energy-driven pattern formation: phase separation in diblock copolymer melts
DESCRIPTION
Energy-Driven Pattern Formation: Phase Separation in Diblock Copolymer Melts. David Bourne. Joint work with Mark Peletier. CASA Day, 11 April 2012. Diblock Copolymer Melts . Diblock Copolymer Melts . Figure from Choksi, Peletier & Williams (2009). Microphase Separation . - PowerPoint PPT PresentationTRANSCRIPT
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Energy-Driven Pattern Formation: Phase Separation in Diblock
Copolymer Melts
David Bourne
CASA Day, 11 April 2012
Joint work with Mark Peletier
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Diblock Copolymer Melts
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Diblock Copolymer Melts
Figure from Choksi, Peletier & Williams (2009)
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Microphase Separation
Figure from MIT OCW
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Microphase Separation
Figure from the Wiesner Group website, Cornell University
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Previous work
CASA:
• Mark Peletier
• Marco Veneroni
• Yves van Gennip
• Matthias Röger
Others:
Alberti, Cicalese, Choksi, Niethammer, Otto, Spadaro, Williams, …..
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Model
phasephase
1 ( )
0 A
vB
x
B
A
A
A
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Model
phasephase
1 ( )
0 A
vB
x
B
A
A
A
1| |
1( )v x dxn
Small volume fraction case:LARGE
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Model
phasephase
( )
0 n A
vB
x
B
A
A
A
1| | ( ) 1v x dx
Small volume fraction limit:
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Model: Energy
1/21) | | ,d( n v
n p nv n v dx vE
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Model: Energy
B
A
A
A
1/21) | | ,d( n v
n p nv n v dx vE
length of the phase interfacesn
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Model: Energy
B
A
A
A
1/21) | | ,d( n v
n p nv n v dx vE
length of the phase interfacesn
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Model: Energy
1/21) | | ,d( n v
n p nv n v dx vE
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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
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Zero volume fraction limit
• Complicated, nonlocal energy
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Zero volume fraction limit
• Complicated, nonlocal energy
• We are interested in the case large, i.e., where the volume fraction of is small
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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
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- Convergence
nE E
nv v Minimisers:
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- 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
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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
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Study of the Limit Energy
• Limit our attention to , square domain
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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
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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
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When
• For , “”
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When
• For , “”
• For fixed finite , the minimising pattern is a
centroidal Voronoi tessellation
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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
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Centroidal Voronoi Tessellations
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1
2( ) (1d , )M
ii
E m
When is in between: Numerical optimisation
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1
2( ) (1d , )M
ii
E m
• To evaluate have to solve an -dimensional linear programming problem
When is in between: Numerical optimisation
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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
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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
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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
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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
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Numerical Results
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Numerical Results
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Numerical Results
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Future Directions
• Comparison with experiments
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Future Directions
• Comparison with experiments
• Numerical exploration of the bifurcation diagram
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Future Directions
• Comparison with experiments
• Numerical exploration of the bifurcation diagram
• What can we prove about the limit pattern?