a non-local model for liquid crystal elastomers

5/25/2005 Modeling the Dynamics of LCEsIMA

1

P. Palffy-Muhoray Liquid Crystal Institute, KSU

L. Malacarne Dept of PhysicsUniversity of Maringa, Brazil

H. Finkelmann Institute für Macromoleculare ChemieAlbert-Ludwigs Universität, Freiburg

M. Shelley CIMSNew York University, NY

work supported by NSF DMR & DMS

A nonA non--local modellocal modelforfor

liquid crystal liquid crystal elastomerselastomers


2

OutlineOutline

• liquid crystal elastomers (LCE)

• effects of illumination on shape

• curious phenomena

• modeling

• summary & prospects


3

Liquid Crystal Liquid Crystal ElastomersElastomers

• key feature: coupling between orientational order and mechanical strain

• free energy:order parameter tensor

( , )f f Qαβ αβε=strain tensor


4

BackgroundBackground


5

Liquid Crystal Liquid Crystal ElastomersElastomers: Behavior : Behavior

• monodomainLCE

• 5cm x 5mm x 0.3mm

• lifts 30g wt. on heating, lowers it on cooling

• large strain!

(H. Finkelmann)

nematic

(400%)


6

NematicNematic liquid crystalsliquid crystals

• order parameter:• symmetric traceless

tensor• measure of mean

squared projection of mesogen axes on principal directions

ilil

1 ˆˆ(3 )2

=< − >Q ll I

• eigenvectors: – direction of alignment:

nematic director

• eigenvalues:– degree of alignment:

order parameter

n

S

1 ˆ ˆ(3 )2

S= −Q nn In 0.8S =


7

NematicNematic free energy densityfree energy density

2 3 2 21 1 1( 1) ( ) ..2 3 4o

c

Ta b cT

= − − + +Q Q QF

• minimizing the free energy with respect to the order parameter:


8

NematicNematic free energy densityfree energy density

• minimizing the free energy with respect to the order parameter:

2 3 2 21 1 1( 1) ( ) .. 2 3 4

12o

c

Ta b cT

ε= − − + + − ΔQ Q Q QEEF

• free energy is minimized if molecules/director align with field


9

Elastic solids:Elastic solids:

• position of a material point before deformation:

• position of a material point after deformation:

• displacement vector:

• strain tensor:

r

+r R

12

RRex x

βααβ

β α

⎛ ⎞∂∂= +⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠

R


10

Elastic solids:Elastic solids:

• for incompressible materials, the free energy density is

• where is Young’s modulus, and is the external stress.

212

extη= −σ eeF

η

Theory of Elasticity, L.D. Landau and E.M. Lifshitz

extσ


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Liquid Crystal Liquid Crystal ElastomersElastomers

• free energy density*:

• sum of free energies of liquid crystal + elastic solid,plus new coupling term !!

• effect of strain on LC order is same as external field• effect of LC order on strain is same as external stress

*P.G. de Gennes, C.R. Seances Acad.Sci.218, 725 (1975)

Qe

2 21 1.. 122 2

exta Uε η− Δ= + − + −QEE ee σQQ eF


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Free energy: another look Free energy: another look

2 3 4 21 1 1 1 ' ...2 3 4 2

extoa b c U Uη= − + − + − −Q Q Q Qe e σ e Q eF

• can also write

2 3 241 1 1 ( )2

'3

..12 4

a b c Uηη

− + +−= + e QQ Q QF

SOFT ELASTICITY!

1442443


13

LCE SamplesLCE Samples


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director

If orientational order increases,

expansion contraction

If orientational order decreases,

expansioncontraction

StructureStructure


15

Appearance of Appearance of nematicnematic LCE samples LCE samples

50mm x 5mm x .3mm

birefringent sample betweencrossed polarizers


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Consequences of coupling between Q and eConsequences of coupling between Q and e

• changing Q changes the shape of the sample

• can change Q with

– temperature

– E & H fields

– light

– impurities


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OptomechanicalOptomechanical effects in effects in nematicnematic LCE LCE

-light induced deformations


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Mechanisms of Mechanisms of OptomechanicalOptomechanical Effects in LCEEffects in LCE

• optomechanical coupling:

light

orientational order

mechanical strain

shape change


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Effects of light on order parameterEffects of light on order parameter• optical field changes order parameter via:

– direct heating– absorption

– disruption of order – photoisomerization

– direct optical torque– angular momentum transfer from light

– indirect optical torque• Landauer’s blowtorch• orientational Brownian ratchet

– no angular momentum transfer from light; – light drives molecular motor

laserray

table

_hp = λlaser

ray

table

_hp = λ


20

Experimental Results (Warner Experimental Results (Warner et alet al.).)

