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

Strain

Cauchy DeformationTensor(A “tangent plane” vector)

Displacementstrain

Invariances

Displacements

( ) ( )= +R x x ux

1;U V -® ®R R x x

ii i i

Rxa a a

a

d h¶

L = = +¶

i iua ah = ¶, = Ref. Spacei,j = Target space

TCL, Mukhopadhyay, Radzihovsky, Xing, Phys. Rev. E 66, 011702/1-22(2002)

5/23/05 IMA

Isotropic and Uniaxial Solid

( )

1

1

2 212

23 2

( ) ( )

( ) ( )

Tr Tr Tr

ff f U V

ff u f V uV

C u DBu u uaa m

-

-

= L = L

= =

= ++ - % %%

13u u uab ab ab ggd= -%

Invariant under

( ) U (V )®R x R x

Isotropic: free energy density f has two harmonic elastic constants

Uniaxial: five harmonic elastic constants Invariant under

uni( ) U (V )®R x R x

Nematic 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

nn nn

nt n

a n

= + +

+ +

=x x

5/23/05 IMA

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

( )

1 12 2

12

( ) ( )T T

k k

u

u u u u uab a b b a a b

d h h= L L - » +

= ¶ +¶ +¶ ¶

2 2 2dR dx u dx dxab a b- =

1

1 1

;

;

U V

U V u V uV

-

- -

® ®

L ® L ®

R R x x

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

5/23/05 IMA

Spontaneous Symmetry Breaking

Phase transition to anisotropicstate as goes to zero

Direction of n0 isarbitrary

0

0 0 13( )

u u

n n

ab ab

a b abd

=

= Y -

% % 2~uaa Y

( )120 0 0

Tu d= L L -

0 02udL = +

Symmetric-Symmetric-TracelessTracelesspartpart Golubovic, L., and Lubensky, T.C.,, PRL

63, 1082-1085, (1989).

5/23/05 IMA

Strain of New Phase

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

u’ is the strain relative to the new state at points x’

u is linearly proportionalto u’( )1 1

2 2' ( )T Tu d h h¢ ¢ ¢ ¢= L L - » +

0( ) ( )

( )

i ij j i

i i

R x u

x u

d= L +

¢ ¢ ¢= +

x x

x

( )0

12 0 0

0 0'

T T

T

u u u

u

d = -

= L L - L L

= L L

0i i k

ij ik kjj jk

xR R

x x x

¢¶¶ ¶¢L = = = L L

¢¶ ¶ ¶

5/23/05 IMA

Elasticity of New Phase

Rotation of anisotropy direction costs no energy

C5=0 because ofrotational invariance

This 2nd order expansionis invariant under all U but only infinitesimal V

( )

( )

11 10 0 0 0

1

14 1 1

' ( )

1 cos2 sin2( 1)

sin2 1 cos2

T

r

rr

u V u V u

rq q

q q

-- -= L - L

æ ö- ÷ç ÷ç= - ÷ç ÷- -ç ÷çè ø

( 1)' ~

4xz

ru

rq

-

2

0||20

r^

L=

L

21 12 21 2 3el

54

zz zz

z z

f C u C u u C u u

C u u C u u

nn nn nn

nt nt n n

¢ ¢ ¢ ¢ ¢= + +

¢ ¢ ¢ ¢+ +

5/23/05 IMA

Soft Extensional Elasticity

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

( )

1

14 1 1

1 cos2 sin2( 1)

sin2 1 cos2

r

rr

u rq q

q q

æ ö- ÷ç ÷ç= - ÷ç ÷- -ç ÷çè ø

1

1( 1 2 )

2

zz xx

xz xx xx

u ur

u u r ur

= -

= - -

5/23/05 IMA

Frozen anisotropy: Semi-soft

( ) ( )hzzf u f u hu= -

( ) ( ) ( 2 )hzz xzf u f u h u uq¢ = - +

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 unn nn nt n= + + + +

Model Uniaxial system:Produces harmonic uniaxial energy 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 uqæ ö- - ÷ç ÷¢® = + ç ÷ç ÷- ÷çè ø

Rotation

5/23/05 IMA

Semi-soft stress-strain

2 2 ( ) 2( )

