Post on 05-Jan-2016
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
shi@mcmaster.cashi@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,
Self-Consistent Field Theory: Simple Self-Consistent Field Theory: Simple ExampleExampleSelf-Consistent Field Theory: Simple Self-Consistent Field Theory: Simple ExampleExample
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,
Self-Consistent Field Theory: Simple Self-Consistent Field Theory: Simple ExampleExampleSelf-Consistent Field Theory: Simple Self-Consistent Field Theory: Simple ExampleExample
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,
The partition function can be written in the form,
n
iirrr
1
)(ˆ
Self-Consistent Field Theory: Simple Self-Consistent Field Theory: Simple ExampleExampleSelf-Consistent Field Theory: Simple Self-Consistent Field Theory: Simple ExampleExample
],[
)()(})({
1
)()()(})({
][][
][][!
][][!
F
nrrrdVn
n
i
ri
rrrdVn
eDD
QeDDn
z
erdeDDn
zZ i
Using the definition,
The partition function can be written in the form,
)(1][ rerd
VQ
constQnrrrdVF ][ln)()(})({],[ where the free energy functional is,
Field theory model corresponds to the particle model
Self-Consistent Field Theory: Simple Self-Consistent Field Theory: Simple ExampleExampleSelf-Consistent Field Theory: Simple Self-Consistent Field Theory: Simple ExampleExample
)(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
Self-Consistent Field Theory of PolymersSelf-Consistent Field Theory of Polymers Self-Consistent Field Theory of PolymersSelf-Consistent Field Theory of Polymers
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
Self-Consistent Field Theory of PolymersSelf-Consistent Field Theory of Polymers Self-Consistent Field Theory of PolymersSelf-Consistent Field Theory of Polymers
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 )]()([))(())(()})(({
Self-Consistent Field Theory of PolymersSelf-Consistent Field Theory of Polymers Self-Consistent Field Theory of PolymersSelf-Consistent Field Theory of Polymers
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
Self-Consistent Field Theory of PolymersSelf-Consistent Field Theory of Polymers Self-Consistent Field Theory of PolymersSelf-Consistent Field Theory of Polymers
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
Self-Consistent Field Theory of PolymersSelf-Consistent Field Theory of Polymers Self-Consistent Field Theory of PolymersSelf-Consistent Field Theory of Polymers
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
Self-Consistent Field Theory of PolymersSelf-Consistent Field Theory of Polymers Self-Consistent Field Theory of PolymersSelf-Consistent Field Theory of Polymers
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
Self-Consistent Field Theory of PolymersSelf-Consistent Field Theory of Polymers Self-Consistent Field Theory of PolymersSelf-Consistent Field Theory of Polymers
Free Energy Expansion
)()()(
)()()()0(
)0(
rrr
rrr
),0,|,(),|,(),|,(),|,(
)()()()|,()'|,(
11112211
1111
)0(
rsrGsrsrGsrsrGsrsrG
rrdsdsrdrdrsrQrsrQ
nnnnnn
nnnn
n
).'|,()()'|,()'|,( )0()0()0(22)0( rsrQrrsrQrsrQs
)|,()(),|,( )0( rssrQsssrsrG
Self-Consistent Field Theory of PolymersSelf-Consistent Field Theory of Polymers Self-Consistent Field Theory of PolymersSelf-Consistent Field Theory of Polymers
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
Self-Consistent Field TheorySelf-Consistent Field TheorySelf-Consistent Field TheorySelf-Consistent Field Theory
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
Self-Consistent Field TheorySelf-Consistent Field TheorySelf-Consistent Field TheorySelf-Consistent Field Theory
)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:
SCMFT: Reciprocal-Space MethodSCMFT: Reciprocal-Space MethodSCMFT: Reciprocal-Space MethodSCMFT: Reciprocal-Space Method
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
SCMFT: Reciprocal-Space MethodSCMFT: Reciprocal-Space MethodSCMFT: Reciprocal-Space MethodSCMFT: Reciprocal-Space Method
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
SCMFT: Reciprocal-Space MethodSCMFT: Reciprocal-Space MethodSCMFT: Reciprocal-Space MethodSCMFT: Reciprocal-Space Method
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
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(,)
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
Symmetry and Fluctuation ModesSymmetry and Fluctuation ModesSymmetry and Fluctuation ModesSymmetry and Fluctuation Modes
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
Symmetry and Fluctuation ModesSymmetry and Fluctuation ModesSymmetry and Fluctuation ModesSymmetry and Fluctuation Modes
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
Systems with Competing InteractionsSystems with Competing InteractionsSystems with Competing InteractionsSystems with Competing Interactions
Modulated phases are ubiquitous in nature
Ripple phase in Lipids
Rayleigh-Bernard Instability
Superconducter Turing patterns
m 7
A240 cm0.1
mm25.0
Systems with Competing InteractionsSystems with Competing InteractionsSystems with Competing InteractionsSystems with Competing Interactions
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
Application: Structural RelationApplication: Structural RelationApplication: Structural RelationApplication: Structural Relation
Hexagonal to lamellar phase
Effect of the most unstable mode
0)()()(01
)0( rarr k
a=0 a=0.5 a=1
Application: Structural RelationApplication: Structural RelationApplication: Structural RelationApplication: Structural Relation
Hexagonal to spherical phase
Effect of the most unstable mode
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, …