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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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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
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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$ancing front.
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Transformation occurring by
nucleation and gro)th.
1=
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Thermodynamics of Transformations
• # phase transformation can occur spontaneously
only )hen the free energy change during thetransformation is negati$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
%*
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
*
( ) 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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*;