• azo-dye incorporated in network

H. Finkelmann, E. Nishikawa, G. G. Pereira and M. Warner, Phys. Rev. Lett. 87,015501 (2001)


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Experimental Results (Warner Experimental Results (Warner et alet al.).)


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Experimental Results (Ikeda et al. )Experimental Results (Ikeda et al. )Yanlei Yu,Makoto Nakano, Tomiki

Ikeda, Nature 425, 125 (2003)

timescale: 10 s

LC + diacrylate network+ functionalizedazo-chromophore

Yanlei Yu,Makoto Nakano, Tomiki Ikeda, Nature 425, 125 (2003)


23

Our experimental resultsOur experimental results

Sample

ArLaser

Sample

ArLaser

sample: nematic elastomer EC4OCH3+ 0.1% dissolvedDisperse Orange 1 azo dye

5mm

300 mμ


24

Force measurementsForce measurements

• schematic of setup

Sample

ArLaserENTRAN

sensor

1.5mm

Computer


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Dynamic ResponseDynamic Response

0 100 200 300 400 5000.0

0.5

1.0

1.5

2.0

2.5

time (ms)

forc

e (m

N)

50 100 150 200 250 300-3

-2

-1

0

1

time (ms)lo

g (fo

rce)

τ=75ms

0 100 200 300 400 5000.0

0.5

1.0

1.5

2.0

2.5

time (ms)

forc

e (m

N)

50 100 150 200 250 300-3

-2

-1

0

1

time (ms)lo

g (fo

rce)

τ=75ms

0.6P W=3d mm=

5 5mm mm×


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Curious PhenomenaCurious Phenomena


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Experimental SetupExperimental Setup

• floating nematic LCE sample illuminated from above

Laser beam

container

waterElastomer


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Experimental Results Experimental Results

• nematic LCE sample floating on water • illuminated from above


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OscillatorOscillator

• CW illumination • irregular sample shape


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Experimental Results Experimental Results

• nematic LCE sample floating on ethylene glycol• illuminated from above


31

the Puzzle:the Puzzle:

energy transfer:- how is light energy converted to kinetic energy?

momentum transfer:- where does momentum come from???

• what is the mechanism of propulsion?


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Motor!Motor!

motion is caused by the transfer of energy,but not momentum:

FF

LASE

RLA

SER

myosinactin

ATP

myosinactin

ATP

myosinactin

myosinactin

ATP


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Novel aspect: information transfer Novel aspect: information transfer

• light carries energy + information to system.

FF• energy source attached to car:

no problem.

LAS

ER

LAS

ER

• laser attached to LCE:no motion!


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the Puzzle:the Puzzle:

• what is the mechanism of propulsion??

light provides energy, but not momentum!

• where does momentum come from??


35

laserlaser

director

smalldisplacement

laserlaser

force on water

force on elastomer

laserlaserlaserlaserlaser

what happens:what happens:


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Locomotion in Locomotion in batoidbatoid fishesfishesAtlantic stingray.

Swims by propagating waves down the pectoral fins from anterior to posterior.

L.J. Rosenberger, J. Exp. Biol. 204, 379-394 (2001).


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Swimming dynamicsSwimming dynamics

• elastomer– swims like a fish– intrinsic instability propagates wave-like deformation in

elastomer

• system is a motor• light provides energy + information

M. Camacho-Lopez, H. Finkelmann, P. Palffy-Muhoray, M. Shelley, Nature Mat. 3, 307, (2004)


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Modeling light induced Modeling light induced deformationsdeformations


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Modeling:Modeling:• want dynamic description:

Jeremy Neal, KSU


40

Modeling the dynamics of Modeling the dynamics of nematicnematic LCEsLCEs

• Order parameters:

– Displacement:

– Orientation:

( )Rα r

( )Qαβ r

r

R 'R

'r


41

StrategyStrategy

• specify free energy density– nematic + elastomer

• specify dissipation – nematic + elastomer

( , )β α= aQ RF F

( , )ααβ

β

∂=

∂

&& RQ

xR R


42

StrategyStrategy

• obtain dynamics from

3 0KEd ddt R R Rα α α

δ δδ δ

⎡ ⎤+ + =⎢ ⎥∂⎣ ⎦

∫ r& &

E F R

3 0αβ αβ

δ δδ δ⎡ ⎤

+ =⎢ ⎥⎢ ⎥⎣ ⎦∫ & d

Q QrF R

momentumconservation:

non-conservedorder parameterdynamics:


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Free energy: Free energy:

• mean field theory

• order parameters and vary in space

• use symmetry allowed squared gradient terms

( , ) ?β α =aQ RF

squared gradient terms may be a problem!

βaQ αR


44

KK1313 problem: problem:

• for nematics, usual Landau-de Gennes free energy density is

• if and is constant, then

2 3 2 2 2 21 2

1 1 1 1 1( ) .. ( ) ( )2 3 4 2 2

αβ αβ

α γ

∂ ∂= − + + + +

∂ ∂Q Q

a b c L Lx x

Q Q QF

1 ˆ ˆ(3 )2

= −SQ nn I S


45

KK1313 problem:problem:

• Frank-Oseen free energy for nematics:

• Saupe1: include symmetry allowed terms

• Oldano2: free energy is unbounded!

2 2 21 2 3

1 1 1ˆ ˆ ˆ ˆ ˆ( ) ( ) ( )2 2 2

K K K= ∇ ⋅ + ⋅∇× + ⋅∇×n n n n nF

2 C. Oldano and G. Barbero, Phys. Lett., 110 A, (4), 213, (1985)1 J. Nehring and A. Saupe, J. Chem. Phys., 64, (1), 337, (1971)

13 24ˆ ˆ ˆ ˆ ˆ ˆ( ) ( )K K∇⋅ ∇ ⋅ + ∇ ⋅ ∇ ⋅ − ⋅∇n n n n n n


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KK1313 problem:problem:

• in index notation

• problem generated great deal of activity– experiments– alternative approaches (2nd order elasticity…)– many papers– saga still continues……..

13 , , , 24 , , , ,( ) ( )K n n n n K n n n nα α β β α β αβ α α β β α β β α+ + −

second derivative

13K


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ResolutionResolution

• non-local free energy is bounded

• problem is due to gradient expansion

• one solution: don’t use expansion!

Avoid gradients;∇

use fully non-local description !


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Elastic free energy Elastic free energy

– for isotropic polymers, distribution of separation of connected crosslinks is

where is the chain length, and is the step length

– the free energy of the polymer chain between the crosslinks is

23( ) ~ exp( )2

ss

RPL

−RL

ln ( )skT P= − RF

L L


49

Elastic free energy* Elastic free energy*

– for anisotropic polymers, distribution of separation of connected crosslinks is

– the matrix of effective step lengths is anisotropic, – for nematic LCEs

– while initially,

13( ) ~ exp( )2

−⎡ ⎤− ⎣ ⎦T

s s sP R R L RL

L

o ol l= + ΔL I Q

* Liquid Crystal Elastomers, M. Warner and E. Terentjev (Cambridge, 2003)

l l= + ΔL I Q


50

NonNon--local elastic free energylocal elastic free energy

– is Lagrangian coordinate,

2 3 31 ( , ') ( , ' ') '2el o oP g d dρ= + +∫∫ r r r R r R r rF

r

( , ' ') ln ( , ' ')g kT P+ + = − + +r R r R r R r R

' ' 1 ' '3 ( ) ( )2 α α αβ β β β β

−= + − − + − −a akTg r R r R L r R r RL

( , )α αβ=el el R QF F

( )=R R r


51

NonNon--local local nematicnematic free energyfree energy

2 3 4

3 36

1 1 1 1[ ( ) ( ) ( ) ..2 2 3 4

( )( ( ) ( ')) ] '

( ' ' )

LC

a

aQ bQ cQ

Q Q QU d dβ αβ αβαρ

= − + +

−+

+ − −

∫∫ r r r

r r rr r

r R r R

F

( , )α αβ=LC LC R QF F


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RayleighRayleigh dissipation functiondissipation function

• viscosities depend on .

