jumps from grain 1 to
grain 2 the boundary locally
advances a small distance
T7008T Phase Transformations 2010 John C. Ion
The kinetics of grain growth
Assume that the mean radius of curvature of grain boundaries is
proportional to the mean grain diameter 𝐷
The mean driving force for grain growth is proportional to 2𝛾
𝐷 giving:
𝜈 = 𝛼𝑀2𝛾
𝐷 =d𝐷
d𝑡
where: 𝜈 = average grain boundary velocity
𝛼 = proportionality constant of the order 1 𝑀 = grain boundary mobility (strongly dependent on temperature)
Integrating, taking 𝐷 = 𝐷0 when 𝑡 = 0:
𝐷 2 = 𝐷02 + 𝑘𝑡
where: 𝑘 = 4𝛼𝑀𝛾
T7008T Phase Transformations 2010 John C. Ion
Pinning of grain boundaries by precipitates (I)
A grain boundary is attached to a particle along a length 2𝜋𝑟 cos 𝜃
It feels a pull of (2𝜋𝑟 cos 𝜃 γ) sin 𝜃
If there is a volume fraction 𝑓 of particles all with a radius 𝑟, the mean
number of particles intersecting unit area of a random plane is 3𝑓
2𝜋𝑟2 such
that the restraining force 𝑃 per unit area of grain boundary is
𝑃 =3𝑓
2𝜋𝑟2. 𝜋𝑟𝛾 =
3𝑓𝛾
2𝑟
Second
phase
particles pin
grain
boundaries
(precipitation
hardening)
T7008T Phase Transformations 2010 John C. Ion
Pinning of grain boundaries by precipitates (II)
The force 𝑃 opposes the driving force for grain growth 2𝛾
𝐷
When 𝐷 is small 𝑃 is relatively insignificant, but as 𝐷 increases the
driving force 2𝛾
𝐷 decreases until
2𝛾
𝐷 =3𝑓𝛾
2𝑟
when the driving force becomes insufficient to overcome the drag, giving:
𝐷 max =4𝑟
3𝑓
T7008T Phase Transformations 2010 John C. Ion
Effect of second phase particles on grain growth
A large volume fraction of stable small particles is required to
stabilise a fine grain grain size during heating at high temperatures
T7008T Phase Transformations 2010 John C. Ion
Interphase interfaces in solids
So far we have considered the structure and properties of
boundaries between crystals of the same solid phase
Now we will consider the boundaries between different solid
phases
We consider adjoining crystals that have:
• different crystal structures
• different compositions
• both
Interphase boundaries in solids may be divided on the basis of
their atomic structure into:
• coherent
• semicoherent
• incoherent
T7008T Phase Transformations 2010 John C. Ion
Interface coherence
A coherent interface arises when the two crystals match perfectly
at the interface plane such that the two lattices are continuous
across the interface
Same crystal structure
Different compositions
Different crystal structures
Different compositions
T7008T Phase Transformations 2010 John C. Ion
Fully coherent interface (I)
Consider Cu-Si alloys in which:
the hcp Si-rich κ phase and
the fcc Cu-rich α matrix
have identical hexagonally close packed planes: (111)fcc: 0001hcp
and identical interatomic distances
111 𝛼// 0001 𝜅
1 10 𝛼// 112 0 𝜅
Orientation relationship:
The only contribution to
interfacial energy is a
chemical component
(1 mJ m-2 for the α-κ
interface)
T7008T Phase Transformations 2010 John C. Ion
Orientation relationships and habit planes
Orientation relationship:
Crystallographic texture is one of the main characteristics of a
polycrystalline material: it determines its functional properties
An orientation relationship between two crystals of the phases α
and β defines the planes and directions that lie in a common plane
between two crystals and is written:
(hkl)α // (hkl)β , [uvw]α // [uvw]β
Habit plane:
The crystallographic plane or system of planes along which certain
phenomena (such as twinning) occur
The habit plane is a common plane between two crystals
T7008T Phase Transformations 2010 John C. Ion
Fully coherent interface (II)
When the distance between the atoms in the interface is not
identical it is still possible to maintain coherency by straining one
or both of the lattices
T7008T Phase Transformations 2010 John C. Ion
Semicoherent interface
Strains at a coherent interface raise the total energy of the system
For sufficiently large atomic misfit, or interfacial area, it becomes
energetically more favourable to replace a coherent interface with a
semicoherent interface containing periodic misfit dislocations (200-
500 mJ m-2)
When more than one
dislocation is present
for every four
interplanar spacings,
regions of poor fit
around the dislocation
cores overlap and the
interface cannot be
considered coherent
any longer
T7008T Phase Transformations 2010 John C. Ion
Incoherent interface
When the interfacial plane has a very different atomic configuration in
the two adjoining phases there is no possibility of good matching
across the interface
The pattern of atoms may either be very different in the two phases or,
if it is similar, the interatomic distances may differ by more than 25%
An incoherent interface then arises
Incoherent interfaces have high energy
(500-1000 mJ m-2)
T7008T Phase Transformations 2010 John C. Ion
Second phase shape: interfacial energy effects
In a two phase microstructure one of the phases is often dispersed
within the other, e.g. β precipitates in an α matrix
Consider for simplicity a system containing one β precipitate
embedded in a single α crystal, and assume for simplicity that both the
precipitate and matrix are strain-free
Such a system will have a minimum free energy when the shape of the
precipitate and its orientation relationship with the matrix are optimised
to give the lowest total interfactial free energy
How may this be achieved for different types of precipitate?
