1. 4/21/05Kent Liquid Crystal Day Liquid Crystal Elastomers Ranjan Mukhopadhyay Leo Radzihovsky Xiangjing Xing Olaf Stenull Andy Lau David Lacoste Fangfu Ye Paul Dalhaimer Dennis Discher Mohammad Islam Arjun Yodh A. Alsayed Z. Dogic J. Zhang M. Nobili

2. 4/21/05Kent Liquid Crystal Day Outline LC elastomers and their properties Lyotropic nematic gels and nanotube gels Nematic membranes Theory of elasticity of nematic elastomers Dynamics of nematic elastomers Other problems, solved and unsolved

3. 4/21/05Kent Liquid Crystal Day Examples of LC Elastomers 1. Liquid Crystal Elastomers - Weakly crosslinked liquid crystal polymers – Finkelmann, Zentel, others 2. Tanaka gels with hard-rod dispersion – Penn group 3. Anisotropic membranes Nematic Smectic-C

4. 4/21/05Kent Liquid Crystal Day Properties I Large thermoelastic effects - Large thermally induced strains - artificial muscles Courtesy of Eugene Terentjev 300% strain

5. 4/21/05Kent Liquid Crystal Day Properties II Large strain in small temperature range Terentjev

6. 4/21/05Kent Liquid Crystal Day Properties III Soft or “Semi-soft” elasticity Vanishing xz shear modulus Soft stress-strain for stress perpendicular to order Warner Finkelmann

7. 4/21/05Kent Liquid Crystal Day Lubensky, Mukhopadhyay, Radzihovsky and Xing PRE 66, 011702 (2002) Lacoste, Lau and Lubensky Euro. Phys. J. E 8, 403 (2002) Volume compression Isotropic (I) Nematic (N) Transitions to Nematic Elastomers ThermotropicLyotropic

8. 4/21/05Kent Liquid Crystal Day Lyotropic Nematic Gels -Theory Phase diagrams calculated from Flory gel theory + Onsager for rods: (Lacoste, Lau,TCL, Europhys. E (2002) ) Gel compression induces nematic state

9. 4/21/05Kent Liquid Crystal Day SWNTs have extraordinary properties: Strength (~100x steel) Tensile strength100-200 GPa Stiffness1.4 TPa Elongation20-30% Electrical conductivity (~Copper) Ballistic electron transport mechanism Highest known current density Thermal conductivity (~3x Diamond) Thermally stable polymer (anaerobic) 100 nm – 10,000 nm ~1 nm ~1 nm Products incorporating SWNTs can benefit from all of these properties simultaneously. Single Wall Carbon Nanotubes

10. 4/21/05Kent Liquid Crystal Day Laser-oven HiPCO Islam, Rojas, Bergey, Johnson, Yodh NanoLett. 3, 269 (2003) 0.5 mg/ml SWNTs: Time: SDS Surfactant: TX-100 0.8 mg/ml 5 days 2 months 20 mg/ml NaDDBS 0 2.505.00 mm 2.50 0 van der Walls attaction: 40 K B T/nm Dispersing SWNTs

11. 4/21/05Kent Liquid Crystal Day Zhou, Islam, Wang, Ho, Yodh, Winey, Fischer Chem. Phys. Lett. 384, 185 (2004) SWNTs: Rigid Rods in Suspension

12. 4/21/05Kent Liquid Crystal Day concentration Isotropic (I) Nematic (N) Onsager Ann. N. Y. Acad. Sci. 51, 627 (1949) SWNTS are Attractive Rods

13. 4/21/05Kent Liquid Crystal Day N-isopropylacrylamide (NIPA) gel: F. Ilmain et al. Nature 349, 400 (1991) Tanaka’s website Temperature Properties of NIPA gel Pelton R., Temperature-sensitive aqueous microgels, Adv. Colloid Interface Sci., 85 (2000) 1-33.

14. 4/21/05Kent Liquid Crystal Day SWNT dispersed in NaDDBS + (NIPA) pre-gel polymerized for 3h at T=22°C 8.25 mg/ml 2.47 mg/ml SWNT-NIPA Gels

15. 4/21/05Kent Liquid Crystal Day Islam, Alsayed, Dogic, Zhang, Lubensky, Yodh PRL 92, 088303 (2004) (A) Temporal and Concentration Dependence

16. 4/21/05Kent Liquid Crystal Day Islam, Alsayed, Dogic, Zhang, Lubensky, Yodh PRL 92, 088303 (2004) O.A.  (P) (A) Isotropic-Nematic Transition: Nematic Nanotube Gels More alignment at surface: more strain – buckling at surface

17. 4/21/05Kent Liquid Crystal Day Defects in Nanotube Nematic Gels “Nematic nanotube gels,” Islam MF, Alsayed AM, Dogic Z, Zhang J, Lubensky TC, Yodh AG, Phys. Rev. Lett. 92 (8): 2004

18. 4/21/05Kent Liquid Crystal Day 4 extinction branches (P) Defects Defects and buckling in nematic lyotropic gels, M. F. Islam, M. Nobili, Fangfu Ye, T. C. Lubensky and A. G. Yodh (submitted to PRL)

19. 4/21/05Kent Liquid Crystal Day Mechanical Properties Percolating network of rods in contact as a result of Van de Waals attraction?

