Self-Consistent Field Theory of Block Copolymers An-Chang Shi McMaster University Hamilton, Ontario, Canada [email protected]

Self-Consistent Field Theory of Self-Consistent Field Theory of

Block CopolymersBlock Copolymers

An-Chang ShiAn-Chang Shi

McMaster UniversityMcMaster UniversityHamilton, Ontario, CanadaHamilton, Ontario, Canada

[email protected]@mcmaster.ca

OutlineOutlineOutlineOutline

Introduction

Why block copolymers self-assemble

Self-Consistent Field Theory

Theoretical framework, derivations, etc

Mean-Field Approximation - SCMFT

SCMFT equations, methods of solution

Gaussian Fluctuations

Stability and kinetic pathways

Nucleation of OOT

ReferencesReferencesReferencesReferences

Doi and Edwards, The Theory of Polymer Dynamics

Toshihiro Kawakatsu, Statistical Physics of Polymers

Schmid, J. Phys.: Condens. Matter 10, 8105 (1998)

Matsen, J. Phys.: Condens. Matter 14, R21 (2002)

Fredrickson, Ganesan and Drolet, Macromolecules 35, 16, (2002)

Shi, in Developments in Block Copolymer Science and Technology, Edited by Hamley (2004)

More references are found in these books and papers

Diblock Copolymers: Complex Phase Diblock Copolymers: Complex Phase BehaviorBehaviorDiblock Copolymers: Complex Phase Diblock Copolymers: Complex Phase BehaviorBehavior

Mesoscopic separation of diblock copolymers

L SCG

Complex structures and phase diagrams

Experiments: Hashimoto, Thomas, Lodge, Bates, ...

Mean-Field theory: Helfand, Whitmore, Matsen and Schick, ...

Fluctuations: Laradji, Shi, Noolandi, Desai, Wang, ...

NA NB

Simple Model System: Diblock Simple Model System: Diblock CopolymersCopolymersSimple Model System: Diblock Simple Model System: Diblock CopolymersCopolymers

Degree of polymerization: N=NA+NB

Entropy: S ~ N-1

Composition: f=NA/N

Segment-segment interaction: AB=(z/2kT)(2AB-AA-BB)

Enthalpy: H ~

Current understanding is based on three parameters

NA NB

Stability of thermodynamic phases

Stable phase: global minimum

Metastable phases: local minima

Unstable phases: local maxima and/or saddle points

Sign of second-order derivatives: fluctuations

Stable, Metastable and Unstable phasesStable, Metastable and Unstable phases Stable, Metastable and Unstable phasesStable, Metastable and Unstable phases

Free energymetastable

stable

unstable

Order parameters

unstable

Self-Consistent Field Theory of PolymersSelf-Consistent Field Theory of Polymers Self-Consistent Field Theory of PolymersSelf-Consistent Field Theory of Polymers

Functional integral approach

(r)

Many-body interaction Fluctuating field

Simple theoretical framework Chain statistics and polymer density (r) determined by (r)

Mean field (r) determined self-consistently by (r)

Flexible framework, applies to many systems

Self-Consistent Field Theory: Simple Self-Consistent Field Theory: Simple ExampleExampleSelf-Consistent Field Theory: Simple Self-Consistent Field Theory: Simple ExampleExample

Monatomic Fluids in Canonical Ensemble A collection of n particles in a volume V

Pairwise interaction potential

Tk

rrrB

n

ii

1,)(ˆ

1

})ˆ({})ˆ({

2

!!1

2

VnV

Tmk

pn

erdn

zepdrd

n

zZ

n

i B

i

)(ˆ|)(|)(ˆ2

1|)(|

2

1})({ rrrVrrdrdrrVrV

jiji

The partition function can be written as,


i

rrrrdeDD

)(ˆ)()(][ˆ,1ˆ][

)(ˆ)()()(})({

)(ˆ)()(})({

})({

})ˆ({

][][!

][][!

ˆ][!

ˆ][!

rrrdrrrdVn

rrrrdVn

Vn

Vn

erdeDDn

z

erdeDDn

z

rdeDn

z

erdDn

zZ

Using the identities,

The partition function can be written in the form,


n

i

ri

rrrdVn

rn

ii

rrrdVn

rrrrdrrrdV

n

rrrdrrrdVn

i

n

ii

n

ii

erdeDDn

z

erdeDDn

z

erdeDDn

z

erdeDDn

zZ

1

)()()(})({

)(

1

)()(})({

)()()()(})({

)(ˆ)()()(})({

][][!

][][!

][][!

][][!

1

1

Using the definition,


n

iirrr

1

)(ˆ


],[

)()(})({

1

)()()(})({

][][

][][!

][][!

F

nrrrdVn

n

i

ri

rrrdVn

eDD

QeDDn

z

erdeDDn

zZ i

Using the definition,


)(1][ rerd

VQ

constQnrrrdVF ][ln)()(})({],[ where the free energy functional is,

Field theory model corresponds to the particle model


)(1][ rerd

VQ

Transformation from particle based theory to field based theory

constQnrrrdVF ][ln)()(})({],[

Partition function of a single particle in a potential

General theoretical framework for many systems

],[})ˆ({

2 ][][!

