Chapter 9 Solid State Phase Transformation

8/10/2019 Chapter 9 Solid State Phase Transformation

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Chapter 9 Solid state phase transformation

9.1 Pure metals

Solid crystal structure with lowest Gibbs free energy

(metals) Eo (binding energy) depends little on atomic arrangement

eg. Na bcc changes to hexagonal at 36 K , heat of fusion = Eo/1000

bonding depends on electronic structure.Allotropy - change in crystal structure.

If the same crystal structure exist in different temperature regimes some physical properties are continuousbetween two regimes. The thermal expansion coefficient of iron fig. 9.1.

9.2 Alloys

9.2.1 Diffusion controlled phase transformation

9.2.1.1 General classificationFrom fig. 9.2 there are several reactions concerning solidstate phase transformation.

1. Dissolution or precipitation of a second phase on

crossing of one phase boundary,

+ (a-b, a-b)

2. Transformation of a crystal structure into another crystal

structure of the same composition on crossing two phase

boundaries: (c-d, c-d)

3. Decomposition of a phase into several new phases on

crossing 3 phase boundaries:

+ (e-f, e-f)

A special case is eutectiod decomposition (e-f)


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At a given temperature and pressure:

G = H - TS

H the enthalpy binding energies between nextneighbor atoms.

HAA, HBB , HABHm the total binding enthalpy or the enthalpy of mixing

Hm = NAAHAA + NBBHBB + NAB HAB

Nij total number of bonds between atom i and j

(i = A, B : j = A, B)

- Hij or Hm is positive

-The enthalpy in this model is not the gain of enthalpy

( H < 0 when gain enthalpy) of an atom during its

transition from the free to the bonded state.-It is the difference between the bonded state and areference state with absolute minimum enthalpy.

-A small value of H refers to a more strongly bonded statethan a large enthalpy.

If N the total number of atoms

z - the number of next neighbor atoms (coordination no)

c - the concentration of B-atomsN.c = no. of B atoms

z.c = each B-atom has B-atom as next neighbors

Atomic concentration of B-atom

NBB = Nzc2

means all B-atom in B-B bonds are counted twice.

9.2.1.2 Thermodynamics of decomposition

Consider: + or is formed from homogeneous

phase

This reaction can be1. has the same crystal structure as but with different

composition (decomposition)

2. has a crystal structure and composition different from. This is the case of precipitation.


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The same as NAA and NBB

Hm = N.z[(1-c)HAA+cHBB+2c(1-c)H0]

H0 = HAB (HAA + HBB)

H0 - exchange energy: H < 0 gained,

H > 0 lost

If AB bonds are generated from AA and BB-bond

H0 - 0 ideal solution binding enthalpy does not

depend on the arrangement of atoms.

S Sv + Sm(the vibration entropy) (the entropy of mixing)

Sv - nearly independent of the atomic arrangement

S Sm 9.4

Sm = k ln m 9.5a

m - the number of possible distinctly different

arrangement

Sm = -Nk [c ln c + (1-c) ln (1-c)] 9.6

Sm - always positive

c = 0.5 symmetric

c = 0 , c =1 infinite slope means

To obtain pure elements from alloys is very difficult

c 0, dG infinitely (large)

From eq. 9.3 and 9.6; the free energy of a regularsolution

Gm (Gibbs free energy of mixing) =. 9.7

Fig. 9.3 shows enthalpy, entropy and the Gibbs free energy

There are 2 different cases1. H0 < 0, Gm (c) has a single minimum2. H0 > 0, Gm (c) has 2 minima (at sufficiently low temp)

at c1 and c2

- The entropy of mixing approaches infinity when A and Bbecome pure elements.

- The change of enthalpy remains finite for all compositions.

- The free energy will always decrease for smallconcentrations.for large concentration: H > TSfor small concentration: H < TS


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- In the range c1 < c < c2 the system can reduce its free

energy if it decomposes into the mixture of 2 phases

with composition c1 and c2- The free energy of the mixture the common tangent

that intersects the G (c) curve at c1 and c2- The points of tangency are usually not the minima of

the Gm (c) curve.

- If the tangent is a horizontal line Gm (c1) = Gm (c2) ,

the points of tangency are minima.

