SO6 Diff Transform 1 Upld
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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
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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
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Development of Development of
MicrostructuresMicrostructures
at Equilibriumat Equilibriumconditionsconditions
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NonNon--EquilibriumEquilibrium
ConditionsConditions
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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
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Phase DiagramsPhase Diagrams -- EutecticEutectic
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MgMg--PbPb Phase DiagramPhase Diagram
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FeFe--C Phase DiagramC Phase Diagram
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Transformations Transformations
Various microstructures could beVarious microstructures could beformed using thermal treatmentsformed using thermal treatments
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A nnealing Processes A nnealing Processes
Stored Energy of Cold Work (SECW)Stored Energy of Cold Work (SECW)
RecoveryRecovery
RecrystallizationRecrystallization Grain GrowthGrain Growth
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Introduction«Introduction«
PrecipitationPrecipitation¶ ¶ ++
EutectoidEutectoidKK ++
OrderingOrdering (disordered)(disordered) ¶ (ordered)¶ (ordered)
MassiveMassive
PolymorphicPolymorphicSingle component systemSingle component system
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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
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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
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Homogeneous Nucleation«Homogeneous Nucleation«
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
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Homogeneous Nucleation«Homogeneous Nucleation«
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
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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.
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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 ((!(
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Homogeneous Nucleation«Homogeneous Nucleation«
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
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Homogeneous Nucleation«Homogeneous Nucleation«
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Homogeneous Nucleation«Homogeneous Nucleation«
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
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Homogeneous Nucleation«Homogeneous Nucleation«
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
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Homogeneous Nucleation«Homogeneous Nucleation«
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)
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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.
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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
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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 (!(
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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
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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
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Nucleation on DislocationsNucleation on Dislocations
Lattice distortionLattice distortionin the vicinity of in the vicinity of dislocations candislocations canassist nucleationassist nucleation
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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
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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
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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
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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
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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
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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
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Relationships for the interfacemigration velocity ³v´ and distance ³x´ defining the region whereadjustment of concentration occurs
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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
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Concentration ProfilesConcentration Profiles
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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.
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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
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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
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The lengthening rate is given by:
k r
C C C
Dv
r
(�
! F
k is numerical constt ~1
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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
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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.
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Thickening of Plate-Like PPts
Thickening proceeds by lateralmovement of linear ledges.
BB
uu
v v