5/23/05 IMA

Elasticity and Dynamics of LC Elastomers

Leo RadzihovskyXiangjing XingRanjan MukhopadhyayOlaf Stenull

5/23/05 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

5/23/05 IMA

StrainDisplacements

( ) ( )= +R x x u x

ii i i

Rxα α αα

δ η∂Λ = = +∂

Cauchy DeformationTensor(A “tangent plane” vector)

Displacementstrain

i iuα αη = ∂α,β = Ref. Spacei,j = Target space

1;U V −→ →R R x xInvariancesTCL, Mukhopadhyay, Radzihovsky, Xing, Phys. Rev. E 66, 011702/1-22(2002)

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Isotropic and Uniaxial SolidIsotropic: free energy density f has two harmonic elastic constants 1

3u u uαβ αβ αβ γγδ= −

( )

1

1

2 212

23 2

( ) ( )

( ) ( )

Tr Tr Tr

f f f U V

f f u f VuV

C u DBu u uαα µ

−

−

= Λ = Λ

= =

= ++ −

Invariant under

( ) U (V )→R x R x

Uniaxial: five harmonic elastic constants Invariant under

uni( ) U (V )→R x R xNematic elastomer: uniaxial. Is this enough?

2 21 12 21 2 3

2 24 5 ;

( , )

zz zz

z

z

f C u C u u C u

C u C u

x

νν νν

ντ ν

α ν

= + +

+ +

=x x

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Nonlinear strainGreen – Saint Venant strain tensor- Physicists’favorite – invariant under U;

2 2 2dR dx u dx dxαβ α β− =

( )

1 12 2

12

( ) ( )T T

k k

u

u u u u uαβ α β β α α β

δ η η= Λ Λ− ≈ +

= ∂ + ∂ + ∂ ∂1

1 1

;

;

U V

U V u VuV

−

− −

→ →

Λ → Λ →

R R x x

u is a tensor is the reference space, and a scalar in the target space

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Spontaneous Symmetry BreakingPhase transition to anisotropicstate as µ goes to zero

( )120 0 0

Tu δ= Λ Λ −

0 02uδΛ = + Direction of n0 isarbitrary

0

0 0 13( )

u u

n n

αβ αβ

α β αβδ

=

= Ψ −

2~uαα ΨSymmetricSymmetric--TracelessTracelesspart Golubovic, L., and Lubensky, T.C.,, PRL

63, 1082-1085, (1989).part

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Strain of New Phaseu’ is the strain relative to the new state at points x’

0( ) ( )

( )i ij j i

i i

R x u

x u

δ= Λ +

′ ′ ′= +

x x

x

0i i k

ij ik kjj jk

xR Rx x x

′∂∂ ∂ ′Λ = = = Λ Λ′∂ ∂ ∂

δu is the deviation ofthe strain relative to the original reference frame R from u0

( )0

12 0 0

0 0'

T T

T

u u u

u

δ = −

= Λ Λ−Λ Λ

= Λ Λδu is linearly proportionalto u’( )1 1

2 2' ( )T Tu δ η η′ ′ ′ ′= Λ Λ − ≈ +

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Elasticity of New Phase

( )

( )

11 10 0 0 0

1

14 1 1

' ( )

1 cos2 sin2( 1)

sin2 1 cos2

T

r

rr

u Vu V u

rθ θ

θ θ

−− −= Λ − Λ

− = − − −

20||20

r⊥

Λ=

Λ

Rotation of anisotropy direction costs no energy

( 1)' ~4xzru

rθ−

C5=0 because ofrotational invariance

This 2nd order expansionis invariant under all Ubut only infinitesimal V

21 12 21 2 3el

54

zz zz

z z

f C u C u u C u u

C u u C u uνν νν νν

ντ ντ ν ν

′ ′ ′ ′ ′= + +

′ ′ ′ ′+ +

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Soft Extensional Elasticity

( )

1

14 1 1

1 cos2 sin2( 1)

sin2 1 cos2r

rr

u rθ θ

θ θ

− = − − −

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

1

1 ( 1 2 )2

zz xx

xz xx xx

u ur

u u r ur

= −

= − −

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Frozen anisotropy: Semi-soft