-If H0 < 0 , - the atoms of each element favor unlike

partners as next neighbors

- the long-range ordering is preferred over

decomposition.

- For c < c1

and c > c2

;

- the solid solution has always a lower free

energy than a phase mixture.

c1 and c2 - solubility limits of the solid solution

c1(T) and c2 (T) - solvus line in binary phase system

A-B

- For the most simple case; HAA = HBBGm is symmetry c = 0.5

c1

and c2

(c2

= 1-c1) minima

The common tangent intersect Gm (c) at minima andthe solubility limit c1 (T) differential eq. 9.7.

d Gm/dc |c=c1 = 0 9.9

c1 (T) exp (- zH0/kT) 9.10

From eq. 9.10 composition of phase depends ontemperature

An example of the phase diagrams are shown in fig. 9.4

and 9.5. which corresponds to the terminal solubility of

solid solution in phase diagram

Fig. 9.4 - the maximum of the curve is not attained

because the alloy becomes liquid at a lower temperature.

depending on the behavior solidus temperature

(monotonic or with a minimum)

(or a peritectic or eutectic)


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If the two phases and have different crystal

structures, there is no common curve Gm (c).

There are separate curves for each phase G (c) and

G (c) that intersect at a certain concentration.

- Since only the phase or phase mixture with the

smallest free energy is thermodynamically stable,

decomposition occurs.

a homogeneous c c1, c c2a phase mixture of and for c1 < c < c2

c1 and c2 points of intersection of the G(c)-curves

with a common tangent fig. 9.6a.

9.2.1.3 Nucleation and spinodal decomposition

The process of decomposition can occur either by

-A nucleation process: a nucleus is formed withequilibrium concentration c2- Spontaneous decomposition (spinodal decomposition)

Fig. 9.7

-Alloys composition c1 decomposes into c1 and c1

The free energy of phase mixture GE tie line betweenG (c1) and G (c1)

GE (c1) G1E > G(c1) G1Such a decomposition is not stable

The system would return to the stable of ahomogeneous solid solution.


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-Alloys composition c3 near the maximum of the

G (c) curve.

decomposition

can proceed spontaneously (free energy gain)

spinodal decomposition

Spinodal decomposition

- diffusion flux runs from low to high concentrations in

contrast to normal diffusion.

- negative diffusion coefficient, owing to a negative

thermodynamic factor.

Fig. 9.8 shows the difference between spinodaldecomposition and regular nucleation.

Nucleation

- a nucleus of is formed by thermal fluctuation with

correct concentration c- grows by diffusion while the composition remains

unchanged (still c )

- the size of nucleus has to exceed the critical size for

stable growth.

- nucleus is preceded by an incubation time

Spinodal decomposition

- proceeds spontaneously without incubation time

- decomposition is amplified until equilibrium is attained


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Nucleation and decomposition

-The terminal states are the same but intermediatemorphologies of the phase mixture are totally different.

Differences

During spinodal decomposition a periodic wave-like

pattern develop during the early stage.the wave length of spinodal decomposed structures

is usually small ( ~ 50 A in Al-37Zn at 100o C, fig. 9.9)

Whereas during nucleation the shape of theprecipitated phase is determined by interfacial energyand elastic energy.

Decomposition is possible if the curve has a concavecurvature d2G/dc2 0

d2G/dc2 = 0

The point of inflection in fig. 9.7, c and c and canbe calculated from second derivative of eq. 9.7.

The dependency c (T) defines the spinodal curve.

For the simple case HAA = HBBc . ( 1 - c ) = kT/ 2zH0 9.11

Fig. 9.10

In the 2 phase regime: there are 2 regions.

1. The center spinodal decomposition can occur.

2. The region between the spinodal curve and thesolubility limits on both sides precipitation can occur bynucleation and nucleus growth.

In solid: the nucleation of precipitates depends on- The differences of volumes of mother phase and the

precipitates.- The difference has to be compensated by elastic

distortion with the elastic energy (Eel).- The elastic energy increases with increasing nucleus

volume V

Eel = el .V

el distortion energy per unit volume = elastic energydensity


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If a spherical nucleus with radius r is formed, the freeenergy increase.

