Chapter 6 Phase Diagrams

8/10/2019 Chapter 6 Phase Diagrams

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PHASE DIAGRAMS

Phase Rule

Types of Phase diagrams

Lever Rule

Alloy Phase EquilibriaA. Prince

Elsevier Publishing Company, Amsterdam (1966)

Advanced Reading

EQUILIBRIUM PHASE DIAGRAMS


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

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 inmaterials science and will be considered in detail in this chapter. Also, we shall restrictourselves 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 tohave 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 curietemperatures are overlaid)


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Atom Structure

Crystal

Electro-magnetic

Microstructure Component

Thermo-mechanicalTreatments

Phases Defects+

Casting Metal Forming Welding Powder Processing Machining

Vacancies Dislocations Twins Stacking Faults Grain Boundaries Voids Cracks

+ Residual Stress

Processing determines shape and microstructure of a component

& their distributions

Since terms like Phase and Microstructure are key to understanding phase diagrams, let ushave a re-look at the figure we considered before

Click here to know more about microstructures
http://../e-book/Microstructure_and_Metallography.ppthttp://../e-book/Microstructure_and_Metallography.ppthttp://../e-book/Microstructure_and_Metallography.ppt


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DEFINITIONSDEFINITIONS

Components of a system

Independent chemical species which comprise the system:These could be: Elements, Ions, Compounds

E.g.

Au-Cu system

: Components Au, Cu (elements)

Ice-water system

: Component H2

O (compound)

Al2

O3

Cr 2

O3

system

: Components Al2

O3

, Cr 2

O3

Let us start with some basic definitions:

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

Note thatcomponents 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 H 2

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!!


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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 asystem.

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

times

OR

The distribution of phases and defects in a material

Grain

The single crystalline part of polycrystalline metal separated by similar entities by agrain 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.

or Map that gives relationship between phases in equilibrium in a system as afunction 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)

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 tographite 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 slowand 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 = CF = C

P + 2P + 2For a system in equilibrium

FF

C +C +

P = 2P = 2

or 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 thedegrees 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 withthe 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 equationconnecting the composition variables and we need to specify only

(C 1) 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

(C 1) 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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It is worthwhile to clarify a few terms at this stage:

Components can

go on to make a phase(ofcourse

one can have single component phases as well-

e.g. BCC iron phase, ferromagnetic iron phase etc.)

Phases can

go on to make a microconstituent

Microconstituents can

go on to make a microstructure

(ofcourse phases can also directly go on to make a microstructure)

For a system in equilibrium the chemical potential of each species is same in all the

phases.

If , , ,

are phases, then: A

( ) = A

( ) = A

( ).

Suppose there are 2 phases ( & phases) and 3 components (A,B,C) in each phase.

Then: A

( ) = A

( ), B

( ) = B

( ), C

( ) = C

( ) i.e

there are 3 equations

For each component there are (P 1) equations and for C components the total number of equations is C(P 1)

In the above example the number of equations is 3(2 1)=3 equations.

F

= (Total number of variables)

(number of relations between variables)

= [P(C

1) + 2] [C(P

1)] = C P + 2

In a single phase system F = no. of variables.

P

F

(for a system with fixed number of components as the number phases increases the degrees of freedom decreases)

The Gibbs Phase Rule

F = C

P + 2F

= C

P + 2


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

F = C + 2 P

Can control the no. ofcomponents added and P & TSystem decided how many

phases to produce given theconditions

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

controland what the system controls

F = CF = C

P + 2P + 2


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No. ofphases

Total variables

P(C

1) +2

Degrees of Freedom

C

P +2

Degrees of Freedom

C

P +1

1 3 3 2

2 4 2 1

3 5 1 0

4 6 0 Not possible

C = 2

No. ofphases

Total variables

P(C

1) +2

Degrees of Freedom

C

P +2

Degrees of Freedom

C

P +1

1 4 4 3

2 6 3 2

3 8 2 1

4 10 1 0

C = 33 components

2 components

Variation of the number of degrees of freedom with number of components and number of phases

Phase rule with pressurefixed (at say 1 atm)

3 phase co-existence is aninvariant point


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The Gibbs phase rule and its geometrical cousin

There is an interesting connection between the Gibbs Phase Rule and the Euler

Rule for Convex Polyhedra .

