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2/4/2005
Hydrodynamic Self Consistent Field Theory:Viscoelastic Microphase Separation
of Diblock Copolymers
David M. Hall LANL T-11 & UCSB Dept. of PhysicsSanjoy Banerjee UCSB, Dept. Chemical EngineeringTurab Lookman LANL Theoretical Division
Hydrodynamic Self Consistent Field Theory:Viscoelastic Microphase Separation
of Diblock Copolymers
HSCFT Overview• HSCFT extends Self Consistent Field Theory to non-equilibrium, hydrodynamic flows
SCFT Diblock Thermodynamic
Model
SCFT Diblock Thermodynamic
Model
“Two Fluid”Hydrodynamic
Model
“Two Fluid”Hydrodynamic
Model
Chemical Potential Gradients
Monomer Concentrations
LTELoop
10−12 −10−9 sec 10−3 −102 sec
Constitutive EquationConstitutive Equation
2/4/2005
Diblock Thermodynamic Model
• Mean field Free Energy
• Local thermal equilibrium conditions
• Monomer concentrations matched by steepest ascent
• Non-equilibrium chemical potentials
F = E − T(SJ + SA + SB )
δFδωA
=nkTV
˜ φ A − φ[ ]= 0 δFδωB
=nkTV
˜ φ B − (1− φ)[ ]= 0
µ = v0δFδφ
=kTN
χN 1− 2φ( )+ ωB −ωA[ ]
ω in +1 = ω i
n + λ δFδω i
˜ φ A =1Q
dsqq+0f∫
Q =1V
drq r,s( )∫ q+ r,s( )
∂q∂s
=Nb2
2d∇2q −ωq
˜ φ B =1Q
dsqq+f1∫
EnkT
= +1V
dr∫ Nχφ 1− φ( )
SA
nk= −
1V
dr∫ ρJ lnq + φωA( )
SB
nk= −
1V
dr∫ ρJ lnq+ + 1−φ( )ωB( )
SJ
nk= +
1V
dr∫ ρJ lnρJ
2/4/2005
Two Fluid Hydrodynamic Model
• Incompressibility
• Monomer Conservation
• Relative Velocity Force Balance
• Momentum Conservation
∇ ⋅ v = 0
∂tφ = −∇ ⋅ φvA( )
w =1
ζ φ( )−φ 1− φ( )∇µ[ ]+ a φ( )∇ ⋅ σ (n )
ρ∂tv = −φ∇µ + ∇ ⋅ σ (n ) − ∇p
v = φvA + 1− φ( )vB
ζ φ( )=ζ Aζ B
ζ A + ζ B
a φ( )= 1− φ( )αA − φαB
α i =ζ i
ζA + ζ B
ζ i = φ N /Ne( )ζ 0i
w = vA − vB
2/4/2005
Linear Maxwell Constitutive Equations
• Shear stress evolution
• Bulk stress evolution
• Concentration dependant moduli
• Stress relaxation times
∂tσ b( i) = −σ b
(i) /τ b(i) φ( )+ K ( i) φ( ) ∇ ⋅ vT( )δ
∂tσ S( i) = −σ S
(i) /τ S(i) φ( )+ G( i) φ( )κT
GA φ( )= G0Aφ 2 K A φ( )= K0
Aθ φ − f( )K B φ( )= K0
Bθ f − φ( )GB φ( )= G0B 1− φ( )2
τ S(i) φ( )= τ S
( i) τ b(i) φ( )= τ b
( i)
κTij = ∂ivT
j + ∂ jvTi −
2D
∇ ⋅ vT( )
vT = αA vA + αB vB
σ (n ) = σ bA + σ b
B + σ SA + σ S
B
2/4/2005
Results• 2D morphology vs. time• W and V curves are most useful for tracking important events• Micro-phase separation consists of three distinct stages
– Early stage has exponential growth– Intermediate stage characterized by rapid transition to defect filled morphology– Late stage characterized by long, slow process of defect reduction and annihilation
events• After very long times, morphologies approach equilibrium states• Length of exponential growth phase has a simple functional form in f• Shearing flows and bulk flows are both important.• Each entropy component has two distinct decay times.• 3D structures are highly defect filled, but correlate well with phase diagram.• Euler characteristics are most useful quantity for classifying 3D structures.• Viscoelastic phase separation occurs much as it does for blends, but with a more
structured network• Junction densities grow exponentially, and are expelled from each phase as it separates.• Viscoelastic phase separation in 3D
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2/4/2005
Time evolution of Morphology vs. f, 2D
f = 0.30
f = 0.35
f = 0.25
f = 0.50
f = 0.45
f = 0.40
time
Velocity Curves• Secondary w peaks: defect annihilations• Secondary v peaks : defect rearrangements
defectmotion defect
annihilation
Early StageConcentration fluctuations grow exponentially
Late StageSlow approach to equilibrium by defect annihilation and rearrangement.
