EQUILIBRIUM PHASE DIAGRAMS

53
EQUILIBRIUM PHASE DIAGRAMS EQUILIBRIUM PHASE DIAGRAMS ANANDH SUBRAMANIAM Materials Science and Engineering INDIAN INSTITUTE OF TECHNOLOGY KANPUR Kanpur- 110016 Ph: (+91) (512) 259 7215, Lab: (+91) (512) 259 7147 [email protected] http://home.iitk.ac.in/~anandh/ 19 Oct 2015 MATERIALS SCIENCE MATERIALS SCIENCE & & ENGINEERING ENGINEERING AN INTRODUCTORY E AN INTRODUCTORY E - - BOOK BOOK http://home.iitk.ac.in/~anandh/E-book.htm A Learner’s Guide A Learner A Learner s Guide s Guide Alloy Phase Equilibria, A. Prince, Elsevier Publishing Company, Amsterdam (1966)

Transcript of EQUILIBRIUM PHASE DIAGRAMS

Page 1: EQUILIBRIUM PHASE DIAGRAMS

EQUILIBRIUM PHASE DIAGRAMSEQUILIBRIUM PHASE DIAGRAMS

ANANDH SUBRAMANIAMMaterials Science and Engineering

INDIAN INSTITUTE OF TECHNOLOGY KANPURKanpur-

110016Ph: (+91) (512) 259 7215, Lab: (+91) (512) 259 7147

[email protected]://home.iitk.ac.in/~anandh/

19 Oct 2015MATERIALS SCIENCEMATERIALS SCIENCE&&

ENGINEERING ENGINEERING AN INTRODUCTORY EAN INTRODUCTORY E--BOOKBOOK

http://home.iitk.ac.in/~anandh/E-book.htmhttp://home.iitk.ac.in/~anandh/E-book.htmA Learner’s GuideA LearnerA Learner’’s Guides Guide

Alloy Phase Equilibria, A. Prince, Elsevier Publishing Company, Amsterdam (1966)

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Phase diagrams are an important tool in the armory of an materials scientist

In the simplest sense a phase diagram demarcates regions of existence of various phases. This is similar to a map which demarcates regions based on political, geographical, ecological etc. criteria.

Phase diagrams are maps*

Thorough understanding of phase diagrams is a must for all materials scientists

Phase diagrams are also referred to as “EQUILIBRIUM PHASE DIAGRAMS”

This usage requires special attention: though the term used is “Equilibrium”, in practical

terms the equilibrium is NOT

GLOBAL EQUILIBRIUM

but MICROSTRUCTURAL LEVEL EQUILIBRIUM (explanation of the same will be considered later)

This implies that any microstructural information overlaid on a phase diagram is for convenience and not implied by the phase diagram.

The fact that Phase Diagrams represent Microstructural Level equilibrium is often not stressed upon.

Phase Diagrams

* there are many other maps that a material scientist will encounter like creep mechanism maps, various kinds of materials selection maps etc.

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Broadly two kinds of phase diagrams can be differentiated* → those involving time and those which do not involve time (special care must be taken in understanding the former class-

those involving time).

In this chapter we shall deal with the phase diagrams not involving time.

This type can be further sub-classified into:

Those with composition as a variable (e.g. T

vs

%Cu)

Those without composition as a variable (e.g. P

vs

T)

Temperature-Composition diagrams (i.e. axes are T and composition)

are extensively used in materials science and will be considered in detail in this chapter. Also, we shall restrict ourselves to structural phases (i.e. phases not defined in terms of a physical property)**

Time-Temperature-Transformations (TTT) diagrams and Continuous-Cooling-

Transformation (CCT) diagrams involve time. These diagrams are usually designed to have an overlay of Microstructural information (including microstructural evolution). These diagrams will be considered in the chapter on Phase Transformations.

* this is from a convenience in understanding point of view** we have seen before that phases can be defined based either on a

geometrical entity or a physical property (sometimes phases based on a physical property are overlaid on a structural phase diagram-

e.g. in a Fe-cementite

phase diagram ferromagnetic phase and curie temperatures are overlaid)

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DEFINITIONSDEFINITIONS

Components of a system

Independent chemical species which comprise the system: These could be: Elements, Ions, CompoundsE.g.

Au-Cu system

: Components → Au, Cu (elements)

Ice-water system

: Component → H2

O (compound)

Al2

O3

– Cr2

O3

system

: Components → Al2

O3

, Cr2

O3

Let us start with some basic definitions:

This is important to note that components need not be just elements!!

Note that components need not

be only elements

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Phase

Physically distinct, chemically homogenous and mechanically separable region of a system (e.g. gas, crystal, amorphous...).