The Eulers formula for convex polyhedra

can be considered as the geometrically

equivalent Gibbs Phase rule. The diagram below brings out the analogous variables.

VV00

VV 11

+ V+ V 22

= 2= 2 VV

E + F = 2E + F = 2or V0

= V Vertices

V1

= E Edges

V2

= F Faces

Edges ~ Components

Vertices ~ degrees of FreedomFaces ~ Phases

FF

C +C +

P = 2P = 2Gibbs Phase Rule

88

12 + 612 + 6

= 2= 2

For the cube

VV

E + F = 2E + F = 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 crystalstructure. (Diagram on next page) .

Note that the units of x-axis are in GPa

(i.e. high pressures are needed in the solid state andliquid 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 ( ) andone in the high T regime ( )) diminish at higher pressures. In fact -

phase field completelyvanishes 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 Pand 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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1535

1410

(BCC)

(HCP)

(FCC)

(BCC)

Liquid

Gas

Pressure (GPa)

T e m p e r a t u r e

( C )

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 (Tor 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 cancoexist in a unary P-T

phase diagram is 3 Note the P is in GPaVery 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)

T e m p e r a

t u r e

( C )

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

and rationalize the same).

These lines slope downward as: Underhigher pressure the phase with higher

packing fraction (lower volume) is preferred

Increase P and gas willliquefy on crossing phaseboundary

Phase fields of non-close packed structuresshrink under higher pressure

Phase fields of close packed structures expandunder higher pressure

Usually (P = 1 atm) the high temperature phase is theloose 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 (unlessofcourse

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 = CF = C

P + 1P + 1Phase rule for condensed phasesFor T

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

These have beenclassified 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 & solidstates

Eutectic

Peritectic

Liquid StateLiquid State Solid State analogueSolid State analogue

Eutectoid

Peritectoid

Monotectic

Syntectic

Monotectoid

Solid state analogue ofIsomorphous


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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 (C L

) and composition (C S

) of the solid.

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

CLB, C S 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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A B

System with complete solid & liquid solubility: ISOMORPHOUS

SYSTEM

%B

T M.P. of B

M.P. of A

C = 1P = 2

F = 0 C = 2P = 2F = 1

A BThe two component region expandswith one degree of freedom i.e. wecan chose one variable

say the T

F = C

P + 1

Let us start with an isomorphous

system with complete liquid and solid solubility.

If we try to draw a straight line connecting the melting point of A and B (the twocomponents)-

we note that the degree of freedom for any intermediate composition is 1

this line expands into a region (which is a two phase mixture of liquid and solid).

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 willcoexist over a range of temperatures when heated.

* Assuming that the components do not decompose or sublime.

melting pointsare fixed!

l h bl d d f f d f h h h d


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A B

%B

T

M.P. of B

M.P. of A

C = 1P = 2

F = 0

C = 2P = 1F = 2

C = 2

P = 1 (liquid)F = 2

Variables T, C S B

2

Variables T, C LB

2

Variables

T, C LB, C S B

3

F = 2

P F= 3

P

C = 2P = 2F = 1

F = 2

P

F = C

P + 1

Now let us map the variables and degrees of freedom in varions

regions of the isomorphous

phase diagram

T and Composition canboth be varied while stillbeing in the single phaseregion

in the two phase region, ifwe fix T (and hence exhaustour DOF), the composition ofliquid and solid in equilibrium

are automatically fixed (i.e. wehave no choice over them).

Alternately we can use our DOF

to chose C L

then Tand C S

are automatically fixed.

For pure components at any T For alloys


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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) T 1

> T A

, (ii) T 2

=T A

, (iii)TA

>T 3

>T B

, (iv) T 4

=T B

, (v) T 5


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Al 2O3 Cr 2O3%Cr 2O3

T ( C )

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, Al 2

O3

-Cr 2

O3.

Note the liquidus

and solidus

lines in the figure.

Schematics

Note that the componentsare 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.

