March 5, 2004The 21st Century COE Program

of Tohoku universityCOE International Symposium

Dynamical Self-Consistent Field Theory for Inhomogeneous Dense Polymer Systems

---- Bridging the Gap between Microscopic Chain Structures and Dynamics of Macroscopic Domains ---

Toshihiro KawakatsuDepartment of Physics, Tohoku University

CONTENTS

1. Inhomogeneity and viscoelasticity of dense polymer systems

2. Macroscopic phenomenological approach3. Introducing microscopic information:

Static self-consistent field (SCF) theory4. Dynamical extension of SCF: Application to

structural phase transition of blockcopolymer5. New SCF approach to dynamics of highly

deformed chains6. Conclusions and future directions

Inhomogeneity of Dense Polymer SystemsPolymer Mesophase/Polymer Nano-Composites

Microphase Separationin Blockcopolymer Melt

Salami Structure in HIPS (PS/PB)

Fischer and Hellman, Macromolecules 29, 2498 (1996)Koubunshi Shashinshu

Chain structure Viscoelasticity

Viscoelasticity of Dense Polymer Systems

Die Swell (extrusion from a channel)

extended

recoil

Viscous fluid Viscoelastic fluid

Purpose of the Study

Viscoelastic fluid + Inhomogeneity

Quantitative prediction ofstructure/dynamics of domains

・ Phase separation (meso/macroscale)・ Chain structure (microscale)

Macroscopic Phenomenological Approach(Jupp, Yuan & Kawakatsu, 2003)

( ) ( ) ( )∫ ∞−

−=t

dsds

sdstGt γσ

Macro/mesoscopic flow simulation(Stokes Equation & Phase separation)

Phenomenological constitutive equation+

( )dt

tdγ( )tγ : deformation

: deformation rate

( )tσ : stress1 γ

Simulation of Viscoelastic Flow Model(Jupp, Yuan & Kawakatsu, 2003)

Dynamics of Viscoelastic Body

Density Field: Two fluid model (Doi & Onuki)

Constitutive Equation: Johnson-Segalman (JS) model

( )BBAA vvvv φφ +≡=⋅∇ 0[ ]σαµ

ςφφ

⋅∇+∇−=− BABA vv

( )( ) ∑=

≡∇+∇=+3

1ii

Ti

i

i

i Gdt

d σσστσ vv

( )⎥⎦

⎤⎢⎣

⎡⋅∇−∇⋅∇+∇⋅−=

∂∂ σαµ

ςφφφφ 22

BAA

A

tv

Simulation of Viscoelastic Phase Separation(Jupp, Yuan & Kawakatsu, 2003)

Segment density Flow field Scattering intensity

・ Two fluid model・ Johnson-Segalman (JS) model

( )( ) ∑=

≡∇+∇=+3

1ii

Ti

i

i

i Gdt

d σσστσ vv

2d Simulation under steady shear ( )const.=γ

Simulation results(shear-induced demixing case)

Stability diagram

( ) ( ) AWe κτ≡≡ force viscousforce elastic2.0=

≡′ AB τττ

0.4=We

Simulation of Viscoelastic Flow Model(Jupp, Yuan & Kawakatsu, 2003)

Shear-induceddemixing

homogeneous

two-phase

χ

φ

Simulation of Viscoelastic Flow Model(Jupp, Yuan & Kawakatsu, 2003)

2d Simulation of shear banding(steady shear )const.=γ

Uniform shear rate Inhomogeneous distributionof shear rate

instability

Introducing information on chain structureinto flow simulation

Relation between Macroscopic and Microscopic Phenomena

Self-Consistent Field (SCF)Theory

Probability distributionof chain conformation

(spatially-folding shape)

Density Functional Theories (SCF vs. GL)

( ){ }[ ]rφFF =( )rφSegment density Fee energy functional

Phase diagram ofblockcopolymer melt

uniformCriticalpoint

Strong segregation

Weak segregation

Self-consistent field (SCF)

Ginzburg-Landau (GL)

( )rV

Basic Formalism of Static SCF Theory

( ) ∑ ⎥⎦

⎤⎢⎣

⎡−=′

onconformati all B

1exp,;, HTk

mnQ rr

( ) ( ) ( )rrrrr ′⎥⎦

⎤⎢⎣

⎡−∇=′

∂∂ ,;,

6,;, 2

2

mnQVbmnQn

β

( )rV Mean field (SCF)Path integral for equilibriumchain conformation

( )rφ

( ){ }[ ] ( )[ ]rrr φFmnQFF =′= ,;, Equilibriumstructures

Static SCF Simulation : Phase Diagram & Domain Structures

Block copolymer melt (Matsen ＆ Schick; Phy. Rev. Lett. 72 (1994) 2660.)

cubic cylinder

lamellar gyroidA BBA

A

NNNf+

=

Periodic structure Fourier space analysis

Prediction of Complex Domain Structures(Morita, Kawakatsu, Doi, Yamaguchi, Takenaka & Hashimoto)

PS-PI diblock copolymer mixture

+experiment

Irregular structure

Real space analysis

3d simulation

8.0:2.0: shortlong =φφ

L + C

L

C

L + D

L

Nlo

ng/

Nsh

ort

Volume fraction of long chain

experiment 2d simulation

w

2 phase 1 phase

5

6

7

8 exptl.

calc.

