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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