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 besame 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 laterin this chapter as a variation of the isomorphous

system (with complete solubility in thesolid 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 phaseregion 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 compositionand 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

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

HUME ROTHERY

RULESClick here to know more about
http://../e-book/Chapter_4a_Structure_of_Solids_Metallic.ppt#393,59,Slide%2059http://../e-book/Chapter_4a_Structure_of_Solids_Metallic.ppt#393,59,Slide%2059http://../e-book/Chapter_4a_Structure_of_Solids_Metallic.ppt#393,59,Slide%2059


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Say the composition C0

is cooled slowly (equilibrium)At T0

there is L + S

equilibriumSolid (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 TieLine ) 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). Thoughthey may be extended to mark the temperature.

To find the fractions of solid and liquid weuse the lever rule.


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%B A B

T

L

L + S

S

C0C1 C2

T0

Cooling

Fulcrum of the lever

Arm of the lever proportional to

the solid


the liquid

Note: strictly speaking

cooling curves cannot beoverlaid on phase diagrams

Tie line

12

10

C C C C f liquid

At T 0The fraction of liquid (f l

) is

(C0

C1

)The fraction of solid (f s

) is

(C2

C0

)

12

02

C C C C f solid

We draw a horizontal line (called the Tie Line ) at the temperature of interest (say T 0

).

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

with thefulcrum at the nominal composition (C 0

).

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 phaseregion and is horizontal.


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Expanded version

Extended tie line

C0C1 C2

T0

Fulcrum of the lever


the solid


the liquid

0 1

2 1liquid

C C f

AB C AC

C

At T0The fraction of liquid (f l

) is proportional to (C 0

C1

) ACThe fraction of solid (f s

) is proportional to

(C2

C0

) CB

2 0

2 1 solid

C C f

AB

C

C

B

C

BA

C

H d th f ti ( %) f lid d li id h g l (i th t h gi )?


31/73

0

20

40

60

80

100

1 2 3 4 5 6 7 8 9Temperature Label

%

o f

S o

l i d o r L

i q u

i d

Fraction of Liquid (%)Fractionof Solid (%)

How does the fraction (or %) of solid and liquid change as we cool (in the two phase region)?

As we cool from T 1

to T 9

in the two phase region:The first solid appears at T 1

(effectively the fraction of solid at T 1

is zero).

The fraction of solid increases according the solid line in the plot below and correspondingly the fraction of liquid decreases.The last bit of liquid solidifies at T 9

.The composition of the solid and liquid in equilibrium with it keeps on changing as we cool (as in upcoming slide).The total percentage (L + S) is always 100%.

P i t t b t d


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For a composition C 0

At T 0

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 (C 1

) with aliquid of definite composition (C 2

)

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

the two phase region (liquid + solidregion) the composition ofthe 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

I h Ph Di


33/73

Isomorphous

Phase Diagrams

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

atlow temperatures (below the peak of the 1

+ 2

phase fielddome)

(we will have to say more about that soon)

Note that

Ag & Au are so similar

that the phase diagrambecomes a thin lens (i.e. any alloy of Au & Ag meltsover a small range of temperatures

as if it werenearly

a pure metal!!).

Any composition melts above the linear interpolatedmelting point.

T 1Below T 1

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

Any compositionmelts above the

linearlyinterpolatedmelting point


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Funda

Check One phase or many phases? the concept of the phase field

We have noted that different crystal structures of the same component are different phases.

Different compositions of the same crystal structure differ in lattice parameter andconstitute different phases.

Sometimes in the region of stability of a phase

(say ), different compositions arereferred to (casually) as -phase.

The region should be called a Phase Field

and different compositions are actually different phases .

The solid solution based on a component is often written in brackets. E.g. (Cu) marked inthe phase diagram implies a solid solution of Cu with another component (say Ag).


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Funda

Check Is the lever rule right? (I.e. does it give the right phase fractions?)

L

1 B x

C 1

C 0

C 2

2 B x

0 B x

L

1 B x

C 1

C 0

C 2

2 B x

0 B x

Let A and B be the two components constituting the binary phase diagram. Consider co-existing phases

(withcomposition C 1

) and

(with composition C 2

) with a meancomposition C 0

. f

denotes the fraction of a particular phase. Let be the compositions expressed in terms of the

percentage of B.