G (r) = (-g +el ) . 4/3 r3 + 4r2 9.12

r the specific energy of the - phase boundary

g free energy gain per unit volume during phasetransformation. (=Gliq-Gsolid /V)

The maximum of the curve G (r) the critical size of

nucleus (r0)

r0 = 2 / (gu - el ) 9.13

The energy of the interface boundary and the elasticdistortion energy play a major role since the small

changes of r0 result in large changes of the nucleationrate.

The elastic distortion energy (per unit volume) for ahard precipitate in a soft matrix can be calculated.

el = {(E 2)/(1-)} (c c )2 . (c/b) 9.14

c , c - concentrations

E

- Youngs modulus of

- Poisson ratio of

- d (ln a)/dc = the atomic size factor

(a, c lattice parameter)

- distortion

- the form factor

If the precipitate has elliptic shape,

c ellipsoidal radius parallel to the rotation axis

b the respective perpendicular ellipsoidal radius.

The relation between shape factor and c/b shows in fig.9.11

The precipitates:

Sphere distortion energy is the largest

- has the smallest surface for a given volume

- the shape of a precipitate is a compromise

between a minimum of distortion energy and

total surface energy.

1) If the precipitate and matrix have the same latticeparameter, ( ~ 0. eq. 9.14) as in the system Al-Ag:

- the elastic energy play a negligible role and the

precipitates have the spherical shape.2) If a precipitate and matrix have a large different crystallattice parameter, >> 0.

- plate-shaped precipitates are preferred as in the systemAl-Cu.

3) If the interfacial energy is anisotropic: different interfacialplane have different energy.

- the area of the interface and its specific energy plays a

role, the plate-shaped precipitates are preferred.- in system Al-Ag

In the beginning, the spherical precipitates, semicoherentinterface with large energy anisotropy, are formed.

During annealing, the precipitates become plate-shaped.


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9.2.1.4 Metastable phases

The nucleation rate ~ exp (-G/RT) 9.15

G0 = G(n) = (16/3) 3 /(gu - el )2

- strongly depends on interface energy - depends on structure

There are 3 types of interface, coherent, partiallycoherent and incoherent, fig. 9.12.

Coherent precipitates

- all lattice planes of matrix continue through the

precipitate- there will be slight elastic distortions because of

differences of the lattice parameter in precipitate andmatrix.

Partially coherent interface

- the difference in lattice parameter between precipitatesand matrix is too large

- the edge dislocations are formed at the interface tocompensate the elastic distortions

- most lattice planes of the matrix are continued through theprecipitate but a few terminate in the interface and create the

interfacial edge dislocations.

Incoherent interface

- crystal structure of precipitate and matrix are different

- large interfacial energy the work of nucleation is high,strongly delays nucleation especially at low temperatureswhere diffusion proceeds slowly.

In this case metastable phases are formed because thegain free energy is not as large as for the equilibrium phase

( phase, fig. 9.13)

metastable phase The gain free energy is small

coherent and partial coherent the energy of interfaceboundary is low

high nucleation rate


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9.2.1.5 Age hardening

The principle of age hardening

precipitate the second phase during annealing of asupersaturated solid solution.

Al-Cu, fig. 9.15

- max solubility of Cu in Al is 5.65% at 548oC

- T < 300oC the solubility decreases to less than 1%

- Composition used for age hardening in fig. 9.15 b

at high temp

+ at low temp - intermediate phase Al2Cu

- tetragonal structure

fcc- single phase, quenched supersaturated

heat increase strength, fig. 9.16

- A metastable phase is thermodynamically unstable, itsfree energy is higher than that of the equilibrium phase

- Frequently, several metastable phases are

successively formed until the equilibrium phase isattained. Consecutive metastable phases have acontinuously lower free energy, fig 9.13.

- The first phases formed are coherent phases (ordecomposition zones) with a size of a few atomiclayers called Guinier-Prestion zone

- Guinier-Prestion zone decreases with increasingtemperature until above a critical temperature, furtherdecomposition zones stop and incoherent phasesappear, fig. 9.14.

- At low temperatures, diffusion effectively stops anddecomposition cannot take place.There is a certain temperature range in which these

metastable phases exist. This is the temperaturerange used for the thermal treatment of agehardened alloys


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Fig. 9.16.

At low annealing temperatures (100o)

- hardness increases slowly but continuously until aplateau is reached.

