Materials Science

•  Historical Perspective

  Stone ® Bronze ® Iron ® Advanced materials

•

•  What is Materials Science and Engineering ?

  Processing ® Structure ® Properties ® Performance

•

•  Classification of Materials

  Metals, Ceramics, Polymers, Semiconductors

•

•  Advanced Materials

  Electronic materials, superconductors, etc.

•

•  Modern Material's Needs, Material of Future

  Biodegradable materials, Nanomaterials, “Smart” materials.

•Beginning of the Material Science - People began to make tools from stone – Start of the Stone Age about two million years ago. 

  Natural materials: stone, wood, clay, skins, etc.

•The Stone Age ended about 5000 years ago with introduction of Bronze in the Far East.  Bronze is an alloy (a metal made up of more than one element), copper + < 25% of tin + other elements.

  Bronze: can be hammered or cast into a variety of shapes, can be made harder by alloying, corrode only slowly after a surface oxide film forms.

•The Iron Age began about 3000 years ago and continues today.  Use of iron and steel, a stronger and cheaper material changed drastically daily life of a common person.

 

•Age of Advanced materials: throughout the Iron Age many new types of materials have been introduced (ceramic, semiconductors, polymers, composites…).  Understanding of the relationship among structure, properties, processing, and performance of materials.  Intelligent design of new materials.

A better understanding of structure-composition-properties relations has lead to a remarkable progress in properties of materials. Example is the dramatic progress in the strength to density ratio of materials, that resulted in a wide variety of new products, from dental materials to tennis racquets.

Introduction to Material Science ..Phase diagrams

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 1

Phase Diagrams

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 2

Microstructure and Phase Transformations in

Multicomponent Systems

Chapter Outline: Phase Diagrams

􀂃 Definitions and basic concepts

􀂃 Phases and microstructure

􀂃 Binary isomorphous systems (complete solid solubility)

􀂃 Binary eutectic systems (limited solid solubility)

􀂃 Binary systems with intermediate phases/compounds

􀂃 The iron-carbon system (steel and cast iron)

Not tested: 9.12 The Gibbs Phase Rule

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Component - chemically recognizable species (Fe and C

in carbon steel, H2O and NaCl in salted water). A binary

alloy contains two components, a ternary alloy – three, etc.

Phase – a portion of a system that has uniform physical

and chemical characteristics. Two distinct phases in a

system have distinct physical or chemical characteristics

(e.g. water and ice) and are separated from each other by

definite phase boundaries. A phase may contain one or

more components.

A single-phase system is called homogeneous,

systems with two or more phases are mixtures or

heterogeneous systems.

Definitions: Components and Phases

Introduction to Materials Science, Chapter 9, Phase Diagrams

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Solvent - host or major component in solution, solute -

minor component (Chapter 4).

Solubility Limit of a component in a phase is the

maximum amount of the component that can be dissolved

in it (e.g. alcohol has unlimited solubility in water, sugar

has a limited solubility, oil is insoluble). The same

concepts apply to solid phases: Cu and Ni are mutually

soluble in any amount (unlimited solid solubility), while C

has a limited solubility in Fe.

Definitions: Solubility Limit

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Microstructure

The properties of an alloy depend not only on proportions

of the phases but also on how they are arranged structurally

at the microscopic level. Thus, the microstructure is

specified by the number of phases, their proportions, and

their arrangement in space.

Microstructure of cast Iron

This is an alloy of Fe with 4 wt.% C. There are several

phases. The long gray regions are flakes of graphite. The

matrix is a fine mixture of BCC Fe and Fe3C compound.

Phase diagrams will help us to understand and predict

the microstructures like the one shown in this page

Introduction to Materials Science, Chapter 9, Phase Diagrams

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A system is at equilibrium if at constant temperature,

pressure and composition the system is stable, not

changing with time.

Equilibrium is the state that is achieved given sufficient

time. But the time to achieve equilibrium may be very long

(the kinetics can be slow) that a state along the path to the

equilibrium may appear to be stable. This is called a

metastable state.

In thermodynamics, equilibrium is described as the state of

system that corresponds to the minimum of the thermodynamic

function called the free energy of the system. Thermodynamics

tells us that:

Equilibrium and Metastable States

• Under conditions of a constant temperature and pressure and

composition, the direction of any spontaneous change is

toward a lower free energy.

• The state of stable thermodynamic

equilibrium is the one with

minimum free energy.

• A system at a metastable state is

trapped in a local minimum of free

energy that is not the global one.

metastable

equilibrium

Free Energy

Introduction to Materials Science, Chapter 9, Phase Diagrams

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A phase diagram - graphical representation of the

combinations of temperature, pressure, composition, or

other variables for which specific phases exist at

equilibrium.

For H2O, a typical diagram shows the temperature and

pressure at which ice (solid),water (liquid) and steam (gas)

exist.

Phase diagram

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A phase diagrams show what phases exist at equilibrium

and what phase transformations we can expect when we

change one of the parameters of the system (T, P,

composition).

We will discuss phase diagrams for binary alloys only and

will assume pressure to be constant at one atmosphere.

Phase diagrams for materials with more than two

components are complex and difficult to represent.

Phase diagram

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Isomorphous system - complete solid solubility of the two

components (both in the liquid and solid phases).

Binary Isomorphous Systems (I)

Three phase region can be identified on the phase diagram:

Liquid (L) , solid + liquid (α +L), solid (α )

Liquidus line separates liquid from liquid + solid

Solidus line separates solid from liquid + solid

α + L

α

L

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Binary Isomorphous Systems (II)

Example of isomorphous system: Cu-Ni (the complete

solubility occurs because both Cu and Ni have the same

crystal structure, FCC, similar radii, electronegativity and

valence).

