thermo/electro-mechanical instabilities - in confined
TRANSCRIPT
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
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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...
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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.
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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
)
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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.
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
Nematic Phase I
Crosslinking in Isotropic Phase
Nematic Phase II
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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.
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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.
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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.
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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.
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
Introduction & Background Knowledge
Nematic Rubber ElasticityNematic Rubber ElasticityShear Waves in Nematic ElastomersArtificial Muscle Effect
Buckling Transition in Nematic Gels
Stripes in Electro-Optics Experiment
Conclusions & Future Works
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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 .
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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.
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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.
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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
).
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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).
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
“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.
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
22.7 42.1
73.1 23.2
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
Introduction & Background Knowledge
Nematic Rubber Elasticity
Buckling Transition in Nematic GelsIntroduction & Description of ExperimentExperimental Observations & ResultsTheoretical Interpretation
Stripes in Electro-Optics Experiment
Conclusions & Future Works
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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).
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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 .
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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.
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
25 µm DMOAP
Temp: 15
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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 .
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
Critical Voltage–Temperature Dependence
10 15 20 25 30 35
20
30
40
50
Voltage (V
)
Temperature
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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;
lij = l⊥(δij + (r − 1)ninj
).
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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;
lij = l⊥(δij + (r − 1)ninj
).
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)
)
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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.
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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.
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
Introduction & Background Knowledge
Nematic Rubber Elasticity
Buckling Transition in Nematic Gels
Stripes in Electro-Optics ExperimentSimple Stripes in a Homogeneous CellExperimental Description & ResultsSimplified Two Dimensional Model
Conclusions & Future Works
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
Simple Stripes in a Homogeneous Cell
Ti
Tf < Ti
n
Note:
The nematic director n is aligned parallel to the surfaces.
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
Simple Stripes in a Homogeneous Cell
d=25 µm, T=25 . Courtesy of J.A. Kornfield
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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.
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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.
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
Thickness Dependence of Stripes
d = 10µm d = 18µm
d = 24µm d = 30µm
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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.
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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).
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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)
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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.
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
Introduction & Background Knowledge
Nematic Rubber Elasticity
Buckling Transition in Nematic Gels
Stripes in Electro-Optics Experiment
Conclusions & Future WorksConclusionsAcknowledgments
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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.
Robert B. Meyer October 1st, 2004 Complex Fluids Group Brandeis University
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.