Transformation in the Solid State

8/9/2019 Transformation in the Solid State

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Metals

V. Transformation in the solid state

J. McCord

(mostly from Gottstein)


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Pure metals allotropic modifications

Crystal structure of a metal is not necessarily stable at all temperatures

Crystal structure determined by the lowest Gibbs free energy GBinding energy E 0 in metals relatively constant

Differences in electronic structure responsible for instability in crystalstructure

Phase transitions

Allotropic modifications

Several phase transformations in the solid state possible

Example thermal expansion coefficient of iron

5.2


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Allotropic modifications

Fig. 5.1 Allotropic modifications of selected elements5.3


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Fig. 5.2 Cooling curve for pure iron(note F).

Fig. 5.3 Temperature dependence of the latticeconstant of iron (note the different scale on theleft and right side).

bcc

fcc

bcc

bcc

bcc

bcc

fcc

Similar temperature dependence for same structures Also phase transitions of different origin magnetic Fe transition

Polymorphism of Iron

5.4


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Diffusion controlled phase transformationsin alloys

Dissolution or precipitation of a secondphase on crossing of one phase boundary

+ (e.g. a' - b')

Transformation of a crystal structureinto another crystal structure of the samecomposition on crossing two phase

boundaries (e.g. c - d, c - d')

Decomposition of a phase into severalnew phases on crossing (up to three)phase boundaries

+ (e.g. e - f, e' - f') Eutectoid decomposition (e" - f")

Fig. 5.4 Phase diagram of a binary alloy AB with phasetransitions in the solid state.

L

L

5.5


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Thermodynamics of decompositionFormation of new phase is from a homogeneous phase

+

Two possibilities depending on

Decomposition has the same crystal structure as , but has a different composition

Precipitation has a crystal structure and a composition different from

Fig. 5.5 Phase diagram of a binary alloy AB with phasetransitions in the solid state.

5.6


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Quasi chemical model (mostly enthalpy H )

Gibbs free energy G = H - TS

Enthalpy H given by the binding enthalpies

H AA

, H BB

, H AB

between next neighbor

AA-, BB-, or AB-atoms

Total binding enthalpy H m = N AA H AA + N BB H BB + N AB H AB

N ij is the total number of bonds between atoms i and j (i = A,B and j = A,B ) N z = total number of atoms coordination number

Exchange energy H 0 < 0 decrease in energy with AB

Exchange energy H 0 > 0 increase in energy with AB

Ideal solution with H 0 = 0

( ) ( )

( ) ( )

2 2

0

11 2 1

2

1 1 2 12

m AA AB BB

AA BB

H N z c H c c H c H

N z c H cH c c H

= + +

= + + [ ]0

12AB AA BB

H H H H = +

5.7


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Mixing entropyS

m

Entropy S = S v + S m S m

Vibrational entropy S v (vibrational modes =independent of atomic arrangement)

Possible arrangements m

Mixing entropy

S m > 0Symmetric with regard to c = 0.5

Approaches c = 0 and c = 1 with infinite slope

( )ln 1 ln(1

!...

n

)

l

! !

m

m m

m A B

S Nk c c

N N

c c

k

N

S

= +

=

= Fig 5.6 Entropy of mixing for ideal solution. Notethat S approaches for c 0 and c 1 with +/-infinite slope. As a result, it is impossible toproduce perfectly pure crystals.

/ m S c

A c B

E n t r o p y o

f m

i x i n g

S m

5.8


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Free energy of solid solution

Fig 5.7 Concentration dependence of free energy of a solidsolution for different heat of fusion H 0.

( ) ( ) ( ) ( )0 ln 1 ln 11 1 2 12 AA BB m G NkT c c c Nz c H cH c c H c + + = + +

( ) ( ) ( )( )11 2 12 1

g m m m c c

G G c G c G c c c

= + +

Rule of common tangent

1 2

0dG dG dc dc

=

1 2

0dG dG

dc dc

= =

5.9


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Phase diagram with a miscibility gap

Fig 5.8 Theoretical phase diagram of a solid solution with amiscibility gap.

Tc

( ) 01 exp zH

c T kT

Depending on the behavior of the solidus temperature

Eutectic phase diagram (shown)Peritectic phase diagram

Eutectic phase diagram

c 1(T )

5.10

solubility limit


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The Au-Ni phasediagramliquid

solid solution

weight% Ni

t e m p e r a

t u r e

( C )

Au-rich solid solution +Ni-rich solid solution

Fig 5.9 Phase diagram of AuNi.