11

2

FV

Tmk

p

eDDepdrdn

Z

n

i B

i

Self-Consistent Field TheorySelf-Consistent Field TheorySelf-Consistent Field TheorySelf-Consistent Field Theory

)2()1()0(

)0(

)0(

],[

)r()r()r(

)r()r()r(

FFFF

Phase behavior described by free energy functional ],[ F

Fluctuations in an ordered state

energy nfluctuatioGaussian

equationsMeanfield

energyfreeMeanfield

)2(

)1(

)0(

0

F

F

F

Self-Consistent Field Theory: Mean-Field Self-Consistent Field Theory: Mean-Field ApproximationApproximationSelf-Consistent Field Theory: Mean-Field Self-Consistent Field Theory: Mean-Field ApproximationApproximation

],[],[ )0()0()0(

][][ FF eeDDZ

Saddle-point approximation:

Ignore higher-order terms leads to,

0)1( F

)(

])[ln()(0

)(

)(

}))({()(0

)(

r

Qnr

r

F

r

Vr

r

F

Conditions for saddle-point are:

Self-consistent mean field equations

Self-Consistent Field Theory: Mean-Field Self-Consistent Field Theory: Mean-Field ApproximationApproximationSelf-Consistent Field Theory: Mean-Field Self-Consistent Field Theory: Mean-Field ApproximationApproximation

Using the relation:

)(

][)(

)(

}))({()(

reVQ

nr

r

Vr

Conditions for saddle-point are:

For a given potential, these equations are solved self-consistently

)(1][ rerd

VQ


Standard Model or Edwards Model Chain statistics modeled by Gaussian chains

Interactions modeled by Flory-Huggins Parameters

Hard-core interaction modeled by incompressibility condition

}ˆ{exp)(1)}({)(!

VrRPDRn

zZ

c

nc

Weiner Measure

Flory-Huggins Interaction

Incompressibility


Standard Model or Edwards Model Chain statistics modeled by Gaussian chains

}ˆ{exp)(1)}({)(!

VrRPDRn

zZ

c

nc

f

ii ds

sRdds

NbAsRp

0

2

20

)(

2

3exp))((

cn

iB

BiA

Ai

Bi

Aii fRfRsRpsRpsRP

1000 )]()([))(())(()})(({


Standard Model or Edwards Model Flory-Huggins monomer-monomer interaction.

}ˆ{exp)(1)}({)(!

WrRPDRn

zZ

c

nc

cn

i

f

i sRrdsN

r1 00

)()(ˆ

)(ˆ)(ˆ})ˆ({})ˆ({ 0 rrdr

Tk

VW BA

B

Model of short range interactions


Standard Model or Edwards Model Hard-core interaction approximated by incompressibility

}ˆ{exp)(1)}({)(!

WrRPDRn

zZ

c

nc

rBA

BA

rr

rr

)(ˆ)(ˆ1

1)(ˆ)(ˆ


Using the identity,

)(ˆ)()(expˆ1 rrrdrDDDi

The partition function can be written as,

Free energy functional

.})({ln)()()()(}){},({3

0

cBA

g QVrrrrNrdN

RF

,1)()(}{}{ }){},({

F

rBA errDDZ


Single Chain Partition function in a field

Propagator representation

Definition of propagator (Green Function)

f

sRds

c esRPsRDV

Q 0

))((

0 ))((})({1

})({

)|,()|,(1

})({ 3221321 rfrQrfrQrdrdrdV

Q BBAAc

rsR

rR

sRds

sRd

Nbds

f

esRDrsrQ

)(

')0(

))(()(

2

3

0

2

2

)()'|,(

Self-Consistent Field TheorySelf-Consistent Field Theory Self-Consistent Field TheorySelf-Consistent Field Theory

Chain statistics specified by Q(r,t|r0)

Probability of finding t-th monomer at r, given the end at r0

)()|0,(

)|,()()|,(6

)|,(

00

002

2

0

rrrrQ

rtrQrrtrQb

rtrQt

p

pppp

p

(0,r0)

(t,r)

End-integrated propagators

They are solutions of the modified diffusion equation with,

,)|,()|,(),(

,)'|,('),(

rfrQrsrQrdrdsrq

rsrQrdsrq

).,()|,()0,(

,1)0,(

frqrfrQrdrq

rq


Free Energy Expansion

)()()(

)()()()0(

)0(

rrr

rrr

),0,|,(),|,(),|,(),|,(

)()()()|,()'|,(

11112211

1111

)0(

rsrGsrsrGsrsrGsrsrG

rrdsdsrdrdrsrQrsrQ

nnnnnn

nnnn

n

).'|,()()'|,()'|,( )0()0()0(22)0( rsrQrrsrQrsrQs

)|,()(),|,( )0( rssrQsssrsrG


Free Energy Expansion

111

)(,,1

,,

)0(

)2()1()0(

,)()(),,()(1

11

1nnn

nn

nc

cccc

rrrrCrdrdV

Q

QQQQ

nn

n

Correlation functions: ),,( 1)(

,,1 nn rrC

n

1 ,,11,,1

)0( ,)()(),,(!