Gases Gaseous state always a single phase

→ mixed at atomic or molecular level

Liquids ►Liquid solution is a single phase

→ e.g. NaCl

in H2

O► Liquid mixture consists of two or more phases

→ e.g. Oil in water

(no mixing at the atomic/molecular level)

Solids

In general due to several compositions and crystals structures many phases are possible

For the same composition different crystal structures represent

different phases.

E.g. Fe (BCC) and Fe (FCC)

are different phases

For the same crystal structure different compositions represent

different phases.

E.g. in Au-Cu alloy

70%Au-30%Cu

& 30%Au-70%Cu

are different phases

This is the typical textbook definition which one would see!!

Three immiscible liquids

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What kinds of Phases exist?

Based on state

Gas, Liquid, Solid

Based on atomic order

Amorphous, Quasicrystalline, Crystalline

Based on Band structure

Insulating, Semi-conducting, Semi-metallic, Metallic

Based on Property

Paraelectric, Ferromagnetic, Superconducting, …..

Based on Stability Stable, Metastable, (also-

Neutral, unstable)

Also sometimes-

Based on Size/geometry of an entity

Nanocrystalline, mesoporous,

layered, …

We have already seen the ‘official’

definition of a phase:

Physically distinct, chemically homogenous and mechanically separable region of a system.

However, the term phase is used in diverse contexts and we list below some of these.

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Microstructure

Structures requiring magnifications in the region of 100 to 1000

timesOR

The distribution of phases and defects in a material

Grain

The single crystalline part of polycrystalline metal separated by similar entities by a grain boundary

Phase Transformation is the change of one phase into another. E.g.:

► Water → Ice► -

Fe (BCC) →

- Fe (FCC)

-

Fe (FCC) → -

Fe (ferrite) + Cementite

(this involves change in composition)

► Ferromagnetic phase → Paramagnetic phase (based on a property)

Phase transformation

Again this is a typical textbook definition which has been included for…!!

An alternate definition based on magnification

(Phases + defects + residual stress) & their distributions

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Phase diagram

Map demarcating regions of stability of various phases. orMap that gives relationship between phases in equilibrium in a system as a function of T, P and composition (the restricted form of the definition sometime considered in materials textbooks)

Variables / Axis of phase diagrams

The axes can be:

Thermodynamic

(T, P, V),

Other possibilities include magnetic field intensity (H), electric field (E) etc.

Kinetic

(t) or

Composition variables

(C, %x) (composition is usually measure in weight%, atom% or mole fraction)

In single component systems (unary systems) the usual variables are T & P

In phase diagrams used in materials science the usual variables are:T & %x

In the study of phase transformation kinetics Time Temperature Transformation (TTT) diagrams or Continuous Cooling Transformation (CCT) diagrams are also used where the axis are T & t

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Phase diagrams are also called Equilibrium Phase Diagrams.

Though not explicitly stated the word ‘Equilibrium’

in this context usually means

Microstructural

level equilibrium and NOT

Global Equilibrium.

Microstructural level equilibrium implies that microstructures are ‘allowed to exist’

and the system is not in the global energy minimum state.

This statement also implies that:

Micro-constituents* can be included in phase diagrams

Certain phases (like cementite

in the Fe-C system) maybe included in phase diagrams, which are not strictly equilibrium phases (cementite

will decompose to

graphite and ferrite given sufficient thermal activation and time)

Various defects are ‘tolerated’

in the product obtained. These include defects

like dislocations, excess vacancies, internal interfaces (interphase

boundaries, grain boundaries) etc.

Often cooling ‘lines/paths’

are overlaid on phase diagrams-

strictly speaking this is not

allowed. When this is done, it is implied that the cooling rate is ‘very slow’

and the system is in ‘~equilibrium’

during the entire process. (Sometimes, even fast cooling paths are also overlaid on phase diagrams!)

Important points about phase diagrams (Revision + extra points)

* will be defined later

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The GIBBS PHASE RULE

F = C F = C

P + 2P + 2For a system in equilibrium

F F

C +C +

P = 2P = 2or

F –

Degrees of Freedom C –

Number of Components

P –

Number of Phases

The Phase rule is best understood by considering examples from actual phase diagrams as shown in some of the coming slides

Degrees of Freedom: A general definition

In response to a stimulus the ways in which the system can respond corresponds to the degrees of freedom of the system

The phase rule connects the Degrees of Freedom, the number of Components in a system and the number of Phases present in a system via a simple equation.

To understand the phase rule one must understand the variables in the system along with the degrees of freedom.

