Nematic colloids for photonic systems with FreeFem++

Transcription

1 Nematic colloids for photonic systems with FreeFem++ Iztok Bajc Adviser: Prof. dr. Slobodan Žumer Fakulteta za matematiko in fiziko Univerza v Ljubljani Slovenija

2 Outline Nematic liquid crystals Colloidal particles Methods/computations Photonic systems Modeling requirements in 3D!

3 Outline Nematic liquid crystals Colloidal particles Methods/computations Photonic systems Modeling requirements in 3D!

4 Most well-known application of liquid crystals?

5 Applications of liquid crystals LCD (Liquid Crystal Displays).

6 Applications of liquid crystals LCD (Liquid Crystal Displays). Eye protecting filters for welding helmets (Balder) Polarizing glasses for 3D vision Liquid crystals have unique optical properties.

7 Liquid crystals properties

8 Liquid crystals properties Liquid crystals are an oily material: flow like a liquid but are also partially ordered - like crystals.

9 Liquid crystals properties Liquid crystals are an oily material: flow like a liquid but are also partially ordered - like crystals. In nematic LC molecules are rodlike. Tend to align in a preferred direction.

10 Liquid crystals properties Liquid crystals are an oily material: flow like a liquid but are also partially ordered - like crystals. In nematic LC molecules are rodlike. Tend to align in a preferred direction. 5CB

11 Liquid crystals properties Liquid crystals are an oily material: flow like a liquid but are also partially ordered - like crystals. Isotropic liquid phase (higher temperature) In nematic LC molecules are rodlike. Tend to align in a preferred direction. 5CB

12 Liquid crystals properties Liquid crystals are an oily material: flow like a liquid but are also partially ordered - like crystals. In nematic LC molecules are rodlike. Tend to align in a preferred direction. Isotropic liquid phase (higher temperature) Low enough temperature Partially ordered mesophase 5CB

13 Liquid crystals properties Liquid crystals are an oily material: flow like a liquid but are also partially ordered - like crystals. In nematic LC molecules are rodlike. Tend to align in a preferred direction. Isotropic liquid phase (higher temperature) Low enough temperature Partially ordered mesophase E

14 Basic example of nematic structure Thin cell (~microns):

15 Description of nematic liquid crystals

16 Description of nematic liquid crystals Basic quantities

17 Description of nematic liquid crystals Basic quantities Director n (r) n 1 Points in preferenced orientation.

18 Description of nematic liquid crystals Basic quantities Scalar order parameter Director S(r ) 1 S 1 2 n (r) n 1 Points in preferenced orientation. Quantifies the degree of order of the director orientation:

19 Description of nematic liquid crystals Basic quantities Scalar order parameter Director S(r ) 1 S 1 2 n (r) n 1 Points in preferenced orientation. Quantifies the degree of order of the director orientation: S=0 isotropic liquid

20 Description of nematic liquid crystals Basic quantities Scalar order parameter Director S(r ) 1 S 1 2 n (r) n 1 Quantifies the degree of order of the director orientation: S=0 isotropic liquid Points in preferenced orientation. S=1 ideally aligned liquid (all molecules parallel)

21 Description of nematic liquid crystals Basic quantities Scalar order parameter Director S(r ) 1 S 1 2 n (r) n 1 Points in preferenced orientation. Quantifies the degree of order of the director orientation: S=0 isotropic liquid S=0.53 a typical intermediate bulk value (for 5CB) S=1 ideally aligned liquid (all molecules parallel)

22 Alternative description with Q-tensor field New quantity: tensor order parameter Q(r) : S Q S 2 P 2 3n n I e e e e its largest eigenvalue and n its corrispondent eigenvector Q traceless: Q Q 0 Q symmetric: Only 5 independent Q Q33 Q 11Q22 Qij Q ji components of Q are required. Q Q 11 Q Q Q Q Q Q 22

23 S Alternative description with Q S 2 Q-tensor field New quantity: tensor order parameter Q(r) : P 2 3n n I e e e e its largest eigenvalue and n its corrispondent eigenvector. Q traceless: Q Q 0 Q symmetric: Only 5 independent 1 1 Q Q33 Q 11Q22 Qij Q ji components of Q are required. Q 2 Q 2 11 Q Q Uniaxial approximation Q Q Q Q 22

24 Alternative description with Q-tensor field New quantity: tensor order parameter Q(r) : S Q S 2 3n n I its largest eigenvalue and n its corrispondent eigenvector. Q traceless: Q Q 0 Q symmetric: Only 5 independent Q Q33 Q 11Q22 Qij Q ji components of Q are required. Q Q 11 Q Q Q Q Q Q 22

25 Nematic LC structures

26 Nematic LC structures Possible nematic structures Minima of the free-energy F

27 Nematic LC structures Possible nematic structures Minima of the free-energy F Landau-de Gennes free-energy functional: F( Q) f ( Q, Q) dv f ( Q, Q) dv bulk bulk border surf

