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phase transformation
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Diffusional Diffusional Transformations Transformations
- 2- 2SOLID STATESOLID STATE
Overall Transformation Overall Transformation KineticsKinetics
Progress of isothermal Progress of isothermal phase transformations phase transformations can be represented by can be represented by plotting plotting Fraction transformed Fraction transformed
(f)(f) as a function of as a function of time (t)time (t) and and temperature (T)temperature (T)
TTTTTT Diagrams Diagrams
αα β β (cellular) (cellular) αα β β + + (cellular) (cellular) αα’ ’ αα + + ββ
Volume fraction ‘f’ Volume fraction ‘f’ varies from 0 to 1varies from 0 to 1
Overall Transformation Overall Transformation KineticsKinetics
Factors that Factors that determine f(t, T)determine f(t, T) Nucleation rateNucleation rate Growth rateGrowth rate Density & Density &
distribution of distribution of nucleation sitesnucleation sites
Overlap of diffusion Overlap of diffusion fieldsfields
Impingement of Impingement of adjacent adjacent transformed transformed volumesvolumes
f depends on nucleation rate and growth rate
f depends on number of nucleation sites and growth rate
Solid-State Transformation Solid-State Transformation KineticsKinetics
Many of the reactions of interest to Many of the reactions of interest to materials scientists involve materials scientists involve transformations in the solid statetransformations in the solid state..
e.g.e.g.
RecrystallizationRecrystallization of a cold of a cold worked materialworked material
PrecipitationPrecipitation of a crystalline of a crystalline polymer from an amorphous polymer from an amorphous phasephase
GrowthGrowth of an equilibrium of an equilibrium phase from a non-equilibrium phase from a non-equilibrium structurestructure
The driving forcedriving force is usually brought about by cooling from one temperature to another.Lets consider the initial phase to be and resulting phase to be .
The total volumetotal volume of the sample is: V = VV = V + V + V
Then the fraction fraction transformedtransformed can be written as:
V
VF
Assume that the Assume that the transformationtransformation of of to to is controlled by is controlled by nucleation and growthnucleation and growth
— — i.e. nucleation of i.e. nucleation of phase within phase within and and then growth of then growth of ..
Let,
eunit volumper rate nucleationN
) of form spherical
(assumingdirection onein rategrowth
dt
drG
The equation relating the The equation relating the fraction fraction transformedtransformed to to nucleation ratenucleation rate, , growth growth raterate, and , and timetime is given by: is given by:
43
3exp1 tNGF
Johnson-Mehl Johnson-Mehl equationequation
eunit volumper rate nucleationN) of form spherical
(assumingdirection onein rategrowth
dt
drG
A similar treatment of the A similar treatment of the subject is given by subject is given by AvramiAvrami..
In general, he expresses the fraction transformed as
nktF exp1
where n is called “ the Avrami nthe Avrami n”
nn may vary from 1-4
k k is equivalent to NG 3
3
nktF exp1
43
3exp1 tNGF
Johnson-Mehl Johnson-Mehl EquationEquation
Avrami Avrami EquationEquation
The variation of The variation of ‘n’‘n’ from 4 (as in Johnson-Mehl eq’n) can from 4 (as in Johnson-Mehl eq’n) can occur for a number of reasons.occur for a number of reasons.
In some solid-state reactions, the In some solid-state reactions, the nucleation rate is a decaying nucleation rate is a decaying function of timefunction of time. In that case the . In that case the Avrami n would be 4 early in the Avrami n would be 4 early in the reaction, but decreasing to 3 as the nucleation decreases as a reaction, but decreasing to 3 as the nucleation decreases as a function of timefunction of time, and the transformation is governed by the growth , and the transformation is governed by the growth rate.rate.
In general, for In general, for 3-dimensional solids3-dimensional solids, the Avrami n is , the Avrami n is between 3 and 4between 3 and 4..
In case of a growth of a phase in In case of a growth of a phase in 2-dimensions 2-dimensions such such as in a sheet or a film, the Avrami n is as in a sheet or a film, the Avrami n is between 2 and 3between 2 and 3..
In the case of wire, a In the case of wire, a 1-dimentional solid1-dimentional solid, the Avrami , the Avrami n is n is between 1 and 2between 1 and 2..