We start with a general definition of the phrase: “degrees of freedom”

The phase rule

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Variables in a Phase Diagram

C

– No. of Components

P

No. of Phases

F

No. of degrees of Freedom

Variables in the system =

Composition variables + Thermodynamic variables

Composition of a phase specified by (C –

1)

variables

(e.g. If the composition is expressed in %ages then the total is 100% there is one equation connecting the composition variables and we need to specify only

(C1) composition variables)

No. of variables required to specify the composition of all Phases: P(C –

1) (as there are P phases and each phase needs the specification of

(C1) variables)

Thermodynamic variables = P + T

(usually considered) = 2 (at constant Pressure (e.g. atmospheric pressure) the thermodynamic variable becomes 1)

Total no. of variables in the system = P(C –

1) + 2

F < no. of variables F < P(C –

1) + 2

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Degrees of Freedom = What you can control What the system controls

F = C + 2 P

Can control the no. of components added and P & T

System decided how many phases to produce given the

conditions

A way of understanding the Gibbs Phase Rule:The degrees of freedom can be thought of as the difference between what you (can)

control

and what the system controls

F = C F = C

P + 2P + 2

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Single component phase diagrams (Unary)

Let us start with the simplest system possible: the unary system

wherein there is just one

component.

Though there are many possibilities even in unary phase diagrams

(in terms of the axis and

phases) , we shall only consider a T-P

unary phase diagram.

Let us consider the Fe unary phase diagram as an illustrative example.

Apart from the liquid and gaseous phases many solid phases are possible based on crystal structure. (Diagram on next page).

Note that the units of x-axis are in GPa

(i.e. high pressures are needed in the solid state and

liquid state to see any changes to stability regions of the phases).

The Gibbs phase rule here is: F = C –

P + 2. (2 is for T & P).

Note that how the phase fields of the open structure (BCC-

one in the low T regime () and

one in the high T regime ()) diminish at higher pressures. In fact -

phase field completely vanishes at high pressures.

The variables in the phase diagram are: T & P (no composition variables here!).

Along the 2 phase co-existence lines

the DOF (F) is 1

→ i.e. we can chose either T or P

and the other will be automatically fixed.

The 3

phase co-existence points

are invariant points with F = 0. (Invariant point implies

they are fixed for a given system).

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15351410

(BCC)

(HCP)

(FCC)

(BCC)

Liquid

Gas

Pressure (GPa) →

Tem

pera

ture

(ºC

) →

Triple points: 3 phase coexistence

F = 1 –

3 + 2 = 0 triple points are fixed points of a phase diagram (we cannot chose T or P)

Two phase coexistence lines

F = 1 –

2 + 2 = 1 we have only one independent variable (we can chose one of the two variables (T or P) and the other is ‘automatically’

fixed by the phase diagram)

Single phase regions

F = 1 –

1 + 2 = 2

T and P can both be varied while still being in the single phase region

F = C – P + 2

The maximum number of phases which can coexist in a unary P-T phase diagram is 3Note the P is in GPa

“Very High pressures are required for things to happen in the solid state”

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Understanding aspects of the iron unary phase diagram

The degrees of freedom for regions, lines and points in the figure are marked in the diagram shown before

The effect of P on the phase stability of various phases is discussed in the diagram below

It also becomes clear that when we say iron is BCC at RT, we mean at atmospheric pressure (as evident from the diagram at higher pressures iron can become

HCP)

(BCC)

(HCP)

(FCC)

(BCC)

Liquid

Gas

Pressure (GPa) →

Tem

pera

ture

(ºC

) →

This line slopes upward as at constant T if we increase the P the gas will liquefy as liquid has lower volume (similarly the reader should draw horizontal lines to understand the effect of pressure on the stability of various phases-

and rationalize the same).

These lines slope downward as: Under higher pressure the phase with higher packing fraction (lower volume) is preferred

Increase P and gas will liquefy on crossing phase boundary

Phase fields of non-close packed structures shrink under higher pressure

Phase fields of close packed structures expand under higher pressure

Usually (P = 1 atm) the high temperature phase is the loose packed structure and the RT structure is close packed.

How come we find BCC phase at RT in iron?

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Binary Phase Diagrams

Binary implies that there are two components.

Pressure changes often have little effect on the equilibrium of solid phases (unless ofcourse

we apply ‘huge’

pressures).

Hence, binary phase diagrams are usually drawn at 1 atmosphere pressure.

The Gibbs phase rule is reduced to:

Variables are reduced to: F = C –

P + 1. (1 is for T).

T & Composition

(these are the usual variables in Materials Phase Diagrams)

F = C F = C

P + 1P + 1Phase rule for condensed phases

For T

In the next page we consider the possible binary phase diagrams.

These have been

classified based on:

Complete Solubility in both liquid & solid states

Complete Solubility in both liquid state, but limited solubility in the solid state

Limited Solubility in both liquid & solid states.