28 Nematic LC structures Possible nematic structures Minima of the free-energy F Landau-de Gennes free-energy functional: F( Q) f ( Q, Q) dv f ( Q, Q) dv bulk bulk surf border f 1 Q L Q 1 ij ij 2 bulk AQijQij BQijQ jkqki C(QijQij) 2 x k x k Elastic energy Thermodynamic energy 1 1 L elastic constants A, B, C material constants W surface energy

29 Nematic LC structures Possible nematic structures Minima of the free-energy F Landau-de Gennes free-energy functional: F( Q) f ( Q, Q) dv f ( Q, Q) dv bulk bulk surf border f 1 Q L Q 1 ij ij 2 bulk AQijQij BQijQ jkqki C(QijQij) 2 x k x k Elastic energy Thermodynamic energy 1 1 L elastic constants A, B, C material constants W surface energy f 1 2 (0) 2 surf W (Qij - Qij ) Surface energy

30 Outline Nematic liquid crystals Colloidal particles Methods/computations Photonic systems

31 One colloidal particle

32 One colloidal particle f 1 Q L Q 1 AQ Q 1 BQ Q 1 ij ij 2 bulk ij ij ij jk ki C(QijQij) 2 x k x k Bulk energy Q f 1 2 (0) 2 surf W (Qij - Qij ) Surface interaction energy L, A, B, C material constants W anchoring (surface) energy

33 One colloidal particle f 1 Q Q 1 ij ij 2 bulk L AQijQij BQijQ jkqki C(QijQij) 2 x k x k Bulk energy 1 f 1 2 (0) 2 surf W (Qij - Qij ) Surface interaction energy L, A, B, C material constants W anchoring (surface) energy

34 One colloidal particle f 1 Q Q 1 ij ij 2 bulk L AQijQij BQijQ jkqki C(QijQij) 2 x k x k Bulk energy 1 f 1 2 (0) 2 surf W (Qij - Qij ) Surface interaction energy L, A, B, C material constants W anchoring (surface) energy

35 Several colloidal particles Inclusion of colloidal particles Photos: I. Muševič group, PRE, PRL, Science,

36 Several colloidal particles Inclusion of colloidal particles Disclination lines (topological defects): Photos: I. Muševič group, PRE, PRL, Science,

37 Several colloidal particles Inclusion of colloidal particles Disclination lines (topological defects): Strong attractive forces Photos: I. Muševič group, PRE, PRL, Science,

38 Several colloidal particles Inclusion of colloidal particles Disclination lines (topological defects): Strong attractive forces Colloidal structures - crystals in nematic. Photos: I. Muševič group, PRE, PRL, Science,

39 Several colloidal particles Inclusion of colloidal particles Disclination lines (topological defects): Strong attractive forces Colloidal structures - crystals in nematic. 1D structures Photos: I. Muševič group, PRE, PRL, Science,

40 Several colloidal particles Inclusion of colloidal particles Disclination lines (topological defects): Strong attractive forces Colloidal structures - crystals in nematic. 2D structures - crystals 1D structures Photos: I. Muševič group, PRE, PRL, Science,

41 Large 3D structures - crystals: Final aim dipolar crystal. Experiment by Andriy Nych, 2010 (to be published).

42 Large 3D structures - crystals: Final aim dipolar crystal. Experiment by Andriy Nych, 2010 (to be published). In the meanwhile:

43 Large 3D structures - crystals: Final aim dipolar crystal. Experiment by Andriy Nych, 2010 (to be published). In the meanwhile: Škarabot, Ravnik et al., PRE, 2008.

44 Outline Nematic liquid crystals Colloidal particles Methods/computations Photonic systems

45 Some already made simulations M. Ravnik, S. Žumer, Soft Matter, 2009.

46 Computational requirements Use of uniform grid becomes impracticable (time/memory) for larger systems or localized resolutions. Nonuniform grid required, for ex. with the finite element method (FEM). Strategy: mesh adaptivity less degrees of freedom (so less memory/time) more details given only where needed (e.g. around defects) A priori Metric based

47 New modeling requirements Mesh adaptivity in 3D, preferably with anisotropic metric. Moving objects (due to nematic elastic forces). Parallel processing (computer clusters). Meshes by Cécile Dobrzynski, Institut de Mathématiques de Bordeaux.

