Entropy Driven Interactions and
Assembly: Spheres, Rods & Polymers
Arjun G. Yodh,
Department of Physics & Astronomy, University of Pennsylvania
Acknowledgements: National Science Foundation, NASA
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Outline
• Entropy, Phase Transitions, Entropic Forces
• Interaction Potential Measurements
(mainly spheres)
• Self-Assembly (mainly spheres)
• Beyond Spheres
– Rods (liquid crystal phases)
– Rods & Polymers
– Rods & Polymer Gels (Carbon Nanotubes)
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73 μm
Particles in Water
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Forces, Potentials ?
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Self-Assembly?
Ludwig Boltzman
S = ENTROPY
W = Number of states (configurations)
Accessible to Thermodynamic
System with Energy E
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N Gas Particles in a Box
LOW ENTROPY
HIGH ENTROPY
Number of Configurations that fill box far exceed the number of
configurations that fill one quarter of the box.
In the absence of external influences systems tend to maximize
entropy (i.e. become more disordered).
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Entropy of N Particles in a Box
Indistinguishable non-interacting particles in a box
V, T, N
S ~ k N ln ( V λ3deBroglie )
N
/
ΔV
If V → V + ΔV: ΔS ≈ kN (
)
V
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Free Energy (F)
r
F = U - TS
r
internal energy
associated with
particle positions
tendency
to
disorder
Phases of Matter (solid, liquid, gas) minimize free energy
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Conventional Solids & Liquids/Gases
increasing
temperature
solid
gas
U large & negative
TS small
U~0
TS large
U dominates S
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S dominates U
Hard Sphere Systems
No attractive energy from U
a
a
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!
F = -TS
r
only depends
on entropy
Monodisperse
Monodisperse Hard
Hard Sphere
Sphere Phase
Phase Behavior
Behavior
Phase diagram – one-component
Real colloidal
crystal
Pusey, P.N., van Megen, W. Nature 320, 340-342 (1986).
Zhu, J.X., Li, M., Rogers, R., Meyer, W., Ottewill, R.H., Russell, W.B., Chaikin, P.M. Nature 387, 883-885 (1997).
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Binary Systems
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Entropic Forces
Depletion Force: (HARD SPHERES)
inaccessible
to
small spheres
U(r) = π(ΦS) ΔV(r,aS,aL)
Osmotic
pressure
Free
Volume Change
Moving 2 large spheres together increases volume accessible to small spheres
Asakura, Oosawa, J. Polym.
Sci. v.33, 1983 (1958)
Vrij, Pure Appl. Chem. v.48,
471 (1976)
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Interaction Measurements
1. Basic Technique
2. Large and Small Particles
Crocker, J.C., Matteo, J.A., Dinsmore, A.D., and Yodh, A.G.,
Physical Review Letters 82, 4352-4355 (1999).
3. Particles and Rods
Lin, K-H., Crocker, J.C., Zeri, A.C., and Yodh, A.G., Physical Review
Letters 87, 088301-1-088301-4 (2001).
Lau, A.W.C., Lin, K-H., and Yodh, A.G., Physical Review E 66,
020401-1-020401-4 (2002).
4. Particles and Polymers
Verma, R., Crocker, J.C., Lubensky, T.C., and Yodh, A.G., Physical Review
Letters 81, 4004-4007 (1998): Macromolecules 33, 177-186 (2000).
Owen, R.J., Crocker, J.C., Verma, R., and Yodh, A.G., Physical Review E
64, 011401-1--011401-6 (2001).
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Optical Micromanipulation
Optical Tweezers
Optical Line Tweezers
Gradiant Force >> Radiation Pressure
• Strongly Focused Beam
Microscope objectives with
high NA provide an easy
solution
• Non-Destructive
Can manipulate small dielectric
particles with piconewton
forces
• Measure Actual 3-Dimensional
Separations
Particles are confined in the yzdirection
• Confine Motion of Particles
Improves Statistics
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A
A Line-scanned
Line-scanned Optical
Optical Tweezer
Tweezer
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Harmonic Potential along the Line:
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Measuring the Interaction
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Isolating the Entropic Effects
of the Background Fluid
Energy Resolution ~ 0.05kT
Spatial Resolution ~ 15-30nm
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An Interaction Measurement Example
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Effect of Adsorbed Polymer
PEO (Polyethylene oxide, [CH2CH20]n)
Silica Microspheres (~1.1μm diameter)
Solvent: Water, pH = 8.0, buffered
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Long Range Potentials
δa is “effective thickness” of polymer layer
λ is an exponential decay length
λ, δa Functions of RG??