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Complete Solubility in both liquid state, but limited solubility in the solid state

Overview of Possible Binary Phase diagrams

Isomorphous

Isomorphous

with phase separation

Isomorphous

with ordering

Complete Solubility in both liquid & solid states

Limited Solubility in both liquid & solid states

Eutectic

Peritectic

Liquid StateLiquid State Solid State analogueSolid State analogue

Eutectoid

Peritectoid

Monotectic

Syntectic

Monotectoid

Solid state analogue of Isomorphous

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We have already seen that the reduced phase rule at 1Atm pressure is: F = C –

P + 1.

The ‘one’

on RHS above is T.

The other two variables are:

Composition of the liquid (CL

) and composition (CS

) of the solid. In a fully solid state reaction:

Composition of one solid (CS1

) and composition of the other solid (CS2

).

The compositions are defined with respect to one of the components (say B):

CLB, CS

B

The Degrees of Freedom (DOF, F) are defined with respect to these variables.

What are the variables/DOF in a binary phase diagram?

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System with complete solid & liquid solubility: ISOMORPHOUS SYSTEM

Let us start with an isomorphous

system with complete liquid and solid solubility Pure

components melt at a single temperature, while alloys in the isomorphous

system melt over a range of temperatures*.

I.e. for a given composition solid and liquid will coexist

over a range of temperatures when heated.

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Model Isomorphous

phase diagram

We mention some important points here (may be/have been reiterated elsewhere!):

Such a phase diagram forms when there is complete solid and liquid solubility.

The solid mentioned is crystalline.

The solid + liquid region is not a semi-solid (like partly molten wax or silicate glass). It is a crystal of well defined composition in equilibrium with a liquid

of well defined

composition.

Both the solid and the liquid and the solid (except pure A and pure B) have both A and B components in them.

A and B components could be pure elements (like in the Ag-Au, Au-Pd, Au-Ni, Ge-Si) or compounds (like Al2

O3

-Cr2

O3

).

At low temperatures the picture may not be ‘ideal’

as presented in the diagram below and we may have phase separation (Au-Ni system) or have compound formation (for some compositions) (Au-Pd system). These cases will be considered later.

Each solid, with a different composition is a different phase. The area marked solid in the phase diagram is a phase field.

If heated further the liquid will vaporize, this part of the phase diagram is usually not shown in the diagrams considered.

Liquid

Liquid (solution)

Solid

Solid (solution)

%B →A B

Solid + Liquid

Solid (solution)

Liquid (solution)

T →

Note that between two single phase regions there is a two phase region (for the alloy) (except for special cases)

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A B%B →

T →

M.P. of B

M.P. of A

C = 1P = 2F = 0

C = 2P = 1F = 2

C = 2P = 1 (liquid)F = 2

Variables → T, CSB

2

Variables → T, CLB

2

Variables →

T, CL

B, CSB

3

F = 2 –

P F= 3 –

P

C = 2P = 2F = 1

F = 2 –

P

F = C – P + 1Now let us map the variables and degrees of freedom in varions

regions of the isomorphous

phase diagram

T and Composition can both be varied while still being in the single phase region

in the two phase region, if we fix T (and hence exhaust our DOF), the composition of liquid and solid in equilibrium are automatically fixed (i.e. we have no choice over them). Alternately we can use our DOF to chose CL

→ then T and CS

are automatically fixed.

For pure components at any T For alloys

Solid

Solid + Liquid

Liquid

Disordered (substitutional) solid solutions

For pure components all transformation

temperatures (BCC to FCC, etc.) are fixed (i.e.

zero ‘F’)

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Gibbs free energy vs

composition plot at various temperatures: Isomorphous

system

As we know at constant T and P the Gibbs free energy determines the stability of a phase. Hence, a phase diagram can be constructed from G-composition (Gmixing

-C) curves at various temperatures. For an isomorphous

system we need to chose 5 sample temperatures: (i) T1

> TA

, (ii) T2

=TA

, (iii) TA

>T3

>TB

, (iv) T4

=TB

, (v) T5

<TB

.

G of L lower than for all compositions and hence L is stable

G of L lower than for all compositions except for pure A.

For compositions between X1

and X2

the common tangent construction gives the free energy of the L+ mixture

5

How to get G versus composition

curves→

Click here to know more

.

How to get G versus composition

curves→

Click here to know more

.

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Al2

O3 Cr2

O3%Cr2

O3

T (ºC

)→

2000

2100

2200

10 30 50 70 90

L

L + S

Solidus

Liquidus

S

Isomorphous

Phase Diagram: an example

A and B must satisfy Hume-Rothery

rules for the formation of ‘extended’

solid solution.

Examples of systems forming isomorphous

systems: Cu-Ni, Ag-Au, Ge-Si, Al2

O3

-Cr2

O3

.

Note the liquidus

(the line separting

L & L+S regions) and

solidus

(the line separating L+S

and S regions) lines in the figure.

Schematics

Note that the components in this case are compounds

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ISOMORPHOUS PHASE DIG.

