DRJ Phase Transformation Cp

8/18/2019 DRJ Phase Transformation Cp

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Thermodynamics of transformations

ntroduction!

• De"nition of phase change

•

#tom mo$ements in phasetransformation

• Types of phase transformations

•

Homogeneous $s. heterogeneoustransformations

• Thermodynamics of transformations

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De"nition of &hase 'hange

• # Phase is a

physically distinct(

chemically homogeneousand

mechanically separable partof the system.

e.g.( ce( )ater( $apor of )ater

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T)o phases are distinguishable from each other( if

• They form di+erent states ofaggregation ,solid( liuid and $apor

• /r in the same state of aggregation(if they ha$e di+erent composition ordi+erent crystal structure

• 0or the same composition and crystalstructure( di+erences in theelectronic structure


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# phase transformation or transition

is de"ned as

• The change from one or more phases ,calledthe parent phases to one or more otherphases ,called the product &hases.

• &hase transformation in$ol$es changes in the1. 2tate of aggregation

%. 'omposition

*. 'rystal structure( or

. 3lectronic structure

•. t may be also a combination of more thanone of the abo$e changes.

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#tom mo$ements in phase

transformtion

• 0or solid state phase transformation()e shall not consider changes in thestate of aggregation.

• No atom mo$ements ta4e placeduring changes in the electronicstructure.

• 'hanges in composition and crystalstructure in the solid state reuirethe mo$ement of atoms )ithin the

solid.


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Three categories of atom mo$ementsduring phase transformation

1. 6o$ements o$er a large number ofinteratomic distances7

%. 6o$ements o$er one or t)ointeratomic distances7 and

*. 6o$ements o$er a fraction of aninteratomic distance.

•. 8ong range and 2hort rangedi+usion ,1 9%

•. :y process of di+usion in solid state.;


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• Di+usion mass


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Di+usionless changes

• 0or category *( the atoms may mo$e onlythrough a fraction of an interatomicdistance.

• 2uch mo$ements bring about crystal

structure changes.• The product crystal structure can be

generated only )hen the atom mo$ementsoccur in a coordinated fashion. /ther)ise(an amorphous product )ill result.

• n the absence of interchange of atompositions by random )al4( the coordinated

transformations are said to be di+usionless. >


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Types of phasetransformation

1. Transformations )ith a change incomposition

%. Transformations )ith a change incrystal structure

*. Transformation )ith bothcomposition and crystal structurechanges

. Transformation )ith a change inorder

5. 3lectronic transitions 1?


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Transformations )ith a change in composition

Al- ZnPhase

diagram

T e m

p e r a t u r e (

@eight percent Ainc

#tomic percent Ainc 11


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1%


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Transformations )ith a change in crystal structure

1392 911

structures changes of Fe under cooling

cool cool

C C BCC BCC FCC Fe Fe Feδ γ α → →o o

1*


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Transformation )ith a change in

order

1


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3lectronic Transitions!

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Homogeneous $s. heterogeneous transformations

• # transformation that ta4es placemore or less simultaneously in allparts of an assembly is regarded as

homogeneous transformation.

• Reactions in the gaseous phases arehomogeneous.

• 'hanges in$ol$ing electronictransitions are homogeneous.

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• # heterogeneous transformation is of thenucleation and gro)th type.

•

Tiny $olumes of the product phase callednuclei( often assumed to be the same instructure and composition as thetransformation product( form "rst.

• # sharp boundary delineates the nucleifrom the surrounding matriB.

• These small regions subseuently gro) bythe out)ard mo$ement of the boundary()ith corresponding changes in composition,and crystal structure behind thead$ancing front.

1;


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Transformation occurring by

nucleation and gro)th.

1=


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Thermodynamics of Transformations

• # phase transformation can occur spontaneously

only )hen the free energy change during thetransformation is negati$e.

• n order to 4no) )hether this condition is satis"edfor a transformation( )e need to 4no) the free

energy of the parent and the product phases.• The free energies of elemental crystals and solid

solutions are discussed here as a function ofcomposition and the eBternal parameters!

temperature and pressure.• The thermodynamic order of transformations is

de"ned.

• t is sho)n ho) a dri$ing force arises for "rst-order

transformations )ith and )ithout compositionalchanges.1>


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0R33 3N3RC /0 38363NT#8 'R2T#82

•

n elements( the composition is not a$ariable.

• #ssume that the elemental crystals areperfect( so that there is no con"gurational

entropy associated )ith crystal imperfections.• The $ariables considered are the eBternal

$ariables( e.g.( temperature and pressure.

• The Cibbs free energy C of the crystal is

gi$en by

%?

(1)G H TS = −)here H and 2 are the enthalpy and the

entropy of the crystal and T is the absolutetem erature in 4el$in .


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The enthalpy term can be )ritten as

)here Ho is the residual enthalpy at ?

K and

cp is the heat capacity at constant

pressure.