Owen, Crocker, Verma, Yodh, Phys. Rev. E. 2001
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Mean Field and Scaling Theories
Fleer, van Male, Johner,
Macromolecules 32, (1999).
Semenov, Joanny, Johner,
Bonet-Avalos,
Macromolecules 30, (1997).
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Entropic (Depletion) Interactions
Dilute Polymer
Solution
Hard Spheres
Verma, Crocker, Lubensky, Yodh,
Physical Review Letters v. 81, 4004 (1998)
& Macromolecules v.33, 177 (2000)
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Crocker, J.C., Matteo, J.A., Dinsmore, A.D., and Yodh,
A.G. Physical Review Letters v. 82, 4352 (1999)
Big Spheres and Little Spheres
FAO(r) = (kT Φs*) (2as*)−3 (2as* + 2aL - r)2 (2as* + 2aL + r/2)
as* = as + δas ; Φs* = Φs ( 1 + δas/as )3
2as = 83 nm
(PS)
2aL = 1100 ± 15 nm
δas = 7 ± 3 nm
Φs from Viscometry
LD-H ≈ 3 nm
(PMMA)
Crocker, Matteo, Dinsmore, Yodh,
Physical Review Letters v. 82, 4352 (1999)
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Concentrated
Suspensions
Theory:
Biben, Bladon, and Frenkel, 1996, Phys. Condens. Matter. 8, 10799.
Chu, Nikolov, and Wasan, 1996, Langmuir 12, 5004.
Dickman, Attard, and Simonian, 1997, J. Chem. Phys. 107, 205.
Gotzelmann, Evans, and Deitrich, 1998, Phys. Rev. E. 57, 6785.
Mao, Bladon, Lekkerkerker, and Cates, 1997, Mol. Phys. 92, 151.
Piasecki, Bocquet, and Hansen, 1995, Physica (Amersterdam) 218A, 125.
Roth R, Evans R, Dietrich S, Phys. Rev E 62 (4): 2000
Roth R, Evans R, Louis AA, Phys. Rev E 64 (5): 2001
Louis AA, Allahyarov E, Lowen H, Roth R, Phys. Rev E 65 (6): 2002
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Rod-sphere Systems
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Depletion Interaction: Rods & Spheres
U(h;R / L) = – kBTnrRL2K(h / L;R / L)
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Model Comparison
Lin, K-H., Crocker, J.C., Zeri, A.C., and Yodh, A.G.,
Physical Review Letters 87, 088301-1-088301-4 (2001).
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Rod (fd-Virus) Depletion Interactions
1.0 μm silica particles
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Rotational Entropies
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Additional Degree of Freedom
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Interactions
Interactions Between
Between Spheres
Spheres &
& Bent
Bent Rods
Rods
Lin, K-H., Crocker, J.C., Zeri, A.C., and Yodh, A.G.,
Physical Review Letters 87, 088301-1-088301-4 (2001).
Lau, A.W.C., Lin, K-H., and Yodh, A.G., Physical
Review E 66, 020401-1-020401-4 (2002).
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Polymer-Sphere Depletion Interactions
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Polymers
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Semi-Dilute
Dilute
INTERACTION
INTERACTION POTENTIAL
POTENTIAL
for
for varying
varying DNA
DNA Concentrations
Concentrations
λ- DNA
• 16.5μm = L
• ~50nm = l*
• Monodisperse
• Non-absorbing
~3nm = LD-H
C* ≈ 30-50μg/ml
Rg ≈ 500nm
~ 1.2μm
Silica
Depth
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Range
POLYMER DEPLETION
Dilute polymer solution
Concentrated polymer solution
Verma, Crocker, Lubensky, Yodh, Physical Review Letters v. 81, 4004 (1998)
Verma, Crocker, Lubensky, Yodh, Macromolecules v.33, 177 (2000)
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Mean Field Depletion Predictions
DNA coils /
DNA coils /
Interaction Measurements Provide Information
about Background Complex Fluid!