20. 4/21/05Kent Liquid Crystal Day Elastomeric Membranes M. Dalhaimer, Dennis Discher, TCL – in preperation Biological systems - spectrin networks Flat membranes- No shear modulus L/D = 11 Xing, X. J., Mukhopadhyay, R., Lubensky, T. C. and Radzihovsky, L., PR E, 021108/1-17, 68 (2003).

21. 4/21/05Kent Liquid Crystal Day Strain Cauchy DeformationTensor (A “tangent plane” vector) Displacement strain Invariances Displacements a,b = Ref. Space i,j = Target space TCL, Mukhopadhyay, Radzihovsky, Xing, Phys. Rev. E 66, 011702/1-22(2002)

22. 4/21/05Kent Liquid Crystal Day Isotropic and Uniaxial Solid Isotropic: two harmonic elastic constants Uniaxial: five harmonic elastic constants Nematic elastomer: uniaxial. Is this enough?

23. 4/21/05Kent Liquid Crystal Day Nonlinear strain Green – Saint Venant strain tensor- Physicists favorite – invariant under U;

24. 4/21/05Kent Liquid Crystal Day Neoclassical Elastomer Theory l=anisotropic step- length tensor l 0 =tensor at time of crosslinking Gel: random walks between crosslinks with probability distribution Free energy density Warner and Terentjev

25. 4/21/05Kent Liquid Crystal Day Spontaneous Symmetry Breaking Phase transition to anisotropic state as m goes to zero Direction of n 0 is arbitrary Symmetric-Tracelesspart Golubovic, L., and Lubensky, T.C.,, PRL 63, 1082-1085, (1989).

26. 4/21/05Kent Liquid Crystal Day Strain of New Phase d u is the deviation of the strain relative to the original reference frame R from u 0 u ’ is the strain relative to the new state at points x’ d u is linearly proportional to u’

27. 4/21/05Kent Liquid Crystal Day Elasticity of New Phase Rotation of anisotropy direction costs no energy C 5 =0 because of rotational invariance This 2nd order expansion is invariant under all U but only infinitesimal V

28. 4/21/05Kent Liquid Crystal Day Soft Extensional Elasticity Strain u xx can be converted to a zero energy rotation by developing strains u zz and u xz until u xx =(r-1)/2

29. 4/21/05Kent Liquid Crystal Day Frozen anisotropy: Semi-soft System is now uniaxial – why not simply use uniaxial elastic energy? This predicts liner stress stain curve and misses lowering of energy by reorientation: Model Uniaxial system: Produces harmonic uniaxial energy for small strain but has nonlinear terms – reduces to isotropic when h=0 f (u) : isotropic Rotation

30. 4/21/05Kent Liquid Crystal Day Semi-soft stress-strain Ward Identity Second Piola-Kirchoff stress tensor; not the same as the familiar Cauchy stress tensor Ranjan Mukhopadhyay and TCL: in preparation

31. 4/21/05Kent Liquid Crystal Day Semi-soft Extensions Not perfectly soft because of residual anisotropy arising from crosslinking in the the nematic phase - semi-soft. length of plateau depends on magnitude of spontaneous anisotropy r. Stripes form in real systems: semi-soft, BC Break rotational symmetry Finkelmann, et al., J. Phys. II 7, 1059 (1997); Warner, J. Mech. Phys. Solids 47, 1355 (1999)

32. 4/21/05Kent Liquid Crystal Day Coupling to Nematic Order Strain u ab transforms like a tensor in the ref. space but as a scalar in the target space. The director n i and the nematic order parameter Q ij transform as scalars in the ref. space but, respectively, as a vector and a tensor in the target space. How can they be coupled? – Transform between spaces using the Polar Decomposition Theorem. Ref->targetTarget->ref

33. 4/21/05Kent Liquid Crystal Day Strain and Rotation Simple Shear Symmetric shear Rotation

34. 4/21/05Kent Liquid Crystal Day Softness with Director Director relaxes to zero

35. 4/21/05Kent Liquid Crystal Day Coupling to Nematic Order The nematic order parameter Q ab transforms like a tensor in Ref space. In equilibrium Q zn is a rotation of the nematic order parameter - costs no energy: it screens the strain dv zn

36. 4/21/05Kent Liquid Crystal Day Free energy with Frank part

37. 4/21/05Kent Liquid Crystal Day NE: Relaxed elastic energy Hydrodynamic modes from effective free energy in terms of strain only

38. 4/21/05Kent Liquid Crystal Day NE: Director-displacement dynamics Director relaxes in a microscopic time to the local shear – nonhydrodynamic mode Stenull-Lubensky PRE (2004), Euro. J. Phys. Tethered anisotropic solid plus nematic

39. 4/21/05Kent Liquid Crystal Day Soft Elastomer Hydrodynamics Same mode structure as a discotic liquid crystal: 2 “longitudinal” sound, 2 columnar modes with zero velocity along n, 2 smectic modes with zero velocity along both symmetry directions Slow and fast diffusive modes along symmetry directions

40. 4/21/05Kent Liquid Crystal Day Other Topics Anomalous Easticity: Xing & Radzihovisky; Stenull, TCL; Europhys. Lett. Fluctuations in Nematic elastomer membranes: Xing, Mukhopadhyay,Radzihovsky, TCL Smectic-A, smectic-C, and biaxial smectic elastomers, soft elasticity and phase transitions: Stenull TCL Remaining Challenges: Origins of semi-softness Random orientational torques and random stresses