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
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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)
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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
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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
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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).
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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
¢¶ ¶ ¶
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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
¢ ¢ ¢ ¢ ¢= + +
¢ ¢ ¢ ¢+ +
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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
( )
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
= -
= - -
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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
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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
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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
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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)
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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
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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
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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
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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
nn aa
nt n
n n nt n t n
n n n
n n n n
e
-
-
= + +
= + +
+ +
= ¶ + ¶ + ¶
= +
= - ¶ - ¶
ò
òò % %
%
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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
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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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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)
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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
= ± -
= -
= -
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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
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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
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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)
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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
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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
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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.
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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
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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
= -
= -
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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
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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
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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.