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Binary Isomorphous Systems (III)

In one-component system melting occurs at a well-defined

melting temperature.

In multi-component systems melting occurs over the range

of temperatures, between the solidus and liquidus lines.

Solid and liquid phases are in equilibrium in this

temperature range.

α + L

α

L Liquid solution

Liquid solution

+

Crystallites of

Solid solution

Polycrystal

Solid solution

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Interpretation of Phase Diagrams

For a given temperature and composition we can use phase

diagram to determine:

1) The phases that are present

2) Compositions of the phases

3) The relative fractions of the phases

Finding the composition in a two phase region:

1. Locate composition and temperature in diagram

2. In two phase region draw the tie line or isotherm

3. Note intersection with phase boundaries. Read

compositions at the intersections.

The liquid and solid phases have these compositions.

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The Lever Rule

Finding the amounts of phases in a two phase region:

1. Locate composition and temperature in diagram

2. In two phase region draw the tie line or isotherm

3. Fraction of a phase is determined by taking the

length of the tie line to the phase boundary for the

other phase, and dividing by the total length of tie

line

The lever rule is a mechanical

analogy to the mass balance

calculation. The tie line in the

two-phase region is analogous to

a lever balanced on a fulcrum.

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The Lever Rule

Mass fractions: WL = S / (R+S) = (Cα - Co) / (Cα- CL)

Wα = R / (R+S) = (Co - CL) / (Cα- CL)

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Derivation of the lever rule

WL = (Cα - Co) / (Cα- CL)

1) All material must be in one phase or the other:

Wα + WL = 1

2) Mass of a component that is present in both phases

equal to the mass of the component in one phase +

mass of the component in the second phase:

WαCα + WLCL = Co

3) Solution of these equations gives us the Lever rule.

Wα = (Co - CL) / (Cα- CL)

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Phase compositions and amounts. An example.

Mass fractions: WL = (Cα - Co) / (Cα- CL) = 0.68

Wα = (Co - CL) / (Cα- CL) = 0.32

Co = 35 wt. %, CL = 31.5 wt. %, Cα = 42.5 wt. %

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Development of microstructure in isomorphous alloys

Equilibrium (very slow) cooling

􀂾 Solidification in the solid + liquid phase occurs

gradually upon cooling from the liquidus line.

􀂾 The composition of the solid and the liquid change

gradually during cooling (as can be determined by the

tie-line method.)

􀂾 Nuclei of the solid phase form and they grow to

consume all the liquid at the solidus line.

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Development of microstructure in isomorphous alloys

Equilibrium (very slow) cooling

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Development of microstructure in isomorphous alloys

Non-equilibrium cooling

• Compositional changes require diffusion in solid and

liquid phases

• Diffusion in the solid state is very slow. ⇒ The new

layers that solidify on top of the existing grains have the

equilibrium composition at that temperature but once

they are solid their composition does not change. ⇒

Formation of layered (cored) grains and the invalidity of

the tie-line method to determine the composition of the

solid phase.

• The tie-line method still works for the liquid phase,

where diffusion is fast. Average Ni content of solid

grains is higher. ⇒ Application of the lever rule gives

us a greater proportion of liquid phase as compared to

the one for equilibrium cooling at the same T. ⇒

Solidus line is shifted to the right (higher Ni contents),

solidification is complete at lower T, the outer part of

the grains are richer in the low-melting component (Cu).

• Upon heating grain boundaries will melt first. This can

lead to premature mechanical failure.

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Solid can’t freeze fast enough:

solidus line effectively shifted

to higher Ni concentrations.

Shift increases with faster

cooling rates, slower diffusion

Complete solidifcation

occurs at lower temp.

and higher Ni conc.

than equilibrium

11

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Development of microstructure in isomorphous alloys

Non-equilibrium cooling

• Note that the center of each grain is rich in

higher mp constituent (freezes first), with

compositional gradient to edge of grain:

segregation

• The resulting microstructure is termed a cored

structure

• On re-heating, GBs will melt first, as they are

rich in lower mp constituent. This can lead to

premature mechanical failure!

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Mechanical properties of isomorphous alloys

Solid solution strengthening

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Binary Eutectic Systems (I)

•Three single phase regions (α - solid solution of Ag in Cu

matrix, β = solid solution of Cu in Ag marix, L - liquid)

•Three two-phase regions (α + L, β +L, α +β)

•Solvus line separates one solid solution from a mixture of

solid solutions. The Solvus line shows limit of solubility

Copper – Silver phase diagram

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Binary Eutectic Systems (II)

Eutectic or invariant point - Liquid and two solid phases

co-exist in equilibrium at the eutectic composition CE and

the eutectic temperature TE.

Eutectic isotherm - the horizontal solidus line at TE.

Lead – Tin phase diagram

Invariant or eutectic point

Eutectic isotherm

TE

CE

E

13

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Binary  Eutectic Systems (III)

General Rules

Eutectic reaction – transition between liquid and mixture of two

solid phases, α + β at eutectic concentration CE.

The melting point of the eutectic alloy is lower than that of the

components (eutectic = easy to melt in Greek).

At most two phases can be in equilibrium within a phase field.

Three phases (L, α, β) may be in equilibrium only only at a few

points along the eutectic isotherm. Single-phase regions are

separated by 2-phase regions.

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Binary Eutectic Systems (IV)

Compositions and relative amounts of phases are

determined from the same tie lines and lever rule, as for

isomorphous alloys

• C

For points A, B, and C calculate the compositions (wt. %)

and relative amounts (mass fractions) of phases present.