The entropy of the crystal is gi$en by

%1

0

0 (2)

T

p H H c dT = + ∫

0

(3)

T

pc dT S

T = ∫


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'ombining 3.,1 through ,*( )eobtain at constant pressure

ntegrating 3., )ithin appropriate

limits( )e can )rite

%%

(4) P

G sT

∂ = − ÷∂

0 0

0

0

0

0 0

(5)

(6)

G T

H

T

T T p

dG SdT

G H SdT

c dT H dT

T

= −

= −

= − ÷

∫ ∫

∫ ∫ ∫


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•0igure 1,a is a schematic plot of the$ariation in the Cibbs free energy as afunction of temperature at constantpressure.

• The cur$e has Aero slope at ? K( the$alue at that temperature being theresidual enthalpy or bond energy Ho.

• The slope becomes more and morenegati$e( as the temperature is

increased.• The rate at )hich the slope decreases)ith temperature is dependent on the$alue of the heat capacity( )hich isusually a function of temperature.

•0rom the thermodynamic la)s( )e cansho) that

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2chematic $ariation of the

Cibbs free energy Cas afunction of ,atemperature at constantpressure( and ,b pressureat constant temperature.

(7)

T

GV

P

∂ = ÷

∂


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• Thus( the Cibbs free energy $ariation)ith pressure at a constanttemperature has a positi$e slope(eual to the molar $olume E( asillustrated schematically in 0ig. 1,b.

• The abo$e description of the freeenergy of elemental crystals alsoapplies to chemical compounds )ith

ideal stoichiometry( for )hich thecon"gurational entropy is Aero.

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0R33 3N3RC /0 2/8D 2/8FT/N2

• Here( the composition is a $ariable.

• 2o( the contribution to the free energy fromthe con"gurational entropy cannot beignored.

The Interaction Energy• The bond energy of a solid solution depends

on the type of bonds bet)een nearest-neighbour atoms in the crystal.

• @e shall ignore the energy contributions tothe bond energy from neBt-nearestneighbours.

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• The types of bonds present in abinary solid solution of # and :

atoms are• #-#( :-: and #- : bonds.

• 8et E##( E:: and E#: be the energies of

these bonds respecti$ely.• @hen pure # is dissol$ed in pure :(

some of the #-# and :-: bonds are

bro4en to create ne) #-: bonds!

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A A A A

B B B B

↔ ⇒ + ↔

b b


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• The interaction energy is de"nedas the energy change that occurs

)hen #-# and :-: bonds are bro4ento produce one #-: bond.

• #s illustrated abo$e( )hen one #-#

and one :-: bond are bro4en( t)o #-: bonds are produced.

• 2o( )e ha$e

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• f the interaction energy is negati$e(unli4e bonds ,#-: bonds are

energetically fa$oured.• f has a large negati$e $alue(

compound formation is fa$oured(

)here e$ery # atom is surrounded by: neighbours and $ice $ersa.

• 0or eBample( in the compound of

silica( E is negati$e and large inmagnitude!

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• f the interaction energy is positi$e(li4e bonds ,#-# and :-: bonds are

energetically fa$oured.• f it is a large positi$e $alue( the t)o

component atoms do not miB )ith

each other and immiscibility results.• n solid solutions( the interaction

energy is either Aero or has a small

positi$e or negati$e $alue.• f E is Aero( )e ha$e a perfectly

random ideal solution.

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• f is negati$e( the tendency )illbe for short range ordering in the

solid solution.• 2hort range ordering refers to

formation of small regions in the

solution( )here #-: bonds arepresent preferentially.

• f & is positi$e( the tendency )ill be

for clustering of li4e atoms in thesolid solution.

• 'lustering refers to the formation ofsmall regions in the solid solution()here #-# or :-: bonds are

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'lustering or 2hort-range /rder &arameter α

• 8et ,: be the probability that a : atom

)ill ha$e an # atom as the nearestneighbour on a speci"ed atomic site.

• n a truly random solid solution( there isno preference for any type /f bonds.

• 2o( )e ha$e here

,: G # ,1@here # is the mole fraction of # in the

solution.

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• @here the tendency is for short-range ordering( the : atom )ill prefer

to ha$e # neighbors.• 2o for this solution(

• @here there is tendency forclustering( the : atom )ill prefer :

neighbors( as :-: bonds lo)er theenergy. 2o( here

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( ) (2) A A B P X >

( ) (3)

A A B P X <


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• The local ordering ,or clusteringparameter α is de"ned as

• The number of #-: bonds( N#:( in one

mole of the solid solution is gi$en by

**

( )

( ) ( )1 1 (4) A B

A A B

A

P P X X

α α = − ∴ = −

( ) 0 (5)

AB B A B N P ZN X =


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@here

I is the coordination number ,number of

nearest-neighbours of an atomN? is #$ogadros number( and

: is the mole fraction of :.

Here gi$es the number of # atomson the I sites surrounding a : atom.

is the number of : atoms in one mole

of the solid solution.2ubstituting for 3. ( )e obtain

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( ) A B P Z

0 B N X

( ) A B P

( )0 1 (6) AB A B N ZN X X α = −


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0ree energy as a function of

temperature

• #t ? K( the bond energy H? )ill be the

sum of the energies of the threetypes of the bonds in the solidsolutions!

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DRJ Phase Transformation Cp