Lecture++3+Crystal+Interfaces+and+Microstructure

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

mailto:[email protected]


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...



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)


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


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


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


Interfacial free energy


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


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


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


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


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


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


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


Lower density of atoms means:

high mobility

high diffusivity

high chemical reactivity

Soap bubble analogy: several grains of varying misorientation


row of

dislocations

(low angle)

disordered structure

(high angle)


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


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


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)


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?


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


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


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


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


Two dimensional grain boundary configurations

Arrows indicate directions of boundary migration during grain growth


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


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


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


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)


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


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


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


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


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)


Orientation relationships and habit planes


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


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


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


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)


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?


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


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



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)


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


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)


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))


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


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)


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


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)


Summary


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

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