• B

• A

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Development of microstructure in eutectic alloys (I)

Several different types of microstructure can be formed in

slow cooling an different compositions.

Let’s consider cooling of liquid lead – tin system at

different compositions.

In this case of lead-rich

alloy (0-2 wt. % of tin)

solidification proceeds in

the same manner as for

isomorphous alloys (e.g.

Cu-Ni) that we discussed

earlier.

L → α +L→ α

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Development of microstructure in eutectic alloys (II)

At compositions between the room temperature solubility

limit and the maximum solid solubility at the eutectic

temperature, β phase nucleates as the α solid solubility is

exceeded upon crossing the solvus line.

L

α +L

α

α +β

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No changes above the eutectic temperature TE. At TE the

liquid transforms to α and β phases (eutectic reaction).

L → α +β

Development of microstructure in eutectic alloys (III)

Solidification at the eutectic composition (I)

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Development of microstructure in eutectic alloys (IV)

Solidification at the eutectic composition (II)

Compositions of α and β phases are very different → the

eutectic reaction involves redistribution of Pb and Sn atoms

by atomic diffusion. This simultaneous formation of α and

β phases result in a layered (lamellar) microstructure that is

called the eutectic structure.

Formation of the eutectic structure in the lead-tin system.

In the micrograph, the dark layers are lead-reach α phase, the

light layers are the tin-reach β phase.

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Development of microstructure in eutectic alloys (V)

Compositions other than eutectic but within the range of

the eutectic isotherm

Primary α phase is formed in the α + L region, and the

eutectic structure that includes layers of α and β phases

(called eutectic α and eutectic β phases) is formed upon

crossing the eutectic isotherm.

L → α + L → α +β

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Development of microstructure in eutectic alloys (VI)

Microconstituent – element of the microstructure having a

distinctive structure. In the case described in the previous

page, microstructure consists of two microconstituents,

primary α phase and the eutectic structure.

Although the eutectic structure consists of two phases, it is

a microconstituent with distinct lamellar structure and

fixed ratio of the two phases.

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How to calculate relative amounts of microconstituents?

Eutectic microconstituent forms from liquid having eutectic

composition (61.9 wt% Sn)

We can treat the eutectic as a separate phase and apply the

lever rule to find the relative fractions of primary α phase

(18.3 wt% Sn) and the eutectic structure (61.9 wt% Sn):

We = P / (P+Q) (eutectic) Wα’ = Q / (P+Q) (primary)

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How to calculate the total amount of α phase (both eutectic

and primary)?

Fraction of α phase determined by application of the lever

rule across the entire α + β phase field:

Wα = (Q+R) / (P+Q+R) (α phase)

Wβ = P / (P+Q+R) (β phase)

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Phase Diagrams with Intermediate Phases

Eutectic systems that we have studied so far have only

two solid phases (α and β) that exist near the ends of

phase diagrams. These phases are called terminal solid

solutions.

Some binary alloy systems have intermediate solid

solution phases. In phase diagrams, these phases are

separated from the composition extremes (0% and 100%).

Example: in Cu-Zn, α and η are terminal solid solutions,

β, β’, γ, δ, ε are intermediate solid solutions.

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Phase Diagrams with Intermetallic Compounds

Besides solid solutions, intermetallic compounds, that

have precise chemical compositions can exist in some

systems.

When using the lever rules, intermetallic compounds are

treated like any other phase, except they appear not as a

wide region but as a vertical line.

This diagram can be thought of as two joined eutectic

diagrams, for Mg-Mg2Pb and Mg2Pb-Pb. In this case

compound Mg2Pb can be considered as a component.

intermetallic

compound

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Peritectic Reactions

A peritectic reaction - solid phase and liquid phase will

together form a second solid phase at a particular

temperature and composition upon cooling - e.g. L + α ↔ β

These reactions are rather slow as the product phase will

form at the boundary between the two reacting phases thus

separating them, and slowing down any further reaction.

Peritectics are not as common as eutectics and eutectiods,

but do occur in some alloy systems. There is one in the Fe-

C system that we will consider later.

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University of Tennessee, Dept. of Materials Science and Engineering 38

Peritectic Reactions

• An invariant point at which on

heating a solid phase transforms

into one solid and one liquid phase

• E.g at point P above:

δ + L ↔ ε

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Congruent Phase Transformations

A congruent transformation involves no change in

composition (e.g., allotropic transformation such as α-Fe to

γ-Fe or melting transitions in pure solids).

For an incongruent transformation, at least one phase

changes composition (e.g. eutectic, eutectoid, peritectic

reactions).

Congruent

melting of γ

Ni-Ti

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Eutectoid Reactions (I)

The eutectoid (eutectic-like in Greek) reaction is

similar to the eutectic reaction but occurs from one

solid phase to two new solid phases.

Invariant point (the eutectoid) – three solid phases

are in equilibrium.

Upon cooling, a solid phase transforms into two

other solid phases (γ ↔ α + β) in the example

below)

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Eutectoid Reactions (II)

The above phase diagram contains both an eutectic

reaction and its solid-state analog, an eutectoid

reaction

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The Iron–Iron Carbide (Fe–Fe3C) Phase Diagram

In their simplest form, steels are alloys of Iron (Fe) and

Carbon (C). The Fe-C phase diagram is a fairly complex

one, but we will only consider the steel part of the diagram,

up to around 7% Carbon.

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Phases in Fe–Fe3C Phase Diagram

􀂾 α-ferrite - solid solution of C in BCC Fe

• Stable form of iron at room temperature.

