Description of Phase Transformation

(i) Equilibrium phase transformations – occurring at the transition

temperature ( e. g. freezing at Tm)

(ii) Non-equilibrium phase transformations

(freezing below Tm)

Equilibrium phase transitions:

Recall the Ehrenfest (“classical”) classification scheme.

An nth order transition is defined by the lowest- order derivative !!"

#$$%

&''

n

n

TG

which becomes discontinuous at the transition temperature.

Modern Scheme of classification:

Classifies transitions as either 1st or 2nd order

1st order transition has a latent heat

2nd order transition has no latent heat

For example consider two order – disorder transitions

G

TTc

G

TTc

Cu3Au (1st order) !-brass (2nd order)

Cp

T

!

Long range order parameter

H

T

H

T

Latent heat

W

T

1.0

Discontinuous change in W at Tc

Cp

T

“"” transition

W

T

1.0

Continuous changein W to Tc

Tc Tc

• 2nd order (continuous) transitions are studied as the “physics of critical phenomena”

Tid bits

• 1st order transition has infinite Cp at Tc

• Cp has a lambda point for 2nd order transition

Non – equilibrium Transformations:

Many important phase transformations do not occur at equilibrium.

(e.g. undercooling during solidification)

Since these transitions do not occur at Tc, we can’t use the previous classification

schemes.

Scheme for classification of non-equilibrium phase transformations:

There are two general mechanisms of non-equilibrium transformation.

(1) Nucleation and growth -

!

"

! nucleates & grows

* motion of a well- defined " / ! interface

* Criteria for 1st order transition

This process is large in degree and small in extent.

Cr

* No movement of a sharp interface – 2nd order non-equilibrium phase transformation

(2) Continuous transformation (e.g., spinodal decomposition)

Phase separation occurs by gradual amplification of composition variations.

Small in degree and large in extent.

Classical Theory of Nucleation

Phase transition requires formation of fluctuation or nucleus. !! !’

!’!

Homogeneous Nucleation

Barrier - interface

critical nucleus size

Foreign substances involved which

serve as nucleation sites.

Heterogeneous Nucleation

!

!/

!/

Heterogeneous Nucleation

Thermodynamics of Nucleation :

(i) Homogeneous Nucleation – (1925 Volmer-Weber)

Series of bimolecular reactions

211 !!! "+

312 !!! "+

ii !!! "+# 11

413 !!! "+

1*1*

+!+ ii """

I

(A)

Here, i* is the critically sized nucleus. Other methods of cluster formation such as simultaneous collisions are less probable.

Volmer’s kinetic analysis considered only the forward rate of reaction (A).

The back reacting rate was considered small and was ignored

Thermodynamic balance of reactions up to (A)

ii !! =1

Mixing of ni clusters of size i increases the entropy of the system ! some

cluster populations are always present.

Dilute solution theory for mixture of n1 ……. ni clusters:

The chemical potential of an i atom cluster :

!+=

ii

iii n

nRT ln0µµ " # per unit volume

in equilibrium : 1 ; i i ii n nµ µ= =

(*chemical potential per mole the same in all clusters ! likely an incorrect assumption!)

10 ln µµ i

nnRT

ii

ii =+

!

!"

#$%

& ''=

( RTi

nn i

ii

i 10

expµµ

Approximation : 1nni

i !" (i.e., # of clusters of i ! 1 is small)

01

1 1exp expi ii

i Gn n nRT kT

µ µ! "# $! "= # = #% & % &' (' (

where ( ) Avogadroii NiG /10 µµ !="

(a) A volume or bulk term describing the free energy change driving the transformation

(b) an interfacial energy term opposing the transformation.

biaiGi +=! 3/2

'!" VgA v#+=

areaassumed independent of i and positive

negativevolume of α’

These assumptions fit a central force nn bond model

each atom has Z bonds of energy !

! bulk energy of solid = -1/2 Z! per atom

The nucleation of "’ creates an interface which costs energy. The free energy of nucleus formation, #Gi, is generally composed of 2 terms :

for spherical clusters :

VGrrG !+=! 32

344 "#"

critical nucleus :

( ) 0=!Gdrd

( )3

* * *2 2 16;

3

VV

r G G rG G! !"= # $ = $ =

$ $

r

4!r2"

r*

#G#G*

#Gv 4/3!r3

The equilibrium conc. of critical nuclei

!!"

#$$%

& '(=kTGnn i

i

*

1* exp

What to use for !Gv?

Pure materials

(1) Gas ! Liquid Gas ! Solid 0

1 lnVPG RT

V P! =

Gas pressure

Equil. vap pressure

(2) Condensed phases

fS T! !

Recall: liq ! sol

1V fG S TV

! = ! !

Entropy of transformation

fs l

e

L TG G G

T!

! = " = " ; Te = equil. transition temp.

Thermodynamic Description of Phase Transitions

1. Component Solidification

m

L TGT!

! " #

T* Tm

GsolidG

Gliquid

s s sG H TS= !l l lG H TS= !

STHTGTGTG ls !"!="=! )()()(

T

@"T="TmGl =Gs

dGl = dGs ; !G = 0

m m

H LΔS = T T!

"

Where L is the enthalpy change of the transition or the heat of fusion (latent heat).

Alloys: !’ nucleus can have a composition different from !. e.g.,binary system, x is atom fraction of one element.

( ) ( )l sliquid x solid x!

A A B BG x xµ µ! = ! + !

; S L S LA A A B B Bµ µ µ µ µ µ! = " ! = "

(ii) Heterogeneous Nucleation –

S-substrate

!"#

!"S !#S

$

"-phase#

$ contact angle%s = !#S - !"S

!"##" cosγγγ SS +=

cos S

γ!"

#$ = %

Superhydrophobic-surface-– Lotus-leaf

No wetting

Perfect wetting! = 0

! < 90

! > 90

! = 180

s !"# $= %

0s! <

0s! >

s! "=

For " > #$% : No wetting ! foreign substrate does not help nucleation

If surface deforms !"#

!"S"#

!#S

$

S Sγ γ cos γ cos! "! "# $+ =

Sγ sin γ sin!" !# $=

%, $

Determination of free energy of formation (&GS) of a nucleus on a substrate:

Shape : assume spherical cap of radius r, contact angle %

%

% r

Volume of cap:

( )!" fr #3 34

( ) ( )( )212 cos 1 cos

4f ! ! != + "

θ

( ) ( ) ( )3

224 2 1 cos sin

3s V srG f G r r!"

# $ # $ % # $ &' = ' + ( +

Cap volume Cap area Contact circle area

!"S - !#S

( ) ( ) ( )( ) !"!" #$$%#$%$% coscos1 cos1 2

3 4 222

3

&&+&+'= rrGfrV

= 4π r3

3f θ( )ΔGV + γ αβπ r2 1− cosθ( ) 2− cosθ − cos2θ( ) ⎡

⎣⎤⎦

f θ( )

( ) ( )!"#!"$% frGfr

V 4 3 4 2

3

+&=

( )!"##$% frGrG Vs &&'

())*

++,=, 4

3 4 2

3

4 f θ( )

( )!fGGs

** "="VG

r!

= "#$2*

( ) 0;0 ;0 * =!== SGf "" !

( ) ** ;1 ;180 GGf S !=!== "" !

(no undercooling)

(no help from foreign substrate)

Concentration of nuclei

( )!"

#$%

& '(=

kTfGnn isis

)** exp

# of single atom sites on substrate