1. Samo Kralj 1,2, Riccardo Rosso 3, Epifanio G. Virga 3 1 Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia 2 Jozef Stefan Institute, Ljubljana, Slovenia 3 Department of Mathematics, University of Pavia, Italy LIQUID CRYSTAL NEMATIC CONFIGURATIONS ON THIN FILMS

2. Liquid crystal phases : Important role in several natural systems. Main advantages: softness (= susceptibility) optical transparency + anisotropy richness of phases & structures Confinement : surface local interactions (affecting translational&orientational ordering) symmetry breaking finite size effects

3. LIQUID CRZSTALS Focus 1) complex behavior in thin nematic hybrid films (frustrations + topological defects) new boojum structure interaction of boojum with elastic distortions boojum dragging towards cell interior defect core enhancement

4. 2) thin nematic shells new 2D Q-tensor mesoscopic approach character of the I-N transition defect structures on ellipsoidal shells

5. Complex behavior : 1)Boojum 2)Frustration 3)Finite size effects 4)External field I) THIN NEMATIC HYBRID FILMS

6. Half of hedgehog ? Well known HEDGEHOG biaxial structure includes order reconstruction Expected boojum structure ?

7. h > h c h < h c Order reconstruction: h<h c

8. Mesoscopic modelling F Uniaxial states : Degree of biaxiality :

9. Cylindrical coordinate system, parametrization (cylindrical symmetry, no twist) Phys. Rev. E 78, 031701 (2008); 81, 021702 (2010).

10. Adequate parametrization for visualization of biaxial states TrQ=0 Phys. Rev. E 81, 021702 (2010). states with negative uniaxiality states with positive uniaxiality states with maximal biaxiality

12. I) RESULS A A B C C ?

13. Half uniaxial hedgehog

14. Half biaxial hedgehog

15. Observed boojum structure

16. “Finger” boojum structure,  2 (r,z)

17. Naively expected biaxial boojum structure

18. Interaction boojum : order reconstruction structure in thin films

19. A B C D

20. boojum can lift the order reconstruction structure

21. Boojum pushed to the top, the order reconstruction structure locally follows it

26. S S=0

31. Width of the elongated boojum ?

32. Planar radial with a negative uniaxial core ER = escaped radial Phys.Rev.E 60, 1858 (1999).

33. Phys.Rev.E 66, 021703 (2002).

34. Dimensionless excess free energy : Phys.Rev.E 60, 1858 (1999). External field contribution Bulk nematic ordering:  =-  /3 = negative uniaxiality

35.  def

37. II) THIN NEMATIC SHELLS eigenframe general frame e i : chosen along the lines of principal curvatures of the surface

38. The surface gradients:

39. Symmetry invariant terms entering the free energy density Condensation term Elastic term, K: Gaussian curvature

40. Uniaxial ellipsoidal surfaces obtained by rotating the ellipse v : meridians u : parallels

41. In agreement with : Director field representation semi-microscopic simulations

43. Oblate surfaces (sphere :  =1)

46. Prolate surfaces (sphere :  =1)

48. Conclusions Complex nematic structures in thin films Rich variety of structures -> interplay among geometrical constraints, elastic forces and finite size effects Of interest for future nanobased electrooptic devices

49. Nelson, Nano. Lett. 2, 1125 (2002) Nematic shells immersed in a solution of an isotropic liquid and flexible linkers = SCALED ATOMA (defect sites > determine valence)