( ) ( )hzzf u f u hu= −

System is now uniaxial – why not simply use uniaxial elastic energy? This predicts linear stress-stain curve and misses lowering of energy by reorientation:

2 2 2 21 12 2 51 2 3 4zz zz zf C u C u u C u C u C uνν νν ντ ν= + + + +

Model Uniaxial system:Produces harmonic uniaxialenergy for small strain but has nonlinear terms – reduces to isotropic when h=0

f (u) : isotropic

2

2xz xx zz

xx zz xz

u u uu u u

u u uθ − − ′→ = + −

Rotation

( ) ( ) ( 2 )hzz xzf u f u h u uθ′ = − +

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Semi-soft stress-strainWard Identity

2 2 ( ) 2( )

( ) 0 or

h

xz xz xx zz zz xx xz

xx xzxz xz

xx zzxx

df hu u u ud

h uu u

u h

σ σ

σσ σ

σθ

=

= − = − −

−= ⇒−

=

+

hfuαβαβ

σ ∂=∂

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 ExtensionsBreak 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-TerentjevNote: Semi-softness only visible in nonlinear propertiesFinkelmann, et al., J. Phys. II 7, 1059 (1997);

Warner, J. Mech. Phys. Solids 47, 1355 (1999)

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Soft Biaxial SmA and SmCFree energy density for a uniaxialsolid (SmA with locked layers)

2 2 2 21 12 21 2

2 2 2 21

2 21 2

5

2 3

3

3 4

ˆ ˆ (

ˆ

ˆ )

ˆzz

zz z z

z

z

zz z

z

zg u u g u u

f C u C u u C

g

v u u v u u

u

v u u

u u C u

u

C

ντ αα ν

ν αα ν ντ

νν

ν τ

τ ντ

νν ντ ν

+ +

= + + + +

+

+

+ +

12u u uντ ντ ντ σσδ= −

Red: Corrections for transition to biaxial SmAGreen: Corrections for trtansition to SmC

C4=0: Transition to Biaxial Smectic with soft in-plane elasticityC5=0: Transition to SmC with a complicated soft elasticity

Olaf Stenull, TCL, PRL 94, 081304 (2005)

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Coupling to Nematic Order• Strain uαβ 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.

T 1/2 T 1/2

T 1/2

T 1/2 1/2

( ) ( )

( ) Rotation Matrix

( ) (1 2 ) Symmetric

OM

O

M u

−

−

Λ = Λ Λ Λ Λ Λ ≡

= Λ Λ Λ =

= Λ Λ = + =T;i i i in O n n O nα α α α= =Ref->target Target->ref

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Strain and Rotation

Simple ShearRotationSymmetric

shear

Λ

n is a reference space vector; it is equal to the

target space vector that is obtained when is

symmetric12 ( )i i i i

i i k k

O u uα α α α

α α

δ

δ ε

≈ + ∂ −∂

≈ − Ω

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Softness with DirectorNα= unit vector along uniaxial direction in reference space; layer normal in a locked SmA phase

( , )zn nα ν=n 2 2 21 ( ) ; , etc.zzn N n c u N u Nν α α ν α αβ β= − ⋅ ≡ =

Red: SmA-SmC transition2 2 21 1

2 21 2 3 4

2 2125 2 1

2 2 21 12 21 2 3 4

2

21

4 214 2

2 21 12 251 2 1 2 1[ ( / ) ] [ ( / )]

zz zz

z z z

zz zz

zz

z z

n u

gn

f C u C u u C u C u

C u D n n u D n

C u C u u C u C u

D n D D u C D u

u

D

n

ν

ν ν ττ

νν νν ντ

ν ν ν ν

νν νν ντ

ν ν ν

λ

λ

+

+ +

= + + +

+ + +

= + + +

+ + + −

+

22

5 51

1 02

R DC C

DSoft= − = ⇒

Director relaxes to zero

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Harmonic Free energy with Frank part

3 2 21 12 21 2 3

2 254

3 2 2 21 1 12 2 21 2 3

3 212 1 2

12

[

]

[ ( ) ( ) ( ) ]

