Beyond-mean-field corrections and effective interactions ...
Beyond the Mean Field - UCSB MRSECghf/cfdc_2006/lennon_cfdc_2006.pdf · Beyond the Mean Field:...
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Beyond the Mean Field:Efficiently Implementing theComplex Langevin Method
Erin M. LennonGlenn H. Fredrickson
Department of Chemical EngineeringUniversity of California, Santa Barbara
Complex Fluids Design Consortium Annual MeetingJanuary 23, 2006
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Disorder in Self-Assembly
Dean et al; Macromolecules 2003
Goal: Increaseepoxy toughness
Amphiphilic blockcopolymers
5% copolymerloading yields 40x to70x increase infracture resistance
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Mean Field Results
Model as a blend ofcopolymer andhomopolymer
Hypothetical micellarregion
Matsen; Phys. Rev. Letters 1995
Homopolymer Volume Fraction
Frac
tion
A in
A-B
Dib
lock
?
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Mean Field Assumption
Assume a single configuration dominates thepartition function
Fluctuations are suppressed
!
Z = D[W ]e"H [W ]# $ e"H [W*]
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Complex Langevin Solution Stochastic sampling of fields
Simultaneous excitation and relaxation
!
"
"tW (r,t) = #$
%H[W ]
%W (r,t)
&
' (
)
* + +,(r,t)
!
"(r,t) = 0
"(r,t)"(r',t') = 2#$(r % r')$(t % t')
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Numerical Limitations
System stability and accuracy require restrictively smallstep sizes
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Novel Method: SDE Splitting
2nd order Semi-Implicit method
!
dx = (A(x) + B(x))dt +"
xn +1 = x
n +#t
2A
xn+1 + B
˜ x + A
xn + B
xn[ ] +"
˜ x = xn + #t A
xn + B
xn[ ] +"
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Novel Method: SDE Splitting
Example:
!
W˜ n = W
n + "t#H
#W
n
+$
ˆ W n +1 =
1
2 + "t ˆ g T
ˆ W n + ˆ $ + "t ˆ g T +1( ) ˆ W
˜ n + "t# ˆ H
#W
˜ n %
& ' '
(
) * *
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Method Comparison
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Diblock Copolymer Model
!
H[W±] = C dr[(2 f "1)W"# " iW+[+W"
2($N)"1]"V lnQ]
Pressure and Difference Fields
01f
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Appearance of Micelles
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Shift in the Order Parameter
C = 240
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Shift in the Order Parameter
C = 60
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Shift in the Order Parameter
C = 30
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Future Work
Continue work on efficient numerical methods
Extend simulations into larger systems
Address physical phenomena intractable bycurrent methodology
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Diblock/Homopolymer Blends
Matsen; Phys. Rev. Letters 1995
Homopolymer Volume Fraction
Frac
tion
A in
A-B
Dib
lock
?A-B Diblock
A Homopolymer
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Micelles in Disorder
New ODT
Unbinding transition
Critical MicelleConcentration
Fluctuations
SCFT Experiments
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Diblock as Surfactant
Ternary Blend
Diblock/Homopolymer Blend
Fluctuations
SCFT ExperimentsA-B Diblock
A HomopolymerB Homopolymer
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Microemulsions
Bates et al; PRL 1997
Fluctuation Destroy Unbinding Transition
Narrow Region!
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Stabilizing Microemulsions
PolydispersityA-B asymmetriesChain Length Variations
Hillmyer et al; J Phys Chem B 1999
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Stabilizing MicroemulsionsDiblocks
Symmetric
Varied Length
Varied Composition
+ =
Homopolymers
?
Hillmyer et al; J Phys Chem B 1999
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Summary
Improved Numerical MethodsFull Complex Langevin Dynamics tractable
Application to Neat Diblock System
Future Work with Diblock Homopolymer Blends
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Acknowledgements
Hector Ceniceros Carlos Garcia-Cervera Kirill Katsov Eric Cochran George Mohler
Happy Birthday Paul Langevin!Born today, 134 years ago