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Yodh Phys295 Entropic Forces

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U n i v e r s i t y o f P e n n s y l v a n i a Entropic Forces & Phase Transitions Entropic Forces & Phase Transitions Arjun G. Yodh, Dept. of Physics & Astronomy
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Page 1: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Entropic Forces &Phase TransitionsEntropic Forces &Phase Transitions

Arjun G. Yodh, Dept. of Physics & Astronomy

Page 2: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

OutlineOutline

• General Motivations

• Entropy, Phase Transitions, Entropic Forces

• Interaction Potential Measurements(mainly spheres)

• Self-Assembly (mainly spheres)

• Beyond Spheres

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U n i v e r s i t y o f P e n n s y l v a n i a

Particles in WaterParticles in Water73

μm

Page 4: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Forces, Potentials ? Self-Assembly?Forces, Potentials ? Self-Assembly?

Page 5: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

(1) Self-Assembly / Collective Properties• Novel Phases (Equilibrium Statistical Physics)

• Role of shape, charge, concentration,conformation, size, ...

• Structure, Dynamics, Rheology, Optical Properties, ...• Beyond Equilibrium: Metastable phases, glasses, …

Templates, Nucleation,…Sedimentation (Microgravity!)

(2) Interactions / Forces• What are the interactions between constituents in What are the interactions between constituents in

suspension?suspension?•• How do these interactions arise?How do these interactions arise?•• How do these interactions affectHow do these interactions affect

selfself--assembly, structure,assembly, structure,dynamics, dynamics, rheologyrheology,,transport properties?transport properties?

MOTIVATIONS / FUNDAMENTALMOTIVATIONS / FUNDAMENTAL

Page 6: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

MOTIVATIONS / PRACTICALMOTIVATIONS / PRACTICALCreation of Novel Structures for “High-Tech” ApplicationsPhotonics, Sensors, MicroArrays, Bragg-Switches, Advanced Composites …

Understanding, CONTROL ofmany “Practical” soft materials

Insight about crowdedenvironments: cellular interiors

Page 7: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Ludwig BoltzmanLudwig Boltzman

S = ENTROPYW = Number of states (configurations)

Accessible to ThermodynamicSystem with Energy E

Page 8: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

N Gas Particles in a BoxN Gas Particles in a Box

E.g.,

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).

Page 9: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Entropy of N Particles in a BoxEntropy of N Particles in a Box

Indistinguishable non-interacting particles in a box

V, T, N

S ~ k N ln ( λ3deBroglie )N

V /

ΔS ≈ kN (ΔVV

)If V → V + ΔV:

Page 10: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Free Energy (F)Free Energy (F)

F = U - TS

internal energyassociated with

particle positions

tendencyto

disorder

Phases of Matter (solid, liquid, gas) minimize free energy

r

r

Page 11: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Conventional Solids & Liquids/GasesConventional Solids & Liquids/Gases

U dominates S S dominates U

solid liquid, gas

increasing

temperature

Page 12: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Hard Sphere SystemsHard Sphere Systems

No attractive energy from U !

F = -TSonly dependson entropy

a

ar

Page 13: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Monodisperse Hard Sphere Phase BehaviorMonodisperse Hard Sphere Phase 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).

Page 14: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Binary SystemsBinary Systems

Page 15: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Entropic ForcesEntropic ForcesDepletion Force: (HARD SPHERES)Depletion Force: (HARD SPHERES)

Moving 2 large spheres together increases volume accessible to small spheres

inaccessibleto

small spheres

Asakura, Oosawa, J. Polym. Sci. v.33, 1983 (1958)

Vrij, Pure Appl. Chem. v.48, 471 (1976)

U(r) = π(ΦS) ΔV(r,aS,aL)

Osmoticpressure

FreeVolume Change

Page 16: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Optical MicromanipulationOptical MicromanipulationOptical TweezersGradiant Force >> Radiation Pressure

• Strongly Focused BeamMicroscope objectives with high NA provide an easy solution

• Non-DestructiveCan manipulate small dielectric particles with piconewtonforces

• Measure Actual 3-Dimensional SeparationsParticles are confined in the yz-direction

• Confine Motion of ParticlesImproves Statistics

Optical Line Tweezers

Page 17: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

A Line-scanned Optical TweezerA Line-scanned Optical Tweezer

Page 18: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Measuring the InteractionMeasuring the Interaction

Page 19: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Isolating the Entropic Effectsof the Background Fluid

Isolating the Entropic Effectsof the Background Fluid

Energy Resolution ~ 0.05kTSpatial Resolution ~ 15-30nm

Page 20: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Big Spheres and Little SpheresBig Spheres and Little Spheres

Crocker, Matteo, Dinsmore, Yodh, Physical Review Letters v. 82, 4352 (1999)

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 (PMMA)δas = 7 ± 3 nmΦs from ViscometryLD-H ≈ 3 nm

Page 21: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Concentrated SuspensionsConcentrated Suspensions

Page 22: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Fluid Phase Crystalline PhaseFluid Phase Crystalline Phase

Increasing Φs

Page 23: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

500μm

Page 24: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Phase DiagramPhase Diagram

Dinsmore, A.D., Yodh, A.G., and Pine, D.J., Physical Review E 52, 4045-4057 (1995).

Page 25: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Entropic Effect Near a WallEntropic Effect Near a WallDepletion 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)

Page 26: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

RANGE OF COMPOSITIONS WHERE “EQUILIBRIUM”COLLOIDAL EPITAXY IS POSSIBLE!