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POLYMER RIGIDITY
τ=1
Schaefer, Joanny, Pincus, Macromolecules 13 (1980) →
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Summary: Interactions
Interaction Measurements
• Useful in many contexts
• Reveal Structural Correlations in
Background Fluids
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Self-Assembly (mainly spheres)
1.
2.
3.
4.
Binary Particle Suspensions (Bulk)
Wall Effects
Wall Structures
Directed Colloidal Assembly
with Grating Templates
Review:
Yodh, A.G., Lin, K-H., Crocker, J.C., Dinsmore, A.D., Verma, R.,
and Kaplan, P.D., The Philosphical Transactions of the Royal
Society of London A 359, 921-937 (2001).
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Fluid Phase
Crystalline Phase
Increasing Φs
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500μm
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Phenomenological Approach
Fluid Phase:
• Gas of Large hard spheres + Gas of
Small hard spheres.
Carnahan-Starling Equation of State.
Solid Phase:
• Close-packed lattice of Large Hard-spheres
permeated by hard-sphere gas of Small Spheres.
Critical Feature Emerging from the Solid Model
The entropy of the Small spheres increases as the crystal becomes
more tightly packed!
Equate Osmotic pressures, and chemical potentials of large and
small spheres within each phase ⇒PHASE DIAGRAM
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Phase Diagram
Dinsmore, A.D., Yodh, A.G., and Pine, D.J., Physical Review E 52, 4045-4057 (1995).
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Entropic Effect Near a Wall
Depletion Forces at Surface: (HARD SPHERES)
Moving large sphere to wall decreases the Free energy even more!
Kaplan, Rouke, Yodh, Pine, Physical Review Letters v.72, 582 (1994)
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RANGE
RANGE OF
OF COMPOSITIONS
COMPOSITIONS WHERE
WHERE “EQUILIBRIUM”
“EQUILIBRIUM”
COLLOIDAL
COLLOIDAL EPITAXY
EPITAXY IS
IS POSSIBLE!
POSSIBLE!
Dinsmore, A.D., Warren, P.B., Poon, W.C.K., Yodh, A.G., Europhys Lett 40, 337-342 (1997).
Dinsmore, A.D., Yodh, A.G., Pine, D.J., Phys Rev E 52, 4045-4057 (1995).
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Entropic
Entropic effects
effects with
with Structure
Structure in
in the
the Walls
Walls
Dinsmore, A.D., Yodh, A.G., Pine, D.J., Nature 383, 239-242 (1996).
Dinsmore, A.D., Wong, D.T., Nelson, P., Yodh, A.G., Phys Rev Lett 80, 409-412 (1998).
Dinsmore, A.D., Yodh, A.G., Langmuir 15, 314-316 (1999).
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Entropic repulsion from a step edge:
Less excludedvolume
overlap
here
Dinsmore, Yodh, Pine, Nature
v.3838, 239 (1996)
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glass terrace
CORNERS
Dinsmore, Yodh, Langmuir v.15, 314 (1999)
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VESICLES
(PARTICLES PUSHED TO WALLS AND REGIONS OF HIGH CURVATURE)
Large Particles Alone
Large and Small Particles
Dinsmore, A.D., Wong, D.T., Nelson, P., Yodh, A.G., Phys Rev Lett 80, 409-412 (1998).
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Controlled Colloidal Epitaxy
•
PMMA beads with Polymer,
index matched for 3D
confocal microscopy.
•
Slight density mis-match for
3D growth (decalin)
Lin, K-H, Crocker, J.C., Prasad, V., Schofield, A.,Lubensky, T.C.,
Weitz, D.A., Yodh, A.G., Physical Review Letters, 85 (2000)
Steven Chou. J.