High solidus temperature

Phase diagram of asystem with limitedsolubility

Miscibility gap

Characteristic s

Common G (c ) curve forboth components

5.11


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System with two different phases and

Fig 5.10 (a) Composition dependence of the free energy in a system with two differentphases and and (b) NbZr phase diagram

weight% Zr

t e m p e r a

t u r e

( C )

atomic% Zr

Two phases and with different crystal structures Separate G (c ) curves for each phase G (c ) and G (c )

Homogeneous or solid solution for c < c 1 or c > c 2 Phase mixture of and for c 1 < c < c 2

(a)

(b)

5.12

c1 c2


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Nucleation and spinodal decompositionProcess of decomposition

by a nucleation process Formation of nucleus is with equilibrium concentration of new phase

by spontaneous or spinodal decomposition Equilibrium composition attained in the course of time Up-hill diffusion

Fig 5.11 Connection ofdecomposition of a solutionwith its free energy. Thecurve's points of inflectionare at c w and c w.

Decomposition wouldlead to an increase offree energy

Decomposition leadsto a gain of energy(spontaneousdecomposition)

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Fig 5.14 Schematic G( x ) curve for a fixedtemperature for the discussion.

in more detail2

2 0d G dc

2

2 0d G dc

Spinodal decomposition

Nucleation and growth

5.14

2

2 0d G dc

2

2 0d G dc


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Nucleation vs. spinodal decomposition

Distance

Distance

Fig 5.12 Schematic diagram of the change in concentration and dimensionsduring decomposition of a solution by (a) nucleation and growth and (b)spinodal decomposition.

(a)

(b)

Nucleus + diffusion

Nucleationandgrowth

Spinodaldecomposition

Continuous process

Equal terminal state

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Concentrationprofiles f (t )

Developing of periodic wave-likepattern

Wave length of spinodallydecomposed structures is small!

Fine lamellae composites

(a)

(b)

(c)

8 min

15 min

23 min

Fig 5.13 Numerically calculated concentrationprofiles for spinodally decomposed Al-37%Znaged at 100C for (a) 8 min., (b) 15 min., and(c) 23 min. The dashed lines indicate theequilibrium concentration of decomposition

5.16


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Fig 5.14 Schematic phase diagram and freeenergy curve of a solid solution when amiscibility gap occurs. G( T a) = free energy ataging temperature T a .

Ta

G(Ta)

Range of spinodaldecomposition

2

2 0d G dc

2

2 0d G dc

( )0

12

AA BB

w w

H H

kT c c

zH

=

=

Spinodal decomposition

Nucleation and growth

Spinodal curve c (T)

5.17

Coherent spinodal line in reality deeperdue to elastic energy contributions.

spinodal line

cw


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Fig 5.15 Schematic phase diagram and freeenergy curve of a solid solution when a

miscibility gap occurs. G( T a) = free energy ataging temperature T (same as 5.14). Thetypical decomposition procedure isillustrated. The sample is quenched from thesolid solution range ( ) to a certaintemperature T below the solubility limit intothe two-phase region ( + ).

Ta

G(Ta)

Range of

nucleation andgrowth

5.18

Region between the spinodal

curve and the solubility limit onboth sides

Undercooling T

Occurrence of precipitation by

1. Nucleation2. Growth

Range ofnucleation

and growth

T


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Decompositionprocedure

T

Fig 5.16 Dependence of thefree energy of a sphericalnucleus on its radius.

( ) 3 24 4( )3t el

g r r G r

= +

0 ,c c G

r G r

=

3

2

16

3( )c t el G

g =

5.19

G c

G ~ r 2

~ r3

rc

2c t el

r g

=

Change of free energy G due to the formationof a spherical nucleus with radius r

specific interfacial energy of the phaseboundary

g t free energy gain per unit volume duringphase transformation

el distortion energy per unit volume or elasticenergy density

Critical radius


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Elastic distortion energy of a precipitate

sphere

needle

disc

s h a p e

f a c

t o r

aspect ratio c/b

low el

high el

Fig 5.17 Elastic shapefactor for a rotationellipsoid with aspectratio c/b.

