Elasticity and Dynamics of LC Elastomers

1. Elasticity and Dynamics of LC Elastomers Leo Radzihovsky Xiangjing Xing Ranjan Mukhopadhyay Olaf Stenull IMA

2. Outline • Review of Elasticity of Nematic Elastomers • Soft and Semi-Soft Strain-only theories • Coupling to the director • Phenomenological Dynamics • Hydrodynamic • Non-hydrodynamic • Phenomenological Dynamics of NE • Soft hydrodynamic • Semi-soft with non-hydro modes IMA

3. Strain Displacements Cauchy DeformationTensor (A “tangent plane” vector) Displacement strain a,b = Ref. Space i,j = Target space Invariances TCL, Mukhopadhyay, Radzihovsky, Xing, Phys. Rev. E66, 011702/1-22(2002) IMA

4. Isotropic and Uniaxial Solid Isotropic: free energy density f has two harmonic elastic constants Uniaxial: five harmonic elastic constants Nematic elastomer: uniaxial. Is this enough? IMA

5. Nonlinear strain Green – Saint Venant strain tensor- Physicists’ favorite – invariant under U; u is a tensor is the reference space, and a scalar in the target space IMA

6. Spontaneous Symmetry Breaking Phase transition to anisotropic state asmgoes to zero Direction ofn0is arbitrary Symmetric- Traceless part Golubovic, L., and Lubensky, T.C.,, PRL 63, 1082-1085, (1989). IMA

7. Strain of New Phase u’ is the strain relative to the new state at pointsx’ duis the deviation of the strain relative to the original reference frameR fromu0 duis linearly proportional tou’ IMA

8. Elasticity of New Phase Rotation of anisotropy direction costs no energy C5=0 because of rotational invariance This 2nd order expansion is invariant under all U but only infinitesimal V IMA

9. Soft Extensional Elasticity Strain uxx can be converted to a zero energy rotation by developing strains uzz and uxzuntil uxx =(r-1)/2 IMA

10. Frozen anisotropy: Semi-soft System is now uniaxial – why not simply use uniaxial elastic energy? This predicts linear 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 IMA

11. 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 IMA

12. 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 the nematic phase - semi-soft. length of plateau depends on magnitude of spontaneous anisotropyr. Warner-Terentjev Note: Semi-softness only visible in nonlinear properties Finkelmann, et al., J. Phys. II 7, 1059 (1997); Warner, J. Mech. Phys. Solids 47, 1355 (1999) IMA

13. Soft Biaxial SmA and SmC Free energy density for a uniaxial solid (SmA with locked layers) Red: Corrections for transition to biaxial SmA Green: Corrections for trtansition to SmC C4=0: Transition to Biaxial Smectic with soft in-plane elasticity C5=0: Transition to SmC with a complicated soft elasticity Olaf Stenull, TCL, PRL 94, 081304 (2005) IMA

14. Coupling to Nematic Order • Strainuab transforms like a tensor in the ref. space but as a scalar in the target space. • The directorni and the nematic order parameterQijtransform 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 Target->ref IMA

15. Strain and Rotation Simple Shear Symmetric shear Rotation IMA

16. Softness with Director Na= unit vector along uniaxial direction in reference space; layer normal in a locked SmA phase Red: SmA-SmC transition Director relaxes to zero IMA

17. Harmonic Free energy with Frank part IMA

18. NE: Relaxed elastic energy Uniaxial solid when C5R>0, including Frank director energy IMA

19. Slow Dynamics – General Approach • Identify slow variables:  Determine static thermodynamics: F() • Develop dynamics: Poisson-brackets plus dissipation • Mode Counting (Martin,Pershan, Parodi 72): • One hydrodynamic mode for each conserved or broken-symmetry variable • Extra Modes for slow non-hydrodynamic • Friction and constraints may reduce number of hydrodynamics variables IMA

20. Preliminaries Harmonic Oscillator: seeds of complete formalism Poisson bracket Poisson brackets: mechanical coupling between variables – time-reversal invariant. Dissipative couplings: not time-reversal invariant friction Dissipative: time derivative of field (p) to its conjugate field (v) IMA

21. Fluid Flow – Navier Stokes Conserved densities: mass:rEnergy: e Momentum: gi = rvi IMA

22. Crystalline Solid I Mass density is periodic Conserved densities: mass:r Energy: e Momentum: gi = rvi Broken-symmetry field: Phase of mass-density field: u describes displacement of periodic part of density Aside: Nonlinear strain is not the Green Saint-Venant tensor Strain Free energy IMA

23. Crystalline Solid II Modes: Transverse phonon: 4 Long. Phonon: 2 Permeation (vacancy diffusion): 1 Thermal Diffusion: 1 permeation Aside: full nonlinear theory requires more care Permeation: independent motion of mass-density wave and mass: mass motion with static density wave IMA

24. Tethered Solid 7 hydrodynamic variables: 1 density,3 momenta, 3 displacements, 1 energy + 1 constraint = 8-1=7 Classic equations of motion for a Lagrangian solid; use Cauchy-Saint-Venant Strain Isotropic elastic free energy + energy mode (1) IMA

25. Gel: Tethered Solid in a Fluid Tethered solid Friction only for relative motion- Galilean invariance Fluid Frictional Coupling Total momentum conserved Fast non-hydro mode: but not valid if there are time scales in G Fluid and Solid move together IMA

26. Nematic Hydrodynamics: Harvard I gis the total momentum density: determines angular momentum Frank free energy for a nematic IMA

27. Nematic Hydrodynamics: Harvard II permeation w – fluid vorticity not spin frequency of rods Symmetric strain rate rotates n Modes: 2 long sound, 2 “slow” director diffusion. 2 “fast” velocity diff. Stress tensor can be made symmetric IMA

28. NE: Director-displacement dynamics Stenull, TCL, PRE 65, 058091 (2004) Tethered anisotropic solid plus nematic Note: all variables in target space Director relaxes in a microscopic time to the local shear – nonhydrodynamic mode Modifications if g depends on frequency Semi-soft: Hydrodynamic modes same as a uniaxial solid: 3 pairs of sound modes IMA

29. 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 IMA

30. Beyond Hydrodynamics: ‘Rouse’ Modes Standard hydrodynamics for wtE<<1; nonanalytic wtE>>1 IMA

31. Rouse in NEs References: Martinoty, Pleiner, et al.; Stenull & TL; Warner & Terentjev, EPJ 14, (2005) Second plateau in G' “Rouse” Behavior before plateau IMA

32. Rheology Conclusion: Linear rheology is not a good probe of semi-softness IMA

33. Summary and Prospectives • Ideal nematic elastomers can exhibit soft elasticity. • Semi-soft elasticity is manifested in nonlinear phenomena. • Linearized hydrodynamics of soft NE is same as that of columnar phase, that of a semi-soft NE is the same as that of a uniaxial solid. • At high frequencies, NE’s will exhibit polymer modes; semisoft can exhibit plateaus for appropriate relaxation times. • Randomness will affect analysis: random transverse stress, random elastic constants will complicate damping and high-frequency behavior. IMA