Elasticity and Dynamics of LC Elastomers

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Elasticity and Dynamics of LC Elastomers

• Leo Radzihovsky
• Xiangjing Xing
• Ranjan Mukhopadhyay
• Olaf Stenull

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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

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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.
E 66, 011702/1-22(2002)
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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?
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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
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Spontaneous Symmetry Breaking
Phase transition to anisotropic state as m goes
to zero
Direction of n0 is arbitrary
Symmetric- Traceless part
Golubovic, L., and Lubensky, T.C.,, PRL 63,
1082-1085, (1989).
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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 u0
du is linearly proportional to u
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Elasticity of New Phase
Rotation of anisotropy direction costs no energy
C50 because of rotational invariance
This 2nd order expansion is invariant under all U
but only infinitesimal V
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Soft Extensional Elasticity
Strain uxx can be converted to a zero energy
rotation by developing strains uzz and uxz until
uxx (r-1)/2
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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 h0
f (u) isotropic
Rotation
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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
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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 anisotropy
r. 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)
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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
C40 Transition to Biaxial Smectic with soft
in-plane elasticity C50 Transition to SmC with
a complicated soft elasticity
Olaf Stenull, TCL, PRL 94, 081304 (2005)
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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-gttarget
Target-gtref
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Strain and Rotation
Simple Shear
Symmetric shear
Rotation
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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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Harmonic Free energy with Frank part
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NE Relaxed elastic energy
Uniaxial solid when C5Rgt0, including Frank
director energy
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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

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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)
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Fluid Flow Navier Stokes
Conserved densities mass r Energy e Momentum
gi rvi
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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
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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
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Tethered Solid
7 hydrodynamic variables 1 density,3 momenta, 3
displacements, 1 energy 1 constraint 8-17
Classic equations of motion for a Lagrangian
solid use Cauchy-Saint-Venant Strain
Isotropic elastic free energy
energy mode (1)
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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
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Nematic Hydrodynamics Harvard I
g is 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
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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
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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
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Beyond Hydrodynamics Rouse Modes
Standard hydrodynamics for wtEltlt1 nonanalytic
wtEgtgt1
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Rouse in NEs
References Martinoty, Pleiner, et al. Stenull
TL Warner Terentjev, EPJ 14, (2005)
Second plateau in G'
Rouse Behavior before plateau
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Rheology
Conclusion Linear rheology is not a good probe
of semi-softness
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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, NEs 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.