Vac. Sci Tech: B
15 No.6 (1997).
Xia, Y., et al,
Science 273,
347-349 (1996).
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2D Assembly
on
Line-gratings
d = mean spacing above groove
2p
scissor angle
tan θ =
d
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2D Assembly
on
Crossedgratings
d = mean spacing above groove
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FCC Crystal
Confocal Image
Reconstruction
20Layer
LayerPortion
PortionWithin
WithinLarger
LargerColloidal
ColloidalCrystal
Crystal
20
Lin, K-H, Crocker, J.C., Prasad, V., Schofield, A.,Lubensky, T.C.,
Weitz, D.A., Yodh, A.G., Physical Review Letters, 85 (2000)
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Summary: Self-Assembly
• Small species plus Geometric Structures
produce Unusual and Useful Entropic
Forces for Materials Synthesis.
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BEYOND SPHERES
• Rods
• Rods & Polymers
• Rods & Polymer Gels
(Carbon Nanotubes)
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Excluded Volume Depends on Phase
2D
L
πD2 2L
~2DL2
isotropic
nematic
Ratio : L/πD
volume fraction at transition
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D
φ =4
L
πD2
Concentration driven IsotropicNematic phase transition in hard rods
increasing
concentration
isotropic phase
D - rod diameter
L – rod length
φI − N - rod concentration
φI − N
at I - N phase transition
D
=4
L
nematic phase
φ(θ)−orientational distribution
functions
order parameter S :
3
1
S = 2π ∫ sin(θ )( cos 2 θ − )φ (θ )dθ
2
2
Onsager, 1949
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colloidal liquid crystals
900 nm
• fd virus : 900 nm length 7 nm diameter
• L/D=130
• higher monodispersity then chemical rod-like colloids
• semiflexible rods – persistence length 2.2 μm
hard core repulsion dominates interaction potential
virus particles – often used to study liquid crystaline behavior
Experiment J. D. Bernal (1936), Onsager (1949)
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Background
Background on
on Lyotropic
Lyotropic Rod
Rod Suspensions
Suspensions
isotropic-nematic (cholesteric)
phase coexistance
smectic phase
four mutants – periodicty 0.3 to 1.2 μm
isotropic
nematic
crossed polarizers
fd virus – model system of monodisperse hard rods
isotropic phase
phase diagram
nematic phase
(cholesteric)
concentration
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smectic phase
Tang and Fraden, Liq. Cryst, 1995
Dogic and Fraden, PRL 1997
Polymers in Nematic Suspensions
Semi-flexible Biopolymer
+
DNA, Neurofilaments, Wormlike
micelle and Actin
Nematic Liquid Crystal
= ?
Aqueous suspension of fd virus
quantitatively understood
• Directly Visualization with
Fluorescence Microscopy
• Quantitative Image Analysis Possible
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Semi-flexible biopolymers
DNA
Neurofilament
16 micron length
2 nm in diameter
40 nm persistence length
5 - 20 micron length
12 nm in diameter
~ 220 nm persistence length
Wormlike Micelle
Actin
( polybutadiene-polyethyleneoxide )
10 – 50 micron length
~ 15 nm in diameter
~ 500 nm persistence length
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2 – 30 micron length
7-8 nm in diameter
~ 16 micron persistence length
Images
Images of
of Polymers
Polymers in
in Isotropic
Isotropic &
& Nematic
Nematic
Suspensions
Suspensions of
of fd
fd Virus
Virus
Isotropic
Nematic
Actin
16 μm
Actin in Nematic Fd
Wormlike
Micelle
500 nm
Neurofilament
200 nm
DNA
50 nm
Hairpin
defects
10 μm
10 μm
Dogic Z, Zhang J, Lau AWC, Aranda-Espinoza H, Dalhaimer P, Discher DE,
Janmey PA, Kamien RD, Lubensky TC, Yodh AG, Phys. Rev. Lett. 92 (12): 2004
