thermo/electro-mechanical instabilities - in confined

Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University

Thermo/Electro-Mechanical Instabilitiesin Confined Samples of Nematic Gels

Robert B. Meyer & Guangnan Meng

Complex Fluids Group, Martin Fisher School of PhysicsBrandeis University, Waltham, Massachusetts

October 1st, 2004


Introduction & Background KnowledgeComplex Fluids — Liquid Crystals & PolymersLiquid Crystalline Elastomer & Gel

Nematic Rubber Elasticity

Buckling Transition in Nematic Gels

Stripes in Electro-Optics Experiment

Conclusions & Future Works


Complex Fluids I — Liquid Crystals

Isotropic Nematic

Smectic Crystal

Liquid Crystalline Phases

Intermediate phases between homogeneous

isotropic liquid and anisotropic crystalline

solid.

Long- & Short-Range Ordering

I Translational Order

I Orientational Order

Driving Forces for Phase Transition

I Temperature — Thermotropic LC;

I Concentration — Lyotropic LC.

Other Phases

Cholesteric, Columnar, Blue, TGB, Banana...


Curvature Elasticity & Fredericks Transition

Curvature Elasticity

Bend Splay Twist

F =1

2

(KB

(n× (∇× n)

)2+ KS(∇ · n)2 + KT (n · ∇ × n)2

)Fredericks Transition

V=0

V = 0

V

V > Vc


Complex Fluids II — Polymers & Elastomers

Polymer Melts Elastomer

Polymeric Materials

I Polymers (poly-mer = many-parts) are the macro-molecules consisting ofmany elementary units (monomers).

I Elastomer (or Rubber) is the crosslinked polymer network with its Tg

below the room temperature.

I Gel is the crosslinked polymer network which is swollen in a solvent.


Free Energy of an Ideal Polymer Chain

R

a1

a2

ai

aN

o

End-to-end Vector:

R = a1 + a2 + · · ·+ aN =N∑

i=1

ai

Probability density:

limN→∞

P(R, N) = (2πNa2

3)−3/2 exp(− 3R2

2Na2)

Configuration Entropy:

S(R, N) = kB ln P(R, N) = −3kBR2

2Na2+ Const.

Helmholtz Free Energy:

F (R, N) = U − TS =3kBTR2

2Na2+ Const.

⇒ δF ∝ δr


Entropic Elasticity of Polymeric Materials

R0

Lx0

Lz0

Ly0

Initial State

R

Lx Lx0

=λx

Lz Lz0

=λz

Ly Ly0

=λy

Deformed State

Affine Deformation:

The relative deformation of each polymer

strand is same as the macroscopic

relative deformation applied on the whole

polymer network.

Cauchy Strain Tensor:

λij =∂Ri

∂R0j

= δij + ηij

Classical Rubber Elasticity:Free energy density (free energy per unit volume) of a general deformation forthe isotropic elastomer becomes:

F (λ) = F (R)− F (R0)

=nskBT

2

(λijλji

)≡ nskBT

2Tr

(λT · λ

)


Liquid Crystalline Elastomers— Solid-Liquid Crystals

Nematic Nematic Elastomer Nematic Gel

Liquid Crystalline Elastomer:The elastic material freezing the liquid crystalline ordering into the crosslinkedpolymeric networks.

I Non-vanishing shear modulus — Solid instead of Fluid;

I Broken Symmetry — Anisotropic instead of Isotropic.


Nematic Phase I

Crosslinking in Isotropic Phase

Nematic Phase II


Soft Elasticity

Solid rubber behaves like a liquid:

To make certain shear strain and to rotate the nematic director n non-

uniformly without any first order elastic energy cost, as long as the entropy of

the polymer network doesn’t change during the transformation, e.g. liquid

crystalline elastomer behaves like a classical liquid.


Semi-Soft Elasticity

Internal field to memorize the crosslinking stage:

I To crosslink the polymer network in an ordered phase, e.g a nematic

mono-domain, cholesteric or other “pre-strained” configurations;

I To produce the “memory” effect of the polymer network about its

thermomechanical history;I e.g. For a uniaxial nematic elastomer crosslinked in the nematic state, there will

be five independent semi-soft elastic modes in the elastic energy expression.

L. Golubovic and T. C. Lubensky, Phys. Rev. Lett. 63, 1082–1085 (1989).

Free Energy Density:

Fsemi =1

2A · µ · f

(θ, θ0(λ)

)where µ is the shear modulus, θ0 and θ are the director angle of the initial

polymerization state and the final deformed state, A is the constant to tell the

magnitude of semi-softness.