Points to be noted:

Pure components (A,B) melt at a single temperature. (General) Alloys melt over a range of temperatures (we will see some special cases soon).

Isomorphous

phase diagrams form when there is complete solid and liquid solubility.

Complete solid solubility implies that the crystal structure of the two components have to be same and Hume-Rothery

rules

have to be followed.

In some systems (e.g. Au-Ni system) there might be phase separation in the solid state (i.e.

the complete solid solubility criterion may not be followed) → these will be considered later in this chapter as a variation of the isomorphous

system (with complete solubility in the

solid and the liquid state).

Both the liquid and solid contain the components A and B.

In Binary phase diagrams between two single phase regions there will be a two phase region → In the isomorphous

diagram between the liquid and solid state there is the

(Liquid + Solid) state.

The Liquid + Solid

state is NOT

a ‘semi-solid’

state → it is a solid of fixed composition

and structure, in equilibrium with a liquid of fixed composition.

In the single phase region

the composition of the alloy is ‘the composition’.

In the two phase region the composition of the two phases is different and is NOT

the nominal composition of the alloy (but, is given by the lever rule). Lever rule is considered next.

HUME ROTHERY RULESClick here to know more about

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Say the composition C0

is cooled slowly (equilibrium) At T0

there is L + S

equilibrium Solid (crystal) of composition C1

coexists with liquid of composition C2

Tie line and Lever RuleGiven a temperature and composition-

how do we find the fraction of the phases present along with the composition?

We draw a horizontal line (called the Tie Line) at the temperature of interest (say T0

).

Tie line is XY.

Note that tie lines can be drawn only in the two phase coexistence regions (fields). Though they may be extended to mark the temperature.

To find the fractions of solid and liquid we use the lever rule.

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%B →A B

T →

L

L + S

SC0C1 C2

T0

Cooling

Fulcrum of the lever

Arm of the lever proportional to

the solid

Arm of the lever proportional to

the liquid

Note: strictly speaking cooling curves cannot be

overlaid on phase diagrams

Tie line

0

0 1

2 1liquid atT

C Cf C C

At T0 The fraction of liquid (fl

) is

(C0

C1

) The fraction of solid (fs

) is

(C2

C0

)

0

2 0

2 1solid atT

C Cf C C

We draw a horizontal line (called the Tie Line) at the temperature of interest (say T0

).

The portion of the horizontal line in the two phase region is akin to a ‘lever’

with the

fulcrum at the nominal composition (C0

).

The opposite arms of the lever are proportional to the fraction of the solid and liquid phases present (this is the lever rule).

Note that tie line is drawn within the two phase region and is horizontal.

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Expanded version

Extended tie line

C0C1 C2

T0

Fulcrum of the lever

Arm of the lever proportional to

the solid

Arm of the lever proportional to

the liquid

0 1

2 1liquid

C Cf AB CAC

C

At T0 The fraction of liquid (fl

) is proportional to (C0

C1

) → AC The fraction of solid (fs

) is proportional to

(C2

C0

) → CB

2 0

2 1solid

C Cf ABC

CB

C

BA

C

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For a composition C0

At T0

→ Both the liquid and the solid phases contain both the components A and B

To reiterate: The state is NOT

semi-solid but a mixture of a solid of a definite composition (C1

) with a liquid of definite composition (C2

)

If the alloy is slowly cooled (maintaining ~equilibrium) then in

the two phase region (liquid + solid region) the composition of the solid will move along the brown line and the composition of the liquid will move along the blue line.

The composition of the solid and liquid are changing as we cool!

Points to be noted

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Isomorphous

Phase Diagrams

Note here that there is solid solubility, but it is not complete

at low temperatures (below the peak of the 1

+ 2

phase field dome)

(we will have to say more about that soon)

Note that

Ag & Au are so ‘similar’

that the phase diagram becomes a thin lens (i.e. any alloy of Au & Ag melts over a small range of temperatures–

as if it were ‘nearly’

a pure metal!!).

Any composition melts above the linear interpolated melting point.

T1Below T1

(820C) for some range of compositions the solid solubility of Au in Ni (and vice-versa) is limited.

Any composition melts above the linearly

interpolated melting point

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Extensions of the simple isomorphous

system: Congruently melting alloys

Congruently melting alloys just like a pure metal

Is the DOF 1? No: in requiring that CL

B

= CSB

we have exhausted the degree of freedom. Hence T is automatically fixed → DOF is actually zero! Tie line has shrunk to a point!

Variables → T, CLB, CS

B

3

C = 2P = 2F = 1?? (see below)

We have seen that a pure metal melts at a single temperature (Why?!!).

An alloy typically melts over a range of temperatures. However, there are special compositions which can melt at a single temperature like a pure metal. One of these

is the congruent melting composition-

in a variation of the isomorphous

phase diagram. Some systems show this type of behaviour.