Hard precipitate in a soft matrix

Concentrations c and c - composition of matrix and precipitate

Young's modulus E and Poisson ratio of the phase

Atomic size factor (distortion depending on difference of lattice parameter with c)

Form factor

2

( )1el E c

c c b

=

5.20


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Structure and shape of phase boundaries

Coherent phase boundaries (low - high el )

Lattice planes of the matrix continue through the precipitateElastic distortions

Semi-coherent phase boundaries (low - lowering of el)Significant difference in lattice parameterInterfacial edge dislocations in the interface compensate the elasticdistortions

Most lattice planes of the matrix continued through the

Fig 5.18 Structure of (a) coherent, (b) semi-coherent, and (c) incoherent phase boundaries.

(a) (b) (c)

5.21


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Structure and shape of phase boundaries

Incoherent phase boundary (large - no el)

Crystal structure of both phases is different

Orientations of matrix and precipitate are different

Different crystal structureIncoherent interphase boundaryLarge interfacial energy hinders nucleation - large r c

Fig 5.19 Structure of phase boundaries (a) coherent, (b) semi-coherent, and (c) incoherent.

(a) (b) (c)

5.22


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Nucleation and growththrough metastablestates

Nucleation rate

with

Increasing nucleation rate by

lowering G c

1 2 3 4 5 6

G 1 > G 2 > G 3 > G 4 > G 5 > G 6

Formation of thermodynamicallyunstable metastable states

Guinier-Preston Zones GP-I, GP-II

-phases

3c if G

exp c G N kT

solid-solution ( )

concentration

f r e e e n e r g y

Fig 5.20 Schematic diagram of the free energy change forsuccessive metastable phases. 5.23


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Existence of GP zones

Limited temperature range for theexistence of coherent GP Zones

Limited to lower temperatures

Elimination of zones through thermal

activation Number of Guinier-Preston zones

decreases with rising temperatures

Above a critical temperature onlyincoherent phases appear

Fig 5.21 Temperature dependence of the equilibriumdensity N of Guinier-Preston zones (existence curve). Dueto insufficient diffusion, the thermodynamic equilibriumcannot be reached at low temperatures.

Density N of GP zones

5.24


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Age hardening in Al-Cu

Age hardening process for strengthening of alloys Age hardened aluminum alloys

Precipitation of a second phase during annealing of a supersaturated solidsolution

Fig 5.22 (a) Al-Cu phase diagram and (b) detail of the Al-Cu phasediagram (Al-rich side) indicating how the annealing takes place forage hardening.

(a)(b)

5.25

L


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H a r d n e s s

H b

Age hardening

curves Characteristic plateaus Increase in hardness

Formation of coherent phasesGP-I - single Cu layers on {100} Allattice planesGP-II or several Cu layers on{100} Al lattice planes

Further increase in hardnessFormation of partially coherentAlCu-phase ( ) Formation of incoherent Al

2Cu

( ) not happening

Decrease in hardness(increase in precipitation size)

time (days)

Fig 5.23 (a) Age hardening curvesof Al-4%Cu-1%Mg. (b) Sectionthrough a GP-I zone in AlCu.

(a)

(b)

GP-I & II

5.26


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Phases in Al-Cu

Fig 5.24 Crystal lattice of thesolid solution (a) , theequilibrium phase (b) , aswell as the metastable phases ' (c) and (d) " of the Al-Cusystem (Cu , Al )

(a) (b)

(c) (d)

equilibrium phase

FCC Al phase

GP-II,

5.27


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Annealing for age hardening of Al-Cu

Fig 5.25 Annealing sequence for age hardening with corresponding change in

microstructure for Al-Cu.5.28

a

b

c (see. Fig. 5.22)


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Time-temperature-transformation - TTT diagram

TTT diagram forAlCu

Supersaturated solid solution

Fig 5.26 Sketch of a TTTdiagram. The time t required to reach aspecified state of precipi-

tation (e.g., a givenvolume fraction) is deter-mined in an isothermalexperiment. In order toprevent precipitation thecooling rate for quenching

has to be large enough toavoid an intersection of theT(t) curve and the kneedescribing the onset ofprecipitation.

5.29


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Growth kinetics of precipitates

Short times (X < 0.2)

r : precipitate radius (assumption of spherical particle)X : precipitate volume fraction

Long times (X > 0.9)

Neighboring precipitates compete for remaining solute atoms

3

B r D t

X(t ) r

3 2X( t ) t

1 2e t / X ( t ) =

Fig 5.27 Schematic diagram of the change in concentration and dimensionsduring decomposition of a solution by nucleation and growth.5.30


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Growth kinetics of

precipitates

Fig 5.28 (a) Sketch of volume fraction of precipitates as function of time at constant temperature. Note thatnucleation gives rise to an incubation time. (b) Growth kinetics of C in -Fe. Solid line - precise theory;dashed line - dependence according to the shown equation fro X (t ) (c k concentration in the precipitate).