• The maximum solubility of C is 0.022 wt%

• Transforms to FCC γ-austenite at 912 °C

􀂾 γ-austenite - solid solution of C in FCC Fe

• The maximum solubility of C is 2.14 wt %.

• Transforms to BCC δ-ferrite at 1395 °C

• Is not stable below the eutectic temperature

(727 ° C) unless cooled rapidly (Chapter 10)

􀂾 δ-ferrite solid solution of C in BCC Fe

• The same structure as α-ferrite

• Stable only at high T, above 1394 °C

• Melts at 1538 °C

􀂾 Fe3C (iron carbide or cementite)

• This intermetallic compound is metastable, it

remains as a compound indefinitely at room T, but

decomposes (very slowly, within several years)

into α-Fe and C (graphite) at 650 - 700 °C

􀂾 Fe-C liquid solution

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A few comments on Fe–Fe3C system

C is an interstitial impurity in Fe. It forms a solid solution

with α, γ, δ phases of iron

Maximum solubility in BCC α-ferrite is limited (max.

0.022 wt% at 727 °C) - BCC has relatively small interstitial

positions

Maximum solubility in FCC austenite is 2.14 wt% at 1147

°C - FCC has larger interstitial positions

Mechanical properties: Cementite is very hard and brittle -

can strengthen steels. Mechanical properties also depend

on the microstructure, that is, how ferrite and cementite are

mixed.

Magnetic properties: α -ferrite is magnetic below 768 °C,

austenite is non-magnetic

Classification. Three types of ferrous alloys:

• Iron: less than 0.008 wt % C in α−ferrite at room T

• Steels: 0.008 - 2.14 wt % C (usually < 1 wt % )

α-ferrite + Fe3C at room T (Chapter 12)

• Cast iron: 2.14 - 6.7 wt % (usually < 4.5 wt %)

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Eutectic and eutectoid reactions in Fe–Fe3C

Eutectoid: 0.76 wt%C, 727 °C

γ(0.76 wt% C) ↔ α (0.022 wt% C) + Fe3C

Eutectic: 4.30 wt% C, 1147 °C

L ↔ γ + Fe3C

Eutectic and eutectoid reactions are very important in

heat treatment of steels

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Development of Microstructure in Iron - Carbon alloys

Microstructure depends on composition (carbon

content) and heat treatment. In the discussion below we

consider slow cooling in which equilibrium is maintained.

Microstructure of eutectoid steel (I)

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When alloy of eutectoid composition (0.76 wt % C) is

cooled slowly it forms perlite, a lamellar or layered

structure of two phases: α-ferrite and cementite (Fe3C)

The layers of alternating phases in pearlite are formed for

the same reason as layered structure of eutectic structures:

redistribution C atoms between ferrite (0.022 wt%) and

cementite (6.7 wt%) by atomic diffusion.

Mechanically, pearlite has properties intermediate to soft,

ductile ferrite and hard, brittle cementite.

Microstructure of eutectoid steel (II)

In the micrograph, the dark areas are

Fe3C layers, the light phase is α-

ferrite

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Compositions to the left of eutectoid (0.022 - 0.76 wt % C)

hypoeutectoid (less than eutectoid -Greek) alloys.

γ → α + γ → α + Fe3C

Microstructure of hypoeutectoid steel (I)

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Hypoeutectoid alloys contain proeutectoid ferrite (formed

above the eutectoid temperature) plus the eutectoid perlite

that contain eutectoid ferrite and cementite.

Microstructure of hypoeutectoid steel (II)

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Compositions to the right of eutectoid (0.76 - 2.14 wt % C)

hypereutectoid (more than eutectoid -Greek) alloys.

γ → γ + Fe3C → α + Fe3C

Microstructure of hypereutectoid steel (I)

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Hypereutectoid alloys contain proeutectoid cementite

(formed above the eutectoid temperature) plus perlite that

contain eutectoid ferrite and cementite.

Microstructure of hypereutectoid steel (II)

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How to calculate the relative amounts of proeutectoid

phase (α or Fe3C) and pearlite?

Application of the lever rule with tie line that extends from

the eutectoid composition (0.75 wt% C) to α – (α + Fe3C)

boundary (0.022 wt% C) for hypoeutectoid alloys and to (α

+ Fe3C) – Fe3C boundary (6.7 wt% C) for hipereutectoid

alloys.

Fraction of α phase is determined by application of the

lever rule across the entire (α + Fe3C) phase field:

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Example for hypereutectoid alloy with composition C1

Fraction of pearlite:

WP = X / (V+X) = (6.7 – C1) / (6.7 – 0.76)

Fraction of proeutectoid cementite:

WFe3C = V / (V+X) = (C1 – 0.76) / (6.7 – 0.76)

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Summary

􀂾 Austenite

􀂾 Cementite

􀂾 Component

􀂾 Congruent transformation

􀂾 Equilibrium

􀂾 Eutectic phase

􀂾 Eutectic reaction

􀂾 Eutectic structure

􀂾 Eutectoid reaction

􀂾 Ferrite

􀂾 Hypereutectoid alloy

􀂾 Hypoeutectoid alloy

􀂾 Intermediate solid solution

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 1

Phase Diagrams

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Microstructure and Phase Transformations in

Multicomponent Systems

Chapter Outline: Phase Diagrams

􀂃 Definitions and basic concepts

􀂃 Phases and microstructure

􀂃 Binary isomorphous systems (complete solid solubility)

􀂃 Binary eutectic systems (limited solid solubility)

􀂃 Binary systems with intermediate phases/compounds

􀂃 The iron-carbon system (steel and cast iron)

Not tested: 9.12 The Gibbs Phase Rule

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 3

Component - chemically recognizable species (Fe and C

in carbon steel, H2O and NaCl in salted water). A binary

alloy contains two components, a ternary alloy – three, etc.

