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


OutlineOutline

• General Motivations

• Entropy, Phase Transitions, Entropic Forces

• Interaction Potential Measurements(mainly spheres)

• Self-Assembly (mainly spheres)

• Beyond Spheres


Particles in WaterParticles in Water73

μm


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


(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


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


Ludwig BoltzmanLudwig Boltzman

S = ENTROPYW = Number of states (configurations)

Accessible to ThermodynamicSystem with Energy E


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).


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:


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


Conventional Solids & Liquids/GasesConventional Solids & Liquids/Gases

U dominates S S dominates U

solid liquid, gas

increasing

temperature


Hard Sphere SystemsHard Sphere Systems

No attractive energy from U !

F = -TSonly dependson entropy

a

ar


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).


Binary SystemsBinary Systems


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


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


A Line-scanned Optical TweezerA Line-scanned Optical Tweezer


Measuring the InteractionMeasuring the Interaction


Isolating the Entropic Effectsof the Background Fluid

Isolating the Entropic Effectsof the Background Fluid

Energy Resolution ~ 0.05kTSpatial Resolution ~ 15-30nm


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


Concentrated SuspensionsConcentrated Suspensions


Fluid Phase Crystalline PhaseFluid Phase Crystalline Phase

Increasing Φs


500μm


Phase DiagramPhase Diagram

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


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)


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).


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).


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)


CORNERSCORNERS

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


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


• 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).


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)


BEYOND SPHERESBEYOND SPHERES

• Rods• Rods & Polymers• Rods & Polymer Gels

(Carbon Nanotubes)


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


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


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


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




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


Polymers Plus RodsPolymers Plus Rods

+ = ?

900 nm

Increasing temp ~32oC


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]


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]


Melting of Lamellar DropletMelting of Lamellar Droplet

smectic droplets (cylindrical shape)

nematic droplets(elongated shape)

isolated 2D membrane

2 μm

5 μm


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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Yodh Phys295 Entropic Forces