Introduction & Background Knowledge

Nematic Rubber ElasticityNematic Rubber ElasticityShear Waves in Nematic ElastomersArtificial Muscle Effect





Effects of LC Ordering on Polymer Network

l||

l⊥

Effective Step Length Tensor:to define the anisotropic Gaussian distributionof the polymer chains:

lij = l⊥δij + (l‖ − l⊥)ninj

= l⊥(δij + (r − 1)ninj

)in which r = l‖/l⊥ is a parameter to describe

how anisotropic the polymer network is: r = 1

for an isotropic shape, r > 1 for a prolate

ellipsoid and r < 1 for an oblate ellipsoid.

Analogy with the dielectric tensor:

D = ε⊥E + (ε‖ − ε⊥)(n · E)n;

Di = εijEj ;

εij = ε⊥δij + (ε‖ − ε⊥)ninj .


Neo-Classical Rubber Elasticity Theory

Assumptions:

I The Gaussian Distribution of the Polymer Chains;

I The Affine Deformation of the Elastomers.

Free Energy Density for Nematic Elastomers:

F =1

2µ · Tr

(l0 · λT · l−1 · λ

)− 1

2A · µ · Tr

(λT · n · λ− n0 · λT · n · λ

)in which µ is the shear modulus, l0 and l are the step length tensor before and

after the deformation since the nematic director n rotates due to the applied

mechanical strain λ, and nij = ninj . A is the constant contribution from semi-

soft elasticity.

Liquid Crystalline Elastomers, M. Warner and E. M. Terentjev, Oxford University

Press, 2003.


Shear Waves

Bend Shear Wave Ω = 0 Splay Shear Wave Ω = π/2

Shearing Displacement & Director’s Rotation

d =(dx sin(kxx + kyy), dy sin(kxx + kyy)

);

n =(1, ξ cos(kxx + kyy)

).

where kx = k cosΩ, ky = k sin Ω; and Ω is the angle measured away from the

x-axis.


Shear Waves in Nematic Elastomers

0 1 2 30

1

2

3

4

5

µeff

Soft Elasticity

Semisoft Elasticity

ky

kx

Effective Shear Modulus:

The vector’s length is the free energy

cost (µeff ) to propagate the shear

waves of unit amplitude along the

vector’s direction (k).

fsoft =µ

2r

(1 + r(r − 2)−

(1 + r(r − 2) + (r2 − 1) cos 2Ω

)2

4(r cos2 Ω + sin2 Ω)2

);

fsemi =µ

2r

(1 + r(A + r − 2)−

(1 + r(A + r − 2) + (r2 − Ar − 1) cos 2Ω

)2

4r2 cos4 Ω + 8r cos2 Ω sin2 Ω + (4 + 4Ar) sin4 Ω

).


Liquid Crystalline Gels— Hyper-Complex Fluids

Liquid Crystalline GelsThe crosslinked polymer network being swollen in the fluidic liquid

crystals as the solvent.

LC Curvature Elasticity:FNematic ∼ 1

2K(

∂n∂x

)2

⇒ [K ] = [Energy ][Length]

Gel Entropic Elasticity:FGel ∼ 1

2µ(

∂d∂x

)2

⇒ [µ] = [Energy ][Length]3

Length Scale of Competition:

New properties not found in either components — co-operation of the gel and

the liquid crystals, on the length scale of√

Kµ

(∼microns).


“Artificial Muscle” Effect

Isotropic Phase, T > TNI Nematic Phase, T < TNI

Shape Change at Nematic-Isotropic Transition —Nematic Ellipsoid vs.

Isotropic Sphere

I Elongation along the direction parallel to the director n;

I Shrinkage along the directions perpendicular to the director n.


22.7 42.1

73.1 23.2


Strain–Temperature Measurements

20 30 40 50 60 70 80

-40

-30

-20

-10

0

10

20

30

40

50Sample 10-wt%

∆L///L

//

0

∆L⊥/L⊥

0

Strain

(%

)

Temperature

I II

III




Buckling Transition in Nematic GelsIntroduction & Description of ExperimentExperimental Observations & ResultsTheoretical Interpretation




Introduction about Material

Molecular Structures:

C

N

5CB

( ) ( )( )n m p

Si

O

Si

O

C

N

ABASiCB4

Material Properties:

I Self-Assembled Liquid Crystalline Gel:insoluble end-blocks & well-solvatedmid-blocks in LC host;

I The gelation temperature (or TNI ) is 37 for 5-wt% nematic gel, which is close to theTNI of 5CB (35.3 ).

Kempe, M.D. et al & Kornfield, J.A. Nature Materials, Vol.3, No.3, 177–182 (2004).


