1. Soft elasticity in nematic elastomers and wetting properties of rough surfaces. Or …Bridging across length scales with mathematical analysis: two case studies Antonio DeSimone SISSA International School for Advanced Studies Trieste, Italy IMA Workshop on Modeling of Soft Matter, Minneapolis 29.09.04 desimone@sissa.it

2. 2 Nematic Elastomers and Rough Wetting, Antonio DeSimone, SISSA (Trieste, ITALY) Theme and Outline Macroscopic material properties whose physics is decided at the micro scale (a microscope reveals underlying mechanism) Derive coarse grained model capturing the decisive microscopic features (take micro model and pass to the macroscopic limit) Joint with: S. Conti, Leipzig; G. Dolzmann, Maryland G. Alberti, Pisa Two case studies: Soft elasticity in nematic elastomersSuper-hydrophobic surfaces

3. 3 Nematic Elastomers and Rough Wetting, Antonio DeSimone, SISSA (Trieste, ITALY) What are Nematic Elastomers? Cross-linked networks of polymeric chains containing nematic mesogens: alignement of mesogens along average direction n induces spontaneous distorsion of chains n n nematic director, | n |=1 a volume preserving uniaxial extension along n of magnitude a -1/3  1 ( a  1) Spontaneous distortion: (H. Finkelmann) (contractions orthogonal to n of magnitude a 1/6  1)

4. 4 Nematic Elastomers and Rough Wetting, Antonio DeSimone, SISSA (Trieste, ITALY) Finkelmann’s stretching experiment Fixed temperature below T IN Initial configuration: s=1, n=e 3. Stretch along e 2 with rigid clamps: n=e 3 2: non-uniform director reorientation (stripe-domain patterns) (H. Finkelmann) 1: stress-strain curve has plateau (soft elasticity) (E. Terentjev)

5. 5 Nematic Elastomers and Rough Wetting, Antonio DeSimone, SISSA (Trieste, ITALY) Energetic interpretation of stripe domain instability, set of zero energy states only ifor if i.e., while n=e 3 n=e 2

6. 6 Nematic Elastomers and Rough Wetting, Antonio DeSimone, SISSA (Trieste, ITALY) Coarse-graining? Coarse-grained model for the red box? A zero-energy (soft) deformation path: cost of imposing local deformations as in the red box, accounting for the possibility of forming microstructures if this is energetically advantageous.

7. 7 Nematic Elastomers and Rough Wetting, Antonio DeSimone, SISSA (Trieste, ITALY) W qc for W after Bladon-Warner-Terentjev Theorem (DS and Dolzmann). if det F = 1, W qc ( F ) = +  if det F  1. Proof: find matching upper and lower bounds for W qc. (polyconvex lowerbound which can be achieved by sequential lamination) where 1 (F)  2 (F)  3 (F) singular values of def. grad. F=  y, and

8. 8 Nematic Elastomers and Rough Wetting, Antonio DeSimone, SISSA (Trieste, ITALY) Coarse-grained energy: Phase Diagram No MicrostructuresMicrostructures K (i.e., states with W=0) Three collective modes of mesoscopic response and corrsponding microstructures W qc = W W qc = 0 W qc = F(t)

9. 9 Nematic Elastomers and Rough Wetting, Antonio DeSimone, SISSA (Trieste, ITALY) A numerical stretching experiment

10. 10 Nematic Elastomers and Rough Wetting, Antonio DeSimone, SISSA (Trieste, ITALY) Theory vs. Experiment Multiscale numerics: (Conti, DS, Dolzmann) X-ray scattering (Zubarev et al.) Direct observation (Terentjev)

11. 11 Nematic Elastomers and Rough Wetting, Antonio DeSimone, SISSA (Trieste, ITALY) References on Nematic Elastomers  A. DeSimone, Energetics of fine domain structures, Ferroelectrics, vol. 222, p. 272 (1999).  A. DeSimone and G. Dolzmann, Material instabilities in nematic elastomers, Physica D, vol. 136, p. 175 (2000).  A. DeSimone and G. Dolzmann, Macroscopic response of nematic elastomers via relaxation of a class of SO(3)-invariant energies, Archive for Rational Mechanics and Analysis, vol. 161, p. 181 (2002).  S. Conti, A. DeSimone, and G. Dolzmann, Soft elastic response of stretched sheets of nematic elastomers: a numerical study, Journal of the Mechanics and Physics of Solids, vol. 50, p. 1431 (2002).  S. Conti, A. DeSimone, and G. Dolzmann, Semi-soft elasticity and director reorientation in stretched sheets of nematic elastomer, Physical Review E, vol. 60, p. 61710 (2002). Many people have contributed to literature on nematic elastomers and gels, including: Lubensky, Finkelmann, Zentel, Brand, Pleiner, de Gennes, Kremer, Ratna, Selinger, Palffy-Muhoray, Meyer, Fried, Silhavy, Ponte Castaneda;................. Monograph: M. Warner and E. Terentjev, Liquid Crystal Elastomers, Oxford University Press, 2003.