Intermediate compounds also have this feature as we shall see later.

Elevation in MP Depression in MPCase A Case B

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AA and BB bonds stronger than AB bonds

Liquid stabilized → Phase separation

in the solid state

AB bonds stronger than AA and BB bonds

Solid stabilized → Ordered solid formation

E.g.

Au-

Ni

Extensions of the simple isomorphous

system: What does this imply w.r.t

the solid state phases?

Elevation in the MP means that the solid state is ‘more stable’

(crudely speaking the

‘ordered state is more stable’) → ordering reaction is seen at low T.

Depression in MP ‘means’

the liquid state (disordered) is more stable → phase separation

is seen at low T. (Phase separation can be ‘thought of’

as the ‘opposite’

of ordering. Ordering (compound formation) occurs for ve

values of Hmix

, while phase separation is favoured

by +ve

values of Hmix

.

Case A Case B

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Examples of isomorphous

systems with phase separation and compound formation

Au-Pd system with 3

compounds

Au-Pt system with phase separation at low temperatures

Au-Ni: model system to understand phase separation

Phase separation in a AlCrFeNi

alloy (with composition Al28.5

Cr27.3

Fe24.9

Ni19.3

) into two BCC phases

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Congruent transformations

We have seen two congruent transformations (transformations which occur without change in composition). The list is as below.

Melting point minimum

Melting point maximum

Order –

disorder transformation

Formation of an intermediate phase

Melting point maximum

Order disorder transformation

Formation of an intermediate phase

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Very few systems exhibit an isomorphous

phase diagram (usually the solid solubility of one

component in another is limited).

Often the solid solubility is severely limited-

though the solid solubility is never zero (due to

entropic reasons).

In a simple eutectic system (binary), there is one composition at which the liquid freezes to two solids at a single temperature. This is in some sense similar to a pure solid which freezes at a single temperature (unlike a pure substance the freezing produces a two solid phases-

both of which contain both the components).

The term EUTECTIC

means Easy Melting

→ The alloy of eutectic composition freezes at a

lower temperature than the melting points of the constituent components.

This has important implications→ e.g. the Pb-Sn*

eutectic alloy melts at 183C, which is lower than the melting points of both Pb

(327C) and Sn

(232C) can be used for

soldering purposes (as we want to input least amount of heat to solder two materials).

In the next page we consider the Pb-Sn

eutectic phase diagram.

As noted before the components need not be only elements. E.g. in the A-Cu system a eutectic reaction is seen between

(Solid solution of Cu in Al) and

(Al2

Cu-

a compound).

Eutectic Phase Diagram

* Actually - eutectic alloy (or (Pb)-(Sn) eutectic alloy)

Page 35: EQUILIBRIUM PHASE DIAGRAMS

Pb Sn%Sn

T (ºC

)→

100

200

300

10 30 50 70 90

L

+

L +

+ L

Liquidus

Solvus

Solidus

18% 62%97%

183C

232C

327C

18362% 18% 97%

Cool

C

LSn Sn Sn

Eutectic reaction (the proper way of writing the reaction)

Eutectic reaction

L →

+

Ceutectic

= CE

Teutectic

= TE

ECEC

ECEC

EC

E FD

ET

Note that Pb

is CCP, while Sn

at RT is Tetragonal (tI4, I41

amd) → therefore complete solid solubility across compositions is ruled out!!

Page 36: EQUILIBRIUM PHASE DIAGRAMS

Note the following points:

and

are terminal solid solutions (usually terminal solid solutions are given symbols ( and )); i.e.

is a solid solution of B (Sn) in A (Pb). (In some systems the terminal solid solubility may be very limited: e.g. the Bi-Cd

system).

has the same crystal structure as that of A (Pb

in the example below) and

has the same crystal structure as B (Sn

in the example below).

Typically, in eutectic systems the solid solubility increases with temperature till the eutectic point (i.e. we have a ‘sloping solvus

line’). In many situations the solubility of component B in A (and vice-

versa) may be very small.

The Liquidus, Solidus

and Solvus

lines are as marked in the figure below.

Pb Sn%Sn →

T (ºC

)→

100

200

300

10 30 50 70 90

L

+

L + + L

Liquidus

Solvus

Solidus

18% 62% 97%

183C

232C

327C

Eutectic reactionL → +

Page 37: EQUILIBRIUM PHASE DIAGRAMS

A B%B →

T (ºC

)

100

200

300

10 30 50 70 90

L

+

L +

+ L

Eutectic reaction

L →

+

Increasing solubility of B in A with ↑T

C = 2P = 3F = 0

At the eutectic point E

(fig. below)→ 3 phases co-exist: L,

&

The number of components in a binary phase diagram is 2 the number of degrees of freedom

F = 0.

This implies that the Eutectic point is an Invariant Point

for a given system it occurs at a fixed composition and temperature.

