Variational analysis of a striped-pattern model

Transcription

1 Variational analysis of a striped-pattern model Mark Peletier TU Eindhoven Show three things: - variational analysis provides additional tools (for singular limit analysis) - show the tools on an example - develop some new maths on the way with: Matthias Röger, Yves van Gennip, Marco Veneroni, David Bourne

2 e microdomain interface between the HS and SI macrophaaes. Heteroarm star Stripes in Block Copolymers a - Y structure b c 200 nm 200 nm d e Koizumi et al 94 Ruzette & Leibler 05

3 Stripes in other systems: rocks

4 Stripes in other systems: rocks

5 Stripes in other systems: rocks

6 Stripes in other systems: rocks

7 Stripes in other systems: Rayleigh-Benard convection rolls c Plapp 1997

8 Two dimensions: numerical simulation Viñals & Boyer Swift-Hohenberg eq.

9 Two dimensions: numerical simulation Viñals & Boyer Swift-Hohenberg eq.

10 Central question Analysis of stripe patterns Study a specific stripe-making energy Understand why (and prove that) it prefers stripes Understand the structure of the stripes and defects and dynamics...

11 Block copolymers d A B 1 μm e 1 μm Ruzette & Leibler 2005

12 Energy functional u {0, 1} a.e. F(u) = u + distance(u,1 u) interfacial energy covalent bond u = 0 u = 1

13 Energy functional u {0, 1} a.e. F(u) = u + distance(u,1 u) interfacial energy covalent bond u = 0 u = 1 Ohta-Kawasaki 86, many others: distance(u,1 u) = u u 2 H 1

14 Energy functional u {0, 1} a.e. F(u) = u + distance(u,1 u) interfacial energy covalent bond u = 0 u = 1 Ohta-Kawasaki 86, many others: distance(u,1 u) = u u 2 H 1 Röger-P 08: distance(u,1 u) = d 1 (u,1 u) = u W 1,1

15 Programme Aim: Understand stripe-forming properties Strategy: Prove an asymptotic development in ε which favours stripes disallows defects penalizes curvature Restrictions: specific model, 2d,...

16 Rescaling: limit of thin layers u {0, 1} a.e. 200 nm F ε (u) = ε u + 1 ε u W 1,1 ε

17 Rescaling: limit of thin layers u {0, 1} a.e. 200 nm F ε (u) = ε u + 1 ε u W 1,1 ε ε ε Length scale ε

18 Rescaling: limit of thin layers u {0, 1} a.e. 200 nm F ε (u) = ε u + 1 ε u W 1,1 ε Basic scaling: we expect L = u 1 ε ε Interface length scales as 1/ε ε u W 1,1 ε F ε 1 Energy scales as 1

19 Gamma-convergence context F ε (u) = ε u + 1 ε u W 1,1 Take sequence ε 0 and functions u ε u ε = 1 2 u ε {0, 1} a.e. Study behaviour of u ε and F ε (u ε ) as ε 0

20 Lower bound in terms of geometry Fix ε > 0 and u Parametrize u by curve γ Ray geometry given by α, β, m ds u = 1 u = 0 β(s) α(s) γ(s) m(s) εds F ε 1 + L 0 2 L 1 m(s) 1 εds + sin β(s) 1 0 εds + ε2 4 L 0 α (s) 2 εds P-Röger 08

21 Energy scales F ε 1 + L 0 2 L 1 m(s) 1 εds + sin β(s) 1 0 εds + ε2 4 L 0 α (s) 2 εds F ε 1 + O(1) F ε 1 + O(ε) F ε 1 + O(ε 2 log ε) F ε 1 + O(ε 2 )

22 If the energy is small... then the geometry is simple F ε 1 ε 2 1 ε 2 L 0 m(s) 1 2 εds m 1: stripe thickness ε + 1 L 1 2 ε 2 sin β(s) 1 εds 0 β 90 : rays become normal L 0 α (s) 2 εds L 2 -norm of curvature

23 Stripes take uniform thickness ε u ε 1 2 in L p µ ε := ε u ε 1 2 L2 as measures

24 Stripes take uniform thickness ε u ε 1 2 in L p µ ε := ε u ε 1 2 L2 as measures No information about angles!

25 Stripes take uniform thickness ε u ε 1 2 in L p µ ε := ε u ε 1 2 L2 as measures No information about angles! On µ ε two vector fields are defined: normal direction ν ε S 1 ray direction θ ε S 1 ν ε ± θ ε 0

26 Limit description: projections As ε 0, stripes are characterized by direction: projection matrix grain boundaries are discontinuities slow variation is curvature

27 Limit description: projections As ε 0, stripes are characterized by direction: projection matrix grain boundaries are discontinuities slow variation is curvature

28 Limit description: projections As ε 0, stripes are characterized by direction: projection matrix grain boundaries are discontinuities Projection P R 2 2 : P 2 = P, rank P = 1, and slow P variation = 1 is curvature P = τ τ with τ = 1 (projection onto τ)

29 Gamma-limit G ε (u) := F ε(u) 1 ε 2 Theorem G ε Γ G 0 G 0 defined on projections P(x) =e(x) e(x) P semi-weak-sol eikonal eq: P div P = 0 curvature penalisation G 0 (P) = 1 4 div P 2 = 1 4 κ 2

30 Eikonal equation Constant-speed front movement Strong formulation: ϕ = 1 Weak formulations: 1. viscosity solutions (Crandall-Lions, Soner) 2. Jabin-Otto-Perthame zero-energy states All these consider vector-valued formulations

31 Eikonal equation Constant-speed front movement Strong formulation: ϕ = 1 Weak formulations: 1. viscosity solutions (Crandall-Lions, Soner) 2. Jabin-Otto-Perthame zero-energy states All these consider vector-valued formulations vector-valued line discontinuity projection-valued: point discontinuity P div P = 0

32 New eikonal equation formulation P projection div P L 2 P div P = 0 stronger than weak formulations: curvature in L 2 Advantages: natural in this variational context correct representation of discontinuities in roll fields

33 Conclusions and outlook System likes stripes, disallows defects Microstructural homogenization result Intermediate results suggest possibilities for point and line defects Powerful inequality in terms of geometry New formulation for the eikonal equation with different discontinuity handling

34 Conclusions and outlook System likes stripes, disallows defects Microstructural homogenization result Intermediate results suggest possibilities for point and line defects Powerful inequality in terms of geometry New formulation for the eikonal equation with different discontinuity handling Extend to defects Extend to SH-type systems Couple to dynamics?