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Energy-Driven Pattern Formation: Phase Separation in Diblock Copolymer Melts

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 Diblock Copolymer Melts

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  1. Energy-Driven Pattern Formation: Phase Separation in DiblockCopolymerMelts David Bourne Joint work with Mark Peletier CASA Day, 11 April 2012

  2. DiblockCopolymer Melts

  3. DiblockCopolymer Melts Figure from Choksi, Peletier & Williams (2009)

  4. Microphase Separation Figure from MIT OCW

  5. Microphase Separation Figure from the Wiesner Group website, Cornell University

  6. Previous work • CASA: • Mark Peletier • Marco Veneroni • Yves van Gennip • Matthias Röger Others: Alberti, Cicalese, Choksi, Niethammer, Otto, Spadaro, Williams, …..

  7. Model B A A A

  8. Model B A A A Small volume fraction case: LARGE

  9. Model B A A A Small volume fraction limit:

  10. Model: Energy

  11. Model: Energy B A A A

  12. Model: Energy B A A A

  13. Model: Energy

  14. Model: Energy B A A A

  15. Zero volume fraction limit • Complicated, nonlocal energy

  16. Zero volume fraction limit • Complicated, nonlocalenergy • Weareinterested in thecaselarge, i.e., wherethevolumefractionofissmall

  17. Zero volume fraction limit • Complicated, nonlocalenergy • Weareinterested in thecaselarge, i.e., wherethevolumefractionofissmall • So wesimplifytheenergybytaking

  18. - Convergence Minimisers:

  19. - Limit Theorem: The -limit ofthefunctionalsis where

  20. Ingredients of the Proof • 2nd Concentrated Compactness Lemma of P.-L. Lions • IsoperimetricInequality • Metrizationoftheweakconvergenceofmeasuresbythe Wasserstein metric

  21. Study of the Limit Energy • Limit ourattentionto, squaredomain

  22. Study of the Limit Energy • Limit ourattentionto, squaredomain • After rescaling so that is the unit square we get • where

  23. 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

  24. When • For , “”

  25. When • For , “” • Forfixedfinite , theminimisingpatternis a • centroidalVoronoitessellation

  26. When • For , “” • Forfixedfinite , theminimisingpatternis a • centroidalVoronoitessellation • i.e., thepointsareatthecentresofmassoftheVoronoicellsthattheygenerate, andtheweightsaretheareasoftheVoronoicells, where

  27. Centroidal Voronoi Tessellations

  28. When is in between: Numerical optimisation

  29. When is in between: Numerical optimisation • To evaluate have tosolvean-dimensionallinearprogrammingproblem

  30. When is in between: Numerical optimisation • To evaluate have tosolvean-dimensionallinearprogrammingproblem • Discretiseusing Gauss quadrature points andweightstoevaluateto at least 6 d.p.

  31. 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

  32. 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

  33. 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

  34. Numerical Results

  35. Numerical Results

  36. Numerical Results

  37. Future Directions • Comparisonwithexperiments

  38. Future Directions • Comparisonwithexperiments • Numericalexploration of the bifurcation diagram

  39. Future Directions • Comparisonwithexperiments • Numericalexploration of the bifurcation diagram • Whatcan we prove about the limit pattern?

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