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IntroductionIntroduction

Phase transformations in solidsPhase transformations in solids ±± bybythermally activated atomicthermally activated atomicmovementsmovements

Different types of transformationsDifferent types of transformations ±±divided into following groupsdivided into following groups PrecipitationPrecipitation

EutectoidEutectoid

OrderingOrdering

MassiveMassive

polymorphicpolymorphic



Chapter 9-5

� Tell us about phases as function of T, Co, P.

� For this course:

--binary systems: just 2 components.

--independent variables: T and Co (P = 1atm is always used).

� PhaseDiagram

for Cu-Ni

system

� 2 phases: L (liquid) E (FCC solid solution)

� 3 phase fields:L

L +E E

wt% Ni20 40 60 80 10001000

1100

1200

1300

1400

1500

1600

T(°C)

L (liquid)

E

(FCC solidsolution)

L +

E l i q u i

d u s

s o l i d u s

Adapted from Fig. 9.2(a), Callister 6e.

(Fig. 9.2(a) is adapted from Phase

Diagrams of Binary Nickel Alloys, P.

Nash (Ed.), ASM International,

Materials Park, OH (1991).

PHASE DIAGRAMS



Development of Development of

MicrostructuresMicrostructures

at Equilibriumat Equilibriumconditionsconditions



NonNon--EquilibriumEquilibrium

ConditionsConditions



Chapter 9-11

� CE

changes as we solidify.� Cu-Ni case:

� Fast rate of cooling:

Cored structure� Slow rate of cooling:

E uilibrium structure

First E to solidify has CE = wt Ni.

ast E to solidify has CE = 5wt Ni.

First E to solidfy:

wt Ni

Uniform CE:

5wt Ni

ast E to solidfy:

< 5wt Ni

CORED VS EQUI IBRIUM PHASES



Phase DiagramsPhase Diagrams -- EutecticEutectic



MgMg--PbPb Phase DiagramPhase Diagram



FeFe--C Phase DiagramC Phase Diagram



Transformations Transformations

Various microstructures could beVarious microstructures could beformed using thermal treatmentsformed using thermal treatments



A nnealing Processes A nnealing Processes

Stored Energy of Cold Work (SECW)Stored Energy of Cold Work (SECW)

RecoveryRecovery

RecrystallizationRecrystallization Grain GrowthGrain Growth



Introduction«Introduction«

PrecipitationPrecipitation¶ ¶ ++

EutectoidEutectoidKK ++

OrderingOrdering (disordered)(disordered) ¶ (ordered)¶ (ordered)

MassiveMassive

PolymorphicPolymorphicSingle component systemSingle component system



Homogeneous NucleationHomogeneous Nucleation

Composition adjustment to formComposition adjustment to form phase in parentphase in parent

Atoms may rearrange intoAtoms may rearrange into crystalcrystalstructurestructure

/ / interface is createdinterface is created ±± leads to anleads to anactivation energy barrieractivation energy barrier



Homogeneous Nucleation«Homogeneous Nucleation«

Free energy change associate withFree energy change associate withthe nucleation process have 3the nucleation process have 3contributionscontributions

Formation of Formation of phasephase causes a volume freecauses a volume freeenergy reductionenergy reduction

Creation of Creation of / / interfacial areainterfacial area will give a freewill give a freeenergy increaseenergy increase

Transformed volume usually doesn¶t fitTransformed volume usually doesn¶t fitperfectly into available spaceperfectly into available space ±± misfit strainsmisfit strains

S V GV AGV G ((!( K




Equation similar to that derived for solidificationEquation similar to that derived for solidification For liquidFor liquid--solid interface,solid interface, KK is roughly the same for allis roughly the same for all

interfacesinterfaces

Nucleation in solidsNucleation in solids KK variesvaries

very low for coherent interfacesvery low for coherent interfaces

high for incoherent interfaceshigh for incoherent interfaces A AKK term should be replaced by a summation over all surfacesterm should be replaced by a summation over all surfaces

of the nucleus, i.e.of the nucleus, i.e.

