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

Disorder in Self-Assembly

Dean et al; Macromolecules 2003

Goal: Increaseepoxy toughness

Amphiphilic blockcopolymers

5% copolymerloading yields 40x to70x increase infracture resistance

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

?

Mean Field Assumption

Assume a single configuration dominates thepartition function

Fluctuations are suppressed

!

Z = D[W ]e"H [W ]# $ e"H [W*]

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')

Numerical Limitations

System stability and accuracy require restrictively smallstep sizes

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[ ] +"

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 %

& ' '

(

) * *

Method Comparison

Diblock Copolymer Model

!

H[W±] = C dr[(2 f "1)W"# " iW+[+W"

2($N)"1]"V lnQ]

Pressure and Difference Fields

01f

Appearance of Micelles

Shift in the Order Parameter

C = 240

Shift in the Order Parameter

C = 60

Shift in the Order Parameter

C = 30

Future Work

Continue work on efficient numerical methods

Extend simulations into larger systems

Address physical phenomena intractable bycurrent methodology

Diblock/Homopolymer Blends

Matsen; Phys. Rev. Letters 1995

Homopolymer Volume Fraction

Frac

tion

A in

A-B

Dib

lock

?A-B Diblock

A Homopolymer

Micelles in Disorder

New ODT

Unbinding transition

Critical MicelleConcentration

Fluctuations

SCFT Experiments

Diblock as Surfactant

Ternary Blend

Diblock/Homopolymer Blend

Fluctuations

SCFT ExperimentsA-B Diblock

A HomopolymerB Homopolymer

Microemulsions

Bates et al; PRL 1997

Fluctuation Destroy Unbinding Transition

Narrow Region!

Stabilizing Microemulsions

PolydispersityA-B asymmetriesChain Length Variations

Hillmyer et al; J Phys Chem B 1999

Stabilizing MicroemulsionsDiblocks

Symmetric

Varied Length

Varied Composition

+ =

Homopolymers

?

Hillmyer et al; J Phys Chem B 1999

Summary

Improved Numerical MethodsFull Complex Langevin Dynamics tractable

Application to Neat Diblock System

Future Work with Diblock Homopolymer Blends

Acknowledgements

Hector Ceniceros Carlos Garcia-Cervera Kirill Katsov Eric Cochran George Mohler

Happy Birthday Paul Langevin!Born today, 134 years ago