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Elasticity and Dynamics of LC Elastomers

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Elasticity and Dynamics of LC Elastomers

Leo Radzihovsky

Xiangjing Xing

Ranjan Mukhopadhyay

Olaf Stenull

IMA

- 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

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: free energy density f has two harmonic elastic constants

Uniaxial: five harmonic elastic constants

Nematic elastomer: uniaxial. Is this enough?

IMA

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

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

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’

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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

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

IMA

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

Ward Identity

Second Piola-Kirchoff stress tensor; not the same as the familiar Cauchy stress tensor

Ranjan Mukhopadhyay and TCL: in preparation

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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

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

- 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

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Simple Shear

Symmetric shear

Rotation

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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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Uniaxial solid when C5R>0, including Frank director energy

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- 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

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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)

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Conserved densities:

mass:rEnergy: e

Momentum: gi = rvi

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Mass density is periodic

Conserved densities:

mass:rEnergy: 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

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

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)

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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

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gis the total momentum density: determines angular momentum

Frank free energy for a nematic

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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

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

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

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Standard hydrodynamics for wtE<<1; nonanalytic wtE>>1

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References: Martinoty, Pleiner, et al.;

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

Second plateau in G'

“Rouse” Behavior before plateau

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Conclusion: Linear rheology is not a good probe of semi-softness

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- 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