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

(n× (∇× n)

)2+ KS(∇ · n)2 + KT (n · ∇ × n)2

)Fredericks Transition

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

End-to-end Vector:

R = a1 + a2 + · · ·+ aN =N∑

Probability density:

limN→∞

P(R, N) = (2πNa2

3)−3/2 exp(− 3R2

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

Initial State

Lx Lx0

Lz Lz0

Ly Ly0

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

(λijλji

)≡ nskBT

(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.

Soft Elasticity

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

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:

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

Soft Elasticity

Semisoft Elasticity

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

(1 + r(r − 2)−

(1 + r(r − 2) + (r2 − 1) cos 2Ω

4(r cos2 Ω + sin2 Ω)2

fsemi =µ

(1 + r(A + r − 2)−

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

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

∂n∂x

⇒ [K ] = [Energy ][Length]

Gel Entropic Elasticity:FGel ∼ 1

∂d∂x

⇒ [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√

(∼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

50Sample 10-wt%

∆L///L

∆L⊥/L⊥

Strain

Temperature

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

Introduction about Material

Molecular Structures:

( ) ( )( )n m p

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).

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

25 µm DMOAP

Temp: 15

25 µm DMOAP

Temp: 20

25 µm DMOAP

Temp: 25

25 µm DMOAP

Temp: 30

25 µm DMOAP

Temp: 35

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

Voltage (V

Temperature

Buckling DiagramInitial Crosslinking State, E > Ec , r 0

n0 = (0, 0, 1);

E = E0z;

l0ij = l0⊥(δij + (r0 − 1)n0

Stressed State, E > Ec , r > r 0

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

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

5.0x108

1.0x109

1.5x109

2.0x109

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

Critical Fie

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 .

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

Critical Fie

10 15 20 25 30 35

Voltage (V

Temperature

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

Simple Stripes in a Homogeneous Cell

Tf < Ti

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:

89-wt% 5CB

O C NOC

9-wt% Monomer

0.8-wt% BME

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

PVA Coating

ITO Coating

PVA rubbed Homogeneous Cell

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

2µ · Tr

(l0 · λT · l−1 · λ

)− 1

2A · µ · Tr

(λT · n · λ− n0 · λT · n · λ

(n× (∇× n)

)2+ KS(∇ · n)2 + KT (n · ∇ × n)2

2 sin2 θ

Other Constrains, Approximations:

I Incompressibility: ∇ · d = 0;

I Average energy density over the space: f = k2π

∫ 2πk

∫ 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

Soft+Semisoft

Soft+Semisoft+Nematic

y (a.u

0 1 2 30

Soft Elasticity

Semisoft Elasticity

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

(µ(A + (1−

r)(r − 1)−

(k2(r − 1)r + q2(1 + (A− 1)r))2

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

)+KSq2 + KBk2−εaE

Data Fitting of 2D Model

5 10 15 20 25 303

8 Experiment

Theory

Wavele

Thickness (µm)

5 10 15 20 25 30 35

Transition Voltage (Experimental)

Voltage (V

5 10 15 20 25 30 35

Critical Voltage (Theory)

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.

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

Voltage (V

Temperature

I Buckling transition observed in homeotropic cell;

I Success of the simple model.

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.

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