Using lever rule: 0 1 0 12 1 2 1

C C x x f

C C x x

, 2 0 2 02 1 2 1

C C x x f

C C x x

Amount of B in C 0 (L) = Amount of B in + Amount of B in

0 2 1 x f x f x = 0 1 2 02 12 1 2 1

x x x x x x

x x x x

= 0 2 1 2 1 2 0 1 02 1

x x x x x x x x x

x x


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Th i h h di i h l i i i b d d


37/73

These isomorphous

phase diagrams with congruent melting compositions can be understood as twosimple isomorphous

diagrams with C 0

as one of the components:

i.e. A-C 0

is one and C 0

-B is the other.

%C 0

Extensions of the simple isomorphous system: What does this imply w r t the solid state phases?


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A A

and B B bonds stronger than A B bonds

Liquid stabilized Phase separation

in the solidstate

A B bonds stronger than A A and B B bonds

Solid stabilized Ordered solid formation

E . g . A

u - N

i

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 theordered state is more stable) ordering reaction is seen at low T.

Depression in MP means

the liquid state (disordered) is more stable phase separationis seen at low T. (Phase separation can be thought of

as the opposite

of ordering. Ordering (compound formation) occurs for ve

values of H mix

, while phase separation is favoured

by +ve

values of H mix

.

Case A Case B

Examples of isomorphous systems with phase


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p p

y pseparation and compound formation

Au-Pd systemwith 3

compounds

Au-Pt system with phaseseparation at low temperatures

Au-Ni: model system tounderstand phase separation

Phase separation in a AlCrFeNi alloy (with compositionAl28.5 Cr 27.3 Fe24.9 Ni 19.3 ) into BCC + B2 phases


40/73

Ti-Zr: With solid

state analogue ofisomorphous

phasediagram


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Congruent transformations

We have seen two congruent transformations. The list is as below.

Melting point minimum

Melting point maximum

Order

disorder transformation

Formation of an intermediate phase

Eutectic Phase Diagram


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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 (dueto entropic reasons).

In a simple eutectic system (binary), there is one composition at which the liquid freezes totwo solids at a single temperature. This is in some sense similar to a pure solid whichfreezes 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 ata lower temperature than the melting points of the constituent components.

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

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

(327 C) and Sn

(232 C) can be used forsoldering 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 aeutectic 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)


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Pb Sn%Sn

T ( C )

100

200

300

10 30 50 70 90

L

+

L +

+ L

Liquidus

Solvus

Solidus

18% 62%97%

183 C

232 C

327 C

18362% 18% 97%

Cool

C

L

Sn Sn Sn

Eutectic reaction (the properway of writing the reaction)

Eutectic reaction

L

+

Ceutectic

= C E

Teutectic

= T E

E C

E C

E C E C

E C

E FD

E T

Note that Pb is CCP, while Sn at RT is Tetragonal (tI4, I4 1amd) therefore complete solid solubility across compositionsis ruled out!!

Note the following points:


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Note the following points:

and

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

is a solid solution ofB (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 crystalstructure 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 ( C )

100

200

300

10 30 50 70 90

L

+

L ++ L

Liquidus

Solvus

Solidus

18% 62% 97%

183 C

232 C

327 C

Eutectic reactionL +

At the eutectic point E (fig. below) 3 phases co-exist: L, &


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A B%B

T ( C )

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

p

( g ) p ,

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 fixedcomposition 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 (forcomposition 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

Further points:


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Further points:

Extension of the boundary line between a single phase region and

adjacent two-phaseregion, should lie in the two phase region ( red dashed lines as in figure below).

The eutectic reaction is a phase reaction

and not

a chemical reaction.

Examples of Eutectic microstructures As pointed out before microstructural information is often overlaid on phasedi Th t i t t hi h l l li


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2 m2 m2 m

diagrams. These represent microstructures which evolve on slow cooling.

Al-Al 2

Cu lamellar eutectic

Sn

Pb

Composition plot across lamellae

Pb-Sn

lamellar eutectic

Though we label the microstructureas Pb-Sn

lamellar eutectic it isactually a - eutectic. Al2

Cu(note that one of the components is a compound!)

