Liquid Crystal Elastomers 42105 Ranjan Mukhopadhyay Leo Radzihovsky
- Slides: 40
Liquid Crystal Elastomers • • • 4/21/05 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 Kent Liquid Crystal Day
Outline • • • 4/21/05 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 Kent Liquid Crystal Day
Examples of LC Elastomers 1. Liquid Crystal Elastomers - Weakly crosslinked liquid crystal polymers – Finkelmann, Zentel, others Nematic Smectic-C 2. Tanaka gels with hard-rod dispersion – Penn group 3. Anisotropic membranes 4/21/05 Kent Liquid Crystal Day
Properties I • Large thermoelastic effects - Large thermally induced strains - artificial muscles Courtesy of Eugene Terentjev 300% strain 4/21/05 Kent Liquid Crystal Day
Properties II Large strain in small temperature range Terentjev 4/21/05 Kent Liquid Crystal Day
Properties III • Soft or “Semi-soft” elasticity Vanishing xz shear modulus Soft stress-strain for stress perpendicular to order 4/21/05 Kent Liquid Crystal Day Warner Finkelmann
Transitions to Nematic Elastomers Thermotropic Lyotropic Volume compression Isotropic (I) Nematic (N) Lacoste, Lau and Lubensky Euro. Phys. J. E 8, 403 (2002) Lubensky, Mukhopadhyay, Radzihovsky and Xing PRE 66, 011702 (2002) 4/21/05 Kent 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 4/21/05 Kent Liquid Crystal Day
Single Wall Carbon Nanotubes SWNTs have extraordinary properties: • Strength (~100 x steel) • Tensile strength • Stiffness • Elongation 100 -200 GPa 1. 4 TPa 20 -30% • Electrical conductivity (~Copper) • Ballistic electron transport mechanism • Highest known current density • Thermal conductivity (~3 x Diamond) • Thermally stable polymer (anaerobic) Products incorporating SWNTs can benefit from all of these properties simultaneously. ~1 nm 100 nm – 10, 000 nm 4/21/05 Kent Liquid Crystal Day
Dispersing SWNTs van der Walls attaction: 40 KBT/nm Laser-oven Hi. PCO 5. 00 Surfactant: SDS SWNTs: 0. 5 mg/ml Time: 5 days TX-100 0. 8 mg/ml 5 days Na. DDBS 20 mg/ml 2 months 2. 50 0 5. 00 mm Islam, Rojas, Bergey, Johnson, Yodh Nano. Lett. 3, 269 (2003)
SWNTs: Rigid Rods in Suspension Zhou, Islam, Wang, Ho, Yodh, Winey, Fischer Chem. Phys. Lett. 384, 185 (2004)
SWNTS are Attractive Rods concentration Isotropic (I) Nematic (N) Onsager Ann. N. Y. Acad. Sci. 51, 627 (1949) 4/21/05 Kent Liquid Crystal Day
Properties of NIPA gel N-isopropylacrylamide (NIPA) gel: F. Ilmain et al. Nature 349, 400 (1991) Temperature Pelton R. , Temperature-sensitive aqueous microgels, Adv. Colloid Interface Sci. , 85 (2000) 1 -33. Tanaka’s website
SWNT-NIPA Gels SWNT dispersed in Na. DDBS + (NIPA) pre-gel polymerized for 3 h at T=22°C 8. 25 mg/ml 2. 47 mg/ml
Temporal and Concentration Dependence (A) Islam, Alsayed, Dogic, Zhang, Lubensky, Yodh PRL 92, 088303 (2004)
Isotropic-Nematic Transition: Nematic Nanotube Gels O. A. a (P) (A) More alignment at surface: more strain – buckling at surface Islam, Alsayed, Dogic, Zhang, Lubensky, Yodh PRL 92, 088303 (2004)
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 4/21/05 Kent Liquid Crystal Day
Defects (P) 4 extinction branches Defects and buckling in nematic lyotropic gels, M. F. Islam, M. Nobili, Fangfu Ye, T. C. Lubensky and A. G. Yodh (submitted to PRL)
Mechanical Properties Percolating network of rods in contact as a result of Van de Waals attraction? 4/21/05 Kent Liquid Crystal Day
Elastomeric Membranes L/D = 11 M. Dalhaimer, Dennis Discher, TCL – in preperation Biological systems spectrin networks Flat membranes. No shear modulus Xing, X. J. , Mukhopadhyay, R. , Lubensky, T. C. and Radzihovsky, L. , PR E, 021108/1 -17, 68 (2003). 4/21/05 Kent Liquid Crystal Day