O

Y

Z

X

V

DMOAP Coating

ITO Coating

Experiments in Homeotropic Cell:

I DMOAP coatings induce the nematicmolecules anchored vertically on thesurfaces;

I The nematic molecules can be aligned alongthe z-axis easily by applying the electric fieldacross the cell, E = Ez;

I The anisotropic properties of the polymernetworks can be controlled by changing thesample’s temperature.

Experimental Steps:

1. Heat up the sample into the isotropic phase(45 ) with the electric field applied(V=50 V);

2. Cool down the sample (at 2.5 /min ) intothe nematic phase (T < 37 ) with theelectric field applied (V=50 V);

3. Decrease the electric field down to zero(V=0 V) at temperature T .


Buckling Under Confined Boundaries

Ti < TNI Tf < Ti < TNI

Notes:

As the nematic gel becomes more ordered in lower temperature, the gel sample

will finally buckle under the confined boundary conditions.


25 µm DMOAP

Temp: 15


Temperature Dependence

25 µm DMOAP

Temp: 10

10

25 µm DMOAP

Temp: 15

15

25 µm DMOAP

Temp: 20

20

25 µm DMOAP

Temp: 25

25

25 µm DMOAP

Temp: 30

30

25 µm DMOAP

Temp: 35

35

Note:

As the temperature increases, the birefringence of the pattern decreases and the

images become uniformly black when the temperature is above 30 .


Critical Voltage–Temperature Dependence

10 15 20 25 30 35

20

30

40

50

Voltage (V

)

Temperature


Buckling DiagramInitial Crosslinking State, E > Ec , r 0

E

n0 = (0, 0, 1);

E = E0z;

l0ij = l0⊥(δij + (r0 − 1)n0

i n0j

).

Stressed State, E > Ec , r > r 0

E

n = (0, 0, 1);

E = E0z;

lij = l⊥(δij + (r − 1)ninj

).

Final Buckled State, E < Ec , r > r 0

n = (ξ cos kx sin qz, 0, 1);

d = (dx cos kx cos qz, 0, dz sin kx sin qz);

E = 0;


).


Theoretical Calculation

Initial State:

n0 = (0, 0, 1);

E = E0z;

l0ij = l0⊥(δij + (r0 − 1)n0

i n0j

).

Final State:

n = (ξ cos kx sin qz, 0, 1);

d = (dx cos kx cos qz, 0, dz sin kx sin qz);

E = 0;


).

Constrains & Approximations:

I Incompressibility: ∇ · d = 0;

I Average energy density over the space;

I Minimize respect to ξ and dz ;

I Ignore higher order terms o(d2x );

I Set k = 0 for the homogeneous birefringence transition as the basic mode.

Free Energy Expression:

f = µ · d2x q2 r 0

(r + (r − 1)r 0 + r 2(A− 1 + εaE

2/µ + KBq2/µ))

4r(1− r 0 + r(A− 1 + r 0 + εaE 2/µ + KBq2/µ)

)


Free Energy Plot of Buckling Transition

0.0 0.5 1.0 1.5 2.0

0.0

5.0x108

1.0x109

1.5x109

2.0x109

Fre

e E

nerg

y (a.u

.)

Electric Field (a.u.)

Notes:

Set q = πd, where d is the

thickness of sample; the curve

is plotted as r0 = 2.0 and

r = 3.0; the free energy

decreases as the electric field

E decreases, and perturbed

state become stable (f < 0)

when the electric field is below

critical field (E < Ec ).

Plotting Parameters:

d = 25µm, µ = 200J/m3, A = 0.1, KS = KB = 10−11J/m, and εa = 20ε0.


Ec − r Dependence

2.5 3.0 3.5 4.0

0.1

0.2

0.3

0.4

0.5

0.6

Critical Fie

ld a

.u.

r

Keeping r0 = 2.0 as fixed, thecritical field decreases as thevalue of r decreases(corresponding to highertemperatures), which agreesqualitatively well withexperimental observation, inwhich the critical voltagedecreases as the temperaturegoes higher.

I r = 2.5, VC = 9.3V ;

I r = 4.0, VC = 46V .

Note:

The quantitative relationship between r and temperature is unknown, which makes it

impossible to fit the critical field and temperature analytically.


2.5 3.0 3.5 4.0

0.1

0.2

0.3

0.4

0.5

0.6

Critical Fie

ld a

.u.

r

10 15 20 25 30 35

20

30

40

50

Voltage (V

)

Temperature





Stripes in Electro-Optics ExperimentSimple Stripes in a Homogeneous CellExperimental Description & ResultsSimplified Two Dimensional Model



Simple Stripes in a Homogeneous Cell

Ti

Tf < Ti

n

Note:

The nematic director n is aligned parallel to the surfaces.