0.2 0.4 0.6 0.8 1.00.0

L + D

LL + LNlo

ng/

Nsh

ort

Volume fraction of long chain

Prediction of Complex Domain Structures(Morita, Kawakatsu, Doi, Yamaguchi, Takenaka & Hashimoto)

Prediction of Complex Domain Structures(Morita, Kawakatsu, Doi, Yamaguchi, Takenaka & Hashimoto)

8.0:2.0: shortlong =φφ

Prediction of exotic structures

Dynamical Extension of SCF Theory

・ Growth of irregular domains・ Structural phase transitions Dynamical model

( ){ }[ ]rφF

( ) ( ) ( ){ } ( )tFLtt

tt

,,,, 2

rrvrr

δφδφφ

∇=⋅∇+∂

∂

( ) ( ) ( )rrrrr ′′⎥⎦

⎤⎢⎣

⎡−∇=′′

∂∂ ,;,

6,;, 2

2

nnQVbnnQn

β

( )t,rφDensity distribution

SCF calculation

Free energy

Local diffusion

Chain structure

Diffusion/Hydrodynamics

Structural Phase Transition between Mesophases

gyroid

cylinder lamellar

・ Growth of unstable modes・ Kinetic pathway・ Conformation change

101

102

103

104

105

106

107

0.05 0.1 0.15 0.2 0.25

Inte

nsity

/(arb

.)

Q (Å-1)

50°C (L)

36°C (MFL)

34°C (R)

30°C (DG)

28°C (C)

R

L

G

H

Diffuse scattering

systemOD /EC 55% 2616

OH)H(OC)(CHCH:EC 6421523616

Modulation fluctuation layer

SAXS Experiment on Microemulsion(Imai, et al., J. Chem. Phys. 119 (2003) 8103.)

GL Model(Weak segregation)

R

L

PL

Stability Analysis on GL Model(Imai, et al., J. Chem. Phys., 119 (2003) 8103.)

003 =−λ

Undulationλ03+ = 2τQ *2

Amplitude modulation

(b)

z (2πQ*)

y(2πQ*) y(2πQ*)

z (2πQ*)

(a)

Bright region : majority phase (surfactant)Dark region: minority phase (water)

Surfactant layers

Dynamics of Structural Phase Transition(M.Nonomura, et al., J. Phys. Condens. Matt. 15 (2003) L423.)

Gyroid → CylinderGinzburg-Landau model

(weak segregation)+

Fourier mode expansion

Changes in symmetry & periodicity (a few %)

Many modes (12 + 6)

Gyroid(1,1,1) → Cylinder: epitaxial growth

Dynamics of Structural Phase Transitionunder Shear Flow (T.Honda and T.Kawakatsu)

Gyroid → Cylinder (Strong segregation)

Shear flowin (1,0,0)direction

Real space dynamic SCF + variable system sizeChain architecture

& stabilityAutomatic adjustment

of periodicity

Dynamics of Structural Phase Transitionunder Shear Flow (T.Honda and T.Kawakatsu)

( ){ }[ ]rφF

( ) ( ) ( ){ } ( )tFLtt

tt

,,,, 2

rrvrr

δφδφφ

∇=⋅∇+∂

∂

( ) ( ) ( )rrrrr ′′⎥⎦

⎤⎢⎣

⎡−∇=′′

∂∂ ,;,

6,;, 2

2

nnQVbnnQn

β

( )t,rφDensity distribution

SCF calculation Chain architecture

Free energy

Local diffusion Hydrodynamic effect

( )α

αα LVFML

t ∂∂

−=∂∂Box size

adjustmentChange in periodicity

Dynamics of Structural Phase Transitionunder Shear Flow (T.Honda and T.Kawakatsu)

A8-B12 blockcopolymer melt, χN = 20Real space 2-d dynamic SCF simulation with weak shear

xL

yL

Dynamics of Structural Phase Transitionunder Shear Flow (T.Honda and T.Kawakatsu)

A7-B13 blockcopolymer melt, χN = 20 → 15 (G → G/C)Real space dynamic SCF simulation with weak shear

Shear direction(1,0,0)

Direct observation of the kinetic pathway

Dynamics of Structural Phase Transitionunder Shear Flow (T.Honda and T.Kawakatsu)

(1,0,0)-growth Non-epitaxial

Real space dynamics Modulated cylinderunder shear (Metastable)

Variable box size Change in periodicity in perpendicular direction

Dissipative Particle Dynamics (DPD)

With hydrodynamic effect

Effect of External Flow & Kinetic Pathway(Groot, Madden & Tildesley, J. Chem. Phys. 110 (1999) 9739.)