1 2e

1 2e

t /

t /

X ( t )

X

=

=

(a)

(b)

incubationtime &

nucleation

3 2X(t ) t

1 2e t / X ( t ) =

growth

Ostwaldripening

5.31

( )1 301 3 2

31 /

B B B /

K

D c c D

c R

=


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Ostwald ripening Competitive" precipitate coarsening Driving force is a decrease of the total interface boundary energy

Flux of atoms from the higher to lower chemical potential (from small

to large particles) Spherical particles : determined by the curvature of the surface

Average particle size

1 2

1 12 B

B

c kT

r r c

= =

Fig 5.29 The contribution of the interfacial energy leads to a chemical potential difference betweenparticles of different size. This causes a solute diffusion current from small to large particles.

diffusion

Radi r 1 and r 2 b equilibrium concentration for a flat

interface boundary (r infinity) atomic volume

30

23 B if B

B D c

r r t T

D t k

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Eutectoid decomposition and discontinuous

precipitation Precipitation processes by nucleation and growth of individual nuclei

Alternatively Solid state phase transformation proceeds by motion of a reaction front

Eutectoid decomposition and discontinuous precipitation1. One phase decomposes into two different phases with different composition

2. Morphology of the phase transformation is a lamellar structure

3. Concurrent precipitation occurs by exchange of atoms via short-rangediffusion

Example

Pearlite reaction in steel

Decomposition of C-rich fcc - -Fe in low-carbon -Fe and cementite (Fe 3C)

5.33


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Lamellar microstructure Decomposition of Fe-0.8%C-X

Microstructure is determined by transformation rate

Lamellar structure ferrite ( -Fe) and cementite (Fe 3C)

Fig. 5.30 Microstructure of eutectoidally decomposed Fe-0.78%C-0.97%Mn. (a) Cooledslowly (solidification at 680C), (b) cooled rapidly (solidification at 639C) and (c) incipientforming of pearlite with advancing carbide lamella in Fe-0.8%C (iron carbide lamella are

light colored, ferrite lamella dark colored).

slow cooling rate fast cooling rate

(a) (b) (c)

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Mechanism of lamella formation

Fig. 5.31 Illustration of the principle of

lamella generation during eutectoiddecomposition.

new -crystal nucleus

Reaction initiated at a grain boundary

Nucleation favored

Assumption: Decomposition of +

phase forms the first nucleus

Local change of concentration

nucleus is generatedLocal change of concentration

Lamellar microstructure

Thickness of the lamellae is determined bythe range of diffusion

Lamellar spacing l decreases with increasingtransformation rate R 1l

R 5.35


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Schematics of laminar growth (Fe-C)

Fig. 5.32 Lamella arrangement and carbon concentration distribution in growthdirection during the pearlite reaction. A depletion or enrichment of C occurs at the

moving phase boundary, leading to a transverse diffusion.

(bcc)

(fcc)

t r a n s v e r s e

d i f f u s i o n

5.36


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Discontinuous precipitation

Fig. 5.33 Reaction front duringdiscontinuous precipitation in Al-2.8%Ag-1%Ca.

Reaction at the transformation front + Only one new phase is formed (precipitation of )

Nucleation starts at grain boundaries

Discontinuous precipitation Generation of lamellar microstructure

Grain boundary migration (similar to recystallization)

Grow through concurrent grain boundarydisplacement in a lamellar morphology

Chemical driving force

Increase by (pre) plastic deformation

Analogous reverse process - dissolution ofprecipitates by moving grain boundaries during the

recrystallization of two-phases5.37


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Martensitic transformation

Increasing cooling rates

Diffusivity slows down

For high cooling rates diffusion fails to accomplish the necessary

concentration changes Suppression of phase transformation

Increase of supercooling of unstable phase

Driving forces for transformation increase drastically. Phase transformation with change of crystal structure (Fe-C from fcc to bcc)

Spontaneous change of crystal structure without change of concentration

Spontaneous phase transformations without concentration change

Martensitic transformations

Despite the name it is not only occurring in the Fe-C system, when the formationof ferrite is suppressed.