)(lnln

1

11n

nnn

n

cc

n

nnrrrrCrdrd

nQVQV

Cumulant Correlation functions: ),,( 1,,1 nrrCn


Cumulant Correlation Functions

.),()|,()|,(),(1

),(

),,()|,(),(1

),()|,(),(1

),(

,),(),(1

)(

0 0

111)0(

0 0)0(

0 0)0(

0)0(

f f

c

f s

c

f s

c

f

c

srqrsfrQrsrQsfrqrdsddsQ

rrC

srqrssrQsfrqsddsQ

srqrssrQsfrqsddsQ

rrC

sfrqsrqdsQ

rC

Computed in the zeroth-order solutions


)2()1()0( FFFF

Free energy expansion

.})({ln)()()()( )0()0()0()0()0()0(3

0)0(

cBA

g QVrrrrNrdN

RF

,)()()()()()( )0()0()0(3

0)1(

rrCrrrrNrd

N

RF g

.)()(),(2

1)()()()(

30)2(

rrrrCrdrdrrrrNrd

N

RF BA

g

.)()(),,(!

)(})({

,,11,,1

30)(

1

11

n

nn nnn

ngn rrrrCrdrd

nN

RF

Self-Consistent Field Theory: Mean-field Self-Consistent Field Theory: Mean-field approximationapproximationSelf-Consistent Field Theory: Mean-field Self-Consistent Field Theory: Mean-field approximationapproximation

Motivation and justification

The functional integral can be evaluated by the largest integrand

}),({ln)()()()(}){},({~

}),{},({~1

}){},({

cBA QVrrrrNrdF

Fh

F

Control parameter:

0/ 2/130 N

g NRNh

}){},({~1

0}){},({~1 )0()0(

][][

F

hhFh eeDDZ

Self-Consistent Mean Field Theory: SCMFTSelf-Consistent Mean Field Theory: SCMFTSelf-Consistent Mean Field Theory: SCMFTSelf-Consistent Mean Field Theory: SCMFT

Technically saddle points determined by F(1)=0, leading to:

Coupled nonlinear equations

Lagrangian multiplier to ensure:

),()()(

,),(),(1

)()(

)0()0()0(

0)0(

)0(

rrNr

sfrqsrqdsQ

rCrf

c

1)()( )0()0( rr BA

Propagators are solutions of:

),()(),(),( )0(22 srqrsrqsrqs

),(1)0(

AAc frqrdV

Q

Single chain partition function is:

),()0,(,1)0,( frqrqrq

Self-Consistent Mean Field Theory (SCMFT)Self-Consistent Mean Field Theory (SCMFT)Self-Consistent Mean Field Theory (SCMFT)Self-Consistent Mean Field Theory (SCMFT)

Mean Field Free Energy Density (per chain)

Within SCMFT, the phase behavior is controlled by the parameters N, f and

}).({ln)()()()(1 )0()0()0()0()0()0()0(

30

)0(

cBAg

QrrrrNrdV

FVR

Nf

SCMFT is controlled by the parameters, N, f and N characterizes the degree of segregation

f and characterize chain structure

Constant shifts in do not change the solution

Self-Consistent Mean Field Theory: SCMFTSelf-Consistent Mean Field Theory: SCMFTSelf-Consistent Mean Field Theory: SCMFTSelf-Consistent Mean Field Theory: SCMFT

Five coupled nonlinear equations for five variables

.1)()(

),()()(

,),(),(1

)(0

rr

rfrNr

sfrqsrqdsQ

r

BA

f

c

),()(),(),( )0(22 srqrsrqsrqs

),(1)0(

AAc frqrdV

Q

),()0,(,1)0,( frqrqrq

}).({ln)()()()(1)0(

30

cBA

g

QrrrrNrdV

FVR

N

Self-Consistent Mean Field Theory (SCMFT)Self-Consistent Mean Field Theory (SCMFT)Self-Consistent Mean Field Theory (SCMFT)Self-Consistent Mean Field Theory (SCMFT)

For a give set of control parameters, the SCMFT equations must be solved to obtain the density profile and free energy

Phase diagram can be constructed from the solutions

Methods of solving SCMFT equations Exact solutions – rarely possible

Approximate solutions Weak segregation limit (WSL) theory

Strong segregation limit (SSL) theory

Numerical techniques Real space method

Reciprocal space method

SCMFT: Exact SolutionSCMFT: Exact SolutionSCMFT: Exact SolutionSCMFT: Exact Solution

Exact solutions are hard to come by!

.0)()(

,1)(

,)(

rr

ffr

ffr

BA

BB

AA

1,1),(),( cQsrqsrq

NffFVR

Nf H

gH

)1(

30

Trivial case: Homogeneous phase

Homogeneous phase is always a solution!