S V GV AGV G ((!( K

ii AK7




Ignoring variation of Ignoring variation of KK andandassuming spherical nucleusassuming spherical nucleus

KT T 234

3

4r GGr G S ((!(

Effect of the misfitstrain energy is toreduce the effectivedriving force fortransformation.

S V GG

r

((

!K2*

2

3*

)(3

16

S GGG

((!(

TK

Differentiation yieldsDifferentiation yields



Driving Force for PrecipitationDriving Force for Precipitation

Alloy XAlloy X00 is solution treated atis solution treated atTT11 and cooled rapidly to Tand cooled rapidly to T22

Supersaturated with BSupersaturated with B ±± will trywill tryto pptto ppt

After completeAfter completetransformation, the freetransformation, the free--energy will decrease byenergy will decrease by GG00

GG00 is the total driving force foris the total driving force fortransformationtransformation

But this (But this (GG00) is NOT the driving) is NOT the driving

force for NUCLEATIONforce for NUCLEATION

- Because the first nuclei to appear do

not significantly change the composition from X0.



Driving Force for PrecipitationDriving Force for Precipitation If small amount of material withIf small amount of material with

nucleus composition (Xnucleus composition (XBB) is) is

removed from theremoved from the phasephase Then total free energy of the systemThen total free energy of the system

will decrease bywill decrease by GG11 (per mole of (per mole of removed)removed)

GG11 is represented by point Pis represented by point P

FE FE Q Q B B A A

X X G !( 1

F F F F

Q Q B B A A

X X G

!( 2

Atoms are then rearranged intoAtoms are then rearranged into crystal structurecrystal structure Per mole of Per mole of formedformed

GG22 is represented by point Qis represented by point Q

Driving force for NucleationDriving force for Nucleation

12GGG

n ((!(




Volume free energyVolume free energydecrease associated withdecrease associated withnucleation event isnucleation event istherefore,therefore,

m

n

V

V

GG

(!(

For dilute solutionsFor dilute solutions

X GV (w(

e

X X X !( 0

wherewhere




AssumedAssumed ±± nuclei are spherical withnuclei are spherical withequilibrium composition & structureequilibrium composition & structure However, in practice, nucleation will beHowever, in practice, nucleation will be

dominated by whatever nucleus has minimumdominated by whatever nucleus has minimumactivation energy barrier.activation energy barrier.

Most effective is to form nucleus with smallestMost effective is to form nucleus with smallesttotal interfacial energytotal interfacial energy

This will mostly dominate nucleationThis will mostly dominate nucleation

Incoherent nuclei have very highIncoherent nuclei have very highenergyenergy ±± virtually impossiblevirtually impossible




Coherent nuclei with orientationCoherent nuclei with orientationrelationship with matrixrelationship with matrix ±± reducedreducedenergyenergy

Coherent nucleiCoherent nuclei ±± high strain energyhigh strain energy Lowering of interfacial energy compensates forLowering of interfacial energy compensates for

the increase in the strain energythe increase in the strain energy




Homogeneous nucleation can occurHomogeneous nucleation can occurin very few systemsin very few systems e.g. Cue.g. Cu--Co systemCo system

Both are FCC with only 2% difference in latticeBoth are FCC with only 2% difference in latticeparameterparameter

Very low coherency strainsVery low coherency strains

NiNi--SuperalloysSuperalloys Precipitation of NiPrecipitation of Ni33AlAl

Most other examples are limited to metastableMost other examples are limited to metastablephases (usually GP zones)phases (usually GP zones)



Gamma-Prime PPts

Extremely smallExtremely small KK' precipitates always' precipitates alwaysoccur as spheres.occur as spheres. In fact, for a given volume of precipitate, a sphere hasIn fact, for a given volume of precipitate, a sphere has

less surface area than a cube, andless surface area than a cube, and thus is the preferred shape to minimize surface energy.thus is the preferred shape to minimize surface energy.

