Phase transitions

• Gas-liquid, liquid-solid, liquid-liquid etc.• Polymer solution-gel• Glass-crystal• Separation of liquids• Nematic-smectic liquid crystal transition• Self assembly• Denaturation of proteins

Origin of a phase transition: (at least one)- order parameter1st order p.t. order parameter changes

discontinuously on a continuous change of interaction parameter

2nd order continuous change

A phase transition involves a decreaseof the free energy:

Helmholtz’s free energy F:F=U-TS

Gibb’s free energy G:G=H-TS

internal energy

entropy

enthalpy

↑

↑

← Constant T and V

Constant T and P

The regular solution model Very simple model - used in a variety ofproblems in physics

A B A+B⇔

FA FB FA+B+

Free energy of mixing

!

Fmix

= FA + B

" (FA

+ FB)

Volume fraction φ

!

"A =VA #moleculesVsystem

!

"A

+ "B

=1Incompressible liquid

Liquid-liquid unmixing regular solution modelAssumptions:

• Molecules occupy lattice sites (z nearestneighbours)

• Probability that site is occupied with A or Bmolecules is independent on what occupies theneighbours-Mean field assumption

• Energy is pairwise additive - molecular interactiononly between nearest neighbours

!

"AA, "

BB, "

AB

Entropy of mixing (per lattice site)

!

S = "kB pi ln pii

#

In the mixed liquid there are only two states for each site.The probabilities are φA and φB.

!

Smix

= SA +B " (SA + S

B) = "k

B(#

Aln#

A+ #

Bln#

B)

SA and SB are naturally =0

Energy of mixing (per lattice site)

!

Umix

=1

2z"

A

2#AA

+ z"B

2#BB

+ 2z"A"B#AB( ) $

1

2z"

A#AA

+ z"B#BB( ) =

=z

2"A

2 $"A( )#AA + "

B

2 $"B( )#BB + 2"

A"B#AB[ ]

Writing:

!

" =z

2kBT2#

AB$#

AA$#

BB( )

we get:

!

Umix

= "#A#B

Interaction parameter↑

Free energy of mixing (per lattice site, in units of kBT)

!

Fmix

kBT

=Umix"SmixT

kBT

= #Aln#

A+ #

Bln#

B+ $#

A#B

χ<2 min@φ=0.5

χ>2 max@φ=0.5

• In (a) the initial composition isstable. Any phase separationleads to increase of F.

• In (b) compositions between φ1and φ2 will lower their F byseparating into thesecompositions.

• Compositions joined by acommon tangent minimise F.These are called coexistingcompositions.

• The locus of these compositions(when χ is changed) is called thecoexisting curve or binodal.

• φa is stable to smallfluctuations, even if notglobally stable. φa is ametastable composition.

• φb is unstable to smallfluctuations. Phaseseparates immediately.

A composition is metastable whenThe locus of this composition asχ is varied is called the spinodal.

!

" 2F

"# 2> 0

We can now construct aphase diagram

There is a critical temperature,Tc (or χc) between temperatureswhere all compositions arestable and T where there existcompositions which will phaseseparate. This is where thespinodal meets the binodal.

Defined by:

!

" 3F

"# 3= 0

For polymer-solvent mixinguse Flory-Huggins theory

• Asymmetric free energycurve.

• Restrictions due to theconnectivity of thepolymer.

• Internally inconsistent• See Hamley,

“Introduction to softmatter”

Examples of phase diagram

Examples arefrom Jones

Kinetics of phase separation

Unstable Metastable

Spinodal decomposition Homogeneous

nucleationHeterogeneousnucleation

Kinetics of phase separation

• Any fluctuation amplifiescontinuously.

• Material flow from regionsof low concentration toregions of high in contrastto normal diffusion.

Spinodaldecomposition low

conc.highconc. ⇐

In equilibrium the chemical potentialhas to be uniform.

highµ

lowµ ⇐

!

µ"#F

#$

!

" 2F

"# 2> 0$

"µ

"#> 0

!

" 2F

"# 2< 0$

"µ

"#< 0

In metastable region: Corresponding tonormal diffusion

In unstable region:

Uphill diffusion

Characteristic lengthscale of fluctuation

Too long wave length isslow due to diffusion overlong distances

Too short wave length yieldsa high cost in surface energy

Quantative analysis: Cahn-Hilliard

Kinetics of phase separation -metastable compounds

• Metastable compounds undergoes thetransition through an activated process callednucleation.

• Compounds are stable to small fluctuationsbut are not in global minima.

• A drop of the coexisting compound must becreated by thermal fluctuations even if thisincreases the free energy.

• This drop, or nucleus, must thereafter growuntil the free energy change is negative.

Kinetics of phase separation -metastable compounds

• The activation is due to the creation of aninterface with a characteristic surface tension,γ associated with a surface energy.

• There is both a positive and a negativecontribution to F

!

"F(r) =4

3#r3"F

v+ 4#r2$

Energy change perunit volume oncomplete separation

Surface energydue to interface

Homogeneous nucleation

Kinetics of phase separation -metastable compounds

!

r = r *

!

"

"r#F = 0

!

r* = "2#

$Fv

$F(r*) = $F* =

=16%# 3

3$Fv

2

Homogeneous nucleation

Kinetics of phase separation -metastable compounds

!

P = exp("#F * /kBT)

Probability of forming a nucleus which can grow is:

Homogeneous nucleation

Example is from Jones. Gibbs free energy of liquid-solid transition.same principles apply

typical values gives significantnucleation growth ~10 K below TmUsually we see crystallisation just Below Tm Why?

Kinetics of phase separation -metastable compounds

!

"F * is decreased by the presence of an interface.Container edge, dust particle (airplanes…) etc.

heterogeneous nucleation

!

"Fhe

#= "F

ho

# (1$ cos%)2(2 + cos%)

4