For a binary system the line DF

is a horizontal line.

Any composition lying between D and F will show eutectic solidification at least in part (for composition E the whole liquid will solidify by the eutectic reaction as shown later).

The percentage of

and

produced by eutectic solidification at E is found by considering DF*

as a lever with fulcrum at E.

EFD

EFD

*

Actually just below DF as tie lines are drawn in a two phase region

Page 38: EQUILIBRIUM PHASE DIAGRAMS

2 μm2 μm2 μm

Examples of Eutectic microstructures As pointed out before microstructural information is often overlaid on phase diagrams. These represent microstructures which evolve on slow cooling.

Al-Al2

Cu lamellar eutectic

Sn

Pb

Composition plot across lamellae

Pb-Sn

lamellar eutectic

Though we label the microstructure as Pb-Sn

lamellar eutectic it is actually a - eutectic.

Al2

Cu(note that one of the

components is a compound!)

(Al)

Page 39: EQUILIBRIUM PHASE DIAGRAMS

C1

C2

C3

C4

The solidification sequence of C4

will be similar to C2

except that the proeutectic

phase will be

Pb-Sn

eutectic

Page 40: EQUILIBRIUM PHASE DIAGRAMS

What is meant by ‘microstructural level equilibrium’?Funda

Check

let us understand the concept using an example considered before.During the eutectic reaction (during slow cooling) a lamellar micro constituent is obtained.

2 μm2 μm2 μm

This results in a huge amount of interfacial area between the two phases (Al, Al2

Cu), which will result in a high value of interfacial energy.

Fig.1: Al-Al2

Cu eutectic

The equilibrium state would correspond to the schematic as shown

below.

Since we ‘tolerate’

the microstructure as in Fig.1 (and do not take the system to the global energy minimum state), the equilibrium considered in typical phase diagrams are microstructural level equilibrium.

Polyhedral crystals

Page 41: EQUILIBRIUM PHASE DIAGRAMS

Peritectic

Phase Diagram

Like the eutectic system, the peritectic

reaction is found in systems with complete liquid

solubility but limited solid solubility.

In the peritectic

reaction the liquid (L) reacts with one solid () to produce another solid ().

L + .

Since the solid

forms at the interface between the L and the , further reaction is

dependent on solid state diffusion. Needless to say this becomes

the rate limiting step and hence it is difficult to ‘equilibrate’

Peritectic

reactions (as compared to say eutectic

reactions). Figure below.

In some peritectic

reactions (e.g. the Pt-Ag system-

next page), the (pure)

phase is not

stable below the peritectic

temperature (TP

= 1186C for Pt-Ag system) and splits into a mixture of (

+ ) just below TP

.

Page 42: EQUILIBRIUM PHASE DIAGRAMS

Pt-Ag Peritectic

system

Peritectic

reaction

L +

Melting points of the components vastly different.

Pt-Ag is perhaps not a good example of a peritectic

system–

obvious looking at the

phase field (not stable below the peritectic

composition).

118666.3% 10.5% 42.4%

Cool

C

LAg Ag Ag

PCPC L

PC

PCPC L

PC

PT

Note that below TP

pure is not stable and splits into ( + )

Formal way of writing the peritectic

reaction

Page 43: EQUILIBRIUM PHASE DIAGRAMS

Funda

Check

Components need not be only elements-

they can be compounds like Al2

O3

, Cr2

O3

.

Phase diagrams usually do not correspond to the global energy minimum-

hence often

microstructures are ‘tolerated’

in phase diagrams.

Phase diagrams give information on stable phases expected for a given set of thermodynamic parameters (like T, P). E.g. for a given composition, T and P the phase diagram will indicate the stable phase(s) (and their fractions).

Phase diagrams do not contain microstructural information-

they are often ‘overlaid’

on

phase diagrams for convenience.

Metastable phases like cementite

are often included in phase diagrams. This is to extend

the practical utility of phase diagrams.

Strictly speaking ‘cooling curves’

(curves where T changes) should not be overlaid on

phase diagrams.

(Again this is done to extend the practical utility of phase diagrams

assuming that the cooling is ‘slow’).

Page 44: EQUILIBRIUM PHASE DIAGRAMS

Precipitation

Page 45: EQUILIBRIUM PHASE DIAGRAMS

The presence of dislocations weakens the crystal → leading to easy plastic deformation.

Putting ‘hindrance’

to dislocation motion increases the strength of the crystal.

Fine precipitates dispersed in the matrix provide such an impediment.

Strength of Al → 100 MPa

Strength of Duralumin with proper heat treatment (Al + 4% Cu + other alloying elements)

→ 500 MPa.

Precipitation Hardening

If a high temperature solid solution is slowly cooled, then coarse (large sized) equilibrium precipitates are produced. These precipitates have a large distance between them. These precipitates have incoherent boundaries with the matrix (incoherent precipitates).