Intermediate StageRapid merger of fluctuations to form defect filled state
2/4/2005
Phase Separation Time vs. f• Structure growth time diverges as f approaches the ODT
twmax ∝1
f − f0( )α 1− f − f0( )α
Giτ i = 0.100f0 = 0.222α =1.30
tmax(w )
f
Giτ i = 0.001f0 = 0.235α =1.00
2/4/2005
Phase Separation Movies, 2D
2/4/2005
QuickTime™ and a decompressor
are needed to see this picture.
f = 0.30
QuickTime™ and a decompressor
are needed to see this picture.
f = 0.40
QuickTime™ and a decompressor
are needed to see this picture.
f = 0.35
f = 0.45 f = 0.50
Euler Characteristic vs. time• Topological invariant, useful in classifying surface geometry
2/4/2005 χE = +24
χE = +60χE = −24
χE = +8
χE t( )
time
f = 0.30
f = 0.25
χE = F + V − E =1
2πKGdA∫∫
2/4/2005
“Viscoelastic” Phase Separation, 2D
KA = 3.0
KA =10.0
KA = 0.0
• Network formation in harder phase, followed by network break up
θ φ, f( )
KA = 3.0
KA =10.0
KA = 0.0
f = 0.5
time
2/4/2005
Phase Inversion, 2D• Minority phase with higher bulk modulus leads to phase inversion
3.0 KA =10.0KA = 2.0 5.0 7.0
θ φ, f( )
KA = 3.0
KA =10.0
KA = 0.0
f = 0.30
time
Viscoelastic Phase Separation,3D
f = 0.30KA = 5.0
f = 0.25KA = 5.0
f = 0.30KB = 5.0
f = 0.30
2/4/2005
EVF and Euler Curves, 3D Visco.
f = 0.30KA = 5.0
f = 0.30KB = 5.0
f = 0.25KA = 5.0
Evf vs. time
f = 0.30KA = 5.0
Euler Characteristic vs. time
2/4/2005
2/4/2005
Viscoelastic Phase Separation Movies, 3Df = 0.30 f = 0.30 KA = 5.0 f = 0.30 KB = 5.0
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
f = 0.25 KA = 5.0
AB Junction Distribution vs. Time
1−< ρJ
2ψ 2 ><ψ 2 >
ψ ≡ φ − f
KA =10.0KA = 0.0
KA =10.0
KA = 0.0
ρJψ
AB Junction BiasA
B
2/4/2005
Research RoadmapMean field Fluctuations
LinearViscoelasticity
Non-linearViscoelasticity
Thermo-Complexity
Hydro Complexity
Complex Channels
Patterned Walls
Diblocks
Multi-BlocksMulti-Block Blends
Nano-particlecomposites
Flat Walls
Tri-blocksDilute Solutions
Polymersomes
Low Freq.Bulk Shear
Low Freq.Thin Film Shear
ConcentratedSolutions
Dendrimers
Confinement Ordering
EpitaxialOrdering
Extrusive Flow Non-isothermal flows
LamellarShear Flipping
Cyclomers
Channel FlowsLarge ParticleComposites
HMXBinders
2/4/2005
Future Research Directions• Examine more complex thermodynamic models
– Multi-block copolymers– Copolymer Blends– Particle, Block Nano-composites– Copolymer solutions, Polymersomes– Highly branched copolymers– Thermal fluctuations: Monte-Carlo sampling of partition function
• Examine more complex hydrodynamic flows:– Shearing flows: thin film shear and bulk shear flows– Verify shear flipping phenomenon?– Phase separation in presence of complex boundaries– Patterned Boundaries– Secondary flows in pressure driven channel simulations
• Extend ability to measure bulk material properties– Material strength vs. various deformations
• Automate adjustment of cell size to minimize external pressure effects
2/4/2005
Comparison of Mesoscale Dynamic TechniquesInstitution Unilever Mesodyn UCSB/LANL
Mesoscale Method
Discrete Particle Dynamics (DPD)
Dynamic Density Functional Theory (DDFT)
Hydrodynamic Self Consistent Field Theory (HSCFT)
Thermo-Model Small N, Bead and Spring modelNewtonian DynamicsSoft core potential
Small N, Bead and Spring modelFlory Huggins interactionsEntropy from Bead Spring model of single chain partition functionSoft core potential
SCFT: large N continuum polymer modelFlory Huggins interactionsEntropy from continuum single chain partition functionHard Core potential
Hydro-Model Newtonian Dynamics Simplified hydrodynamics:Diffusion equation andDarcy’s law
Full hydrodynamics: Two fluid Navier-Stokes with viscoelastic constitutive eqn.
Viscoelasticity Limited energy storage implicit in bead positions
Elastic dumbbells in thermo model
Maxwell Stress constitutive eqn.:Bulk and Shear Elastic stressesCaptures Dynamic Asymmetry
Approximations Short polymer chainsSoft core potentialResolution limited by # particles
Mean field,Incompressible,Short polymer chains
Mean field,Incompressible,Long polymer chains
2/4/2005
Peak wave number vs. Time• Spinodal decomposition:
– early on, a single wave number is selected to grow,– high freqs create too much area, – low freqs take more time to develop– Self similar exponential growth means the peak q should stay about the same…– At the transition, structures sharpen up, so high frequencies should appear.– In the late stages, as defects work themselves out, the structures become somewhat
more coarse,– so there should be so shift toward lower peak freq.
2/4/2005
Microphase-Microphase Transitions• Examine transitions between microphases in 3D as the temperature is changed.
• Sphere-Cylinder, Cylinder-Gyroid, Gyroid-Lamellae• Sphere-Gyroid, Sphere-Lamellae• Cylinder-Lamellae• And the reverse of all of these!
• Are the forward and reverse paths the same? Or is there some sort of morphological hysteresis?
• How does the rate of quenching affect all these transitions?• Can this process be used to form novel/useful structures?• Use Euler characteristics as well as other topological invariants to quantify
phase changes.
2/4/2005
Validation by Grid-Size Refinement• Show that the simulation results do not change as the grid point resolution is
increased, for a fixed set of parameters (lx, etc.)
2/4/2005
Effect of periodicity vs. Domain Size• Determine what size is necessary for measured quantities to be independent of
lx, ly, lz• For example tmax(w) seems to stabilize after lx=6 or so?
2/4/2005
Runtime vs. Number of Processors• Measure parallelizability of the algorithm.• Determine the optimal number of Grendels processors to use for a run of a
given size.
2/4/2005