5.38


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Martensitic transformations

Structural change Metal or alloy

fcc hcp Co, Fe-Mn

fcc bcc Fe-Ni

fcc tetragonal bcc Fe-C, Fe-(Mn, Cr, Ni,...)-C

fcc tetragonal fcc In-Tl, Mn-Cu

bcc hcp Li, Ti, Zr, many Ti and Zr alloys

bcc distorted hcp Cu-Al

bcc tetragonal fcc Cu-Zn, Cu-Sn

bcc orthorhombic Au-Cd

tetragonal orthorhombic U-Cr

5.39

Table 5.1 Important

martensitic transformations


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Mixed microstructureSpontaneous change of the crystal structure Change of volume and shape Suppressed change of composition

Transformation proceeds by successive displacive transformations of needleor lath shaped regions into the new crystal structure

Consecutively transformed regions are limited in their spatial extent by priortransformed regions

Fig 5.34 Martensite andretained austenite in anFeNiAl-alloy

5.40

austenitization-temperature 850C


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Existence range of

martensite Volume fraction of the martensite

phase depends on the temperatureto which the material was quenched

Not dependent on time

Ms - martensite start1% martensitic volume fraction in

the microstructure Mf - martensite finish

99% martensitic volume fraction

Range of existence of martensitedepends on composition Fig. 5.35 Existence curve of martensite in Fe-0.45%C.

Ac3 is the temperature of beginning transformation toaustenite during heating. In practice M s , and M f are

determined by martensite contents of 1% and 99%,respectively.

austenitization temperature 850C

t e m p e r a

t u r e

( C )

martensite content (%)

5.41


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Composition range

Fig. 5.36 Concentration dependence of the range ofexistence of martensite in FeNi.

Ms - martensite start As austenite start

Ms decreases with increasingconcentration

Fe-C Decomposition on annealing into

ferrite and cementite Not retransforming to austenite

By plastic deformation the differencebetween the transformation starttemperatures M s and A s can bereduced (M d and A d).Martensitic transformationaccompanied by deformation

(calculated)

5.42


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Bain model of martensitic formation

Fig. 5.37 The Bain-model of martensitic formation in the FeC system and the correspondingorientation correlation of austenite and martensite after Kurdjumov-Sachs. The open circles indicatepossible positions of carbon.

Crystallographic relationships observed during martensitic transformation in thesystem Fe-C - Bain model

Transformation from fcc to bcc bcc unit cell tetragonally distorted in the presence of carbon

a) Center of two next neighbor fcc unit cells (lattice parameter a 0) contains a

tetragonal body-centered unit cell (bct)

(a) (b) (c)

5.43


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Bain model of martensitic formation

Fig. 5.38 The Bain-model of martensitic formation in the FeC system and the correspondingorientation correlation of austenite and martensite after Kurdjumov-Sachs. The open circles indicatepossible positions of carbon.

b) Obtaining a bcc unit cell from a bct unit cell by compression along the c directionand stretching in the plane perpendicular to the c direction (volume change ofabout 3 to 5%)During martensite transformation the atomic positions change only slightly andnext neighbors remain next neighbors after phase transformation.

c) Bain correspondence is substantiated by X-ray crystallography investigations that

confirm an orientation relationship Fe-C {111} || {110} and ||

(a) (b) (c)

5.44


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Complicated irrationallyindexed Habit planes

(not described by simpleBain model)

Fig. 5.39 Stereographic representation of the Kurdjumov-Sachs orientation relationship. The stereogram is centeredon (111) fcc || (011) bcc . The unmarked neighboring pairs ofpoles would superpose exactly for the Bain orientation. Theydo not do so for the Kurdjumov-Sachs orientation.

{111} || {110}

||

5.45

Kurdjumov-Sachsorientation relationship

L i


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Martensitic transformation of Fe-C -lattice transformation from fcc to bct

C stabilizes phase C austenite is located on the

octahedral interstitial sites of the ironlattice

After transformation the carbon

atoms are located only on the c axis lattice parameter in the c-directionincreases, i.e. tetragonal martensite(bct) is formed

Fig. 5.40 Dependence of the lattice parameters of austenite and martensite on carbon content

Lattice parameters

0

surfaceinterior

austenite

tetragonal

martensite

weight % C

a x i s r a

t i o c

/ a

a a x

i s l e n g

t h ( 1 0 - 1 0 m

)

a o r c a x

i s l e n g

t h ( 1 0 - 1 0 m

)

5.46


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Mechanism of martensite transformation

Fig. 5.41 (a) Illustration of a martensitic transformation fcc hcp by shear deformation, i.e., stackingsequence changes from ABAB... to ABCABC... . (b) Shearing within a matrix. (c) Internal plasticdeformation of the transformed phase to keep the original shape. (d) Martensite crystal a M betweentwo grain boundaries.

