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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 .
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Energy-Driven Pattern Formation: Phase Separation in DiblockCopolymerMelts David Bourne Joint work with Mark Peletier CASA Day, 11 April 2012
DiblockCopolymer 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 B A A A
Model B A A A Small volume fraction case: LARGE
Model B A A A Small volume fraction limit:
Model: Energy B A A A
Model: Energy B A A A
Model: Energy B A A A
Zero volume fraction limit • Complicated, nonlocal energy
Zero volume fraction limit • Complicated, nonlocalenergy • Weareinterested in thecaselarge, i.e., wherethevolumefractionofissmall
Zero volume fraction limit • Complicated, nonlocalenergy • Weareinterested in thecaselarge, i.e., wherethevolumefractionofissmall • So wesimplifytheenergybytaking
- Convergence Minimisers:
- Limit Theorem: The -limit ofthefunctionalsis where
Ingredients of the Proof • 2nd Concentrated Compactness Lemma of P.-L. Lions • IsoperimetricInequality • Metrizationoftheweakconvergenceofmeasuresbythe Wasserstein metric
Study of the Limit Energy • Limit ourattentionto, squaredomain
Study of the Limit Energy • Limit ourattentionto, squaredomain • After rescaling so that is the unit square we get • where
Study of the Limit Energy • Limit ourattentionto, squaredomain • After rescaling so that is the unit square we get • where • The parameter determinesfor the minimiserand the minimisingpattern
When • For , “”
When • For , “” • Forfixedfinite , theminimisingpatternis a • centroidalVoronoitessellation
When • For , “” • Forfixedfinite , theminimisingpatternis a • centroidalVoronoitessellation • i.e., thepointsareatthecentresofmassoftheVoronoicellsthattheygenerate, andtheweightsaretheareasoftheVoronoicells, where
When is in between: Numerical optimisation • To evaluate have tosolvean-dimensionallinearprogrammingproblem
When is in between: Numerical optimisation • To evaluate have tosolvean-dimensionallinearprogrammingproblem • Discretiseusing Gauss quadrature points andweightstoevaluateto at least 6 d.p.
When is in between: Numerical optimisation • To evaluate have tosolvean-dimensionallinearprogrammingproblem • Discretiseusing Gauss quadrature points andweightstoevaluateto at least 6 d.p. • Minimise in MATLAB usingfminconforfixed, thenfindoptimalM
When is in between: Numerical optimisation • To evaluate have tosolvean-dimensionallinearprogrammingproblem • Discretiseusing Gauss quadrature points andweightstoevaluateto at least 6 d.p. • Minimise in MATLAB usingfminconforfixed, thenfindoptimalM • Good news: CVT is a verygoodinitialguess. Easy tocomputeusingLloyd’salgorithm
When is in between: Numerical optimisation • To evaluate have tosolvean-dimensionallinearprogrammingproblem • Discretiseusing Gauss quadrature points andweightstoevaluateto at least 6 d.p. • Minimise in MATLAB usingfminconforfixed, thenfindoptimalM • Good news: CVT is a verygoodinitialguess. Easy tocomputeusingLloyd’salgorithm • Bad news: CVT is a verygoodinitialguess. Needtoworkto high accuracytoseethat the minimiserisn’t a CVT
Future Directions • Comparisonwithexperiments
Future Directions • Comparisonwithexperiments • Numericalexploration of the bifurcation diagram
Future Directions • Comparisonwithexperiments • Numericalexploration of the bifurcation diagram • Whatcan we prove about the limit pattern?