SO7 Diff Transform 2

29
Diffusional Diffusional Transformations Transformations - 2 - 2 SOLID STATE SOLID STATE

description

phase transformation

Transcript of SO7 Diff Transform 2

Page 1: SO7 Diff Transform 2

Diffusional Diffusional Transformations Transformations

- 2- 2SOLID STATESOLID STATE

Page 2: SO7 Diff Transform 2

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

Page 3: SO7 Diff Transform 2

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

Page 4: SO7 Diff Transform 2

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

Page 5: SO7 Diff Transform 2

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

Page 6: SO7 Diff Transform 2

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

Page 7: SO7 Diff Transform 2

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

Page 8: SO7 Diff Transform 2

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

Page 9: SO7 Diff Transform 2

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..

Page 10: SO7 Diff Transform 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

Page 11: SO7 Diff Transform 2

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

Page 12: SO7 Diff Transform 2

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

Page 13: SO7 Diff Transform 2
Page 14: SO7 Diff Transform 2

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

Page 15: SO7 Diff Transform 2

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.

Page 16: SO7 Diff Transform 2

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 "

Page 17: SO7 Diff Transform 2

Precipitation in Precipitation in Age-Hardening Age-Hardening AlloysAlloys

Activation Activation energy energy barrierbarrier

Page 18: SO7 Diff Transform 2

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

Page 19: SO7 Diff Transform 2
Page 20: SO7 Diff Transform 2

Precipitation in Precipitation in Age-Hardening Age-Hardening AlloysAlloys Nucleation sitesNucleation sites

Page 21: SO7 Diff Transform 2

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

Page 22: SO7 Diff Transform 2

Precipitate Free Zone Precipitate Free Zone (PFZ)(PFZ)

Page 23: SO7 Diff Transform 2

Precipitation HardeningPrecipitation Hardening

Under-agedUnder-aged Peak-agedPeak-aged Over-agedOver-aged

Page 24: SO7 Diff Transform 2

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

Page 25: SO7 Diff Transform 2

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

Page 26: SO7 Diff Transform 2

Spinodal Decomposition vs Spinodal Decomposition vs N & GN & G

Page 27: SO7 Diff Transform 2

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

Page 28: SO7 Diff Transform 2

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,

Page 29: SO7 Diff Transform 2

Rate of ppt CoarseningRate of ppt Coarsening