With a coherent particle, however, theWith a coherent particle, however, theinterfacial energy can be minimized byinterfacial energy can be minimized byforming cubes and allowing theforming cubes and allowing the

crystallographic planes of the cubic matrixcrystallographic planes of the cubic matrixand precipitate to remain continuous.and precipitate to remain continuous. Thus as theThus as the KK' grows, the morphology can change from' grows, the morphology can change from

spheres to cubes (as shown in this figure) or platesspheres to cubes (as shown in this figure) or platesdepending on the value of the matrix/precipitate latticedepending on the value of the matrix/precipitate lattice

mismatch.mismatch.



Heterogeneous NucleationHeterogeneous Nucleation

Nucleation is mostly heterogeneousNucleation is mostly heterogeneous

Suitable nucleation sitesSuitable nucleation sites Defects like excess vacancies, dislocations,Defects like excess vacancies, dislocations,

grain boundaries, stacking faults, inclusionsgrain boundaries, stacking faults, inclusionsand free surfacesand free surfaces

Creation of a nucleus results in the destructionCreation of a nucleus results in the destructionof a defectof a defect ±± some free energy (some free energy (GGdd) will be) will bereleasedreleased

This will reduce the activation energy barrierThis will reduce the activation energy barrier

S V het G AGGV G (((!( K



Nucleation on GBsNucleation on GBs

Optimum embryo shape would be one thatOptimum embryo shape would be one thatminimize total interfacial free energyminimize total interfacial free energy

EF

EE

K

K U

2cos !

EEEEEFEF K K A AGV G V (!(



Nucleation on GBs«Nucleation on GBs«

Critical radius of theCritical radius of thespherical capsspherical caps

V G

r

(

!EFK 2

*

Activation energyActivation energybarrierbarrier

US V

V

G

Ghet het

!!(

(*

hom

*

*

hom

*

w here

2cos1cos2

2

1UUU !S



Nucleation on GBs«Nucleation on GBs«

If matrix and precipitate are sufficientlyIf matrix and precipitate are sufficientlycompatiblecompatible Low energy (coherent) facets can be formedLow energy (coherent) facets can be formed

Nucleus may have orientation relationship with one of theNucleus may have orientation relationship with one of the

grainsgrains



Nucleation on DislocationsNucleation on Dislocations

Lattice distortionLattice distortionin the vicinity of in the vicinity of dislocations candislocations canassist nucleationassist nucleation



Nucleation on Dislocations /Nucleation on Dislocations /

Stacking FaultsStacking Faults

In FCC crystals, unit dislocations

usually dissociate to produce two

partial-dislocations Creates a ribbon of stacking fault

Stacking fault is, in fact, 4 close-packed layers

of HCP crsystal (ABAB stacking)

This is a very potent site for nucleation of an HCPprecipitate



Rate of Heterogeneous NucleationRate of Heterogeneous Nucleation

In order of decreasingIn order of decreasing((GG**

Homogeneous sitesHomogeneous sites

VacanciesVacancies DislocationsDislocations

Stacking faultsStacking faults

Grain boundaries /Grain boundaries /

interphaseinterphase boundariesboundaries Free surfacesFree surfaces

Overall rate of Overall rate of transformation alsotransformation also

depends on thedepends on therelativerelativeconcentration of concentration of these sitesthese sites



Rate of Heterogeneous Nucleation«Rate of Heterogeneous Nucleation«

If concentrationIf concentrationof heterogeneousof heterogeneoussites is Csites is C11 per unitper unitvolume, thenvolume, then

¹¹ º

¸©©ª

¨ (¹

º

¸©ª

¨ (!

k

G

k

GC N

m

het

*

1 expexp[

Nuclei m-3 s-1



Rate of HeterogeneousRate of Heterogeneous

Nucleation«Nucleation«

Relative magnitudesRelative magnitudesof heterogeneousof heterogeneousand homogeneousand homogeneousvolume nucleationvolume nucleationratesrates

(Differences in(Differences in [[andand ((GGmm are not soare not so

important and soimportant and so

ignored.)ignored.)