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Obtain orientational distribution
function (ODF) from real space images
Extract S
from ODF:
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3
1
2
S = 2π ∫ sin(θ )( cos θ − )φ (θ )dθ
2
2
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Actin Order Parameter Vs. Length
41 mg/ml
28 mg/ml
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Tangent – Tangent Correlations
t(s’)
t(s’+s)
h(s)
actin in nematic phase
Isotropic phase – quasi 2D
r
r
−s / 2 L
⟨t ( s '+ s ) ⋅ t ( s ' )⟩ = e
p
Orientational correlations
decay exponentially
Lp – persistence length
actin in
isotropic phase
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Tangent tangent correlation function
for different fd concentrations
<tx(0)tx(s)>
0.010
fd concentration
39 mg/ml
50 mg/ml
98 mg/ml
0.005
0.000
0
1
2
s [μm]
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3
Free energy of a polymer in a
nematic liquid crystal
Bending Energy
Coupling Energy
Elastic Energy
L
r 2
l p ⌠ ⎛ ∂t ⎞
r
r2 3
1
ΓL r
2
β F = ⎮ ⎜ ⎟ dz + ∫ (t ( z ) − δn (0, z )) dz + K ∫ ∇δn d r
2 ⌡ ⎝ ∂z ⎠
20
2
0
splay
bending
low energy
Twist
lp
high energy Γ
K
r
δn
Persistence Length
Coupling Constant
Elastic Constant
Local Fluctuation of
Nematic Director
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*R.D. Kamien et al, Phys.
Rev. A. 45, 8727(1992).
Theoretical prediction for tangenttangent correlation function
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Extract physical parameters
from the data fitting
fd conc.
Odijk def. l.
[μm]
gamma
[kT/μm]
K
[dyne 10-8]
39
0.080
78
1.23
50
0.050
200
1.64
98
0.036
386
2.17
*K ≈ 3×10-8 (dyne/cm)
*Z. Dogic and Seth Fraden, Langmuir 16, 7820(2000).
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Summary:
Summary: Biopolymers
Biopolymers in
in Lyotropic
Lyotropic Nematics
Nematics
• Polymers (except DNA) couple strongly to
nematic and stretch out.
• Polymers more “ordered” than nematic
• Theory + Experiment yield coupling
parameter, Odijk length
Dogic Z, Zhang J, Lau AWC, Aranda-Espinoza H, Dalhaimer P, Discher DE,
Janmey PA, Kamien RD, Lubensky TC, Yodh AG, Phys. Rev. Lett. 92 (12): 2004
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Rods in Polymer Solutions
• fd/NIPA mixtures
– a thermotropic suspension
• Melting of Lamellar phases
• Single-layer rod membranes
A. Alsayed, Z. Dogic and A.G. Yodh, to be published in Physical Review Letters (2004).
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Rods and Polymer in Good Solvent
N
• Hard to observe melting directly
Z. Dogic and S. Frade, Phil. Trans. R. Soc. Lond. A v. 359 (2001).
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Temperature-Sensitive NIPA Polymer
N-isopropyl acrylamide
acylamide group
(hydrophilic)
Propyl group
(hydrophobic)
Mw~250k
at 10oC C*~ 0.01 gm/ml
Increasing temp
Rg~ 55 nm
Tanaka T. et al , Nature, 325 (1987) 796-798.
C. Wu and X. Wang, Phys. Rev. Lett. 80, 4092 (1998)
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~32oC
Our Experiment
Solvent quality tunes interactions
fd rods
NIPA
Repulsive
polymer-polymer Steric
- Low Temperature rod-rod
rod-polymer
Homogeneous
Samples
heat
polymer-polymer attractive → Inhomogeneous
- High Temperature rod-rod
repulsive
Samples
rod-polymer
repulsive
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NIPA
NIPA polymer
polymer &
& fd
fd rods
rods at
at Low
Low Temperatures
Temperatures
7.5 mg/ml fd + 37.5 mg/ml NIPA
50 mg/ml fd + 7.5 mg/ml NIPA
lamellar
T < 5 oC
T < 17 oC
isotropic
•Low temperature, low rod
concentration is miscible with
polymer.
