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