(1)

(2)

(3)

2

( )

TS Q Q

RR Q Qx x

RRx x

αβγδ αβ γδ

γααβγδ γδ αβ

β δ

γααβγδ

β δ

ν

ν

ν

= = +

∂∂+ +

∂ ∂

∂∂∂ ∂

& & &

&&& &

&&

R

( , )α αβ= &&LC R QR R

αβγδναβQ


53

Dynamics Dynamics

• for material points

• for nematic order parameter

2(2) (3)γδ γ

α αβγδ αβγδα β β δ

δρ ν νδ

∂ ∂= − + +

∂ ∂ ∂

& &&& Q RR

R x x xF

(1) (2) γαβγδ γδ γδαβ

αβ δ

δν νδ

∂= − −

∂

&& R

QQ xF

R. Ennis, L. Malacarne, P. Palffy-Muhoray, M. Shelley, http:www.e-lc.org/docs/2004_12_11_00_45_55


54

DynamicsDynamics

• for the surrounding fluid:

• boundary conditions:

on

( 2 )D pDt

ρ η= ∇⋅ − +v I D

0∇⋅ =v

fl el⋅ = ⋅σ n σ n

fl el= =V v v∂Ω


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Discrete ModelDiscrete Model

-10

-5

0

5

10

-5

0

5-2

0

2

4

-10

-5

0

5

-10 -5 5 10

1

2

3

4

5

-7.5 -5 -2.5 2.5 5 7.5

1

2

3

4

5

-10

-5

0

5

10

-5

0

50

2

4

-10

-5

0

5

Rectangular Plate: Rectangular Plate: Uniform illumination from belowUniform illumination from below

Cross-section


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Discrete ModelDiscrete Model

10 20 30 40

-6

-4

-2

2

4

10 20 30 40

0 10 20

-5

0

5

10 20

10 20 30 40

0

10

20

-5

0

5

10

20

illumination

5 10 15 20 25 30 35

0.2

0.4

0.6

0.8

1

Intensity distribution

Rectangular Plate: Gaussian Rectangular Plate: Gaussian illumination from belowillumination from below


57

Discrete ModelDiscrete ModelRectangular Plate: Rectangular Plate: Gaussian illumination from belowGaussian illumination from below

10 20 30 40

-10

-7.5

-5

-2.5

2.5

5

10 2030

40 010

20

-10

-5

0

5

10 2030

5 10 15 20 25

-12

-10

-6

-4

10 20 30 40

0

10

20

-10

-5

0

5

10

20


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DynamicsDynamics

0.0 0.50.0

0.5

1.0

1.5

2.0

2.5

3.0

Dis

plac

emen

t (m

m)

Time (s)

experiment

Field OFF

0.0 0.50.0

0.5

1.0

1.5

2.0

2.5

3.0

Dis

plac

emen

t (m

m)

Time (s)

experiment

Field OFF0 10000 20000 30000 40000 50000 60000 70000

0.0

0.5

1.0

1.5

2.0

2.5

Dis

plac

emen

t

Time

simulation fit

Field OFF

0.0 0.50.0

0.5

1.0

1.5

2.0

2.5

3.0

Dis

plac

emen

t (m

m)

Time (s)

experiment

Field OFF

0.0 0.50.0

0.5

1.0

1.5

2.0

2.5

3.0

Dis

plac

emen

t (m

m)

Time (s)

experiment

Field OFF0 10000 20000 30000 40000 50000 60000 70000

0.0

0.5

1.0

1.5

2.0

2.5

Dis

plac

emen

t

Time

simulation fit

Field OFF0 10000 20000 30000 40000 50000 60000 70000

0.0

0.5

1.0

1.5

2.0

2.5

Dis

plac

emen

t

Time

simulation fit

Field OFF


59

-7.5-5

-2.52.5

57.5

-6 -4 -2 2 4 6

10

20

10

20

1

2

3

4

5

10

20

10

20

Saddle shapeSaddle shape


60

Summary:Summary:

• fast and large photoinduced deformations in dye doped nematic LCEs

• novel swimming phenomena: – mechanism: swims like a fish – motor: requires energy + information

• modeling: – developed non-local continuum model– agreement with existing results – numerics: under construction

-10 -5 5 10

123456

a non-local model for liquid crystal elastomers

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