At slightly higher temperatures

- another rise of hardness is observed after the plateau.

- the second hardness approach maximum

At even higher temperatures

- the plateau at the maximum will be reached earlier butthe value of the plateau hardness decreases.

At 300oC

- the plateau is not obtained anymore, only the secondhardness occurs.

During heat treatment: the following phases may occur.- GP I zones are single layers of copper atoms on {100}

aluminium lattice planes.

- GP II zones ( phase) are comprised of several parallelcopper layers on {100} planes cause atetragonal distortion of the second of the matrix fig.9.17 and 9.18.

- intermediate phase

(CaF2

structure, coherent with the matrix on {100}planes

- the equilibrium phase


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The first hardening step

The plateau

- the formation of GP I and GP II zones

- GP zones are decomposition products. The number ofGP zones decreases with rising temperature.

- the hardness of the plateau decreases at higher

temperature.

The second hardening stage is obtained only at highertemperatures

- artificial aging

- caused by the formation of the phase. Due toprecipitate coarsening at longer annealing times. Theprecipitate spacing increases and the strength decreases,the precipitate circumvented by dislocation.

Overaging

- the equilibrium phase


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9.2.1.6 Growth kinetics of precipitates

The growth kinetics of nucleus are diffusion controlled

In binary alloy AB

c0 - the initial concentration

B the average concentration of the matrix,

changes with time

jB - flux of B atomr0 - precipitate size

4/3 r3 . d B/dt = jB (r0) . 4 r02 9.16

R the average matrix volume per precipitate, a sphere

For steady state growth of the precipitates: the

concentration profile in front of the particle remainsunchanged stationary diffusion eq.

cB (r) = c0 (c0 cB) . (r0 /r) 9.17

cB is the equilibrium concentration at r = r0From Ficks law

jB (r0) = - DB (c0 - cB) / r0 9.18

For the conservation of the no. of B atoms

{Conc change in matrix = conc of precipitate}

4/3 r3 (c0 - B ) = 4/3 ck r03 9.19

ck is the concentration in the precipitate

The precipitated volume fraction is

9.20

For a short timeX(t) = (2t/3)3/2 9.21

After a long time

X(t) = 1 2exp( t/)

- the time constant, determined by the diffusionconstant

1/ = 3 DB (c0 cB)1/2 / ck1/3. R2 9.23

For a long annealing times

r0 ~ DB . t

a ~ DB . t

X ~ r03

X ~ t3/2

Fig. 9.21 shows good agreement with experiments,(elastic distortion was ignored). In some cases, lessagreement, the growth kinetics can be controlled bymechanism other than the atomic transport. (atoms Bmoving across phase boundary)

particleofconcc

matrixofconcc

cc

cctX

B

B

__

__)(

0

0

'

0

0

r0

R


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For spherical particles

Chemical potential is determined by the curvature of thesurface.

The chemical potential difference between 2 particles withdifferent r1 and r2

= 2 (1/r1 1/r2 ) = kT (cB/B) 9.24

B - equilibrium concentration of the matrix for a flat

interface boundary (r )

- interface boundary energy

- atomic volume

The chemical potential difference is related to aconcentration gradient.

c = c(r1) c(r2) fig. 9.22 which cause diffusionflux from the small to the large particle. The large particlegrows while the small particle dissolves. The averageparticle size,

3 - 03 ~ DB t 9.25

Fig. 9.22

- This model, elastic distortions were not accounted for.

- If the precipitates are plates or needles, the shapefactor needed to be considered.

- r ~ t1/3 is observed. For each edge of a cuboidal particlealthough with different kinetic coefficients (DB) fordifferent spatial directions.

-During aging: max hardness and overaging are due toOstwald ripening.

During annealing

- small precipitates are dissolved.- large precipitates continue to grow Ostwald ripening

The driving force decrease of the total interface

boundary energy.

The total energy smallest, small no. of big particles

instead of the large no. of small

particles.

- the precipitate is to dissolve or to grow depends on the

chemical potential of atoms in the vicinity of a precipitate.- there is always a flux of atoms from the higher to lowerchemical potential.

- In equilibrium, the chemical potential is constanteverywhere.


