Elasticity and Dynamics of LC Elastomers

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5/23/05

IMA

Elasticity and Dynamics of LC
Elastomers

Leo Radzihovsky

Xiangjing Xing

Ranjan Mukhopadhyay

Olaf 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)

Displacement

strain

Invariances

Displacements

( ) ( )
= +
R x x u x
1
;
U V
-
® ®
R R x x
i
i i i
R
x
a a a
a
d h

L = = +

i i
u
a a
h
= ¶
a
,
b

= Ref. Space

i,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
2
1
2
2
3 2
( ) ( )
( ) ( )
Tr
Tr Tr
f f f U V
f f u f V uV
C u D
Bu u
u
aa
m
-
-
= L = L
= =
=
+
+
-
% %
%
1
3
u u u
ab ab ab gg
d
= -
%
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 2
1 1
2 2
1 2 3
2 2
4 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 strain

Green


Saint Venant strain tensor
-

Physicists’

favorite


invariant under
U;

(
)
1 1
2 2
1
2
( ) ( )
T
T
k k
u
u u u u u
ab a b b a a b
d h h
= L L - » +
= ¶ + ¶ + ¶ ¶
2 2
2
dR dx u dx dx
ab 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 anisotropic

state as

m

goes to zero

Direction of

n
0

is

arbitrary

0
0 0
1
3
( )
u u
n n
ab ab
a b ab
d
=
= Y -
% %
2
~
u
aa
Y
(
)
1
2
0 0 0
T
u
d
= L L -
0 0
2
u
d
L = +
Symmetric
-

Traceless

part

Golubovic, L., and Lubensky, T.C.,
, PRL

63
, 1082
-
1085, (1989).

5/23/05

IMA

Strain of New Phase

d
u

is the deviation of

the strain relative to the
original reference frame

R
from

u
0

u


is the strain relative
to the new state at
points

x



d
u

is linearly proportional

to

u


(
)
1 1
2 2
'( )
T
T
u
d h h
¢ ¢ ¢ ¢
= L L - » +
0
( ) ( )
( )
i ij j i
i i
R x u
x u
d
= L +
¢ ¢ ¢
= +
x x
x
(
)
0
1
2
0 0
0 0
'
T T
T
u u u
u
d
= -
= L L - L L
= L L
0
i i k
ij
ik kj
j j
k
x
R R
x x x
¢

¶ ¶
¢
L = = = L L
¢
¶ ¶ ¶
5/23/05

IMA

Elasticity of New Phase

Rotation of anisotropy

direction costs no energy

C
5
=0 because of

rotational invariance

This 2nd order expansion

is invariant under all
U


but only infinitesimal
V

(
)
(
)
1
1
1
0 0 0 0
1
1
4
1 1
'( )
1 cos2 sin 2
( 1)
sin 2 1 cos2
T
r
r
r
u V u V u
r
q q
q q
-
-
-
= L - L
æ ö
-
÷
ç
÷
ç
= -
÷
ç
÷
- -
ç
÷
ç
è ø
( 1)
'~
4
xz
r
u
r
q
-
2
0| |
2
0
r
^
L
=
L
2
1 1
2 2
1 2 3
el
5
4
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
u
xx

can be converted to a
zero energy rotation by
developing strains
u
zz

and
u
xz

until
u
xx

=(
r
-
1)/2

(
)
1
1
4
1 1
1 cos2 sin 2
( 1)
sin 2 1 cos2
r
r
r
u r
q q
q q
æ ö
-
÷
ç
÷
ç
= -
÷
ç
÷
- -
ç
÷
ç
è ø
1
1
( 1 2 )
2
zz xx
xz xx xx
u u
r
u u r u
r
= -
= - -
5/23/05

IMA

Frozen anisotropy: Semi
-
soft

( ) ( )
h
zz
f u f u hu
= -
( ) ( ) ( 2 )
h
zz xz
f u f u h u u
q
¢
= - +
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 2
1 1
2 2
5
1 2 3 4
zz zz z
f C u C u u C u C u C u
nn 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
2
xz xx zz
xx zz xz
u u u
u u u
u u u
q
æ ö
- -
÷
ç
÷
¢
® = +
ç
÷
ç
÷
-
÷
ç
è ø
Rotation

5/23/05

IMA

Semi
-
soft stress
-
strain

2 2 ( ) 2( )
( )
0 or
h
xz xz xx zz zz xx xz
xx xz
xz
xz
xx zz
xx
df
hu u u u
d
h u
u u
u h
s s
s
s
s
s
q
=
= - = - -
-
= Þ
-
=
+
h
f
u
ab
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 2
1 1
2 2
1 2
2 2 2 2
1
2 2
1 2
5
2 3
3
3 4
ˆ ˆ
(
ˆ
ˆ
)
ˆ
zz
zz z z
z
z
zz z
z
z
g 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
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)