{ { }111 0001fcc hcp =Shear deformation (diffusionless transformation)

Simplest case Co: fcc hcp

Invariant plane (Habit plane)

(a)(b) (c) (d)

5.47

ConstrainedtransformationUnconstrained

transformation


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Shape change

Shape changes lead to elastic compatibility strains in the immediateenvironment of the martensite plates

Reduced by plastic deformation within the martensite via glide and twinning

Overall deformation corresponds to a shear parallel to an undistorted plane Undistorted plane {111}fcc = (0001) hex during the cobalt transformation

Habit plane irrational for Fe-C

Fig. 5.42 The formation of martensite changes the shape of a the original crystal (a) by shear(b) slip (c) or twinning (d) in martensite can largely compensate for this shape change.

(a) (b) (c) (d)

5.48

f


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Plastic deformation during martensitic

transformation

Fig. 5.43 (a) Formation of a characteristic surface texture by martensitic transformation.(b) Plate martensite in Fe-33.2%Ni with internal twinning.

Martensitic transformation causes a substantial plastic deformation Compression and stretching of the unit cell by shear deformation

Shape change of the transformed region

Surface relief

(a)

(b)

surface

5.49

TTT diagram and


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TTT diagram andmartensitictransformation

Fig. 5.41 TTT-diagram of an alloy steel (Fe-C-Cr) with cooling curves. Different cooling rates yieldmartensite (1), pearlite (2) or bainite (3). Only for very low cooling rates occurs the transformation atequilibrium temperature (F - ferrite; P - pearlite; M, - martensite start)

Special material properties areachieved by martensitictransformation

Not isothermal!

Cooling rate must be high enough

to avoid the pearlite or bainitereaction.

1. Martensite

2. Pearlite

3. Bainite time (sec)

t e m p e r a

t u r e

( C )

bainite

perlite-start

end of trans.

martensitic start

5.50

austenization temperature 970C


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Shape memory effect

Characteristics Regaining original after deformation by a respective heat treatment

Requirements1. Martensitic transformation with M s As (very little hysteresis)

2. Deformation only by martensitic transformation and twinning, i.e.,suppression of dislocation slip

Alloys: AuCd, Fe 3Pt, NiAl, NiTi (Nitiniol)

Cooling through the transformation temperature M f Crystallographically equivalent variants are adjusted in such a way that the

shape of the specimen remains unchanged

After deformation - material heated to temperature A f Martensite disappears and original shape is restored

5.51


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Shape memory effect

singlecrystal

-phaseheat

treatment

quenched inwater

(T < M f)reaction to

appliedstress

reversiblestrain limit

reheatinginto the -zone

martensiteand suitable

variants

stress inducedgrowth ofvariants austeniteformation and

shaperecovery

martensitesinglevariantstate

Fig. 5.44 Schematic illustration of the shape-memory-effect works in a single crystal.

5.52


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Pseudo- or superelasticity

Similar effect to shape memory effect (same materials)

Introduction of martensitic phase by applied mechanical stress

Large strain

Good damping properties - energy of mechanical impact consumed by themartensitic transformation

Fig. 5.45 Typical stress strain curve of a superelastic material (in comparison to steel).

strain

stress(arb. units)

austenite to martensitetransformation

5.53

Comparison (a)


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Comparison

shape-memory effectand superelasticity

Fig. 5.46 (a) (c) Schematic illustration of themechanism of the shape-memory effect andsuperelasticity, in which solid lines represent theshape-memory path and dotted lines representthe superelasticity path. Martensite forms upon

cooling below M f . However, the macroscopicshape of the sample does not change becausedifferent crystallographic equivalent forms occurwith the same frequency. Application of anexternal stress induces the growth of thecrystallographic variant that causes the largest

shear deformation in the direction of the appliedstress. At high stress only this variant survives.Heating above Af gives rise to a transformationinto the original phase and hence to shaperecovery.

(a)

(b) (c)

5.54

Transformation in the Solid State

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Transcript of Transformation in the Solid State