T7008T Phase Transformations 2010 John C. Ion
Fully coherent precipitates
A zone with no misfit
e.g. ⃝ Al and ∙ Ag
Ag-rich zones (GP) zones in an
Al-4 at% Ag alloy (TEM)
Since the two crystal structures match across all interfacial
planes the zone may be any shape and remain coherent
T7008T Phase Transformations 2010 John C. Ion
Partially coherent precipitates (I)
Coherent plate of θ’ in
Al-3.9wt%Cu alloy
Unit cell of θ’ precipitate in
Al-Cu alloys
Unit cell of
matrix in Al-
Cu alloys
001 𝜃′// 001 𝛼
100 𝜃′// 100 𝛼
Orientation relationship:
T7008T Phase Transformations 2010 John C. Ion
Partially coherent precipitates (II)
When the precipitate and matrix have different crystal structures it is
usually difficult to find a lattice plane that is common to both phases
Nevertheless for certain phase combinations there may be one
plane that is common to both phases
By choosing the correct
orientation relationship
orientation a low energy coherent
or semicoherent interface to be
formed
Widmanstätten morphology of γ’ precipitates in Al-4at% Ag alloy
(TEM, H = GP zone)
T7008T Phase Transformations 2010 John C. Ion
Widmanstätten morphologies
Widmanstätten patterns (also called Thomson structures) are
microstructural features characterised by a cross-hatched
appearance due to one phase having formed along certain
crystallographic planes
Crystalline intergrowth of
two Fe-Ni alloys,
kamacite and taenite
T7008T Phase Transformations 2010 John C. Ion
Incoherent precipitates
When the two phases have completely different crystal structures, or
when the two lattices are in a random orientation, it is unlikely that
any coherent or semicoherent interfaces form, and the precipitates
are said to be incoherent
Incoherent
precipitates of θ in
an Al-Cu alloy
(TEM)
T7008T Phase Transformations 2010 John C. Ion
Solid / liquid interfaces
Two types of atomic structure in solid / liquid interfaces:
• Atomically flat close packed (as solid / vapour interfaces, a))
• Atomically diffuse (transition over several atom layers, b) and c))
T7008T Phase Transformations 2010 John C. Ion
Examples of solid / liquid interfaces in metallic systems
Nonfaceted dendrites of
silver in a Cu-Ag eutectic
matrix
Faceted cuboids of β’ SnSb compound in a
matrix of Sn-rich material
T7008T Phase Transformations 2010 John C. Ion
Interface migration
Many phase transformations occur by a process known as
nucleation and growth
The new phase (β) first appears at certain sites in the metastable
parent (α) phase (nucleation), which grow into the surrounding
matrix
An interface is created during nucleation, which migrates into the
surrounding parent phase during growth
Nucleation is important, but most of the transformation product is
formed during the growth stage by the transfer of atoms across a
moving parent/product interface
There are two basic types of interface:
• glissile (migrate by dislocation glide, athermal, military)
• non-glissile (migrate by random jumps of atoms, thermal,
civilian)
T7008T Phase Transformations 2010 John C. Ion
Military transformations
Nearest neighbours of any atom are essentially unchanged
Parent and product phases have the same composition (no
diffusion)
Examples:
martensite forming from austenite in steels
formation of mechanical twins
T7008T Phase Transformations 2010 John C. Ion
Civilian transformations
Parent and product phases may or may not have the same
composition
If there is no change in composition, e.g. ferrite (α) → austenite (γ)
in pure iron, the new phase grows as fast as atoms cross the
interface (interface controlled)
If the parent and product phases
have different compositions growth of
the new phase will require long range
diffusion:
The growth of a B-rich β phase into
an A-rich α phase can only occur if
diffusion is able to transport A away
from the interface, and B towards the
advancing interface (diffusion
controlled growth)
T7008T Phase Transformations 2010 John C. Ion
Summary
T7008T Phase Transformations 2010 John C. Ion
The three most important interfaces in metals and alloys:
free surfaces of a crystal (solid/vapour interface)
grain boundaries (α/α interfaces)
interfaces between phases (interphase interfaces, α/γ)
Equilibrium in polycrystalline materials is achieved by minimising
surface energy
Coherent, semicoherent and incoherent interfaces may be formed
between phases
Atomic migration resulting from differences in free energy control
kinetic transformations
Phase transformations may be classified in many ways
Military or civilian
Diffusionless or diffusion-controlled
Athermal or thermally activated
Interfaces play an important role in all types of phase
transformation