( )0 or

h

xz xz xx zz zz xx xz

xx xzxz xz

xx zzxx

dfhu u u u

dh u

u uu h

s s

ss s

sq

=

= - = - -

-= Þ

-=

+

hfuab

ab

s¶

=¶

Ward Identity

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

Ranjan Mukhopadhyay and TCL: in preparation

5/23/05 IMA

Semi-soft Extensions

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

Stripes form in real systems: semi-soft, BC

Break rotational symmetry

Finkelmann, et al., J. Phys. II 7, 1059 (1997);Warner, J. Mech. Phys. Solids 47, 1355 (1999)

Note: Semi-softness only visible in nonlinear properties

5/23/05 IMA

Soft Biaxial SmA and SmC

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

zgu u gu u

f C u C u u C

g

vu u vu u

u

v u u

u u C u

u

C

nt aa n

n aa n nt

nn

n t

t nt

nn nt n

+ +

= + + + +

+

+

+ +

Free energy density for a uniaxial solid (SmA with locked layers)

12u u unt nt nt ssd= -

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

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

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

5/23/05 IMA

Coupling to Nematic Order• Strain u 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

-

-

L = L L L L L º

= L L L =

= L L = + =T;i i i in O n n O na a a a= =% %Ref->target Target->ref

5/23/05 IMA

Strain and Rotation

L

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

target space vector that is obtained when is

symmetric

Simple ShearSymmetric shear

Rotation

12( )i i i i

i i k k

O u ua a a a

a a

d

d e

» + ¶ - ¶

» - W

5/23/05 IMA

Softness with Director

2 2 21 12 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

n

n n tt

nn nn nt

n n n n

nn nn nt

n n n

l

l

+

+ +

= + + +

+ + +

= + + +

+ + + -

+

%

% %

%

L%% %

22

5 51

10

2R D

C CD

Soft= - = ÞDirector relaxes to zero

( , )zn n n

Nunit vector along uniaxial direction in reference space; layer normal in a locked SmA phase

2 2 21 ( ) ; , etc.zzn N n c u N u N

Red: SmA-SmC transition

5/23/05 IMA

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

nn aa

nt n

n n nt n t n

n n n

n n n n

e

-

-

= + +

= + +

+ +

= ¶ + ¶ + ¶

= +

= - ¶ - ¶

ò

òò % %

%

5/23/05 IMA

NE: Relaxed elastic 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

= + +

+ + + ¶ + ¶

= -

æ ö æ ö÷ ÷ç ç÷ ÷= + = -ç ç÷ ÷ç ç÷ ÷ç

= ¹

çè ø è ø

ò

Soft : Semi - Soft :

Uniaxial solid when C5R>0,

including Frank director energy

5/23/05 IMA

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

5/23/05 IMA

PreliminariesHarmonic Oscillator: seeds of complete formalism

221; { , } 1

{ , }

2

,

2

{ }

p x

Hp x kx

x

pH kx

m

p x v

Hx p

pp

xm

p mv

=

¶- -

¶¶

-

= +

= - G = - G

= =¶

=

& &

& friction

Poisson bracket

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)

5/23/05 IMA

Fluid Flow – Navier StokesConserved densities:

mass:Energy:

Momentum: gi = vi2

0

0

t

t i i ij i

t i

i i

i

ig v

j

g

pe

r

s h

e

¶

-

¶ + =

¶ = ¶ = + Ñ

¶ +¶ =

¶

212 ( / ) [ ]d dF d x g d xfr r= +ò ò

2

2

2

2(2 modes)

3

(2 modes)

(1 mode)p

cq iq

i q

i qC

hw

rh

wrk

w

= ± -

= -

= -

5/23/05 IMA

Crystalline Solid I

Conserved densities:

mass:Energy:

Momentum: gi = vi

Broken-symmetry field:

Phase of mass-density field: u describes displacement of periodic part of density

21 12 2( / ) [ ]d

ii ijkl ij klF d x g f u K u ur r l dré ù= + - +ê úë ûò

( )/ 2ij i j j i

u u u= ¶ +¶

( )ier r r × -® +å G x uG

G

Mass density is periodic

Strain

Free energy

Aside: Nonlinear strain is not the Green Saint-Venant tensor

5/23/05 IMA

Crystalline Solid II

2

0

1t

t ii

t i

i i

i

i

ii

Fu

u

g

up

v

g vFd

r

g

d

dd

h

¶ + =

¶ = -

-

¶

+¶ Ñ-¶ =

permeation

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

mass-density wave and mass: mass motion with static density wave

Aside: full nonlinear theory requires more care

5/23/05 IMA

Tethered Solid

/t

dr r

¶ =

= - Ñ ×

u v

u

No permeation :