48 Newton iteration of tensor fields

49 Newton iteration of tensor fields F(Q) min: F( Q) F' ( Q) Q 0 Euler-Lagrange equations

50 Newton iteration of tensor fields F(Q) min: F( Q) F' ( Q) Q 0 Euler-Lagrange equations F( Q) f ( Q, Q) dv f Q r LQ ij f QdV ( Q) ij f QdV ( Q) 0 AQ BQ Q CQ Q Q dv W Q Q da ij ik kj ik kl lj ij ij ij ij

51 Newton iteration of tensor fields F(Q) min: F( Q) F' ( Q) Q 0 Euler-Lagrange equations F( Q) f ( Q, Q) dv f Q r LQ ij f QdV ( Q) ij f QdV ( Q) 0 AQ BQ Q CQ Q Q dv W Q Q da ij ik kj ik kl lj ij LQ W (Q ij ij ij ij ij - Q 0 ij )

52 Newton iteration of tensor fields F(Q) min: F( Q) F' ( Q) Q 0 Euler-Lagrange equations F( Q) f ( Q, Q) dv f Q r LQ ij f QdV ( Q) ij f QdV ( Q) 0 AQ BQ Q CQ Q Q dv W Q Q da ij ik kj ik kl lj ij LQ W (Q ij ij ij ij ij - Q 0 ij ) F '' ( Q k ) v F' ( Q ) k ( - test functions) k Newton iteration equation Q k1 Q k v k (next iteration step)

53 Outline Nematic liquid crystals Colloidal particles Methods/computations Photonic systems

54 1. Two particles (dimer) 3D geometry: Interior Profile cross section Particle diameter: 1 um Boundary condition: strong radial (energy density W=1e-2 J/m^2). Cell dimensions: side: [um]. Floor, bottom, walls: strong vertical BC Inbetween particles and box: nematic liquid crystal (5CB).

55 Tetrahedral mesh (cross section) ~ mesh points. Mesh adaptivity used: metric based. At the moment isotropic adaptivity but anisotropic being setting up. (prooved to be globally quasi-optimal) Director field (cross section) Code written in FreeFem++: ~ 2000 lines.

56 2. One large + two small particles 3D geometry: Interior Cross section 1 large particle: 1 um 2 small particles: 0,1 um Boundary conditions: strong radial (energy density W=1e-2 J/m^2). Cubical simulation cell: side = 2 um. Floor, bottom, walls: strong vertical BC. Inbetween particle and box: nematic liquid crystal (5CB).

57 2. Tetrahedral mesh profile cross section ~ mesh points. Mesh generator: TetGen Metric: mshmet

58 2. Director field profile cross section ~ mesh points. Mesh generator: TetGen Metric: mshmet

59 Scalar order parameter S Topological defects

60 Scalar order parameter S Topological defects

61 Outline Nematic liquid crystals Colloidal particles Methods/computations Photonic systems

62 New potential applications: metamaterials, microresonators

63 New potential applications: metamaterials, microresonators Photonic crystals:

64 New potential applications: metamaterials, microresonators Photonic crystals: Solid state metamaterials:

65 New potential applications: metamaterials, microresonators Photonic crystals: Solid state metamaterials: Soft metamaterials?

66 New potential applications: metamaterials, microresonators Photonic crystals: Nematic droplet. Solid state metamaterials: Soft metamaterials? Figures: I. Muševič, CLC Ljubljana Conference, 2010.

67 New potential applications: metamaterials, microresonators Photonic crystals: Nematic droplet. Whispering Gallery Modes (WGM ) in a microresonator. Solid state metamaterials: Soft metamaterials? Figures: I. Muševič, CLC Ljubljana Conference, 2010.

68 New potential applications: metamaterials, microresonators Photonic crystals: Nematic droplet. Whispering Gallery Modes (WGM ) in a microresonator. Solid state metamaterials: Soft metamaterials? Figures: I. Muševič, CLC Ljubljana Conference, 2010.

69 Nematic droplet E 0 By tuning electric field we switch between optical modes. Figures: I. Muševič, CLC Ljubljana Conference, 2010.

70 Computational photonics Detail dimensions comparable with wavelength. Ray optics not adequate. Numerical solution of full Maxwell equations Time-harmonic expansion 1) Frequency-domain eigenproblems 2) Frequency-domain response

71 Frequency domain eigenproblems ( r) 1 H 0 H c 2 H Eigenequation (+ condition) Reduces to a matrix eigenproblem: Ax 2 Bx Škarabot, Ravnik et al., PRE, 2008.

72 Future work Movement of particles to stationary positions movement of mesh. Initial mesh and starting guess for general setting of spherical particles (should be good enough in order to converge). Anisotropic mesh adaptivity (code module mmg3d). Visualization. EM code for solving Maxwell eqns in nonhomogeneously anisotropic media.

73 Acknowledgments: Slobodan Žumer (adviser), Faculty of Mathematics and Physics, Ljubljana, Slovenija. Frédéric Hecht, Laboratoire Jacques-Louis Lions, UPMC, Paris. Miha Ravnik, Faculty of Mathematics and Physics, Ljubljana. Igor Muševič & experimental group, Jožef Stefan Institute, Ljubljana. Pascal Frey, Laboratoire Jacques-Louis Lions, UPMC, Paris. Cécile Dobrzynski, Institut de Mathématiques de Bordeaux. Work has been finansed by EU: Hierarchy Project, Marie-Curie Actions