SCMFT: Weak Segregation TheorySCMFT: Weak Segregation TheorySCMFT: Weak Segregation TheorySCMFT: Weak Segregation Theory

Homogeneous phase as the zeroth-order solution

)()()()(),,()2(!4

)()()(),()2(!3

)()()()2(!2

)1(

32132132143213

3

2121213212

2

1111

10

qqqqqqqqqqdqdqdV

qqqqqqqdqdV

qqqSqdV

NffFTVk

N

B

rqiA efrd

Vq

1)(

NNaqFqS 2)6/()( 221

Solve the SCMFT equation for in terms of

Similar to Landau Theory – more laterLeibler 1980

SCMFT: Strong Segregation TheorySCMFT: Strong Segregation TheorySCMFT: Strong Segregation TheorySCMFT: Strong Segregation Theory

A and B segments are completely separated and the chains are strongly stretched

BA VB

VA

Bst

Astin

drzVNaf

drzVNaf

Sa

FFFF

222

22

22

2

0 8

3

8

3

6

Interaction and stretching energies treated separately

Extremely valuable tool to understand structures

Semenov 1985

Microphase Separation: Selection of Length Microphase Separation: Selection of Length ScaleScaleMicrophase Separation: Selection of Length Microphase Separation: Selection of Length ScaleScale

)-()()(2

1 1)2( qqSqqdF

Another view: spinodal point of the disordered phase (Leibler 1980)

Chain ConnectivityOrder-disorder transition (ODT) occurs at finite q0: Microscopic Phase Separation!

495.10)( 0 ODTAB NqS

2 4 6 8 10

1

2

3

4

5

NA NB

Phase Separation: Polymer BlendsPhase Separation: Polymer BlendsPhase Separation: Polymer BlendsPhase Separation: Polymer Blends

)-()()(2

1 1)2( qqSqqdF

Spinodal point of the disordered phase – Polymer Blend

Critical point occurs at zero q: Macroscopic Phase Separation!

2)0( CAB NqS

1 2 3 4 5q

2

4

6

8

10

S

NA NB

SCMFT: Analogy with Quantum MechanicsSCMFT: Analogy with Quantum MechanicsSCMFT: Analogy with Quantum MechanicsSCMFT: Analogy with Quantum Mechanics

Modified diffusion equation is Schrödinger equation with imaginary time!

)(

),(),(

22 rH

srqHsrqs

)()( rrH nnn

n

ns

n reqsrq n )(),(

)(2

),(),(

22

rVm

H

trHtrt

i

Many ideas and techniques in QM can be applied to polymers!

SCMFT: Numeric SolutionsSCMFT: Numeric SolutionsSCMFT: Numeric SolutionsSCMFT: Numeric Solutions

SCMFT equations are solved self-consistently

Recent progresses include fast Fourier algorithm

Real Space Method The space is discretized in to grids

An initial guess of the mean fields

The modified diffusion equations are solved to obtain

the propagators, and the results are used to compute

the densities

The fields for the next iterations are obtained using a

linear mixture of the new and old fields

The iteration is repeated until the solution becomes

self-consistent

SCMFT: Numeric SolutionsSCMFT: Numeric SolutionsSCMFT: Numeric SolutionsSCMFT: Numeric Solutions

SCMFT equations are solved self-consistently

The SCMFT equations can be cast in these Fourier components

Reciprocal-Space Method The functions are periodic

For an ordered phase, the reciprocal lattice vectors are

completely specified by the space group of the

structure

The plane waves corresponding to the reciprocal

lattice vectors can be used as basis functions

G

riG

G

riG

eGr

eGr

)()(

)()(

SCMFT: Reciprocal-Space MethodSCMFT: Reciprocal-Space MethodSCMFT: Reciprocal-Space MethodSCMFT: Reciprocal-Space Method

.)()(

),()()(

,),(),(1

)(

0,

0,

0

GBA

G

G

f

c

GG

GfGNG

sfGGqsGqdsQ

G

),()0,(,)0,(

)(),(

),(),(),(

0,

,22

fGqGqGq

GGGGGH

sGqGGHsGqs

G

GG

G

In Fourier Space:


GGn

nn

nmG

mn

nnG

n

GG

GG

GGGGH

,

*

*

)()(

)()(

)()(),(

m Gnmm

fn

nn

nn

sn

nn

sn

GGeq

q

GeqsGq

GeqsGq

m

n

n

)()()0()0(

)0()0(

)(),(

)(),(

**

*

Construct eigenvalues and eigenfunctions

The propagators are solved in terms of the eigenfunctions


mn

fBm

G

Bm

An

An

fc

BBmA

An eGGeQ

,

** )0()()()0(

The single chain partition function and density are

mnm

Gmnn

f

nm

f

c

qGGGef

e

Q

fG n

nm

,

* )()()0(1

)(

Differential equations become algebra ones


ni

rGini

n

n

nieS

Nrf

1)(

Further simplification: Point group symmetry

SCMFT equations in terms of the expansion coefficients

New Basis Functions for equivalent lattice vectors

nn

nn

rfr

rfr

)()(

)()(

Diblock Copolymers: SCMFT Phase DiagramDiblock Copolymers: SCMFT Phase Diagram

Controlling parameters:N

NfandN A

Diblock Copolymers: Polydispersity EffectDiblock Copolymers: Polydispersity Effect