1
2
ˆ
u u u
nt nt nt s s
d
= -
C
4
=0: Transition to Biaxial Smectic with soft in
-
plane elasticity

C
5
=0: Transition to SmC with a complicated soft elasticity

Red
: Corrections for transition to biaxial SmA

Green
: Corrections for trtansition to SmC

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

5/23/05

IMA

Coupling to Nematic Order



Strain

u
ab

transforms like a
tensor

in the
ref. space

but
as a
scalar

in the
target space
.



The
director

n
i

and the
nematic order parameter

Q
ij

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
( ) ( )
( ) Rot at ion Mat rix
( ) (1 2 ) Symmet ric
OM
O
M u
-
-
L = L L L L L º
= L L L =
= L L = + =
T
;
i i i i
n O n n O n
a a a a
= =
% %
Ref
-
>target

Target
-
>ref

5/23/05

IMA

Strain and Rotation

L
n
%
is a reference space vector; it is equa
l to the
target space vector that is obtained whe
n is
symmetric
Simple Shear

Symmetric
shear

Rotation

1
2
( )
i i i i
i i k k
O u u
a a a a
a a
d
d e
» + ¶ - ¶
» - W
5/23/05

IMA

Softness with Director

2 2 2
1 1
2 2
1 2 3 4
2 2
1
2
5
2 1
2 2 2
1 1
2 2
1 2 3 4
2
2
1
4 2
1
4
2
2 2
1 1
2 2
5
1 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 Dn
C u C u u C u C u
D n D D u C D u
u
D
n
n
n n t t
nn nn nt
n n n n
nn nn nt
n n n
l
l
+
+ +
= + + +
+ + +
= + + +
+ + + -
+
%
% %
%
L
%% %
2
2
5 5
1
1
0
2
R
D
C C
D
Soft
= - = Þ
Director relaxes to zero

(,)
z
n n
a 

n
N
a

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

2 2 2
1 ( );, et c.
zz
n N n c u N u N
 a a  a
ab b
    
Red: SmA
-
SmC transition

5/23/05

IMA

Harmonic Free energy with Frank part

3 2 2
1 1
2 2
1 2 3
2 2
5
4
3 2 2 2
1 1 1
2 2 2
1 2 3
3 2
1
2
1 2
1
2
[
]
[ ( ) ( ) ( ) ]
[ ]
( )
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 Dn 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
ò
ò
% %
%
5/23/05

IMA

NE: Relaxed elastic energy

eff 3 2 2
1 1
2 2
1 2 3
2 2 2 2 2 2
1 1
2 2
5
4 1 3
2
2
5 5
1
2 2
2 2
1 1
5
3
5
1 1 3
[
( ) ( ) ]
;
2
1 1
1;
0
1
0
4 4
;
u zz zz aa
ii
R
R
R R
az a z z a
ab
R
R R
R
F d x C u C u u C u
C u C u K u K u
D
C C
D
C
D D
K K K
C
K
D D
= + +
+ + + ¶ + ¶
= -
æ ö æ ö
÷ ÷
ç ç
÷ ÷
= + = -
ç ç
÷ ÷
ç ç
÷ ÷
ç
= ¹
ç
è ø è ø
ò
Soft:Semi - Soft:
Uniaxial solid when
C
5
R
>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

Preliminaries

Harmonic Oscillator
: seeds of complete formalism

2
2
1
;
{,} 1
{,}
2
,
2
{ }
p x
H
p x kx
x
p
H kx
m
p x v
H
x p
p
p
x
m
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 Stokes

Conserved densities:

mass
:

r

Energy:
e

Momentum:
g
i

=
r
v
i

2
0
0
t
t
i i ij i
t
i
i i
i
i
g v
j
g
p
e
r
s h
e

-
¶ + =
¶ = ¶ = + Ñ
¶ + ¶ =

2
1
2
(/) [ ]
d d
F d x g d xf
r r
= +
ò ò
2
2
2
2
(2 modes)
3
(2 modes)
(1 mode)
p
cq iq
i q
i q
C
h
w
r
h
w
r
k
w
= ± -
= -
= -
5/23/05

IMA

Crystalline Solid I

Conserved densities:

mass
:

r

Energy:
e

Momentum:
g
i

=
r
v
i

Broken
-
symmetry field
:


Phase of mass
-
density field:
u

describes displacement of
periodic part of density

2
1 1
2 2
(/) [ ]
d
ii ijkl ij kl
F d x g f u K u u
r r l dr
é ù
= + - +
ê ú
ë û
ò
( )/2
ij i j j i
u u u
= ¶ + ¶
( )
i
e
r r r
× -
® +
å
G x u
G
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
1
t
t
i
i
t
i
i i
i
i
i
i
F
u
u
g
u
p
v
g v
F
d
r
g
d
d
d
h
¶ + =
¶ = -
-

+

Ñ
-
¶ =
permeation

Modes:

Transverse phonon: 4

Long. Phonon: 2

Permeation (vacancy
diffusion): 1

Thermal Diffusion: 1

Permeation: 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
i
t
i i
F
u
u
v
h
d
r
d
¶ = - + Ñ
(
)
2 2
1
2
2
d
ii ij
F d x u u
l m
= +
ò
Isotropic elastic free energy

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

2
2
(4)
2
2 2
(2)
3
T
L
q i q
q i q
m h
w
r r
l m h
w
r r
= ± +
+
= ± +
+ energy mode (1)

5/23/05

IMA

Gel: Tethered Solid in a Fluid

Tethered solid

2 2
( )
s s
i
i
t
i
i
i
F
u u
u
u v
d
r h
d
¶ = - + Ñ -
G -
&
&
2
( )
i
i
i
t
i i
g p v
v u
h
¶ = - ¶ Ñ
-
+ -
G
&
Fluid

Frictional Coupling

( )
T
s
t
i i j ij
g u
r s
¶ + = ¶
&
Total momentum conserved

2
( ) ( )
s s
i i
i
F
u u
u
d
r r h h
d
+ = - + + Ñ
&& &
1 1 1
( )
s
F
i i
w t r r
- - -
= - = - + G
Fast non
-
hydro mode: but not
valid if there are time scales in
G

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

2
1
2
(/) [,]
d d
F d x g d xf
r r
= +
ò ò
n
g

is the total momentum density:
determines angular momentum

= ´
x
l
g
(
)
2
2
1 2
2
3
1 1
[,] ( ) ( )[ ( )]
2 2
1
( )[ ( )]
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 II

1
t
i
i
t
i
j
ijk k
j
k
i
ij
ij
k
j
F
n
n
g p
v
F
n
d
g
d
d
l
d
l
s
¶ = -
¢
¶ = - ¶ +

æ ö
÷
ç
÷

ç
÷
ç
÷

è
+
ç
ø
(
)
(
)
1 1
2 2
;
ij
ijkl kl
T T T T
ij j ij j
ijk k ik k ik
A
n n n n
s h
l d d l d d
¢
=
= - + +
(
)
1
2
1
2
ij i j j i
i j
ijk k
A v v
v
w e
= ¶ + ¶
= ¶
:
1
t
w l
g
¶ =
+
+
´
n A
n
n
h
w



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
i
i
i i
t
i
j
ijk k
j
ki
i
i
j
j j
k
i
F
g
F
F
n
n
u g
v
n
g
F
u
l
d
l
d
g d
s
d
d
r
d
d
d

æ ö
÷
ç
÷

ç
÷
ç
÷
ç
è ø
¶ = -
= =
¢
¶ = + ¶
-
&
1
1
f
D
i
i
w
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
g

depends on frequency

5/23/05

IMA

Soft Elastomer Hydrodynamics

eff
u
i j
ijkl l k
i
F
u v
u
d
r h
d
= - + ¶ ¶
&&
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
2
5
2
2
s
f
K
i q
i q
w
h
h
w
r
= -
= -
5/23/05

IMA

Beyond Hydrodynamics: ‘Rouse’
Modes

( ) ( )
E
G i
w m ww
 
2 1
1
2
1
( ) ( )
( )
( )
1, 0
3/(2 ),
E
N
N
f i
p x
f x
p
x
x x
w  w



 












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

5/23/05

IMA

Rouse in NEs

2
2
5
1
2
5
2 2
2 2
1
( )
2
[ ( ) (/2) ( )]
[1 ( )]
1
2 1 ( )
n
D
G C
D
i
D i
D i
w
w w  gw
w w
w w
 
 

 
 
 

 
1
2 1 2
5 5
( ) ( )/;
( ) (/) ( )
( ) ( );
( ) ( )
n
n
n n
E
E
D
D D
f i
f i
 w gw
 w   w
 w  w
 w  w

 
 
 
References: Martinoty, Pleiner, et al.;

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

5
1
5
2
2
( ) 1
1
( ) | | 1 | |;1
2
R
n
R
n
E
G C
D
G C D
D
w w
w   w w


 

  
 
 
or
n n
E 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.