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• Low temperature, small
amount of polymer in
concentrated rod suspension
destabilizes nematic phase and
stabilizes smectic layers
Behavior of fd/NIPA mixture:
Large [fd] and Low [NIPA]
50 mg/ml fd + 0.7 % NIPA in 20 mM trizma buffer solution, pH 8.15.
Temperature increase leads to unbinding of smectic layers and creation of
nematic droplet when NIPA polymer is fully expelled
16oC
5oC
7oC
12oC
15oC
scale bar is 5 microns
17oC
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Behavior of fd/NIPA mixture:
Large
Large [fd]
[fd] and
and Low
Low [NIPA]
[NIPA]
50 mg/ml fd + 0.7 % NIPA in 20 mM trizma buffer solution, pH 8.15.
lamellar
dislocation
swollen lamellar
isotropic
nucleation of nematic nematic swollen lamellar
droplet at the
dislocation position
Temperature
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nematic
isotropic droplet
Behavior
Behavior of
of fd/NIPA
fd/NIPA mixture:
mixture:
low
low [fd]
[fd] and
and high
high [NIPA]
[NIPA]
isotropic
T=15oC
7mg/ml fd +
3.75% NIPA in
20 mM trizma
buffer solution,
pH 8.15.
smectic
T=20oC
5 μm
20 - 31oC
5 μm
5 μm
nematic
T=29oC
5 μm
5 μm
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Behavior
Behavior of
of fd/NIPA
fd/NIPA mixture:
mixture:
low
low [fd]
[fd] and
and high
high [NIPA]
[NIPA]
7mg/ml fd + 3.75% NIPA in 20 mM trizma buffer solution, pH 8.15.
isotropic
nematic droplet
smectic droplet
membrane
membrane
membrane melting
Temperature
5 μm
isotropic
T=15oC
smectic
T=20oC
nematic
T=29oC
5 μm
5 μm
20 - 31oC
5 μm
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5 μm
Melting of lamellar droplet
22oC
26oC
5 μm
5 μm
28oC
5 μm
29oC
5 μm
7mg/ml fd + 3.75% NIPA in 20 mM trizma buffer solution, pH 8.15.
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Melting of Colloidal Membranes
31oC
after 1 minute
after 1.5 minutes
after 2 minutes
Kinetic barrier to create nematic 3d droplet
out of 2d membrane
Melted totally to nematic droplet
isolated 2D membrane
Colloidal membranes can be prepared in a metastable state - easily superheated
nucleation barrier for melting 2D smectic layer into 3D nematic droplet !!!
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Disturbing
Disturbing colloidal
colloidal membrane
membrane mechanically
mechanically
(colloidal
(colloidal membrane
membrane is
is aa stable
stable object)
object)
• temperature is 28 oC (near melting temperature)
• scale bar is 5 microns
time
pull membrane
many times with
tweezer and
silica bead
metastable
nematic
droplet
(tweezer off)
Condensing smectic
back to
layers
smectic
coalesce
onto
membrane
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Two colloidal membranes slide
and join.
Summary: Rods in NIPA Solutions
• Solvent effects lead to Temperature-dependent phase
transitions in Lyotropic Systems.
• Solvent effects at low fd-concentration lead to
heterogeneous nucleation of nematic/smectic
droplets. (Isotropic - Smectic/Lamellar – Nematic)
• Solvent effects at high fd-concentration lead to
swelling of Lamellar phase, defects/nematic
droplets, and eventually isotropic droplets in
Nematic background.
• Melting of Lamellar structures: Kinetic barriers
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Rods in Polymer Gels
“Nematic nanotube gels,” Islam MF, Alsayed AM, Dogic Z, Zhang J, Lubensky
TC, Yodh AG, Phys. Rev. Lett. 92 (8): 2004
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An
An Important
Important Rod:
Rod: Single
Single Wall
Wall Carbon
Carbon Nanotubes
Nanotubes
SWNTs have extraordinary properties:
• Strength (~100x steel)
• Tensile strength 100-200 GPa
• Stiffness
1.4 TPa
• Elongation
20-30%
• Electrical conductivity (~Copper)
• Ballistic electron transport mechanism
• Highest known current density
• Thermal conductivity (~3x Diamond)
• Thermally stable polymer (anaerobic)
Products incorporating SWNTs can benefit from all of these properties simultaneously.