[ ]

( )

u n u n

u zz zz

z

n z

u n z

z z

F F F F

F d x C u C u u C u

C u C u

F d x K n K n K n

F d x D n D n u

n n u u

νν αα

ντ ν

ν ν ντ ν τ ν

ν ν ν

ν ν ν ν

ε

−

−

= + +

= + +

+ +

= ∂ + ∂ + ∂

= +

= − ∂ −∂

∫

∫∫

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NE: Relaxed elastic energyUniaxial solid when C5

R>0, including Frank director energy

eff 3 2 21 12 21 2 3

2 2 2 2 2 21 12 254 1 3

22

5 51

2 2

2 2

1 1

5

3

5

1 1 3

[

( ) ( ) ]

; 2

1 11 ;

0

1

0

4 4

;

u zz zz aa ii

R

R

R Raz a z z aab

R

R R

R

F d x C u C u u C u

C u C u K u K u

DC C

DC

D DK K K

C

KD D

= + +

+ + + ∂ + ∂

= −

= + = −

= ≠

∫

Soft : Semi - Soft :

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Slow Dynamics – General Approach• Identify slow variables: ϕ Determine static

thermodynamics: F(ϕ)• 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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PreliminariesHarmonic Oscillator: seeds of complete formalism

Poisson bracket2

21 ; , 1

,

2

,

2

p x

Hp x kxx

pH kxm

p x v

Hx pp

pxm

p mv

=

∂− −∂∂−

= +

= −Γ = −Γ

= =∂

=

friction

Poisson brackets: mechanical coupling between variables –time-reversal invariant.Dissipative couplings: not time-reversal invariant

Dissipative: time derivative of field (p) to its conjugate field (v)

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Fluid Flow – Navier Stokes21

2 ( / ) [ ]d dF d x g d x fρ ρ= +∫ ∫Conserved densities:

mass: ρ Energy: ε

Momentum: gi = ρvi 2

0

0

t

t i i ij i

t i

i i

i

ig v

j

g

pε

ρ

σ η

ε

∂

−

∂ + =

∂ = ∂ = + ∇

∂ + ∂ =

∂

2

2

2

2 (2 modes)3

(2 modes)

(1 mode)p

cq iq

i q

i qC

ηωρ

ηωρκω

= ± −

= −

= −

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Crystalline Solid I( )ieρ ρ ρ ⋅ −→ +∑ G x u

GG

Mass density is periodic

Conserved densities:mass: ρ Energy: ε

Momentum: gi = ρvi

Broken-symmetry field:

Phase of mass-density field: udescribes displacement of periodic part of density

( )/2ij i j j iu u u= ∂ + ∂ Aside: Nonlinear strain is not the Green Saint-Venant tensorStrain

Free energy

21 12 2( / ) [ ]d

ii ijkl ij klF d x g f u K u uρ ρ λδρ = + − + ∫

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Crystalline Solid II

2

0

1t

t ii

t i

i i

i

i

ii

Fuu

g

up

v

g vFδ

ρ

γ

δ

δδ

η

∂ + =

∂ = −

−

∂

+∂ ∇−∂ =

permeation

Modes:Transverse phonon: 4Long. Phonon: 2Permeation (vacancy diffusion): 1Thermal Diffusion: 1

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

/t

δρ ρ

∂ =

= −∇⋅

u v

u

No permeation :

Density locked : 7 hydrodynamic variables: 1 density,3 momenta, 3 displacements, 1 energy + 1 constraint = 8-1=7Classic equations of motion for a Lagrangiansolid; use Cauchy-Saint-Venant StrainIsotropic elastic free energy

( )2 212 2d

ii ijF d x u uλ µ= +∫2 2

it i i

Fuu

vηδρδ

∂ = − + ∇

2

2

(4)2

2 2 (2)3

T

L

q i q

q i q

µ ηωρ ρ

λ µ ηωρ ρ

= ± +

+= ± +

+ energy mode (1)

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Gel: Tethered Solid in a FluidTethered solid

2 2 ( )s si

it i ii

Fu uu

u vδρ ηδ

∂ = − + ∇ −Γ −Friction only for relative motion-Galilean invariance

2 ( )ii it i ig p v v uη∂ = −∂ ∇ −+ −ΓFluid Frictional Coupling

( ) Tst i i j ijg uρ σ∂ + = ∂

1 1 1( )sF i iω τ ρ ρ− − −= − = − + Γ1 : ωτ Effective Tethered Hydro.

