Energy-Driven Pattern Formation: Phase Separation in Diblock

Copolymer Melts

David Bourne

CASA Day, 11 April 2012

Joint work with Mark Peletier

Diblock Copolymer Melts

Diblock Copolymer Melts

Figure from Choksi, Peletier & Williams (2009)

Microphase Separation

Figure from MIT OCW

Microphase Separation

Figure from the Wiesner Group website, Cornell University

Previous work

CASA:

• Mark Peletier

• Marco Veneroni

• Yves van Gennip

• Matthias Röger

Others:

Alberti, Cicalese, Choksi, Niethammer, Otto, Spadaro, Williams, …..

Model

phasephase

1 ( )

0 A

vB

x

B

A

A

A

Model

phasephase

1 ( )

0 A

vB

x

B

A

A

A

1| |

1( )v x dxn

Small volume fraction case:LARGE

Model

phasephase

( )

0 n A

vB

x

B

A

A

A

1| | ( ) 1v x dx

Small volume fraction limit:

Model: Energy

1/21) | | ,d( n v

n p nv n v dx vE

Model: Energy

B

A

A

A

1/21) | | ,d( n v

n p nv n v dx vE

length of the phase interfacesn

Model: Energy

B

A

A

A

1/21) | | ,d( n v

n p nv n v dx vE

length of the phase interfacesn

Model: Energy

1/21) | | ,d( n v

n p nv n v dx vE

Model: Energy

1/21) | | ,d( n v

n p nv n v dx vE

Wasserstein distancep

B

A

A

A

Minimal cost of transporting every po to somei

w

nt

ith cost | |pB y A

x y

x

Zero volume fraction limit

• Complicated, nonlocal energy

Zero volume fraction limit

• Complicated, nonlocal energy

• We are interested in the case large, i.e., where the volume fraction of is small

Zero volume fraction limit

• Complicated, nonlocal energy

• We are interested in the case large, i.e., where the volume fraction of is small

• So we simplify the energy by taking

n

- Convergence

nE E

nv v Minimisers:

- Limit

Theorem: The -limit of the functionals is

where

1/2( ) 2 (d 1, )i pi

mE

, | |, ii x i i

i i

m mx

1x2x

3x

Ingredients of the Proof

• 2nd Concentrated Compactness Lemma of P.-L. Lions

• Isoperimetric Inequality

• Metrization of the weak convergence of measures by the Wasserstein metric

Study of the Limit Energy

• Limit our attention to , square domain

Study of the Limit Energy

• Limit our attention to , square domain

• After rescaling so that is the unit square we get

where 12( ) (1d , )

M

ii

E m

1 1

[0,1, ] [0,1], 1 i

M M

i x i ii i

xm m

Study of the Limit Energy

• Limit our attention to , square domain

• After rescaling so that is the unit square we get

where

• The parameter determines for the minimiser and the

minimising pattern

1 1

[0,1, ] [0,1], 1 i

M M

i x i ii i

xm m

12( ) (1d , )

M

ii

E m

 

 

When

• For , “”

When

• For , “”

• For fixed finite , the minimising pattern is a

centroidal Voronoi tessellation

When

• For , “”

• For fixed finite , the minimising pattern is a

centroidal Voronoi tessellation

i.e., the points are at the centres of mass of the Voronoi cells that they generate, and the weights are the areas of the Voronoi cells, where

{ : | | | | }i i jx x x x xV j i

Centroidal Voronoi Tessellations

1

2( ) (1d , )M

ii

E m

When is in between: Numerical optimisation

1

2( ) (1d , )M

ii

E m

• To evaluate have to solve an -dimensional linear programming problem

When is in between: Numerical optimisation

1

2( ) (1d , )M

ii

E m

• To evaluate have to solve an -dimensional linear programming problem

• Discretise using Gauss quadrature points and weights to evaluate to at least 6 d.p.

When is in between: Numerical optimisation

1

2( ) (1d , )M

ii

E m

• To evaluate have to solve an -dimensional linear programming problem

• Discretise using Gauss quadrature points and weights to evaluate to at least 6 d.p.

• Minimise in MATLAB using fmincon for fixed, then find optimal M

When is in between: Numerical optimisation

1

2( ) (1d , )M

ii

E m

• To evaluate have to solve an -dimensional linear programming problem

• Discretise using Gauss quadrature points and weights to evaluate to at least 6 d.p.

• Minimise in MATLAB using fmincon for fixed, then find optimal M

• Good news: CVT is a very good initial guess. Easy to compute using Lloyd’s algorithm

When is in between: Numerical optimisation

1

2( ) (1d , )M

ii

E m

• To evaluate have to solve an -dimensional linear programming problem

• Discretise using Gauss quadrature points and weights to evaluate to at least 6 d.p.

• Minimise in MATLAB using fmincon for fixed, then find optimal M

• Good news: CVT is a very good initial guess. Easy to compute using Lloyd’s algorithm

• Bad news: CVT is a very good initial guess. Need to work to high accuracy to see that the minimiser isn’t a CVT

When is in between: Numerical optimisation

Numerical Results

Numerical Results

Numerical Results

Future Directions

• Comparison with experiments

Future Directions

• Comparison with experiments

• Numerical exploration of the bifurcation diagram

Future Directions

• Comparison with experiments

• Numerical exploration of the bifurcation diagram

• What can we prove about the limit pattern?