Phase – a portion of a system that has uniform physical

and chemical characteristics. Two distinct phases in a

system have distinct physical or chemical characteristics

(e.g. water and ice) and are separated from each other by

definite phase boundaries. A phase may contain one or

more components.

A single-phase system is called homogeneous,

systems with two or more phases are mixtures or

heterogeneous systems.

Definitions: Components and Phases

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 4

Solvent - host or major component in solution, solute -

minor component (Chapter 4).

Solubility Limit of a component in a phase is the

maximum amount of the component that can be dissolved

in it (e.g. alcohol has unlimited solubility in water, sugar

has a limited solubility, oil is insoluble). The same

concepts apply to solid phases: Cu and Ni are mutually

soluble in any amount (unlimited solid solubility), while C

has a limited solubility in Fe.

Definitions: Solubility Limit

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 5

Microstructure

The properties of an alloy depend not only on proportions

of the phases but also on how they are arranged structurally

at the microscopic level. Thus, the microstructure is

specified by the number of phases, their proportions, and

their arrangement in space.

Microstructure of cast Iron

This is an alloy of Fe with 4 wt.% C. There are several

phases. The long gray regions are flakes of graphite. The

matrix is a fine mixture of BCC Fe and Fe3C compound.

Phase diagrams will help us to understand and predict

the microstructures like the one shown in this page

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 6

A system is at equilibrium if at constant temperature,

pressure and composition the system is stable, not

changing with time.

Equilibrium is the state that is achieved given sufficient

time. But the time to achieve equilibrium may be very long

(the kinetics can be slow) that a state along the path to the

equilibrium may appear to be stable. This is called a

metastable state.

In thermodynamics, equilibrium is described as the state of

system that corresponds to the minimum of the thermodynamic

function called the free energy of the system. Thermodynamics

tells us that:

Equilibrium and Metastable States

• Under conditions of a constant temperature and pressure and

composition, the direction of any spontaneous change is

toward a lower free energy.

• The state of stable thermodynamic

equilibrium is the one with

minimum free energy.

• A system at a metastable state is

trapped in a local minimum of free

energy that is not the global one.

metastable

equilibrium

Free Energy

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 7

A phase diagram - graphical representation of the

combinations of temperature, pressure, composition, or

other variables for which specific phases exist at

equilibrium.

For H2O, a typical diagram shows the temperature and

pressure at which ice (solid),water (liquid) and steam (gas)

exist.

Phase diagram

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 8

A phase diagrams show what phases exist at equilibrium

and what phase transformations we can expect when we

change one of the parameters of the system (T, P,

composition).

We will discuss phase diagrams for binary alloys only and

will assume pressure to be constant at one atmosphere.

Phase diagrams for materials with more than two

components are complex and difficult to represent.

Phase diagram

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 9

Isomorphous system - complete solid solubility of the two

components (both in the liquid and solid phases).

Binary Isomorphous Systems (I)

Three phase region can be identified on the phase diagram:

Liquid (L) , solid + liquid (α +L), solid (α )

Liquidus line separates liquid from liquid + solid

Solidus line separates solid from liquid + solid

α + L

α

L

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 10

Binary Isomorphous Systems (II)

Example of isomorphous system: Cu-Ni (the complete

solubility occurs because both Cu and Ni have the same

crystal structure, FCC, similar radii, electronegativity and

valence).

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 11

Binary Isomorphous Systems (III)

In one-component system melting occurs at a well-defined

melting temperature.

In multi-component systems melting occurs over the range

of temperatures, between the solidus and liquidus lines.

Solid and liquid phases are in equilibrium in this

temperature range.

α + L

α

L Liquid solution

Liquid solution

+

Crystallites of

Solid solution

Polycrystal

Solid solution

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 12

Interpretation of Phase Diagrams

For a given temperature and composition we can use phase

diagram to determine:

1) The phases that are present

2) Compositions of the phases

3) The relative fractions of the phases

Finding the composition in a two phase region:

1. Locate composition and temperature in diagram

2. In two phase region draw the tie line or isotherm

3. Note intersection with phase boundaries. Read

compositions at the intersections.

The liquid and solid phases have these compositions.

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 13

The Lever Rule

Finding the amounts of phases in a two phase region:

1. Locate composition and temperature in diagram

2. In two phase region draw the tie line or isotherm

3. Fraction of a phase is determined by taking the

length of the tie line to the phase boundary for the

other phase, and dividing by the total length of tie

line

The lever rule is a mechanical

analogy to the mass balance

calculation. The tie line in the

two-phase region is analogous to

a lever balanced on a fulcrum.

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 14

The Lever Rule

Mass fractions: WL = S / (R+S) = (Cα - Co) / (Cα- CL)

Wα = R / (R+S) = (Co - CL) / (Cα- CL)

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 15

Derivation of the lever rule

WL = (Cα - Co) / (Cα- CL)

1) All material must be in one phase or the other:

Wα + WL = 1

2) Mass of a component that is present in both phases

equal to the mass of the component in one phase +

mass of the component in the second phase:

WαCα + WLCL = Co

3) Solution of these equations gives us the Lever rule.

Wα = (Co - CL) / (Cα- CL)

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 16

Phase compositions and amounts. An example.