(Al)

Gibbs free energy vs composition plot at various temperatures: Eutectic system


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gy

p p p : y

Isomorphous to Eutectic A A and B B bonds stronger than A B bonds


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A B

L

L +

A B

L

L +

1 2

1

+ 2

A B

L

1 2

L +

1 + 2 A B

L

1 2

L + 1

1

+ 2

L + 2

p

Liquid stabilized

A eutectic system can be visualized as arising from an isomorphous

system with depression inMP, as below.

Solidification of Eutectic and Off-eutectic compositions


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To reiterate an important point: phase diagrams do not

contain microstructural information (i.e. they cannot tell youwhat is the microstructure produced by cooling). Often microstructural information is overlaid on phase diagrams forconvenience. Hence, strictly cooling is not in the domain of phase diagrams-

but we can overlay such informationkeeping in view the assumptions involved.

For the following explanations refer to the Pb-Sn

eutectic diagram and the diagram considered next. Solidification ofthree range(s) of compositions need to be understood: (i)

to the left of C E , (ii)

between C E

and C E

and (iii) CE.

(iii)

On cooling an eutectic composition

(CE

or

C3E

in the next page), at the eutectic temperature ( TE

), both theconstituents (say

& ) of the eutectic reaction will simultaneously

form from the liquid. In practice (based on

other factors) one of the constituents may form before the other. The entire solidification takes place at a singletemperature-

TE

.

On slow cooling, the microstructure produced by the eutectic reaction could be lamellar, Chinese script

(like), etc.This distribution of phases (say lamellar) is called a microconstituent .

(i) On cooling a composition to the left of CE Th h ill lidif i f h li id d b 1 d 1b h ill b i f d L


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The

phase will start solidifying from the liquid and between 1a and

1b there will be a mixture of

and L.

Between 1b and 1c there will be only

phase.

Below 1c (as the solubility of B in

reduces)

phase will precipitate out of

(and we will have a mixture of

+).

(ii)

On cooling a composition between CE

and C E

Between 2a and 2b, the

phase will start solidifying from the liquid and we will have a

mixture of L +

(this

isknown as the proeutectic-

implying that it is pre -eutectic ). On cooling towards T E

the composition of theliquid travels along the 2a-E curve.

On reaching T E

(i.e. point 2b) the liquid remaining after the solidification of proeutectic-

will solidify as a eutectic

mixture (according to the eutectic reaction).

For a composition between C E and C E

the situation is similar to case (ii) above except that we will

get proeutectic-instead of proeutectic- . And for a composition beyond C E

the situation is similar to case (i) above (except that wewill get

instead of ).

The important point to note that even for off-eutectic compositions between CE

and CE

part of the liquid solidifies by the eutectic reaction

C2


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C1

2

C3

C4

The solidification sequence of C 4

will be similar to C 2

exceptthat the proeutectic

phase will be

Special eutectic diagrams


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Eutectic system without terminal solidsolubility: Bi-Cd

Technically this is incorrect:there has to be some terminal

solid solubility

Technically it is incorrect

to draw

eutectic phase diagrams with zero

solidsolubility.

This would imply that a pure component

(say Bi in the example considered) meltsover a range of temperatures (from P

to271.4 C) which is wrong.

Also, let us consider an example of a point P

(which lies on the eutecticline

PQ). At P

the phase rule becomes:

F = CP+1 = 13+1 = 1 !!!

Note that the above is an alternate wayof arriving at the obvious contradictionthat at P

on one hand we are sayingthat there is a pure component and on theother hand we are considering a three

phase equilibrium (which can happen

only for Bi-Cd

alloys).

P Q

Correct

version of the diagram with some terminal solid solubility.

Eutectic system with terminal solidl b l d

Special eutectic diagrams


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solubility on one side: Ag-Ge

With terminalsolid solubilityon both sides

SolvedDuring the solidification of a off eutectic (Pb-Sn) composition (C 0), 90 vol.% of the solid

i d f h i i d 10 l % f h i h Wh i h l


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SolvedExample

consisted of the eutectic mixture and 10vol.% of the proeutectic

phase. What is the valueof C 0

?

