Progress on a 'spectral-filtering' algorithm:increased speed in numerical SCFT simulations

Scott Sides, Glenn Fredrickson

≈1014 “atoms” τ>10 s

Why Field-Theoretic Computer Simulations ?

AB + A + B microemulsionPDMS melt: MD simulation~20,000 atoms

Few nanoseconds takesseveral days on tensof processors (Sandia)!

6 nm

≈1014 “atoms” τ>10 s

Why Field-Theoretic Computer Simulations ?AB + A + B microemulsion numerical SCFT simulation

A,B density fields

*Field theoretic formulation is amenable to a

battery of analytic and efficient numerical

solution methods...

Mean-Field Diblock Phase Diagram

lm3m HEX la3d LAM

f

“Spectral SCFT”Matsen-Bates (1995)compare known morphologies

GHF (12/98)

GHF (12/98)

“Numerical SCFT”search for new morphologies

Outline of self-consistent field theory (SCFT):Example: diblock copolymers

s

rA

rB

A

B

Partition function of n diblock chains

Fully-flexible chain model

“Numerical SCFT”search for new morphologies

After field theory transformation...

A

B

s

Transformed total partition function

* Mean-field approximation: find saddle-point configuration of fields *

Single-chain partition function

Compose monomer densities

Solve diffusion equation for q(r,s)

*Problem reduced to calculationof single-chain partition functionin chemical potential field

Algorithm outline / Simulation steps0. Initialize ω fields w/ random values

3. Find new ω fields

...continue until fields reach equilibrium configurations

1. Solve diffusion equation for q(r,s)

F

F

2. Compose monomer densities

GHF (12/98)

GHF (12/98)

“Numerical SCFT”search for new morphologies

χAB= 0.05χAC= 0.20χBC= 0.05

A B C

fA = fC = 0.25 fB = 0.50

iterations = 2000

Example: numerical SCFT simulation results

χAB= 0.05χAC= 0.20χBC= 0.05

A B C

fA = fC = 0.25 fB = 0.50

iterations = 4000

Example: numerical SCFT simulation results

?

f A = f B = 0.50 χAB N = 14.0

...without spectral filtering

Instead of complex ABC linear triblocklets try a simple AB linear diblock...

global minimum

“typical” update scheme

“better” update scheme ?

Topological defects in a real-space SCFT algorithmrandom initialcondition

local minimum

f A = f B = 0.50 χAB N = 14.0 ....with spectral filtering

Algorithm outline / Simulation stepsInitialize ω fields w/ random values

Find new ω fields

...continue until fields reach equilibrium configurations

Solve diffusion equation for q(r,s)

F

F

Compose monomer densities

Spectral filtering

Spectral filteringevery x (e.g. x=100) SCFT iterations

SCFT algorithmrefinefields normally

1D example: single spectral filtering step

fcut

f A = f B = 0.50 χAB N = 14.0

fE = -0.176 fE = -0.189

2D-examples

f A = 0.70 χAB N = 18.0

fE = -0.174 fE = -0.175

2D-examples

fA = 0.67

Ipdi = 1.5

fixed polydisperse

χNn = 18.0fA = 0.67

Ipdi = 1.0

fixed fixed

1. Equilibrate LAM, HEX etc for new modelse.g polydispersity

χNn = 18.0

quenched annealed

Why no defects?

2. Verify new space groupse.g. ABC triblock (FDDD)

Why no defects? with defects!

3. Block copolymer chains inconfinement:

Why few defects?

Lateral Confinement of Block CopolymerThin Films in Numerical SCFT:August Bosse: 2pm

f A = f B = 0.50 χAB N = 14.0

fE = -0.171 Lamellae

3D-examples

fE = -0.190

f A = 0.70 χAB N = 18.0

fE = -0.169 fE = -0.181HEX cylinders

3D-examples

f A = 0.80 χAB N = 25.0

fE = -0.081 fE = -0.086BCC spheres

3D-examples

f A = 0.60 χAB N = 13.5

fE = -0.069 Gyroid? fE = -0.068

3D-examples

Gyroid structure is “delicate” ?

LAM

gyroid

Sakurai et.al. J. Chem. Phys., 108 8 1998

f A = 0.60 χAB N = 13.5

fE = -0.069 Gyroid? fE = -0.068

3D-examples

fcut1

|k|-dependent spectral filter cutoff ? fcut --> fcut(|k|)

fcut2

fcut3

Evaporation Induced SelfEvaporation Induced Self--AssemblyAssembly

dip-coating

Precursor solution:1.38g P12310g ethanol

2.7g pH=2 H2O5.2g TEOS

•Aging (25oC, 65-75% RH):--Evaporization of ethanol,H2O, HCl.

--Liquid-crystalline mesophase locks in.•Calcined at 400oC for 3hr.

hydrophilic hydrophobic hydrophilic

EO EOPO

20 70 20

500nm

* Expt. cata courtesy of Prof. Stucky group at UCSB

3. Block copolymer chains in confinement:

Confined block copolymers:e.g. silicon nanowires:

Experimental results

SCFT results

Nature Materials 3, 816 (2004)

fE = -0.307 fE = -0.305 fE = -0.314

no filtering “annealing” filtering on

Spectral filtering & confinement

f A = 0.70 χAB N = 18.0

Summary for numerical SCFT code:1. Parallelization --> big

2. Spectral filtering --> fast

Thanks for helpful discussions:Dr. Eric CochranDr. Kirill Katsov

http://www.mrl.ucsb.edu/~swsides/Download_pscft/Code available formembers of Complex FluidsDesign Consortium (CFDC)

National Laboratory Partners include:

* Sandia National Laboratoryo Dr. Gary Gresto Dr. John Curro

* Los Alamos National Laboratory (LANL)Theoretical Division

o Dr. Tony Redondo o Dr. Turab Lookman

Current Corporate Partners include:

* Rhodia* General Electric* Atofina Chemicals* Mitsubishi Chemical Corporation* Nestle Corporation* Dow Chemical Corporation