Self-Assembly on the Sphere: A Route to Functional Colloids and

Semiflexible PolymersTanya L. Chantawansri

Glenn H. Fredrickson, Hector D. Ceniceros, and Carlos J. García-Cervera

February 4, 2008

CFDC Annual Meeting 2008

Contents

2D Diblock Copolymer Melt Model/Results

3D Diblock Copolymer MeltModel/Results

3D Binary Homopolymer Brush MeltModel/Status

Semiflexible Polymers

Motivation

2D SCFT Model

Thin but finite filmComposition only varies parallel to the film surfaceθ : Colatitude є [0,π]Φ: Longitude є [0,2π)

A B

Thomson Problem

Attempts to find the ground state (lowest-energy) arrangement of N Coulomb charges confined to the surface of a sphere.

Bausch et al. (2003) www.flmnh.ufl.edu/pollen geology.er.usgs.gov

Gaussian Chain Model

Matsen, J.Physics: Condens. Matter (2002).

Used to model flexible polymersContains a harmonic energy penalty for local chain stretching

Internal coil structure of each coarse grain segment b→length of segment can vary# of internal configurations decreases as chain stretches

Cylindrical Phase

2D SCFT Model: Cylindrical Phase χN=25.0, f = 0.8

SSL Model: Cylindrical Phase χN=25.0, f = 0.8

Analytic Strong Segregation Limit CalculationAbsence of 11 and 13 domain due to topological constraints on interfacial and stretching free energies.

Higher symmetry solutions (12 and 14 domains) have lower free energy unit cell configurations.Low symmetry solutions (11, 13, 15 domains) are characterized by high-energy unit cells.

Lamellar phase

Spiral

Hedgehog

Quasibaseball

2D SCFT Model: Lamellar PhaseχN=12.5, f = 0.5

2D SCFT Model: Lamellar PhaseχN=12.5, f = 0.5

Analogy with Smectic-A Liquid Crystals

Small sphere radius (Elastic LC theory below the short-length-scale cutoff/ does not apply)Larger sphere radius

Competition between the bending and compression degrees of freedomHedgehog morphology

Minimum compression when the circumference is an integer multiple of lamellar spacing, while compression can be quite large for intermediate values

Spiral morphologyIntroduces area of curvature (bend) to relieve compression for intermediate values of radius to lower overall free energy

Functional Colloids

Can be used to build ordered materials on small length scales (micrometer/submicrometer)

Particles at this length scale are mostly spheres

Important to control the packing of spheres

Manoharan and Pine, MRS Bulletin (2004).

Multivalent NanoparticlesArrangement of microdomains qualitatively similar to packing structures.Self-assembly behavior of immiscible polymers may be used to produce similar clusters.

Microdomains on different spheres tend to overlap →interfacial energy

Assumptions/Experimental Validity

Uniform and thin film in the radial directionDifficult to experimentally realize, in the form of colloids/ nanoparticles coated with a thin layer of block copolymer

Neutralize inner and outer surface of layer so that the film is compositionally homogeneous in the radial direction

Invest in a full 3D SCFT model

3D SCFT model

Thin, finite filmComposition varies both parallel to the film surface and in the radial direction

θ : Colatitude є [0,π]Φ: Longitude є [0,2π)r: Radius є [R0,Rf]

A B

Solving the 3D Modified Diffusion EquationModified Diffusion Equation:

→Solved with a fourth order accurate (O(Δs4)) Backwards Differentiation Formula (BDF4) / Adams-Bashford

→ Orientational portion of Laplacian: spherical harmonics

→ Radial portion: 2nd order accurate finite difference (O(Δr2)) and Robbins Boundary Conditions

Cochran, Garcia-Cervera, and Fredrickson Macromolecules (2006).

Robbins Boundary Conditions

Incompressibility constraint:

Neumann BC: Suitable for neutral surfaces

Robbins BC: Surface has a preferential attraction to one component

G H. Fredrickson, Oxford University Press ( 2006).

3D SCFT Model: Lamellar Phase χN=15.0, f = 0.5

→ Ro = 3 Rgo, Rf = 5 Rgo

→ Neumann BC at Ro and Rf (neutral surface)

3D SCFT Model: Lamellar PhaseχN=15.0, f = 0.5

→ Ro = 4 Rgo, Rf = 5 Rgo

→ Robbins BC at Ro, Neumann BC at Rf

3D Brush ModelA/B homopolymers tethered at one end to the surface of the sphere.Polymers are randomly and permanently bonded to the surface of the sphere.Composition varies both parallel to the film surface and in the radial directionUtilize same numerical methods: BDF4, spherical harmonics, finite difference.

A B

1D Brush ModelOur test model:

Fixed problems with incompressibilityFunctional Code

1 component caseBinary blend when χN is below the ODT.

Currently we observe convergence problems when χN is above the ODT.

May be due to chain stretching near the wall.A B

Semiflexible Polymers

Most polymeric systems have some degree of structural rigidity.Biological Examples

DNA and RNA Tobacco Mosaic VirusActin filaments

Liquid Crystalline PolymersRod-Coil Polymers

http://www.accessexcellence.org/RC/VL/GG/images/rna.gif

Phases: Rod-Coil Block Copolymer Thin Films

Exhibit interesting phase morphologies not observed in flexible coil-coil block copolymers.

Olsen et al., Macromolecules ( 2007).

Poly(alkoxyphenylenevinylene-b-isoprene) (PPV-b-PI) rod-coil block copolymers

Lamellae Structure: Coil-Coil vs. Rod-Coil

Olsen and Segalman, Macromolecules ( 2005).

Rods: Poly(alkoxyphenylene vinyleneCoil: polyisoprene

Coil-Coil Block Copolymer:Poly(styrene-block methyl methacrylate)

Kim et al., Nature, (2003).

Worm-like Chainno local energy penalty for stretching

Was included for continuous Gaussian chain model

local energy penalty for bendingAdditional parameter: segment orientation u=(θ, φ), in addition to spatial dimension r= (x,y,z)

persistence length (λ)Rigid Rod limit Lc/λ « 1Flexible limit Lc/λ » 1

Matsen, J.Physics: Condens. Matter. 2002, 14, R21-R47

Semiflexible: Tool Belt

A tool that we utilized for the sphere system can be used to model semiflexible systems:

SPHEREPACK 3.1: Spherical HarmonicsAnother Relevant Tool

FFTW: Fourier Transforms

http://www.cisl.ucar.edu/css/software/spherepack/http://www.fftw.org/

Isotropic-Nematic Transition

Holyst et al, Macromol. Theory Simul. 2001, 10, 1-16

IsotropicHomogeneous: no preferred direction

Nematics:Long-ranged orientational order [tend to be parallel to a common axis called the director], short ranged positional order.

Conclusion/ Future PlansCurrently implemented

A self-assembly model for a free AB diblock copolymer thin film on the surface of a sphere

Surface can prefer either the A or B componentParallel: Domain decomposition/MPI communication calls

Future PlansContinue to develop the model for the grafted AB Binary Blend

Can compare the different self-assembled patterns obtained from the grafted/free systems

Continue to develop our code to study the Isotropic-Nematic transition

Acknowledgements

Glenn H. Fredrickson, Hector D. Ceniceros, Carlos García-Cervera, Ed Kramer, August Bosse, Alexander Hexemer, Kirill Katsov, Erin M. Lennon, Won Bo Lee, and Jonghoon Lee.NSF IGERT grant DGE02-21715MRL CSP Technologies FellowshipMRL Central Facilities: MRSEC Program NSF DMR05-20415