12. 12 Nematic Elastomers and Rough Wetting, Antonio DeSimone, SISSA (Trieste, ITALY) Part 2: Wetting of rough surfaces

13. 13 Nematic Elastomers and Rough Wetting, Antonio DeSimone, SISSA (Trieste, ITALY) Nelumbo nucifera (lotus)

14. 14 Nematic Elastomers and Rough Wetting, Antonio DeSimone, SISSA (Trieste, ITALY) The sacred purity of lotus leaves 10  m

15. 15 Nematic Elastomers and Rough Wetting, Antonio DeSimone, SISSA (Trieste, ITALY) Artificial surfaces: nanograss of vertically aligned carbon nanotubes 50 µm (G. McKinley) 1  m

16. 16 Nematic Elastomers and Rough Wetting, Antonio DeSimone, SISSA (Trieste, ITALY) Water droplet on nanograss: superhydrophobicity 3 mm (G. McKinley)

17. 17 Nematic Elastomers and Rough Wetting, Antonio DeSimone, SISSA (Trieste, ITALY) Fakir droplets (K. Hashimoto)

18. 18 Nematic Elastomers and Rough Wetting, Antonio DeSimone, SISSA (Trieste, ITALY) Coarse graining: wetting properties of a hairy planar surface hom     min!  contact angle  (A) (B) hom  min!   hom Equivalent energy (hydrophobic case):

19. 19 Nematic Elastomers and Rough Wetting, Antonio DeSimone, SISSA (Trieste, ITALY) Hydrophobic case continued (A)(B)(C) (A) (B) Alberti & DS, Proc. Roy. Soc. London, 2004  where SL.... too bold claims are for loosers... Wenzel Cassie-BaxterMixed Mode

20. 20 Nematic Elastomers and Rough Wetting, Antonio DeSimone, SISSA (Trieste, ITALY) Statics predicts “liquid marbles” Do these marbles roll? Do they bounce?  = 170° (G. McKinley)

21. 21 Nematic Elastomers and Rough Wetting, Antonio DeSimone, SISSA (Trieste, ITALY) Self cleaning of superhydrophobic surfaces Lotusan TM 50  m (Lotus-Effect ®, C. Neinhuis, W. Barthlott)

22. 22 Nematic Elastomers and Rough Wetting, Antonio DeSimone, SISSA (Trieste, ITALY) Rolling droplets (experiments by P. Aussillous, D. Quèrè, D. Richard) Trajectory of a marker particle moving with a droplet which is rolling on a horizontal super-hydrophobic surface

23. 23 Nematic Elastomers and Rough Wetting, Antonio DeSimone, SISSA (Trieste, ITALY) Liquid drops or solid balls? (G. McKinley) (D. Quèrè) Different impact velocities: low highvery high

24. 24 Nematic Elastomers and Rough Wetting, Antonio DeSimone, SISSA (Trieste, ITALY) References on wetting of rough surfaces  C. Neinhuis, W. Barthlott, Characterization and distribution of water-repellent self-cleaning plant surfaces. Ann. Bot. 79, 667-677, 1997.  L. Mahadevan, Y. Pomeau, Rolling droplets. Phys. Fluids 11, 2449-2453, 1999.  R. Blossey, Self-cleaning surfaces. Nature Mat. 2, 301-306, 2003.  K.K. Lau, J. Bico, K. Teo, M. Chowalla, G. Amaratunga, W. Milne, G. McKinley, K. Gleason, Superhydrophobic carbon nanotube forests. Nano Lett. 3, 1701-1705, 2003.  G. Alberti, A. DeSimone, Wetting of rough surfaces - a homogenization approach. Proc. Royal Society of London, in press, 2004. Monograph: P.-G. De Gennes, F. Brochard-Wyart, D. Quéré, Capillarity and wetting phenomena. Springer, 2004.

25. 25 Nematic Elastomers and Rough Wetting, Antonio DeSimone, SISSA (Trieste, ITALY) Conclusions Macro models of phenomena whose physics is decided at micro scale: coarse graining through analysis for two case studies. What have we learned?  soft elasticity in nematic elastomers rich bevavior (both for phase diagram and for BVP‘s) from simple model  wetting properties of rough surfaces classical models give bounds for minimal energy energy minimality=most conservative estimate of effects of roughness