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9.2.1.7 Eutectoid decomposition and discontinuousprecipitate

Solid state phase transformation proceeds by motion ofreaction front

Eutectoid: +

and - phase transformation proceed by motion of a

reaction front.

- occur simultaneously

- different compositions, possibly different

crystal structure

- lamellar structure, atoms diffuse in a short

range

- i.e. + Fe3C in steel

Eutectiod, fig. 9.24

- initiate at grain boundary

- if phase forms first and enrich in B, therefore, itsimmediate environment is enriched in A (but deplete inB) nucleus is generated.

- lamellar structure formed.

- lamellar spacing decreases with increasingtransformation rate, fig. 9.23.


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Discontinuous precipitation

+

-Nucleation begins at grain boundary

- precipitation is not homogeneous in the grain interior. Itoccurs at a few preferred locations, mostly at grainboundaries discontinuous precipitation.

- connect to grain boundary migration- diffusion proceeds faster in a grain boundary. Behindthe moving grain boundary the new phase isprecipitated

- plastic deformation can further increase the grainboundary migration in addition to the chemical drivingforce for phase transformation.

9.2.2 Martensitic transformations

- the spontaneous phase transformations withoutconcentration change.

- in Fe-C system (ferrite is suppressed)

- If the spontaneous change of the crystal structure isassociated with a volume change or change of shape(which may be further complicated by the suppressedchange of composition, transformation proceeds bysuccessive displacive transformation of) needle or lath

shape (regions into the new crystal structure) will beformed.

- Consecutively transformed regions will be limited intheir spatial extent by prior transformed regions.

- The volume fraction of martensite phase usually doesnot depend on time but only on the temperature to whichthe material was quenched.

-Volume fraction of martensite increases with decreasingtemperature, fig. 9.27.

-Ms (1% martensite volume fraction)

-Mf (99% martentsite volume fraction)

-Ms Mf depends on composition, fig. 9.28

-Ms decreases with increasing concentraion.

-On heating, martensite does not transform at Ms into fccaustenite but superheating to As (austenite start).

As > Ms and As depends on composition, fig. 9.28

- In special case of Fe-C, martensite usuallydecomposes on annealing into ferrite and cementiteinstead of retransforming to austenite.


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Martensite transformation in Fe-C system was given byBain, fig 9.29.

- pure iron: fcc bcc

- but with carbon: fcc tetragonal

- a0 lattice parameter of fcc

- the center of 2 next neighbor fcc unit cell contains atetragonal body-centered unit cell (bct) with latticeparameter

a = a0/2 and c = a0- to obtain bcc, a bct has to be compressed in c direction

and stretched in the plane perpendicular to c direction, bywhich distortion the volume will be change by about 3-5%.

-This martensite transformation model, the atomicpositions do not change much and the next neighbors

remain next neighbors after phase transformation.- Fe-C system: martensite transformation fcc bct

- c/a ratio in martensite increases with increasing carbonconcentration whereas the magnitude of the a-axisindependent of carbon content (or decreases slightly), fig.9.30

-In fcc- phase a0 increases with increasing carboncontent. C in fcc located the octahedral interstitial sites ofiron lattice (which are the center of the cube or on the

center of its edges).-- After transformation, C are located only on c-axis. Catoms are larger than the octahedral interstitial sites. Thisleads to a slight increase of the lattice parameter of fcc inc direction.

-By plastic deformation the difference between Ms andAs can be reduced.

-Assume T0 the thermodynamic equilibrium temp forboth phase (same free energy).

T0 = (Ms + As)/2


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-Because of all C atoms are in c direction of martensitec direction increases. Tetragonal martensite is formed.

-During tempering of martensite, C atom is allowed todiffuse precipitate ; bcc martensite or ferrite is obtained.

-Solubility of C in is much larger than in

bct martensite is a strongly supersaturated solid

solution.

Tetragonal distortion reduces the dislocation mobility

(increase Peierls stress)

-The martensite transformation causes a substantialplastic deformation besides the change of lattice structure.

- the compression and stretching of unit cell sheardeformation a shape change of the transformed regionand form relief on the surface, fig 9.31.

- the shape changes cause large elastic strain to theimmediate environment of the martensite which can bereduced by plastic deformation within the martensite viaglide and twinning, fig 9.31 and 9.32.

Chapter 9 Solid State Phase Transformation

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