RANGE OF COMPOSITIONS WHERE “EQUILIBRIUM”COLLOIDAL EPITAXY IS 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).

Page 27: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Entropic effects with Structure in the WallsEntropic effects with Structure in the 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).

Page 28: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Entropic repulsion from a step edge:Entropic repulsion from a step edge:

Less excluded-volumeoverlap

here glass terrace

Dinsmore, Yodh, Pine, Nature v.3838, 239 (1996)

Page 29: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

CORNERSCORNERS

Dinsmore, Yodh, Langmuir v.15, 314 (1999)

Page 30: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

VESICLES VESICLES

(PARTICLES PUSHED TO WALLS ANDAND REGIONS OF HIGH CURVATURE)

Dinsmore, A.D., Wong, D.T., Nelson, P., Yodh, A.G., Phys Rev Lett 80, 409-412 (1998).

Large Particles Alone Large and Small Particles

Page 31: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

• 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)

Controlled Colloidal EpitaxyControlled Colloidal Epitaxy

Steven Chou. J. Vac. Sci Tech: B 15 No.6 (1997).Xia, Y., et al, Science 273,347-349 (1996).

Page 32: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

FCC CrystalFCC CrystalConfocal Image Reconstruction

20 Layer Portion Within Larger Colloidal Crystal

Lin, K-H, Crocker, J.C., Prasad, V., Schofield, A.,Lubensky, T.C., Weitz, D.A., Yodh, A.G., Physical Review Letters, 85 (2000)

Page 33: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

BEYOND SPHERESBEYOND SPHERES

• Rods• Rods & Polymers• Rods & Polymer Gels

(Carbon Nanotubes)

Page 34: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Rods: colloidal liquid crystalsRods: colloidal liquid crystals

Experiment J. D. Bernal (1936), Onsager (1949)

• fd virus : 900 nm length 7 nm diameter • L/D=130

• semiflexible rods – persistence length 2.2 μm

• higher monodispersity then chemical rod-like colloids

virus particles – often used to study liquid crystaline behavior

hard core repulsion dominates interaction potential

900 nm

Page 35: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

isotropic nematic

πD2 2L

πD2

~2DL2

2D

L

Ratio : L/πD

Excluded Volume Dependson Rod Orientation

Excluded Volume Dependson Rod Orientation

Orientational Entropy ↔ Packing Entropy

Page 36: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Concentration driven Isotropic-Nematic phase transition in hard rods

Concentration driven Isotropic-Nematic phase transition in hard rods

isotropic phase nematic phase

increasingconcentration

Onsager, 1949

LD

NI 4=−φ

D - rod diameterL – rod length

n transitiophase N-Iat ionconcentrat rod -NI−φ

order parameter S :

f(θ)−orientational distribution functions

∫ −= θθfθθπ 2 )()21cos

23)(sin(2 dS

Page 37: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Phases of Lyotropic Rod SuspensionsPhases of Lyotropic Rod Suspensions

isotropic

nematic

crossed polarizers

isotropic-nematic (cholesteric) phase coexistance

smectic phasefour mutants – periodicty 0.3 to 1.2 μm

fd virus – model system of monodisperse hard rods

phase diagram isotropic phase nematic phase smectic phase

concentration(cholesteric)

Tang and Fraden, Liq. Cryst, 1995Dogic and Fraden, PRL 1997

Page 38: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Concentration driven Isotropic-Nematic phase transition in hard rods

Concentration driven Isotropic-Nematic phase transition in hard rods

isotropic phase nematic phase

increasingconcentration

Onsager, 1949

LD

NI 4=−φ

D - rod diameterL – rod length

n transitiophase N-Iat ionconcentrat rod -NI−φ

order parameter S :

φ(θ)−orientational distribution functions

∫ −= θθφθθπ 2 )()21cos

23)(sin(2 dS

Page 39: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Polymers Plus RodsPolymers Plus Rods

+ = ?

900 nm

Increasing temp ~32oC

Page 40: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

lamellar swollen lamellar

nematic

nematicisotropic

isotropic dropletswollen lamellardislocation nucleation of nematic droplet at the

dislocation position

Temperature

50 mg/ml fd + 0.7 % NIPA in 20 mM trizma buffer solution, pH 8.15.

Behavior of fd/NIPA mixture:Large [fd] and Low [NIPA]

Behavior of fd/NIPA mixture:Large [fd] and Low [NIPA]

Page 41: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Temperature

isotropic smectic droplet

membrane

nematic droplet

membrane

membrane melting

7mg/ml fd + 3.75% NIPA in 20 mM trizma buffer solution, pH 8.15.

5 μm

isotropic T=15oC

smecticT=20oC

5 μm5 μm

nematic T=29oC

20 - 31oC

5 μm

Behavior of fd/NIPA mixture:low [fd] and high [NIPA]

Behavior of fd/NIPA mixture:low [fd] and high [NIPA]

Page 42: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

Melting of Lamellar DropletMelting of Lamellar Droplet

smectic droplets (cylindrical shape)

nematic droplets(elongated shape)

isolated 2D membrane

2 μm

5 μm

Page 43: Yodh Phys295 Entropic Forces

U n i v e r s i t y o f P e n n s y l v a n i a

SummarySummary

• Entropy is a pervasive effect in Condensed Matter Physics.

• In this talk we have used Model Systems to illustrate its effect.

• In practice Spheres & Rods can be small Molecules.


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