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### Elasticity and Dynamics of LC Elastomers

Leo Radzihovsky

Xiangjing Xing

Ranjan Mukhopadhyay

Olaf Stenull

IMA

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

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

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

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

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

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

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

Soft Extensional Elasticity

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

IMA

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

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

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

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

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

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

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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

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

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

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

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

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

Nematic Hydrodynamics: Harvard I

gis the total momentum density: determines angular momentum

Frank free energy for a nematic

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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

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

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

Rouse in NEs

References: Martinoty, Pleiner, et al.;

Stenull & TL; Warner & Terentjev, EPJ 14, (2005)

Second plateau in G\'

“Rouse” Behavior before plateau

IMA

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

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