Fast non-hydro mode: but not valid if there are time scales in Γ

Total momentum conserved

2( ) ( )s si ii

Fu uuδρ ρ η ηδ

+ = − + + ∇Fluid and Solid move together

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Nematic Hydrodynamics: Harvard I21

2 ( / ) [ , ]d dF d x g d x fρ ρ= +∫ ∫ n

g is the total momentum density: determines angular momentum = ×x g

( )2 21 2

23

1 1[ , ] ( ) ( )[ ( )]2 21 ( )[ ( )]2

f K K

K

ρ ρ ρ

ρ

= ∇⋅ + ⋅ ∇×

+ × ∇×

n n n n

n n

Frank free energy for a nematic

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Nematic Hydrodynamics: Harvard II( )1

2

12

ij i j j i

i jijk k

A v v

vω ε

= ∂ + ∂

= ∂

1t i

i

t i

jijk k

j ki ij ijk

j

Fnn

g p

v

Fn

δγ

δ

δλ

δλ σ

∂ = −

′∂ = −∂ +

∂

∂ ∂

+

permeation

( ) ( )1 12 2

;ij ijkl kl

T T T Tij j ij jijk k ik k ik

A

n n n n

σ η

λ δ δ λ δ δ

′ =

= − + +

ω – fluid vorticity not spin frequency of rods

Symmetric strain rate rotates n

: 1t ω λ

γ∂ = + +× n An n h 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 dynamicsStenull, TCL, PRE 65, 058091 (2004)

Tethered anisotropic solid plus nematic

1

1

t ii

i i

t i

jijk k

j ki

i

ij j j

ki

Fg

F

Fnn

u g

v

ng F

u

λ

δλ

δγ δ

σ

δ

δ

ρ

δδ

δ

∂

∂

∂ = −

= =

′∂ = + ∂−1

1f

D iiωγ τ

= − = −

Semi-soft: Hydrodynamic modes same as a uniaxial solid: 3 pairs of sound modes

Note: all variables in target space Director relaxes in

a microscopic time to the local shear –nonhydrodynamicmode

Modifications if γdepends on frequency

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Soft Elastomer Hydrodynamicseff

ui jijkl l k

i

Fu vu

δρ ηδ

= − + ∂ ∂

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

2

5

25

2

2

s

f

Ki q

i q

ωηη

ωρ

= −

= −

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Beyond Hydrodynamics: ‘Rouse’ Modes

( ) ( )EG iω µ ωη ωτ= −

2 11

21

( ) ( )

( )( )

1, 0

3/(2 ),

E

N

N

f i

p xf x

p

x

x x

η ω η ωτ

π

−

−

= −

+=

→→ ∞

∑∑

Standard hydrodynamics for ωτE<<1; nonanalyticωτE>>1

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Rouse in NEs References: Martinoty, Pleiner, et al.;Stenull & TL; Warner & Terentjev, EPJ 14, (2005)

22

51

25

2 22 2

1

( )2

[ ( ) ( /2) ( )]

[1 ( )]1

2 1 ( )n

DG C

D

i

D iD i

ω

ω ν ω λ γ ω

ωτ ωωτ ω

= −

− +

− − − −

1

2 1 2

5 5

( ) ( )/ ;

( ) ( / ) ( )

( ) ( );

( ) ( )

n

n

n n E

E

D

D D

f i

f i

τ ω γ ω

τ ω λ τ ω

τ ω τ ωτ

ν ω ν ωτ

=

= −

= −

= −

n Eτ τ Second plateau in G'

5

15 2

2

( ) 1

1( ) | | 1 | | ; 12

Rn

RnE

G C

DG C D

D

ω ωτ

ω λ λ ωτ ωτ

′ =

′ = + −

or n nE Eτ τ τ τ<∼ “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, 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.