Mass fractions: WL = (Cα - Co) / (Cα- CL) = 0.68

Wα = (Co - CL) / (Cα- CL) = 0.32

Co = 35 wt. %, CL = 31.5 wt. %, Cα = 42.5 wt. %

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 17

Development of microstructure in isomorphous alloys

Equilibrium (very slow) cooling

􀂾 Solidification in the solid + liquid phase occurs

gradually upon cooling from the liquidus line.

􀂾 The composition of the solid and the liquid change

gradually during cooling (as can be determined by the

tie-line method.)

􀂾 Nuclei of the solid phase form and they grow to

consume all the liquid at the solidus line.

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 18

Development of microstructure in isomorphous alloys

Equilibrium (very slow) cooling

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Development of microstructure in isomorphous alloys

Non-equilibrium cooling

• Compositional changes require diffusion in solid and

liquid phases

• Diffusion in the solid state is very slow. ⇒ The new

layers that solidify on top of the existing grains have the

equilibrium composition at that temperature but once

they are solid their composition does not change. ⇒

Formation of layered (cored) grains and the invalidity of

the tie-line method to determine the composition of the

solid phase.

• The tie-line method still works for the liquid phase,

where diffusion is fast. Average Ni content of solid

grains is higher. ⇒ Application of the lever rule gives

us a greater proportion of liquid phase as compared to

the one for equilibrium cooling at the same T. ⇒

Solidus line is shifted to the right (higher Ni contents),

solidification is complete at lower T, the outer part of

the grains are richer in the low-melting component (Cu).

• Upon heating grain boundaries will melt first. This can

lead to premature mechanical failure.

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 20

Solid can’t freeze fast enough:

solidus line effectively shifted

to higher Ni concentrations.

Shift increases with faster

cooling rates, slower diffusion

Complete solidifcation

occurs at lower temp.

and higher Ni conc.

than equilibrium

11

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 21

Development of microstructure in isomorphous alloys

Non-equilibrium cooling

• Note that the center of each grain is rich in

higher mp constituent (freezes first), with

compositional gradient to edge of grain:

segregation

• The resulting microstructure is termed a cored

structure

• On re-heating, GBs will melt first, as they are

rich in lower mp constituent. This can lead to

premature mechanical failure!

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 22

Mechanical properties of isomorphous alloys

Solid solution strengthening

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Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 23

Binary Eutectic Systems (I)

•Three single phase regions (α - solid solution of Ag in Cu

matrix, β = solid solution of Cu in Ag marix, L - liquid)

•Three two-phase regions (α + L, β +L, α +β)

•Solvus line separates one solid solution from a mixture of

solid solutions. The Solvus line shows limit of solubility

Copper – Silver phase diagram

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 24

Binary Eutectic Systems (II)

Eutectic or invariant point - Liquid and two solid phases

co-exist in equilibrium at the eutectic composition CE and

the eutectic temperature TE.

Eutectic isotherm - the horizontal solidus line at TE.

Lead – Tin phase diagram

Invariant or eutectic point

Eutectic isotherm

TE

CE

E

13

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 25

Binary  Eutectic Systems (III)

General Rules

Eutectic reaction – transition between liquid and mixture of two

solid phases, α + β at eutectic concentration CE.

The melting point of the eutectic alloy is lower than that of the

components (eutectic = easy to melt in Greek).

At most two phases can be in equilibrium within a phase field.

Three phases (L, α, β) may be in equilibrium only only at a few

points along the eutectic isotherm. Single-phase regions are

separated by 2-phase regions.

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University of Tennessee, Dept. of Materials Science and Engineering 26

Binary Eutectic Systems (IV)

Compositions and relative amounts of phases are

determined from the same tie lines and lever rule, as for

isomorphous alloys

• C

For points A, B, and C calculate the compositions (wt. %)

and relative amounts (mass fractions) of phases present.

• B

• A

14

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 27

Development of microstructure in eutectic alloys (I)

Several different types of microstructure can be formed in

slow cooling an different compositions.

Let’s consider cooling of liquid lead – tin system at

different compositions.

In this case of lead-rich

alloy (0-2 wt. % of tin)

solidification proceeds in

the same manner as for

isomorphous alloys (e.g.

Cu-Ni) that we discussed

earlier.

L → α +L→ α

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 28

Development of microstructure in eutectic alloys (II)

At compositions between the room temperature solubility

limit and the maximum solid solubility at the eutectic

temperature, β phase nucleates as the α solid solubility is

exceeded upon crossing the solvus line.

L

α +L

α

α +β

15

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 29

No changes above the eutectic temperature TE. At TE the

liquid transforms to α and β phases (eutectic reaction).

L → α +β

Development of microstructure in eutectic alloys (III)

Solidification at the eutectic composition (I)

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 30

Development of microstructure in eutectic alloys (IV)

Solidification at the eutectic composition (II)

Compositions of α and β phases are very different → the

eutectic reaction involves redistribution of Pb and Sn atoms

by atomic diffusion. This simultaneous formation of α and

β phases result in a layered (lamellar) microstructure that is

called the eutectic structure.

Formation of the eutectic structure in the lead-tin system.

In the micrograph, the dark layers are lead-reach α phase, the

light layers are the tin-reach β phase.

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Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 31

Development of microstructure in eutectic alloys (V)

Compositions other than eutectic but within the range of

the eutectic isotherm

Primary α phase is formed in the α + L region, and the

eutectic structure that includes layers of α and β phases

(called eutectic α and eutectic β phases) is formed upon

crossing the eutectic isotherm.

L → α + L → α +β

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 32

Development of microstructure in eutectic alloys (VI)

Microconstituent – element of the microstructure having a

distinctive structure. In the case described in the previous

page, microstructure consists of two microconstituents,

primary α phase and the eutectic structure.