¹¹ º

¸©©ª

¨ ((!

kT

GG

C

C

N

N het het

**

hom

0

1

hom

exp

A lthough A lthough ((GG** is alwaysis always

smallest for heterogeneoussmallest for heterogeneous

nucleation , the factornucleation , the factor CC11/C/C00

must also be taken intomust also be taken into

accountaccounti.e. the number of atoms on thei.e. the number of atoms on the

heterogeneous sites relative toheterogeneous sites relative to

the number of atoms withinthe number of atoms within

the matrixthe matrix



Precipitate GrowthPrecipitate Growth

Coherent interfacesCoherent interfaces ±± SlowSlow

Incoherent interfacesIncoherent interfaces ±± FastFast Shape is determined by the relativeShape is determined by the relative

migration rates of the interfacesmigration rates of the interfaces Formation of disk or plate shapeFormation of disk or plate shape ±± WidmanstattenWidmanstatten

morphologymorphology

Slow Slow

V V ** FastFast



Growth Behind Planar IncoherentGrowth Behind Planar Incoherent

InterfacesInterfaces

Assume a slabAssume a slabof of --phasephaseforms byforms bycombiningcombiningseveral nucleiseveral nucleiformed on aformed on a

grain boundarygrain boundary



Relationships for the interfacemigration velocity ³v´ and distance ³x´ defining the region whereadjustment of concentration occurs



Note the following:

(i)

(ii)

(iii)

Dt x w

Dt x

w

0 X v (w

t

Dv w

ppt thickening obeys a parabolic gro w

th la w

For a given time the gro w th rate is

proportional to the supersaturation



Concentration ProfilesConcentration Profiles



Grain Boundary DiffusionGrain Boundary Diffusion

Usually g.b. ppts do not form asUsually g.b. ppts do not form ascontinuous layercontinuous layer

Grain boundary diffusion can lead toGrain boundary diffusion can lead torapid lengthening and thickening of rapid lengthening and thickening of ppts.ppts.



DiffusionDiffusion--Controlled Lengthening of Controlled Lengthening of

Plates or NeedlesPlates or Needles

Consider the ppt is a plate withConsider the ppt is a plate with Constant thickness andConstant thickness and

curved incoherent edge with radiuscurved incoherent edge with radius rr

rr



Due to Gibbs-Thomson effect, theequilibrium concentration in thematrix adjacent to the edge will beincreased to CCrr.

The concentrationThe concentration

gradient avialablegradient avialableto drive the diffusionto drive the diffusionto the advancingto the advancingedge is thereforeedge is thereforereduced toreduced to ((C/LC/L

Where,Where,

((C = CC = C00 ±± CCrr

L = CharacteristicL = Characteristicdiffusion distancediffusion distance



The lengthening rate is given by:

k r

C C C

Dv

r

(�

! F

k is numerical constt ~1



Composition difference available toComposition difference available todrive diffusion will depend on the tipdrive diffusion will depend on the tip

radiusradius With certain simplifying assumptions,With certain simplifying assumptions,

it can be shown thatit can be shown that

¹¹ º

¸©©ª

¨(!(r

r X X

*

0 1

W here,

(X=X 0 ² X r

(X 0= X

0 ² X e

r* = critical nucleus

radius



Assuming constant molar volume, weAssuming constant molar volume, wecan write:can write:

¹¹ º

¸©©ª

¨ �

(!

r

r

r X X k

X Dv

r

*0 1

1

F

This equation appliesas long as there is no

decrease in

supersaturation far

from the interface

due to other ppts.



Thickening of Plate-Like PPts

Thickening proceeds by lateralmovement of linear ledges.

BB

uu

v v

so6 diff transform 1 upld

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