Density data for

and :

= 10300 Kg/m 3

= 7300 Kg/m 3

Let us start with some observations:

Pb

is heavier

than Sn

and hence the density of

is more than that of .

Since the proeutectic

phase is the composition is hyperpeutectic

(towards the Sn

side).

The volumefractions (in %) are usually calculated by taking the area fractions by doing metallography (microstructure) andthen converting it into volume fractions (usually volume fraction is assumed to be equal to area fractions).

Using the fact that there is 10 vol% phase:

eutectic mix

Wt. of 0.1 7300Wt. fraction of proeutectic =

Wt. of the alloy (0.1 7300) (0.9 )

(1)

Where, eutectic mix 397 62 62 18

10300 7300 4563 4066 862997 18 97 18

Kg m

Substituting in equation (1): Wt. fraction of proeutectic = 0.086

Using lever rule: 00.62

0.0860.97 0.62C

, C 0 = 0.650 = 65.0%

Eutectic Data:

183 C

62 wt.% Sn 18362% 18% 97%

Cool

C

L

Sn Sn Sn

Peritectic Phase Diagram


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Like the eutectic system, the peritectic

reaction is found in systems with complete liquidsolubility 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 andhence it is difficult to equilibrate

Peritectic

reactions (as compared to say eutecticreactions). 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

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

+ ) just below T P

.

Pt-Ag Peritectic system


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Peritectic

reaction

L +

Melting points of thecomponents vastly different

118666.3% 10.5% 42.4%

Cool

C

L

g Ag Ag

P C

P C L

P C

P C

P C L

P C

P T

Note that below T P

pure is not stable and splits into ( + )

Formal way of writingthe peritectic

reaction

Isomorphous to Peritectic


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A B

L

L +

A B

L

L + 1

( 2)

1 2 1 + 2 A B

L

L + 1

1 2 1 + 2

L

L +

1 21 + 2

Examples: Some important phase diagrams


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Some phase diagrams are technologically important and via these further aspects of theutility of phase diagrams can be learnt.

The Fe-C (or more precisely the Fe-

Fe3

C) diagram is an important one. Cementite

is ametastable phase and strictly speaking

should not be included in a phase diagram. But thedecomposition rate of cementite

is small and hence can be thought of as stable enough

to be included in a phase diagram. Hence, we typically consider the

Fe-

Fe3

C part of the Fe-C

phase diagram.

Another technologically (and from a perspective of physical Metallurgy) important phasediagram is the Al-Cu system

(especially the Al rich end of the phase diagram).

The Fe-Cementite phase diagram


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A portion of the Fe-C diagram-

the part from pure Fe to 6.67 wt.% carbon (correspondingto cementite, Fe 3

C)-

is technologically very relevant.

Cementite

is not a equilibrium phase and would tend to decompose into Fe and graphite.This reaction is sluggish and for practical purpose (at the microstructural level) cementitecan be considered to be part of the phase diagram. Cementite

forms as it nucleates readilyas compared to graphite.

Compositions up to 1.5%C are called steels

and beyond 2%C are called cast irons . In realitythe classification should be based on castability

and not just on carbon content.

Heat treatments can be done to alter the properties of the steel

by modifying themicrostructure we will learn about this in the chapter on Phase Transformations. This

may involve production of metstable

phases like martensite

(not found in the equilibrium phase diagram).

As before we will use slow

cooling curves to see

the microstructures produced. The partof the phase diagram of interest is: (i) with less than ~2% C and (ii) less that ~1100 C.Phases of interest are listed in the table below.

Phase Structure

Austenite ( ) FCC (CCP)

Ferrite ( ) BCCCementite (Fe 3C) Orthorhombic

Crystal structure of Cementite

Fe-Cementite diagram
http://simple_to_complex_crystal_structures.ppt/#19.%20Slide%2019http://simple_to_complex_crystal_structures.ppt/#19.%20Slide%2019


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%C

T

Fe Fe3

C6.7 4.30.80.16

2.06

PeritecticL +

Eutectic

L

+ Fe 3

C

Eutectoid

+ Fe3

C

L

L +

+ Fe 3

C

1493C

1147C

723C

0.025

RT~0.008

Three reactions are seen in the Fe rich side: (i) Peritectic, (ii) Eutectic & (iii) Eutectoid.