Perturbation Theory:

1n

w

M

M

Diblock Copolymers: Polydispersity EffectDiblock Copolymers: Polydispersity Effect

Mechanism of stabilizing non-lamellar phases

Interfacial and entropic contributions to free energy

Hex phase, N = 15, f = 0.35

Full free energy

Non-Centrosymmetric StructuresNon-Centrosymmetric StructuresNon-Centrosymmetric StructuresNon-Centrosymmetric Structures

Lamellar Structures from ABC and ac block copolymer blends

Comparison with Recent ExperimentsComparison with Recent ExperimentsComparison with Recent ExperimentsComparison with Recent Experiments

2 = 0.18 2 = 0.4

ABC: polystyrene-b-polybutediene-b-poly(tert-butyl methacrylate)TEM Staining: A-light grey, B-black, C-white

T. Goldacker, V. Abetz, R. Stadler, I. Erukhimovich and L. Leibler, Nature 398, 137 (1999).

L. Leibler, C. Gay and I. Erukhimovich, Europhys. Lett. 46, 549 (1999).

Pure polymeric NCS Structure is still illusive

Phase Diagram: Phase Diagram: ZZ22 – – 22 plane planePhase Diagram: Phase Diagram: ZZ22 – – 22 plane plane

Z3/Z2 = 1.5

NCS Structure stable at 2=0.4 Wickham and Shi, Macromolecules, 2001

ABCD Tetrablock MeltABCD Tetrablock MeltABCD Tetrablock MeltABCD Tetrablock Melt

Phase Diagram with NCS phase: Possibility of pure NCS

Experiments: Matsushita and coworkers (2003)

Karim, Wickham and Shi, Macromolecules, 2004


)2()1()0(

)0()0(

)(

)r()r()r(),r()r()r(

FFFF

Phase behavior described by free energy functional F(,)

Expansion around an ordered state

energy nfluctuatioGaussian

equationsMeanfield

energyfreeMeanfield

)2(

)1(

)0(

0

F

F

F

Gaussian FluctuationsGaussian FluctuationsGaussian FluctuationsGaussian Fluctuations

)0()'r()r()'r,r(ˆ

)r'()'r,r(ˆ)r('rr2

1

2

)2(

F

C

CddF

Lowest order free energy cost of fluctuations

)'r,r(C Linear Hermitian operator

)q()r,r(ˆ)r,r(1

SCS

Fluctuation ModesFluctuation ModesFluctuation ModesFluctuation Modes

2)2(

2

1

)r()r(

)r()'r()'r,r(ˆr'

F

Cd

Eigenfunction representation

Nature of eigenmodes (r)?

eigenmode

eigenvalue

Free energy cost

Fluctuation Modes: Simple ExampleFluctuation Modes: Simple ExampleFluctuation Modes: Simple ExampleFluctuation Modes: Simple Example

20212

1akF

kk

kkC

2100

2100

2

1,2

2

1,0

k

Meanfield solution: 1-2=a0

Fluctuation modes:

2

21

2

21)2(

22

20

2

1 kF

Fluctuation ModesFluctuation ModesFluctuation ModesFluctuation Modes

The operator C(r,r’) quantifies free energy cost of fluctuations

The ordered phase is unstable if the smallest eigenvalue of C(r,r’) is negative

Scattering function = Fourier transform of C(r,r’)

Fluctuation modes classified by eigenfunctions of C(r,r’)

Elastic moduli can be obtained from C(r,r’)

Direct calculation of C(r,r’) impossible!Direct calculation of C(r,r’) impossible!

New techniques needed

Symmetry and Fluctuation ModesSymmetry and Fluctuation ModesSymmetry and Fluctuation ModesSymmetry and Fluctuation Modes

O

OCOOC

OCCO

OCOC

ˆ

)ˆ()ˆ(ˆ)ˆ(ˆ

ˆˆˆˆ

ˆˆˆˆ 1

Ordered structure:

invariant under symmetry operationsinvariant under symmetry operations

are simultaneous eigenfunctions of are simultaneous eigenfunctions of O O andand C C

can be constructed and catalogued using symmetriescan be constructed and catalogued using symmetries


Discrete translation symmetry

rqRqrqˆ

)Rr()r(ˆ

iii

R

R

eeeT

ffT

Fluctuation modes classified by Bloch waves

ijji

nmlG

nmlR

2ba

bbb

aaa

321

321

q and q+G degenerate RiqR)Gq( eei

)r()G( krk

G

r)Gk(k n

iin ueeC

unk(r+R)= unk(r), k within the 1st Brillouin zone


Point group symmetry:

(rotation, mirror reflection, inversion)

