Liquid%20Crystal%20Elastomers

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Liquid Crystal Elastomers

• 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

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Outline

• 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

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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
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Properties I

• Large thermoelastic effects - Large thermally
  induced strains - artificial muscles

Courtesy of Eugene Terentjev
300 strain
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Properties II
Large strain in small temperature range
Terentjev
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Properties III

• Soft or Semi-soft elasticity

Vanishing xz shear modulus
Warner Finkelmann
Soft stress-strain for stress perpendicular to
order
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Transitions to Nematic Elastomers
Thermotropic
Lyotropic
Volume compression
Nematic (N)
Isotropic (I)
Lacoste, Lau and Lubensky Euro. Phys. J. E 8, 403
(2002)
Lubensky, Mukhopadhyay, Radzihovsky and Xing PRE
66, 011702 (2002)
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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
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Single Wall Carbon Nanotubes

• SWNTs have extraordinary properties
• Strength (100x steel)
• Tensile strength 100-200 GPa
• Stiffness 1.4 TPa
• Elongation 20-30
• Electrical conductivity (Copper)
• Ballistic electron transport mechanism
• Highest known current density
• Thermal conductivity (3x Diamond)
• Thermally stable polymer (anaerobic)

Products incorporating SWNTs can benefit from all
of these properties simultaneously.
100 nm 10,000 nm
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Dispersing SWNTs
van der Walls attaction 40 KBT/nm
SDS
Surfactant
TX-100
NaDDBS
Laser-oven HiPCO
SWNTs
0.5 mg/ml
0.8 mg/ml
20 mg/ml
Time
5 days
5 days
2 months
5.00
2.50
0
0
2.50
5.00
mm
Islam, Rojas, Bergey, Johnson, Yodh NanoLett. 3,
269 (2003)
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SWNTs Rigid Rods in Suspension
Zhou, Islam, Wang, Ho, Yodh, Winey, Fischer
Chem. Phys. Lett. 384, 185 (2004)
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SWNTS are Attractive Rods
concentration
Isotropic (I)
Nematic (N)
Onsager Ann. N. Y. Acad. Sci. 51, 627 (1949)
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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.
Tanakas website
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SWNT-NIPA Gels
SWNT dispersed in NaDDBS (NIPA) pre-gel
polymerized for 3h at T22C
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Temporal and Concentration Dependence
(A)
Islam, Alsayed, Dogic, Zhang, Lubensky, Yodh PRL
92, 088303 (2004)
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Isotropic-Nematic TransitionNematic Nanotube
Gels
More alignment at surface more strain buckling
at surface
Islam, Alsayed, Dogic, Zhang, Lubensky, Yodh PRL
92, 088303 (2004)
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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
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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)
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Mechanical Properties
Percolating network of rods in contact as a
result of Van de Waals attraction?
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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).
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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 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
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Neoclassical Elastomer Theory
Gel random walks between crosslinks with
probability distribution
Warner and Terentjev
Free energy density
lanisotropic step-length tensor l0tensor at
time of crosslinking
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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 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 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.
Finkelmann, et al., J. Phys. II 7, 1059
(1997) Warner, J. Mech. Phys. Solids 47, 1355
(1999)
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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
Director relaxes to zero
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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
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Free energy with Frank part
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NE Relaxed elastic energy
Hydrodynamic modes from effective free energy in
terms of strain only
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NE Director-displacement dynamics
Stenull-Lubensky PRE (2004), Euro. J. Phys.
Tethered anisotropic solid plus nematic
Director relaxes in a microscopic time to the
local shear nonhydrodynamic mode
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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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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