Determine the value of the Avrami n.Determine the value of the Avrami n.
nktF exp1
nktF exp1
nktF 1ln
tnkF
lnln1
1lnln
Thus the Avrami n is the slope of the plot of the ln ln 1/(1 – F) versus ln t
Overall Transformation Overall Transformation KineticsKinetics
TTTTTT Diagrams Diagrams
αα ββ αα’ ’ αα + + ββ Volume fraction ‘f’ varies Volume fraction ‘f’ varies
from 0 to 1from 0 to 1
nktf exp1
33
3
4
3
4vtrV
43
3exp1 tNvf
Value of n is numerical exponent that varies from ~1 to 4
Precipitation in Age-Precipitation in Age-Hardening AlloysHardening Alloys
Al-4Cu (1.7 Al-4Cu (1.7 at%) alloyat%) alloy αα-phase-phase θθ-phase-phase
Precipitation in Age-Precipitation in Age-Hardening Alloys – Hardening Alloys – Transition PhasesTransition Phases GP zonesGP zones
Fully coherentFully coherent Very low interfacial energyVery low interfacial energy Two atomic layers thickTwo atomic layers thick 10 nm diameter10 nm diameter
Precipitation in Age-Precipitation in Age-Hardening Alloys – Hardening Alloys – Transition Phases…Transition Phases…
GP zones are GP zones are formed as first formed as first ppt during low ppt during low temperature temperature aging of many aging of many technologically technologically important alloys.important alloys.
Precipitation in Age-Precipitation in Age-Hardening Alloys – Hardening Alloys – Transition Phases…Transition Phases…
Precipitation Precipitation processprocess θθ” are fully ” are fully
coherent plate-coherent plate-like pptslike ppts
Visible through Visible through coherency-strain coherency-strain fieldsfields
Orientation Orientation relationship with relationship with the matrixthe matrix
4'
3"
21 GPzoneso
Go G1 G2 G3 G4
001001 "
100100 "
Precipitation in Precipitation in Age-Hardening Age-Hardening AlloysAlloys
Activation Activation energy energy barrierbarrier
Precipitation in Age-Precipitation in Age-Hardening AlloysHardening Alloys
θθ” Tetragonal” Tetragonal
θθ’ Tetragonal’ Tetragonal Composition Composition
approx CuAlapprox CuAl22
θθ Body- Body-centered centered tetragonaltetragonal
Precipitation in Precipitation in Age-Hardening Age-Hardening AlloysAlloys Nucleation sitesNucleation sites
Precipitation in Age-Precipitation in Age-Hardening AlloysHardening Alloys Effect of aging temperature on the Effect of aging temperature on the
sequence of precipitatessequence of precipitates
Fastest Fastest transformation transformation rates are rates are associated withassociated with
highest highest nucleation rates nucleation rates
and therefore and therefore the finest ppt the finest ppt distributionsdistributions
Precipitate Free Zone Precipitate Free Zone (PFZ)(PFZ)
Precipitation HardeningPrecipitation Hardening
Under-agedUnder-aged Peak-agedPeak-aged Over-agedOver-aged
Spinodal DecompositionSpinodal Decomposition
Transformations having Transformations having no barrier to nucleationno barrier to nucleation
Phase diagram with a Phase diagram with a miscibility gapmiscibility gap
Temperature lowered Temperature lowered from Tfrom T11 to T to T22
Alloy will immediately Alloy will immediately become “become “unstable”unstable”
Small fluctuation in Small fluctuation in composition can produce A-composition can produce A-rich and B-rich regionsrich and B-rich regions
Up-hill diffusion takes placeUp-hill diffusion takes place
02
2
dX
GdFree-energy Free-energy has a negative has a negative curvaturecurvature
Spinodal DecompositionSpinodal Decomposition
For spinodal decomposition For spinodal decomposition the alloy must lie between the alloy must lie between the two points of inflectionthe two points of inflection
Locus of the points on the phase Locus of the points on the phase diagram is known as the diagram is known as the chemical spinodalchemical spinodal
For alloys outside spinodalFor alloys outside spinodal Small variation in composition Small variation in composition
will lead to an increase in free-will lead to an increase in free-energyenergy
Thus alloy is “Thus alloy is “metastablemetastable”” Nucleation & growth processNucleation & growth process Down-hill diffusion occursDown-hill diffusion occurs
02
2
dX
GdFree-energy Free-energy has a positive has a positive curvaturecurvature
Spinodal Decomposition vs Spinodal Decomposition vs N & GN & G
Particle CoarseningParticle Coarsening
Microstructure of a 2-Microstructure of a 2-phase alloy is not phase alloy is not completely stable completely stable unless the total unless the total interfacial free energy interfacial free energy is minimumis minimum
High density of fine High density of fine ppt will tend to ppt will tend to coarsen into a lower coarsen into a lower density of larger pptdensity of larger ppt Reduces overall Reduces overall
interfacial areainterfacial area
Gibbs-Thomson effectGibbs-Thomson effect
Total number of ppts Total number of ppts decreases and the decreases and the mean radius ‘mean radius ‘rr’ ’ increases with timeincreases with time
If If rr00 is the mean radius is the mean radius at at t=0t=0 then then
ktrr 30
3
eXDk Where,
Rate of ppt CoarseningRate of ppt Coarsening