)k()k()r()r(ˆ

)r(ˆ)k()r(ˆˆ

)r()r(ˆ

k)(k

kk

1

nnnn

nnn

O

OOC

ffO

Fluctuation modes classified by Bloch waves

with a wave vector k within the irreducible BZ

Nature of Fluctuation ModesNature of Fluctuation ModesNature of Fluctuation ModesNature of Fluctuation Modes

Space group symmetry ensures:

Eigenvalues n(k) form a band structure

n(k) labeled by wave vector k within the 1st BZ

Eigenmodes are Bloch functions nk(r)=eikrunk(r)

unk(r+R)= unk(r) periodic functions

Fluctuation modes = Eigenmodes n(k) of )r,r(ˆ C

Fluctuation modes determined by

)r()k()r'()rr,(ˆr kk nnnCd

Band Structure of Fluctuation ModesBand Structure of Fluctuation ModesBand Structure of Fluctuation ModesBand Structure of Fluctuation Modes

0,0,0 zyx kkk3

22,

3

1,

3

1 zyx kkk0,0,0 zyx kkk

Example: Cylindrical structure

First BZ: hexagonal prism

ApplicationsApplicationsApplicationsApplications

Fluctuation modes provide:

n(k) = free energy cost of fluctuation modes n(k) < 0: unstable phaseminnk[n(k)] = 0: spinodal line Most unstable mode: structural relations Scattering functions Elastic moduli

General symmetry argument provides:

A powerful technique for classifying the anisotropic fluctuation modes in an ordered structure

Spinodal Decomposition vs. Spinodal Decomposition vs. NucleationNucleationSpinodal Decomposition vs. Spinodal Decomposition vs. NucleationNucleation

spinodal decomposition nucleation

near spinodal: near phase boundary:minnk[n(k)] < 0 minnk[n(k)] > 0

Free energymetastable

stable

unstable

Order parameters

Saddle-point

Weak Segregation Theory RevisitedWeak Segregation Theory RevisitedWeak Segregation Theory RevisitedWeak Segregation Theory Revisited

Model of systems with short wavelength instability Microphases of diblock copolymers Nematic to smectic-C transition in liquid crystals Weakly anisotropic antiferromagnets Onset of Rayleigh-Benard convection Pion condensates in neutron stars

Studied extensively Mean field approximation Hartree approximation for fluctuations

Theory of weak crystallization (Landau-Brasovsky)

432220

240

2

)(!4

1)(

!3)(

2)(

2rrrrq

qdrF

Microphase Separation: Selection of Length Microphase Separation: Selection of Length ScaleScaleMicrophase Separation: Selection of Length Microphase Separation: Selection of Length ScaleScale

)-()()(2

1 1)2( qqSqqdF

Another view: spinodal point of the disordered phase (Leibler 1980)

Chain ConnectivityOrder-disorder transition (ODT) occurs at finite q0: Microscopic Phase Separation!

495.10)( 0 ODTAB NqS

2 4 6 8 10

1

2

3

4

5

NA NB

Systems with Competing InteractionsSystems with Competing InteractionsSystems with Competing InteractionsSystems with Competing Interactions

Competing interactions & modulated structures

Interactions of long rangeDipolar interactionsNonlocal interactions associated with:

polarization, magnetization, elastic strainPolymer connectivity and/or entropy effects

Coupled order parameters

Membranes, amphiphilic monolayers, liquid crystal films

Spontaneous selection of primary length scales

Fluctuation spectrum has a minimum at q=q0


Modulated phases are ubiquitous in nature

Ripple phase in Lipids

Rayleigh-Bernard Instability

Superconducter Turing patterns

m 7

A240 cm0.1

mm25.0


Phases and phase transitions

Ferromagnetic film

Ferrofluid

Stripes

Bubbles

Landau-Brasovsky TheoryLandau-Brasovsky TheoryLandau-Brasovsky TheoryLandau-Brasovsky Theory

Free energy expansion

)2()1()0(

)0( )()(

FFFF

rr

4)4(

3)0()3(

2)0()0(220

240

2)2(

3)0(2)0()0(20

240

2)1(

4)0(3)0(2)0(2)0(20

240

2)0(

)(!4

1

)()(!3

1

)()(2

1)()(

2

1

)()(!3

1)(

2)(

)(!4

1)(

!3)(

2)(

2

rdrF

rrdrF

rrrqq

rdrF

rrrrqq

drF

rrrrqq

drF

Mean Field Theory: Phase DiagramMean Field Theory: Phase DiagramMean Field Theory: Phase DiagramMean Field Theory: Phase Diagram

0)(!3

1)(

2)(0

3)0(2)0()0(20

240

2)1(

rrrq

qF

Mean field equation

Mean field phase diagram

G

riGeGr )()( )0()0(

Kats et al. 1993, Podneks and Hamley 1996, Shi 1999

Gaussian FluctuationsGaussian FluctuationsGaussian FluctuationsGaussian Fluctuations

Fluctuation modes determined by:

G

riGeGVrrrV

rVqq

rCrrCrdrF

)()(2

1)()(

)()(ˆ),()(ˆ)(2

1

2)0()0(

220

240

2)2(

)r()r()r(22

02

40

2

Vqq

G

riGnk

riknk eGuer )()(

Space group symmetry ensures:

Eigenvalues n(k) form a band structure

n(k) labeled by wave vector k within the 1st BZ

Eigenmodes are Bloch functions nk(r)=eikrunk(r)

unk(r+R)= unk(r) periodic functions

Nature of Fluctuation ModesNature of Fluctuation Modes

2)2(

2

1

)r()r(

F

Fluctuation modes = Eigenmodes n(k) of )r,r(ˆ C

)r()k()r'()rr,(ˆr kk nnnCd

Shi. Yeung, Laradji, Desai, Noolandi

)0()'r()r()'r,r(ˆ);r'()'r,r(ˆ)r('rr

2

1 2)2(

F

CCddF

Anisotropic Fluctuation ModesAnisotropic Fluctuation ModesAnisotropic Fluctuation ModesAnisotropic Fluctuation Modes

Fluctuation modes determined by:

G

GG

nknG

nk

GGGGGV

GGVqGkq

GGC

GukGuGGC

)()(2

1)()(

)'()',(ˆ

)()()'()',(ˆ

)0()0()0(

',

220

2

40

2

'

G

riGnk

riknk eGuer )()(

Fluctuation modes as Bloch functions:

Band Structure of Fluctuation ModesBand Structure of Fluctuation ModesBand Structure of Fluctuation ModesBand Structure of Fluctuation Modes

0,0,0 zyx kkk3

22,

3

1,

3

1 zyx kkk0,0,0 zyx kkk

Example: Cylindrical structure

First BZ: hexagonal prism

Stability of the Ordered PhasesStability of the Ordered PhasesStability of the Ordered PhasesStability of the Ordered Phases

Spinodal lines determined by 0)k(mink

n

Fluctuation Modes of Lamellar PhaseFluctuation Modes of Lamellar PhaseFluctuation Modes of Lamellar PhaseFluctuation Modes of Lamellar Phase

Fluctuation modes from perturbation theory

)()(22

02

40

2

GVqkq

k For k2=(k-G)2

q0

q0/2kx

kz

Most unstable modes at two rings defined by:

020

22

2

1,

4

3qkqkk zyx

Hexagonal symmetry in x-z plane Infinity degenerate in x-y plane Two layer periodicity in z-direction

Scattering Functions of Lamellar PhaseScattering Functions of Lamellar PhaseScattering Functions of Lamellar PhaseScattering Functions of Lamellar Phase

kx

kz

Hexagonal symmetry in x-z plane Infinity degenerate in x-y plane

qz qy

qx qx

Application: Structural RelationApplication: Structural RelationApplication: Structural RelationApplication: Structural Relation

Lamellar to hexagonal phase

Effect of the most unstable mode

0)()()(01

)0( rarr k

a=0 a=1 a=2


Hexagonal to lamellar phase


0)()()(01

)0( rarr k

a=0 a=0.5 a=1


Hexagonal to spherical phase


0)()()(01

)0( rarr k

a=0 a=0.25 a=0.75

Applications: Block CopolymersApplications: Block CopolymersApplications: Block CopolymersApplications: Block Copolymers

Hexagonally packed cylinder to sphere transition

•Good agreement with experimentally observed epitaxies

•Reference: Ryu, Vigild, Lodge PRL 81, 5354, 1998

NucleationNucleationNucleationNucleation

Decay of a metastable phase into a stable, equilibrium phase vianucleation

Sota et al., Macromolecules (2003)

Nucleation rate =

F(R)

R

RC

CF

(e.g. nucleation of liquid from supersaturated vapour via thermal fluctuations)

= liquid/vapour interfacial free-energy

R

/kTFeΓ C

(quasi-equilibrium)

attempt frequency

23 43

4)( RfRRF

0 vapourliquid fff

fRC

2

2

3

)(3

16

fFC

Droplet free-energy:

drop shrinks

drop grows

Classical Nucleation TheoryClassical Nucleation TheoryClassical Nucleation TheoryClassical Nucleation Theory

(micro)structured phases in/out of droplet

account for length scales

symmetry of microstructure leads to:

•anisotropic interfacial free-energies

•anisotropic droplet shapes

•complicated interfacial structure

needs to be modified

Key quantity:

23 43

4)( RfRRF

Nucleation at order-order transitions: Nucleation at order-order transitions: challengeschallengesNucleation at order-order transitions: Nucleation at order-order transitions: challengeschallenges

Landau-Brazovskii field-theory

single-mode, slowly varying envelopeapproximations

Amplitude (phase-field) model for interfaces between eg. lamellae and cylinders

profile for planar interfacebetween e.g. lamellae and cylinders

Anisotropic interfacial free-energy: (,)

Non-spherical droplet shape

Critical droplet size and nucleation barrier

Wulff construction

modified classical nucleation theory

Outline of the Theoretical MethodOutline of the Theoretical MethodOutline of the Theoretical MethodOutline of the Theoretical Method

Model and AssumptionsModel and AssumptionsModel and AssumptionsModel and Assumptions

Assume separation of length-scales:

microstructure period < droplet interfacial width << droplet size

Weak segregation: single mode approximation to order parameter

)]cos()cos())[cos((2

)]cos())[cos((2)cos()(2)(

6543

32211

rGrGrGra

rGrGrarGrar

: reciprocal lattice vectors

: spatially-varying amplitudes

amplitude-only model: )](),([ raraf ii

<111>iG

)(rai

spheres: saaaa 321

cylinders: 0; 321 aaaa c

lamellae: 0; 321 aaaa l

MethodMethodMethodMethod

1) Use amplitude model to compute the interfacial free-energy (,) for an interface of arbitrary orientation (,) between coexisting, epitaxial cylinder and sphere phases.