Such (coarse) precipitates, which have a large inter-precipitate distance, are ‘not the best’

in terms of the increase in the hardness.

Hence, we device a 3 step process to obtain a fine distribution of precipitates, which have a low inter-precipitate distance, to obtain a good increase in hardness.

Philosophy behind the process steps in Precipitation Hardening

Coarse incoherent precipitates, with large

inter-precipitate distance

Multi-step process used to obtain a fine distribution of

precipitates (with small inter-precipitate distance)

Not good Better

Page 46: EQUILIBRIUM PHASE DIAGRAMS

Al-Cu phase diagram: the sloping solvus

line and the design of heat treatments

The Al-Cu system is a model system to understand precipitation hardening (typical composition chosen is Al-4 wt.% Cu).

Primary requirement (for precipitation hardening) is the presence of a sloping solvus

line

(i.e. high solubility at high temperatures and decreasing solubility with decreasing temperature). In the Al rich end, compositions marked with a shaded box can only be used for precipitation hardening.

Sloping Solvus

line: high T → high solubility

low T → low solubility of Cu in Al

AlCu

Page 47: EQUILIBRIUM PHASE DIAGRAMS

4 % Cu

+

+

Slow equilibrium cooling gives rise to coarse

precipitates which is not good in impeding dislocation motion.*

RTCu

TetragonalCuAl

RTCu

FCC

CCu

FCCcoolslow

o

% 52)(

% 5.0)(

550 % 4

)( 2

*Also refer section on Double Ended

Frank-Read Source in the chapter on plasticity: max

= Gb/L

Page 48: EQUILIBRIUM PHASE DIAGRAMS

C

A

B

Heat (to 550oC) → solid solution

Quench (to RT) →

Age (reheat to 200oC)

→ fine precipitates

4 % Cu

+

CA

B

To obtain a fine distribution of precipitates the cycle A

→ B

→ C is used

Note:

Treatments

A,

B,

C are for the same composition

Supersaturated solution

Increased vacancy concentration

Heat treatment steps to obtain a fine distribution of precipitates

Assume that we start with a material having coarse equilibrium precipitates (which has been obtained by prior slow cooling of the

sample)

.

A: We heat the sample to the single phase region () in the phase diagram (550C).

B: We quench (fast cooling) the sample in water to obtain a metastable supersaturated solid solution (the amount of Cu in the sample is more than that allowed at room temperature according to the

phase diagram).

C: We reheat the sample to relatively low temperature (~180C/200C) get a fine distribution of precipitates. We have noted before that at ‘low’

temperatures nucleation is dominant over growth.

Page 49: EQUILIBRIUM PHASE DIAGRAMS

Log(t) →

Har

dnes

s → 180oC

100oC

20oC

Higher temperature

less time of aging to obtain peak hardness

Lower temperature

increased peak hardness optimization between time and hardness required

Schematic curves →

Real experimental curves are in later slides

Note: Schematic curves shown-

real curves considered later

Page 50: EQUILIBRIUM PHASE DIAGRAMS

Log(t) →

Har

dnes

s →

180oC

OveragedUnderaged

Peak-aged

Region of solid solution strengthening→ Hardness is higher than that of Al

(no precipitation hardening)

Region of precipitation hardening

(but little/some solid solution strengthening)

Dispersion of fine precipitates

(closely spaced)

Coarsening of precipitates

with increased

inter-precipitate spacing

Not zero of hardness scale

Page 51: EQUILIBRIUM PHASE DIAGRAMS

Log(t) →

Har

dnes

s →

180oC Peak-aged

Particle radius (r)

CR

SS In

crea

se→

21

r r1

Particle shearing

Particle By-pass

)(tfr

Cohe

rent

(GP

zone

s) In-coherent (precipitates)

Section of GP zone parallel to (200) plane

Page 52: EQUILIBRIUM PHASE DIAGRAMS

Cu rich zones fully coherent with the matrix → low interfacial energy

(Equilibrium

phase has a complex tetragonal crystal structure which has incoherent

interfaces)

Zones minimize their strain energy by choosing disc-shape

to the elastically soft <100>

directions in the FCC matrix

The driving force (Gv

Gs

) is less but the barrier to nucleation is much less (G*)

2 atomic layers thick, 10nm in diameter with a spacing of ~10nm

The zones seem to be homogenously nucleated (excess vacancies seem to play an important role in their nucleation)

GP Zones

Page 53: EQUILIBRIUM PHASE DIAGRAMS

Atomic image of Cu layers in Al matrix

Bright field TEM micrograph of an Al-4% Cu alloy (solutionized and aged) GP zones.

5nm

5nm

Selected area diffraction (SAD) pattern, showing streaks arising from the zones.