~1 nm
100 nm – 10,000 nm
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Dispersing SWNTs
van der Walls attaction: 40 KBT/nm
Laser-oven
HiPCO
Surfactant:
SDS
SWNTs: 0.5 mg/ml
Time:
5 days
TX-100
0.8 mg/ml
5 days
NaDDBS
20 mg/ml
2 months
5.00
2.50
0
2.50
0
5.00
μm
Islam, Rojas, Bergey, Johnson, Yodh NanoLett. 3, 269 (2003)
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Surfactant Adsorption on Nanotubes
Popular belief
Our suggestion
Recent experimental evidence
Richards, Balavoine, Schultz, Ebbesen, and Mioskowski Science 300, 775 (2003)
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SWNTs Behave like Rigid Rods
100
Q
-2
I
I
I(0.5% NaDDBS/D2O)
I
I(1.0% NaDDBS/D2O)
10
Q
-1
(a)
1
0.1
0.1% HiPco/D2O
with 1% NaDDBS
1E-3
0.01
0.1
Q (Å -1 )
Zhou, Islam, Wang, Ho, Yodh, Winey, Fischer Chem. Phys. Lett. 384, 185 (2004)
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SWNTS are Attractive Rods
concentration
Isotropic (I)
Nematic (N)
Onsager Ann. N. Y. Acad. Sci. 51, 627 (1949)
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Nematic Elastomers
Volume
compression
Isotropic (I)
Nematic (N)
Lacoste, Lau and Lubensky Euro. Phys. J. E 8, 403 (2002)
Lubensky, Mukhopadhyay, Radzihovsky and Xing PRE 66, 011702 (2002)
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Properties of NIPA gel
N-isopropylacrylamide (NIPA) gel:
F. Ilmain et al. Nature 349, 400 (1991)
Temperature
Pelton R., Temperature-sensitive aqueous microgels,
Adv. Colloid Interface Sci., 85 (2000) 1-33.
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Tanaka’s website
fd-virus
fd-virus in
in Gel between Crossed
Crossed Polarizers
Polarizers
Before shrinking
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After shrink
SWNT-NIPA Gels
SWNT dispersed in NaDDBS + (NIPA) pre-gel polymerized for 3h at T=22°C
8.25 mg/ml
2.47 mg/ml
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(P)
Temporal
Temporal and
and Concentration
Concentration Dependence
Dependence
(A)
Islam, Alsayed, Dogic, Zhang, Lubensky, Yodh PRL 92, 088303 (2004)
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Determination of Nematic Order
O.A.
(P)
(A)
σ ||
Stronger
absorption
σ⊥
O.A.
α
⎡I ⎤
ln ⎢ // ⎥ = − NLSΔσ
⎣ I⊥ ⎦
S = ∫ f(α) 1(3cos2 α − 1)dΩ
2
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Independently
Determined
Isotropic-Nematic Transition:
Nematic Nanotube Gels
O.A.
α
(P)
⎡ I // ⎤
ln ⎢ ⎥ = − NLSΔσ
⎣ I⊥ ⎦
Islam, Alsayed, Dogic, Zhang, Lubensky, Yodh PRL 92, 088303 (2004)
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(A)
Defects
(P)
(A)
4 extinction branches
Defects and buckling in nematic lyotropic gels, M. F. Islam, M. Nobili,
T. C. Lubensky and A. G. Yodh (in preparation)
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Mechanical Properties
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Theory
6
2
σ (10 cm /mole C)
Experiment
6
2
σ (10 cm /mole C)
Direct Measurement of Polarized
Absorption Cross-Section
2.0
1.5
E22
E33
σ ||
1.0
σ⊥
0.5
0.0
5
4
3
2
1
0
0.5
E11
No Depolarization
E12
1.0
E22
E33
1.5
2.0
2.5
Energy (eV)
σ ||
σ⊥
3.0
3.5
With Depolarization
Direct Measurement of the Polarized Absorption Cross-Section of Single-Wall Carbon
Nanotubes, M. F. Islam, D. E. Milkie, C. L. Kane, A. G.Yodh and J. M. Kikkawa (to be
published in Phys. Rev. Lett. (2004)).