Of the three reactions the eutectoid reaction istechnologically important

A1

A3

Acm

Note: we can also draw thetrue Fe-C diagram (not shown here). Sometime oneof these diagrams is shownin dashed lines and overlaidon a single diagram.

L

(BCC)

(FCC)

(BCC)912C

For Pure Fe

1394C

1538C

The portion of the phase diagram, which is technologically relevant is shown in the figure below


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Ferrite

FerriteFerrite


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C1 C2C3

C1

C2

C3

Pearlite

a micro-constituent(Not a phase)

PearlitePearliteAustenite

Similar* to C 1

butthe proeutectoid

phase is Cementite*Similar in solidification sequence not in properties

Pro-eutectoidCementite

along prior Austenite

grain boundaries

Fe3

C

Grain boundary

Pearlite is a micro-constituent with alternating lamellae of cementite and ferrite.


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SEM

micrograph of pearlite

inEutectoid composition (0.8%C)

steel

g

SEM

micrograph of pearlite

inHyper-eutectoid composition(1.0%C) steel

Pro-eutectoidCementite

along prior Austenitegrain boundaries

Pearlite

Structure of Etched Pearlite in AFM


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Composition of EN9 alloy in wt.%: C-0.5, Si-0.25, Mn-0.70, S-0.05, P-0.05

Processing: Hot rolled steel sample annealed at 700 C and followed by slow cooling

Hardness: 180-230 BHN

(Avg. = 205 BHN)

Images taken on Solver Pro NT-MDT-

semi contact mode.

Images courtesy: Prof. Sandeep Sangal and K. Chandra Sekhar (MSE, IITK).

Funda Check


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Funda

Check

Components need not be only elements-

they can be compounds like Al 2

O3

, Cr 2

O3

.

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

hence oftenmicrostructures 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 phasediagram 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 extendthe 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).

Solved Two separate alloys are cooled from the phase field: (i) one hyper-eutectoid and other(ii) h id A i h i b h h id h f l h


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6%

94%

6%

94%

Example (ii) hypoeutectoid. Assuming that in both cases the pro-eutectoid phase forms along thegrain boundaries and its phase fraction is 6%, determine the carbon composition of the twoalloys.

Given: Case-1: 6% ferrite ( , proeutectoid phase) along grain boundaries (GB),

Case-2: 6% cementite (Fe 3C, proeutectoid phase) along GB.

Case-1: referring the Fe-C diagram 10.80.060.8 0.025

C , C 1 = 0.75% C

Case-2: referring the Fe-C diagram 20.8

0.066.67 0.8C

, C 2 = 1.15% C

%C

T

Fe 0.8C 1

+ Fe 3C723C

0.025

RT~0.008

C 2

At 723 C


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End

Solved


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Example

Data for Cu:

H f

(Cu vacancy) = 120

103

J/mole

R (Gas constant) = 8.314 J/mole/K

F = C P + 1


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A B

%B

T

M.P. of B

M.P. of A

C = 1P = 2F = 0

C = 2

P = 1F = 2

C = 2P = 1F = 2

Variables T, CS

B

2

Variables T, C LB

2

Variables T, C LB, C S B

3

2 P 3 P

C = 2P = 2F = 1

2

P

Say T 1

is chosen

The compositions C LB

& C S B

are automatically chosen by the system

Liquid (melt)


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Cooling

Pb Sn

100

200

300

10 30 50 70 90

L

+

L + + L

Hypereutecticcomposition

Proeutectic

+

EutecticMixture of

Grain boundary

Liquid (melt)


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Cooling

C1 C2

%C

T

0.80.02

Eutectoid

+ Fe 3

C

+ Fe 3

C

+

+ Fe 3

C

Fe

C3


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T ( C )

900

1300

1500

Ag Pt10 30 50 70 90

L

+

L +

1100

1700

Melting points ofthe componentsvastly different

Peritectic

reaction

L +

Documents

Chapter 6 Phase Diagrams