Although the eutectic structure consists of two phases, it is

a microconstituent with distinct lamellar structure and

fixed ratio of the two phases.

17

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How to calculate relative amounts of microconstituents?

Eutectic microconstituent forms from liquid having eutectic

composition (61.9 wt% Sn)

We can treat the eutectic as a separate phase and apply the

lever rule to find the relative fractions of primary α phase

(18.3 wt% Sn) and the eutectic structure (61.9 wt% Sn):

We = P / (P+Q) (eutectic) Wα’ = Q / (P+Q) (primary)

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 34

How to calculate the total amount of α phase (both eutectic

and primary)?

Fraction of α phase determined by application of the lever

rule across the entire α + β phase field:

Wα = (Q+R) / (P+Q+R) (α phase)

Wβ = P / (P+Q+R) (β phase)

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Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 35

Phase Diagrams with Intermediate Phases

Eutectic systems that we have studied so far have only

two solid phases (α and β) that exist near the ends of

phase diagrams. These phases are called terminal solid

solutions.

Some binary alloy systems have intermediate solid

solution phases. In phase diagrams, these phases are

separated from the composition extremes (0% and 100%).

Example: in Cu-Zn, α and η are terminal solid solutions,

β, β’, γ, δ, ε are intermediate solid solutions.

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 36

Phase Diagrams with Intermetallic Compounds

Besides solid solutions, intermetallic compounds, that

have precise chemical compositions can exist in some

systems.

When using the lever rules, intermetallic compounds are

treated like any other phase, except they appear not as a

wide region but as a vertical line.

This diagram can be thought of as two joined eutectic

diagrams, for Mg-Mg2Pb and Mg2Pb-Pb. In this case

compound Mg2Pb can be considered as a component.

intermetallic

compound

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Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 37

Peritectic Reactions

A peritectic reaction - solid phase and liquid phase will

together form a second solid phase at a particular

temperature and composition upon cooling - e.g. L + α ↔ β

These reactions are rather slow as the product phase will

form at the boundary between the two reacting phases thus

separating them, and slowing down any further reaction.

Peritectics are not as common as eutectics and eutectiods,

but do occur in some alloy systems. There is one in the Fe-

C system that we will consider later.

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 38

Peritectic Reactions

• An invariant point at which on

heating a solid phase transforms

into one solid and one liquid phase

• E.g at point P above:

δ + L ↔ ε

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Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 39

Congruent Phase Transformations

A congruent transformation involves no change in

composition (e.g., allotropic transformation such as α-Fe to

γ-Fe or melting transitions in pure solids).

For an incongruent transformation, at least one phase

changes composition (e.g. eutectic, eutectoid, peritectic

reactions).

Congruent

melting of γ

Ni-Ti

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 40

Eutectoid Reactions (I)

The eutectoid (eutectic-like in Greek) reaction is

similar to the eutectic reaction but occurs from one

solid phase to two new solid phases.

Invariant point (the eutectoid) – three solid phases

are in equilibrium.

Upon cooling, a solid phase transforms into two

other solid phases (γ ↔ α + β) in the example

below)

21

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 41

Eutectoid Reactions (II)

The above phase diagram contains both an eutectic

reaction and its solid-state analog, an eutectoid

reaction

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 42

The Iron–Iron Carbide (Fe–Fe3C) Phase Diagram

In their simplest form, steels are alloys of Iron (Fe) and

Carbon (C). The Fe-C phase diagram is a fairly complex

one, but we will only consider the steel part of the diagram,

up to around 7% Carbon.

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Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 43

Phases in Fe–Fe3C Phase Diagram

􀂾 α-ferrite - solid solution of C in BCC Fe

• Stable form of iron at room temperature.

• The maximum solubility of C is 0.022 wt%

• Transforms to FCC γ-austenite at 912 °C

􀂾 γ-austenite - solid solution of C in FCC Fe

• The maximum solubility of C is 2.14 wt %.

• Transforms to BCC δ-ferrite at 1395 °C

• Is not stable below the eutectic temperature

(727 ° C) unless cooled rapidly (Chapter 10)

􀂾 δ-ferrite solid solution of C in BCC Fe

• The same structure as α-ferrite

• Stable only at high T, above 1394 °C

• Melts at 1538 °C

􀂾 Fe3C (iron carbide or cementite)

• This intermetallic compound is metastable, it

remains as a compound indefinitely at room T, but

decomposes (very slowly, within several years)

into α-Fe and C (graphite) at 650 - 700 °C

􀂾 Fe-C liquid solution

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 44

A few comments on Fe–Fe3C system

C is an interstitial impurity in Fe. It forms a solid solution

with α, γ, δ phases of iron

Maximum solubility in BCC α-ferrite is limited (max.

0.022 wt% at 727 °C) - BCC has relatively small interstitial

positions

Maximum solubility in FCC austenite is 2.14 wt% at 1147

°C - FCC has larger interstitial positions

Mechanical properties: Cementite is very hard and brittle -

can strengthen steels. Mechanical properties also depend

on the microstructure, that is, how ferrite and cementite are

mixed.

Magnetic properties: α -ferrite is magnetic below 768 °C,

austenite is non-magnetic

Classification. Three types of ferrous alloys:

• Iron: less than 0.008 wt % C in α−ferrite at room T

• Steels: 0.008 - 2.14 wt % C (usually < 1 wt % )

α-ferrite + Fe3C at room T (Chapter 12)

• Cast iron: 2.14 - 6.7 wt % (usually < 4.5 wt %)

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Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 45

Eutectic and eutectoid reactions in Fe–Fe3C

Eutectoid: 0.76 wt%C, 727 °C

γ(0.76 wt% C) ↔ α (0.022 wt% C) + Fe3C

Eutectic: 4.30 wt% C, 1147 °C

L ↔ γ + Fe3C

Eutectic and eutectoid reactions are very important in

heat treatment of steels

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 46

Development of Microstructure in Iron - Carbon alloys

Microstructure depends on composition (carbon

content) and heat treatment. In the discussion below we

consider slow cooling in which equilibrium is maintained.