2) Compute droplet shape using Wulff construction droplet size >> interface width (near coexistence)

non-spherical droplet shape

3) Compute nucleation barrier, critical droplet size using classical nucleation theory

w

sh

asa

w

sh

aaaasasa

s

cscs

12

)(

22)()(

3

21<111>

Theory predicts that bcc symmetry leads to:

isotropic interfacial free-energy

spherical droplet

Mean-field criticalbehaviour

3|2/1~| Af

Sota et al. (2003)

Interfacial free-energy

BCC phase nucleating from disorderBCC phase nucleating from disorderBCC phase nucleating from disorderBCC phase nucleating from disorder

Interfacial Free Energy: Cylinders in Interfacial Free Energy: Cylinders in lamellaelamellaeInterfacial Free Energy: Cylinders in Interfacial Free Energy: Cylinders in lamellaelamellae

Lowest when: cylinders perpendicular to interface or lamellae parallel to surface

Two fold symmetry in x-y planeWickham et al., J. Chem. Phys. 118, 10293 (2003).

Disorder –to- bcc sphere

Sphere –to- hex cylinder

Lamellar -to- hex cylinder

Disorder -to- hex cylinder

(Disorder -to- lamellar: Fredrickson & Binder, Hohenberg & Swift)

All these phases can be studied near the mean-field critical point within the single Fourier mode approximation

Work near phase coexistence lines

Order-order and disorder-order transitions studiedOrder-order and disorder-order transitions studiedOrder-order and disorder-order transitions studiedOrder-order and disorder-order transitions studied

Nucleation barrier Droplet Radius

fA = 0.4fA = 0.4N = 1000

TkFTk BcB 1060 Consider barriers in the range:

Critical radius ~ 25 cubic lattice spacings

Suggests observed droplets have grown beyond critical size, but kept initial shape

BCC phase nucleating from disorderBCC phase nucleating from disorderBCC phase nucleating from disorderBCC phase nucleating from disorder

0

2

<111>

Shape elongated along cylinder axis

Aspect ratio ~ 1.45

Critical diameter ~ 40 cylinders across

45.0Af

Droplet Shape

high

low

Hex cylinders nucleating from BCC Hex cylinders nucleating from BCC phasephaseHex cylinders nucleating from BCC Hex cylinders nucleating from BCC phasephase

Droplet is lens-shaped, flattened along the cylinder axis

Aspect ratio ~ 1/4 at fA = 0.45

Diameter of critical droplet ~ 30 cylinders at fA = 0.45

Aspect ratio -> 0 as fA -> ½

f A= 0.49

f A= 0.45

= 0o

= 90o

Droplet shape

high

low

[Wickham, Shi, Wang, J. Chem. Phys. (2003).]

Hex cylinders nucleating from lamellar Hex cylinders nucleating from lamellar phasephaseHex cylinders nucleating from lamellar Hex cylinders nucleating from lamellar phasephase

(PS-PI) (fPI~ 0.2)/ homopolymer (PS) blend

Cylinder/disorder coexistence region

Koizumi et al. Macromolecules (1994)

Aspect ratio ~ 0.29 Diameter ~ 20 cylinders

But a different system!

ExperimentsExperimentsExperimentsExperiments

Sota et al. Macromolecules, (2003).

Metastable disorder –to- metastable cylinder (in our model)

Droplet is lens-shaped, flattened along the cylinder axis

Aspect ratio ~ 0.17 at fA = 0.4

Diameter of critical droplet ~ 60 cylinders at fA = 0.4

Aspect ratio -> 0 as fA -> ½

fAA=0.4=0.4

Hex cylinders nucleating from Hex cylinders nucleating from disorderdisorderHex cylinders nucleating from Hex cylinders nucleating from disorderdisorder

Discussions and ConclusionsDiscussions and ConclusionsDiscussions and ConclusionsDiscussions and Conclusions

Self-consistent field theory provides a useful

theoretical framework for polymeric systems

SCFT is capable of studying the ordered phases in

block copolymer systems

SCFT provides a general framework for any

statistical mechanical systems

More challenges ahead: complex structures,

phase transition kinetics, rod-coil copolymers,

associating polymers, micelles, membranes, …

Documents

Self-Consistent Field Theory of Block Copolymers An-Chang Shi McMaster University Hamilton, Ontario, Canada [email protected]