Simple Stripes in a Homogeneous Cell

d=25 µm, T=25 . Courtesy of J.A. Kornfield


Synthesis of Material

Molecular Structures:

C N

89-wt% 5CB

O C NOC

O

9-wt% Monomer

C

O

O

0.8-wt% BME

C

O

O

OOC

O

O

C

O

O O C

O

1.2-wt% Crosslinker

Synthesis:

I Mixing the chemical components on vortex shaker for about four hours;

I Slow in-situ-polymerization under the UV light for four hours.


Stripe Pattern

O

Y

Z

X

PVA Coating

ITO Coating

V

PVA rubbed Homogeneous Cell

Note:

An AC Sine voltage with frequency of 1 kHz

applied.


Thickness Dependence of Stripes

d = 10µm d = 18µm

d = 24µm d = 30µm


Director & Displacement Field of 2D Model

θ = ξ cos qz sin kx , φ = 0 d =(dx sin qz sin kx , 0, dz cos qz cos kx

)Note:

where θ, φ are respectively the tilt and azimuthal angles, into which the nematic

director n pointing, as n = (cos θ cos φ, cos θ sin φ, sin θ); and q is determined by the

sample thickness d as q = πd.


Free Energy Calculation of 2D Model

Free Energy Density Expression:

F = FElastomer + FNematic + FDielectric

=1

2µ · Tr

(l0 · λT · l−1 · λ

)− 1

2A · µ · Tr

(λT · n · λ− n0 · λT · n · λ

)+

1

2

(KB

(n× (∇× n)

)2+ KS(∇ · n)2 + KT (n · ∇ × n)2

)−1

2εaE

2 sin2 θ

Other Constrains, Approximations:

I Incompressibility: ∇ · d = 0;

I Average energy density over the space: f = k2π

qπ

∫ 2πk

0

∫ π2q

− π2q

Fdzdx ;

I Minimize respect to dx and dz ;

I Ignore the higher order terms o(ξ2).


Free Energy Plot of 2D Model

0 2 4 6 8 100.0

0.5

1.0

1.5

Soft

Soft+Semisoft

Soft+Semisoft+Nematic

Energ

y (a.u

.)

k

0 1 2 30

1

2

3

4

5

µeff

Soft Elasticity

Semisoft Elasticity

ky

kx

Notes:

The total free energy is

minimum at a finite value

of k; The critical field

can be found as the free

energy at the minimum

point reaches zero.f = felastomer + fnematic + fDielectric

=ξ2

8

(µ(A + (1−

1

r)(r − 1)−

(k2(r − 1)r + q2(1 + (A− 1)r))2

r(2k2q2r + k4r2 + q4(1 + Ar))

)+KSq2 + KBk2−εaE

2)


Data Fitting of 2D Model

5 10 15 20 25 303

4

5

6

7

8 Experiment

Theory

Wavele

ngth

(µm

)

Thickness (µm)

5 10 15 20 25 30 35

20

30

40

50

60

70

80

90

Transition Voltage (Experimental)

Voltage (V

)

5 10 15 20 25 30 35

4

6

8

10

12

14

16

Critical Voltage (Theory)

Volta

ge (V

)

Thickness (µm)

Fitting Parameters:

r = 3.17, A = 0.05, µ = 160J/m3, KS = 8.0× 10−12J/m, KB = 7.5× 10−12J/m,

and εa = 20ε0.

Note:

The predicted values of critical voltage can not match with the experimental values,

roughly five times smaller than the experimental ones.






Conclusions & Future WorksConclusionsAcknowledgments


25 µm DMOAP

Temp: 10

25 µm DMOAP

Temp: 25

25 µm DMOAP

Temp: 35

10 15 20 25 30 35

20

30

40

50

Voltage (V

)

Temperature

I Buckling transition observed in homeotropic cell;

I Success of the simple model.

n

I Simple stripes observed in homogeneous cell;

I Stripe patterns observed in EO Experiments;

I Success of simple 2D calculation & Difficulty of 3D model.


Acknowledgments

I Dmitri Konovalov, Jong-Bong Lee, Seth Fraden @BrandeisUniversity;

I Julia A. Kornfield, and Rafael Verduzco @California Institute ofTechnology;

I Dirk J. Broer @Philips Research & Liang-Chy Chien @LCI of KentState University;

I M. Warner, E. Terenjev @Cambridge Univeristy & Tom C. Lubensky@University of Pennsylvania;

I National Science Foundation & Brandeis University.

thermo/electro-mechanical instabilities - in confined

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