Without hydrodynamic effect

Brownian Dynamic (BD)

Artificial stabilization of interconnected transient structure

Kinetic Pathway(Groot, Madden & Tildesley, J. Chem. Phys. 110 (1999) 9739.)

DPD simulation of kinetic pathwayin block copolymer melt

0.1 0.40.30.2

time

f

Viscoelastic Properties of Mesophases

Sheared mesophase Sheared brushes

・ Highly-deformed conformation・ Entanglements・ External/internal flow

Standarddynamic SCF

Limitations of standard SCF

Basic assumption of DSCF

( ) ( )⎥⎦

⎤⎢⎣

⎡−≡′′ ∑∑

kkVnnQ rrr βexp,;,

onconformati all

Local equilibrium(Canonical ensemble)

Non-equilibrium conformation

(Viscoelasticity, etc.) Sheared polymer brushes

Viscoelasticity and Reptation Motion in Concentrated Polymer Systems

Polymer melt

Step sheardeformation

Stress relaxation by reptation

1 γPolystyrene

t

Large deformationγ

(Osaki, et al., Macromolecules 15 (1982) 1068.)

Extension of SCF to Highly Deformed Chains(Shima, Kuni, Okabe, Doi, Yuan, and Kawakatsu, 2003)

Path Integral

( ) ( ) ( )[ ] ( )rrrrSrr ′−∇∇=′∂∂ ,;,:,,;, mnQVnmnQn

β

Bond orientation tensor ( )rS ,n

+reptation, flow deformation, constraint release

( )rS ,nBond orientation

Extension of SCF to Highly Deformed Chains(Shima, Kuni, Okabe, Doi, Yuan, and Kawakatsu, 2003)

Path Integral

( ) ( ) ( )[ ] ( )rrrrSrr ′−∇∇=′∂∂ ,;,:,,;, mnQVnmnQn

β

Reptation dynamics for ( )tn ,, rS( ) ( ) ( ) ( )[ ]

( ) ( ) ( ) ( )[ ] ( )release constraint

,,,,,,,,

,,,,,,,

tnt

ttntnt

tnttnn

tnt

T rSrrSrSr

rSrvrjrS

∂∂

+⋅+⋅+

⋅⋅∇−∂∂

−=∂∂

κκ

( ) ( )tt ,, rvr ∇≡κ Velocity gradient tensor

Simulation of Polymer Melt UnderStep Shear Deformation

(Shima, Kuni, Okabe, Doi, Yuan, and Kawakatsu, 2003)1 γ

Comparison with reptation theory

for uniform systems

experiment simulation

t0.500.10

0.50.11.001.0=γ

Stre

ss

rela

xatio

n

Polystyrene

t

Large deformationγ

Simulation of Polymer BrushesUnder Steady Shear Deformation

(Shima, Kuni, Okabe, Doi, Yuan, and Kawakatsu, 2003)

x

y

D

Single Chain Conformation )0.1( =yxκ

0.10=t0.0=t 5.0=t

Segment Distribution

0

0.5

1

0.20,0.10,0.5,0.2,0.0=t

Overlappingregion

0.1=yxκ

Dxsegm

ent v

olum

e fr

actio

n

0.0 0.1

0.1=yxκ

Dx

end

segm

ent d

ensi

ty

0.0 0.10.0

0.2

End Segment Distribution

0.20,0.10,0.5,0.2,0.0=t

x

y

D

Simulation of Polymer BrushesUnder Steady Shear Deformation

(Shima, Kuni, Okabe, Doi, Yuan, and Kawakatsu, 2003)

Coincide microscopic Monte Carlo simulation resultsDynamical properties that cannotbe reproduced by standard SCF

Conclusion and Future Direction

Polymer Mesophase/Polymer Nano-Composites

Predictions on structural transition/viscoelastic behavior

based on microscopic chain structure

Numerical Analysis ofViscoelastic Equations

+Dynamic Extension of

SCF Theory

Macroscopicflow

Chainstructure

Collaborators

T.Shima (Ericsson Co., Ltd.)H.Kuni (IBM Co., Ltd.)

Y.Okabe (Tokyo Metropolitan Univ.)M.Doi (Nagoya Univ.)

X.F.Yuan (Kings College)

M.Imai & K.Nakaya (Ochanomizu Univ.)

T. Honda (Nippon Zeon Co. Ltd.)

T.Hashimoto, M.Takenaka, D.Yamaguchi(Kyoto Univ.)

H.Morita (JST,CREST)

END