Strain Displacements Cauchy Deformation. Tensor (A “tangent plane” vector) Displacement a, b = Ref. Space strain i, j = Target space Invariances TCL, Mukhopadhyay, Radzihovsky, Xing, Phys. Rev. E 66, 011702/1 -22(2002) 4/21/05 Kent Liquid Crystal Day
Isotropic and Uniaxial Solid Isotropic: two harmonic elastic constants Uniaxial: five harmonic elastic constants Nematic elastomer: uniaxial. Is this enough? 4/21/05 Kent Liquid Crystal Day
Nonlinear strain Green – Saint Venant strain tensor- Physicists favorite – invariant under U; 4/21/05 Kent Liquid Crystal Day
Neoclassical Elastomer Theory Warner and Terentjev Gel: random walks between crosslinks with probability distribution Free energy density l=anisotropic step-length tensor l 0=tensor at time of crosslinking 4/21/05 Kent Liquid Crystal Day
Spontaneous Symmetry Breaking Phase transition to anisotropic state as m goes to zero Direction of n 0 is arbitrary Symmetric. Traceless part Golubovic, L. , and Lubensky, T. C. , , PRL 63, 1082 -1085, (1989). 4/21/05 Kent Liquid Crystal Day
Strain of New Phase u’ is the strain relative to the new state at points x’ du is the deviation of the strain relative to the original reference frame R from u 0 du is linearly proportional to u’ 4/21/05 Kent Liquid Crystal Day
Elasticity of New Phase Rotation of anisotropy direction costs no energy C 5=0 because of rotational invariance This 2 nd order expansion is invariant under all U but only infinitesimal V 4/21/05 Kent Liquid Crystal Day
Soft Extensional Elasticity Strain uxx can be converted to a zero energy rotation by developing strains uzz and uxz until uxx =(r-1)/2 4/21/05 Kent 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 Rotation 4/21/05 Kent Liquid Crystal Day f (u) : isotropic
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 4/21/05 Kent Liquid Crystal Day
Semi-soft Extensions Break rotational symmetry Stripes form in real systems: semi-soft, BC Not perfectly soft because of residual anisotropy arising from crosslinking in the nematic phase - semi-soft. length of plateau depends on magnitude of spontaneous anisotropy r. Finkelmann, et al. , J. Phys. II 7, 1059 (1997); Warner, J. Mech. Phys. Solids 47, 1355 (1999) 4/21/05 Kent Liquid Crystal Day
Coupling to Nematic Order • Strain uab transforms like a tensor in the ref. space but as a scalar in the target space. • The director ni and the nematic order parameter Qij 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->target 4/21/05 Target->ref Kent Liquid Crystal Day
Strain and Rotation Symmetric shear 4/21/05 Simple Shear Kent Liquid Crystal Day Rotation
Softness with Director relaxes to zero 4/21/05 Kent Liquid Crystal Day
Coupling to Nematic Order In equilibrium The nematic order parameter Qab transforms like a tensor in Ref space. Qzn is a rotation of the nematic order parameter - costs no energy: it screens the strain dvzn 4/21/05 Kent Liquid Crystal Day
Free energy with Frank part 4/21/05 Kent Liquid Crystal Day
NE: Relaxed elastic energy Hydrodynamic modes from effective free energy in terms of strain only 4/21/05 Kent Liquid Crystal Day
NE: Director-displacement dynamics Tethered anisotropic solid plus nematic 4/21/05 Kent Liquid Crystal Day Stenull-Lubensky PRE (2004), Euro. J. Phys. Director relaxes in a microscopic time to the local shear – nonhydrodynamic mode
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 4/21/05 Kent 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 4/21/05 Kent Liquid Crystal Day
