EntropyEntropy DrivenDriven InteractionsInteractions andand Assembly:Assembly: Spheres,Spheres, RodsRods && PolymersPolymers Arjun G. Yodh, Department of Physics & Astronomy, University of Pennsylvania Acknowledgements: National Science Foundation, NASA 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 • 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) U n i v e r s i t y o f P e n n s y l v a n i a ParticlesParticles inin WaterWater m μ 73 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,Forces, PotentialsPotentials ?? Self-Assembly?Self-Assembly? U n i v e r s i t y o f P e n n s y l v a n i a LudwigLudwig Boltzman Boltzman S = ENTROPY W = Number of states (configurations) Accessible to Thermodynamic System with Energy E U n i v e r s i t y o f P e n n s y l v a n i a NN GasGas ParticlesParticles inin aa BoxBox 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). U n i v e r s i t y o f P e n n s y l v a n i a EntropyEntropy ofof NN ParticlesParticles inin aa BoxBox Indistinguishable non-interacting particles in a box V, T, N S ~ k N ln (V λ3 ) N / deBroglie ΔV If V → V + ΔV: ΔS ≈ kN ( ) V U n i v e r s i t y o f P e n n s y l v a n i a FreeFree EnergyEnergy (F)(F) r F = U - TS internal energy tendency r associated with to particle positions disorder Phases of Matter (solid, liquid, gas) minimize free energy U n i v e r s i t y o f P e n n s y l v a n i a ConventionalConventional SolidsSolids && Liquids/GasesLiquids/Gases increasing temperature solid gas U large & negative U ~ 0 TS small TS large U dominates S S dominates U U n i v e r s i t y o f P e n n s y l v a n i a HardHard SphereSphere SystemsSystems No attractive energy from U ! a F = -TS only depends on entropy r a U n i v e r s i t y o f P e n n s y l v a n i a MonodisperseMonodisperse Hard Hard SphereSphere PhasePhase BehaviorBehavior 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). U n i v e r s i t y o f P e n n s y l v a n i a BinaryBinary SystemsSystems U n i v e r s i t y o f P e n n s y l v a n i a EntropicEntropic Forces Forces Depletion Force: (HARD SPHERES) inaccessible U(r) = π(ΦS) ΔV(r,aS,aL) to small spheres Osmotic Free pressure 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) U n i v e r s i t y o f P e n n s y l v a n i a InteractionInteraction MeasurementsMeasurements 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). U n i v e r s i t y o f P e n n s y l v a n i a OpticalOptical MicromanipulationMicromanipulation 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 • Measure Actual 3-Dimensional particles with piconewton Separations forces Particles are confined in the yz- direction • Confine Motion of Particles Improves Statistics U n i v e r s i t y o f P e n n s y l v a n i a AA Line-scannedLine-scanned OpticalOptical TweezerTweezer U n i v e r s i t y o f P e n n s y l v a n i a HarmonicHarmonic PotentialPotential alongalong thethe Line:Line: U n i v e r s i t y o f P e n n s y l v a n i a MeasuringMeasuring thethe InteractionInteraction U n i v e r s i t y o f P e n n s y l v a n i a IsolatingIsolating thethe Entropic Entropic Effects Effects ofof thethe BackgroundBackground FluidFluid Energy Resolution ~ 0.05kT Spatial Resolution ~ 15-30nm U n i v e r s i t y o f P e n n s y l v a n i a AnAn InteractionInteraction MeasurementMeasurement ExampleExample U n i v e r s i t y o f P e n n s y l v a n i a EffectEffect ofof AdsorbedAdsorbed PolymerPolymer PEO (Polyethylene oxide, [CH2CH20]n) Silica Microspheres (~1.1μm diameter) Solvent: Water, pH = 8.0, buffered U n i v e r s i t y o f P e n n s y l v a n i a LongLong RangeRange PotentialsPotentials 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 U n i v e r s i t y o f P e n n s y l v a n i a MeanMean FieldField andand ScalingScaling TheoriesTheories Fleer, van Male, Johner, Macromolecules 32, (1999). Semenov, Joanny, Johner, Bonet-Avalos, Macromolecules 30, (1997). U n i v e r s i t y o f P e n n s y l v a n i a EntropicEntropic (Depletion) (Depletion) InteractionsInteractions Dilute Polymer Solution Hard Spheres Verma, Crocker, Lubensky, Yodh, Physical Review Letters v. 81, 4004 (1998) Crocker, J.C., Matteo, J.A., Dinsmore, A.D., and Yodh, & Macromolecules v.33, 177 (2000) A.G. Physical Review Letters v. 82, 4352 (1999) U n i v e r s i t y o f P e n n s y l v a n i a BigBig SpheresSpheres andand LittleLittle SpheresSpheres −3 2 FAO(r) = (kT Φs*) (2as*) (2as* + 2aL -r) (2as* + 2aL + r/2) 3 as* = as + δas ; Φs* = Φs ( 1 + δas/as ) 2as = 83 nm (PS) 2aL = 1100 ± 15 nm (PMMA) δas = 7 ± 3 nm Φs from Viscometry LD-H ≈ 3 nm Crocker, Matteo, Dinsmore, Yodh, Physical Review Letters v. 82, 4352 (1999) U n i v e r s i t y o f P e n n s y l v a n i a ConcentratedConcentrated SuspensionsSuspensions 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 U n i v e r s i t y o f P e n n s y l v a n i a Rod-sphereRod-sphere SystemsSystems U n i v e r s i t y o f P e n n s y l v a n i a DepletionDepletion Interaction:Interaction: RodsRods && SpheresSpheres 2 U(h;R / L) = – kBTnrRL K(h / L;R / L) U n i v e r s i t y o f P e n n s y l v a n i a ModelModel ComparisonComparison Lin, K-H., Crocker, J.C., Zeri, A.C., and Yodh, A.G., Physical Review Letters 87, 088301-1-088301-4 (2001). U n i v e r s i t y o f P e n n s y l v a n i a RodRod (fd-Virus)(fd-Virus) DepletionDepletion InteractionsInteractions 1.0 μm silica particles U n i v e r s i t y o f P e n n s y l v a n i a RotationalRotational EntropiesEntropies U n i v e r s i t y o f P e n n s y l v a n i a AdditionalAdditional DegreeDegree ofof FreedomFreedom U n i v e r s i t y o f P e n n s y l v a n i a InteractionsInteractions BetweenBetween SpheresSpheres && BentBent RodsRods Lin, K-H., Crocker, J.C., Zeri, A.C., and Yodh, A.G., Physical Review Letters 87, 088301-1-088301-4 (2001).