Microstructure of eutectoid steel (I)

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When alloy of eutectoid composition (0.76 wt % C) is

cooled slowly it forms perlite, a lamellar or layered

structure of two phases: α-ferrite and cementite (Fe3C)

The layers of alternating phases in pearlite are formed for

the same reason as layered structure of eutectic structures:

redistribution C atoms between ferrite (0.022 wt%) and

cementite (6.7 wt%) by atomic diffusion.

Mechanically, pearlite has properties intermediate to soft,

ductile ferrite and hard, brittle cementite.

Microstructure of eutectoid steel (II)

In the micrograph, the dark areas are

Fe3C layers, the light phase is α-

ferrite

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 48

Compositions to the left of eutectoid (0.022 - 0.76 wt % C)

hypoeutectoid (less than eutectoid -Greek) alloys.

γ → α + γ → α + Fe3C

Microstructure of hypoeutectoid steel (I)

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Hypoeutectoid alloys contain proeutectoid ferrite (formed

above the eutectoid temperature) plus the eutectoid perlite

that contain eutectoid ferrite and cementite.

Microstructure of hypoeutectoid steel (II)

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 50

Compositions to the right of eutectoid (0.76 - 2.14 wt % C)

hypereutectoid (more than eutectoid -Greek) alloys.

γ → γ + Fe3C → α + Fe3C

Microstructure of hypereutectoid steel (I)

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Hypereutectoid alloys contain proeutectoid cementite

(formed above the eutectoid temperature) plus perlite that

contain eutectoid ferrite and cementite.

Microstructure of hypereutectoid steel (II)

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 52

How to calculate the relative amounts of proeutectoid

phase (α or Fe3C) and pearlite?

Application of the lever rule with tie line that extends from

the eutectoid composition (0.75 wt% C) to α – (α + Fe3C)

boundary (0.022 wt% C) for hypoeutectoid alloys and to (α

+ Fe3C) – Fe3C boundary (6.7 wt% C) for hipereutectoid

alloys.

Fraction of α phase is determined by application of the

lever rule across the entire (α + Fe3C) phase field:

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Example for hypereutectoid alloy with composition C1

Fraction of pearlite:

WP = X / (V+X) = (6.7 – C1) / (6.7 – 0.76)

Fraction of proeutectoid cementite:

WFe3C = V / (V+X) = (C1 – 0.76) / (6.7 – 0.76)

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 54

Summary

􀂾 Austenite

􀂾 Cementite

􀂾 Component

􀂾 Congruent transformation

􀂾 Equilibrium

􀂾 Eutectic phase

􀂾 Eutectic reaction

􀂾 Eutectic structure

􀂾 Eutectoid reaction

􀂾 Ferrite

􀂾 Hypereutectoid alloy

􀂾 Hypoeutectoid alloy

􀂾 Intermediate solid solution

􀂾 Intermetallic compound

􀂾 Invariant point

􀂾 Isomorphous

􀂾 Lever rule

􀂾 Liquidus line

􀂾 Metastable

Make sure you understand language and concepts:

􀂾 Microconstituent

􀂾 Pearlite

􀂾 Peritectic reaction

􀂾 Phase

􀂾 Phase diagram

􀂾 Phase equilibrium

􀂾 Primary phase

􀂾 Proeutectoid cementite

􀂾 Proeutectoid ferrite

􀂾 Solidus line

􀂾 Solubility limit

􀂾 Solvus line

􀂾 System

􀂾 Terminal solid solution

􀂾 Tie line

28

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Reading for next class:

Chapter 10: Phase Transformations in Metals

􀂾 Kinetics of phase transformations

􀂾 Multiphase Transformations

􀂾 Phase transformations in Fe-C alloys

􀂾 Isothermal Transformation Diagrams

􀂾 Mechanical Behavior

􀂾 Tempered Martensite

Optional reading (Parts that are not covered / not tested):

10.6 Continuous Cooling Transformation DiagramsIntermetallic compound

􀂾 Invariant point

􀂾 Isomorphous

􀂾 Lever rule

􀂾 Liquidus line

􀂾 Metastable

Make sure you understand language and concepts:

􀂾 Microconstituent

􀂾 Pearlite

􀂾 Peritectic reaction

􀂾 Phase

􀂾 Phase diagram

􀂾 Phase equilibrium

􀂾 Primary phase

􀂾 Proeutectoid cementite

􀂾 Proeutectoid ferrite

􀂾 Solidus line

􀂾 Solubility limit

􀂾 Solvus line

􀂾 System

􀂾 Terminal solid solution

􀂾 Tie line

28

Introduction to Materials Science, Chapter 9, Phase Diagrams

University of Tennessee, Dept. of Materials Science and Engineering 55

Reading for next class:

Chapter 10: Phase Transformations in Metals

􀂾 Kinetics of phase transformations

􀂾 Multiphase Transformations

􀂾 Phase transformations in Fe-C alloys

􀂾 Isothermal Transformation Diagrams

􀂾 Mechanical Behavior

􀂾 Tempered Martensite

Optional reading (Parts that are not covered / not tested):

10.6 Continuous Cooling Transformation Diagrams