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Background
Energy
0.6
Optical Absorption
at E11, E22, E33, …
0
1
2
3
-0.6, E
E11, E22
33
WaveKp
Vector
Lin, Phys. Rev. B 62, 13153 (2000)
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Linear Absorption Anisotropy
• Linear Absorption is a fundamental
optical quantity
σ ||
• Needed for proper interpretation of
optical spectra
σ⊥
• Comparison to theory
• Offers new methods to measure
alignment in suspensions and
composites
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Magnetic Alignment
Combine SWNT with
Pre-Gel ingredients
Monomer
N-isopropylacrylamide (NIPA)
Cross-linker
Align Sample in 9T
Magnetic Field
UV Polymerization
Locks in NT Alignment
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G-Band Peak Intensity
(Arbitrary Units)
Quantify Alignment (Raman Scattering)
θ̀
1.2
VV
1.0
0.8
0.6
0.4
0.2
0.0
f (θ)
-2
5.68x10 mg/ml
-3
1.43x10 mg/ml
Fit
0
90
180
270
360
Rotation (Degrees)
S≡∫
1
−1
⎛ 3 cos 2 θ − 1 ⎞
⎟⎟ d (cos θ )
f (θ )⎜⎜
2
⎠
⎝
NEMATIC Order Parameter
Concentration
5.68x10-2 (mg/ml)
1.43x10-3 (mg/ml)
5.85x10-2 (mg/ml)
2.93x10-3 (mg/ml)
Order Parameter
S = 0.18
S = 0.11
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Combine
Combine Raman
Raman Scattering
Scattering with
with Polarized
Polarized Optical
Optical
Absorption
Absorption to
to Obtain
Obtain Absorption
Absorption Cross-Section
Cross-Section
σ ||
T⊥ = e − nα ⊥ d
T|| = e
Independently
measure using
Raman
scattering
− nα || d
σ⊥
2
1
(σ || − σ ⊥ )S + (σ || + 2σ ⊥ )
α || =
3
3
1
1
α ⊥ = − (σ || − σ ⊥ ) S + (σ || + 2σ ⊥ )
3
3
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Measurement & Theory
EIN
σ (10 cm /mole C)
2
1.5
E22
E33
σ ||
1.0
σ⊥
6
2
Theory
critical in calculation.
6
•Depolarization Effect
2.0
0.5
σ (10 cm /mole C)
model agrees within
factor of 2.
Experiment
•No free parameter
ED
0.0
5
4
3
2
1
0
0.5
E11
No Depolarization
E12
1.0
E22
σ ||
σ⊥
E33
1.5
2.0
2.5
Energy (eV)
3.0
3.5
With Depolarization
C.L. Kane and E.J. Mele, cond-mat/0403153 (2004).
H. Ajiki and T. Ando, Physica B 201, 349 (1994).
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Summary: Beyond Spheres
• Measurements of Isolated Biopolymers in Hard
Rod Suspensions.
• Hard Rods in Polymer Solutions (NIPA)
– Temperature-dependent phase transitions in lyotropic systems
– Melting of Smectic Phases
• Hard Rods in Cross-linked Polymer Solutions (NIPA)
– Nematic Nanotube Gels
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Collaborators
Mohamad Islam,
Jian Zhang,
Tony Dinsmore,
Ritu Verma,
Peter Kaplan,
Larry Hough,
Daniel Chen,
Paul Dalhaimer,
Enrique Rojas,
Zvonimir Dogic,
Keng-hui Lin,
John Crocker,
Andy Lau,
Ahmed Alsayed,
Joe Matteo,
Jenifer Rouke,
Helim Aranda-Espinoza,
Daniel Bergey
Tom Lubensky (PENN)
Dave Pine (UCSB)
Dave Weitz (Harvard)
Dennis Disher (PENN)
Paul Janmey (PENN)
Charlie Johnson (PENN)
Randy Kamien (PENN)
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...
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