Density locked :

2 2

it i i

Fu

uvh

dr

d¶ = - + Ñ

( )2 212 2d

ii ijF d x u ul m= +ò

Isotropic elastic free energy

7 hydrodynamic variables: 1 density,3 momenta, 3 displacements, 1 energy + 1 constraint = 8-1=7Classic equations of motion for a Lagrangian solid; use Cauchy-Saint-Venant Strain

2

2

(4)2

2 2 (2)

3

T

L

q i q

q i q

m hw

r r

l m hw

r r

= ± +

+= ± +

+ energy mode (1)

5/23/05 IMA

Gel: Tethered Solid in a FluidTethered solid

2 2 ( )s s

iit i ii

Fu u

uu v

dr h

d¶ = - + Ñ - G -& &

2 ( )ii it i i

g p v v uh¶ = - ¶ Ñ -+ - G &Fluid Frictional Coupling

( ) Tst i i j ij

g ur s¶ + = ¶&Total momentum conserved

2( ) ( )s si ii

Fu u

ud

r r h hd

+ = - + + Ñ&& &

1 1 1( )sFi iw t r r- - -= - = - + G

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

1: wt = Effective Tethered Hydro.

Fluid and Solid move together

Friction only for relative motion- Galilean invariance

5/23/05 IMA

Nematic Hydrodynamics: Harvard I

212 ( / ) [ , ]d dF d x g d xfr r= +ò ò n

g is the total momentum density: determines angular momentum = ´xl g

( )2 21 2

23

1 1[ , ] ( ) ( )[ ( )]

2 21

( )[ ( )]2

f K K

K

r r r

r

= Ñ × + ×Ñ ´

+ ´ Ñ ´

n n n n

n n

Frank free energy for a nematic

5/23/05 IMA

Nematic Hydrodynamics: Harvard II1

t ii

t i

jijk k

j ki ij ijk

j

Fn

n

g p

v

Fn

dg

d

dl

dl s

¶ = -

¢¶ = - ¶ +

¶

æ ö÷ç ÷¶ ç ÷ç ÷¶

è+

ç ø

( ) ( )1 12 2

;ij ijkl kl

T T T Tij j ij jijk k ik k ik

A

n n n n

s h

l d d l d d

¢=

= - + +

( )12

12

ij i j j i

i jijk k

A v v

vw e

= ¶ +¶

= ¶

:1

tw l

g¶ = + +´ n An n h

– fluid vorticity not spin frequency of rods

Symmetric strain rate rotates n

permeation

Stress tensor can be made symmetric

Modes: 2 long sound, 2 “slow” director diffusion.2 “fast” velocity diff.

5/23/05 IMA

NE: Director-displacement dynamics

1

1

t ii

i i

t i

jijk k

j ki

i

ij j j

ki

Fg

F

Fn

n

u g

v

ng

Fu

l

dl

dg d

s

d

d

r

dd

d

¶

æ ö÷ç ÷¶ ç ÷ç ÷çè ø

¶ = -

= =

¢¶ = +¶-

&

1

1f

D iiw

g t= - = -

Director relaxes in a microscopic time to the local shear – nonhydrodynamic mode

Stenull, TCL, PRE 65, 058091 (2004)Tethered anisotropic solid plus nematic

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

Note: all variables in target space

Modifications if depends on frequency

5/23/05 IMA

Soft Elastomer Hydrodynamicseff

ui jijkl l k

i

Fu v

ud

r hd

= - + ¶ ¶&&

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

wh

hw

r

= -

= -

5/23/05 IMA

Beyond Hydrodynamics: ‘Rouse’ Modes

( ) ( )E

G i

2 1

12

1

( ) ( )

( )( )

1, 0

3/ (2 ),

E

N

N

fi

p xf x

p

x

x x

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

5/23/05 IMA

Rouse in NEs22

51

25

2 22 2

1

( )2

[ ( ) ( / 2) ( )]

[1 ( )]1

2 1 ( )n

DG C

D

i

D i

D i

1

2 1 2

5 5

( ) ( )/ ;

( ) ( / ) ( )

( ) ( );

( ) ( )

n

n

n n E

E

D

D D

fi

fi

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

5

15 2

2

( ) 1

1( ) | | 1 | | ; 1

2

Rn

RnE

G C

DG C D

D

or n nE E

Second plateau in G'n E

“Rouse” Behavior before plateau

5/23/05 IMA

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

5/23/05 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.