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

Transcription

1 Self-Assembly on the Sphere: A Route to Functional Colloids Tanya L. Chantawansri Glenn H. Fredrickson, Hector D. Ceniceros, and Carlos J. García-Cervera January 23, 2007 CFDC Annual Meeting 2007

2 Contents Motivation 2D Sphere Model/Results Preliminary 3D Sphere Model/Results Current Status Conclusion / Future Plans

3 Functional Colloids Stable suspension of nanometer-sized particles Particles exhibiting novel surface functionalities Structural elements fashioned from several different materials Quantum-scale dimensions produce electronic, optical, catalytic properties different from the bulk material Offers a route to the simple assembly of complex structures Can be used to create a variety of electronic and sensor components A.N. Shipway, E. Katz, and I. Willner, CHEMPHYSCHEM, 1, 18 (2000).

4 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 V.N. Manoharan and DJ Pine, MRS Bulletin, (2004).

5 Multivalent Nanoparticles Nanoparticles with precisely controlled number and location of functional sites Phase separation between immiscible polymers nanostructures that comprise of isolated domains Simulations of end-grafted homopolymers Produced structures similar to those found in small clusters of colloidal microspheres (Roan 2006) Experimentally realizable: (Li et al, 2005) Grafted blend of polymer brushes on silica particles Obtained nanoparticles with equilibrium structures induced by chemical incompatibility

6 2D Model Thin but finite film Thin Block Copolymer Film A B Composition only varies parallel to the film surface θ : Colatitude є [0,π] Φ: Longitude є [0,2π) Can be experimentally viable for films where the radius of the sphere is much larger than the thickness

7 Thomson Problem Attempts to find the ground state (lowestenergy) arrangement of N Coulomb charges confined to the surface of a sphere. Bausch et al. (2003) geology.er.usgs.gov

8 Diblock Copolymers in Flat Space Field Theory Representation of the Hamiltonian H[W +,W - ] W + : Pressure Field W - : Exchange Chemical Potential Field f: Fraction of A monomers in the polymer chain χn: AB Flory Parameter, Index of Polymerization Q[w a,w b ]: Partition function Matsen, (2002) M.W. Matsen and M. Schick, PRL 72, 2660 (1994).

9 Diblock Copolymers in Curved Space Effective Hamiltonian: Mean field (saddle point) solution (SCFT) Solving for Q[w a,w b ]: most numerically expensive step Modified Diffusion Equation:

10 Spherical Harmonics (SH) Approximate a function as: SH are Eigenfunctions of Laplacian operator: We can easily transform between l,m space and real space using SPHEREPACK 3.1

11 Solving the 2D Modified Diffusion Equation Modified Diffusion Equation: Operator Splitting Method: Real space l,m space Real space K. O. Rasmussen and G. Kalosakas, Journal of Polymer Science B: Polymer Physics, 2002, 40, 1777

12 Basic Schematic

13 Defects in the Cylindrical and Lamellar Phase Flat Space Self Assembly: Ordered Lattices with few or no defects Sphere: Topology requires defects to occur Cylindrical Phase defect charge = 12 Always 12 more 5-fold than 7-fold disclinations.

14 Cylindrical Phase 12 (5-fold disclinations) 69 (5-fold), 350 (6-fold), and 57 (7-fold)

15 2D SCFT Model: Cylindrical Phase χn=25.0, f = 0.8

16 Grain Boundaries on Spheres Properties: High angled (30º) and freely terminates within the sphere. Consists of 3-5 dislocations and one excess 5-fold disclination. Total of 12 per a sphere Present when: R/a 5 [ a = mean domain spacing] # of domains 360 A.R. Bausch et al. Science, 2003, 299,

17 2D SCFT Model: Grain Boundary Scars R=20 Rg, χn=25.0, f=0.8 Total of 446 domains: 69 (5-fold), 350 (6-fold), and 57 (7-fold)

18 Lamellar phase Spiral Hedgehog Quasibaseball

19 2D SCFT Model: Lamellar Phase χn=12.5, f = 0.5

20 2D SCFT Model: Lamellar Phase χn=12.5, f = 0.5

21 3D model Thin Block Copolymer Film Thin, finite film Composition 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 [R 0,R f ] A B

22 Solving the 3D Modified Diffusion Equation Modified Diffusion Equation: Backwards Differentiation Formula (BDF4) / Adams-Bashford: Orientational portion of Laplacian: spherical harmonics Radial portion: 2 nd order accurate finite difference (O(Δr 2 )) and Robbins Boundary Conditions E. W. Cochran, C. J. Garcia-Cervera, and G. H. Fredrickson Macromolecules 39, 2449

23 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

24 Current Status

25 3D Preliminary Results χn=15.0, f = 0.5 φ A r (R g0 ) R o = 3 R go, R f = 4 R go Robbins BC at R o, Neumann BC at R f

26 3D Preliminary Results χn=15.0, f = 0.5 R o = 4 R go, R f = 6.1 R go r = 4, 4.51, 5.07, 5.59, 6.1 R g0 Neumann BC at R o and R f (neutral surface)

27 Conclusion/ Future Plans Currently 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 component Future Plans Develop a self-assembly model for grafted AB diblock copolymers on the surface of the sphere Can compare the different self-assembled patterns obtained from the grafted/free systems Parallelization Domain decomposition and MPI communication calls

28 Acknowledgements August Bosse and Alexander Hexemer Kirill Katsov, Richard Elliot, David R. Nelson, Vincenzo Vitelli, Erin M. Lennon, Won Bo Lee. Funding: NSF IGERT grant DGE MRL Central Facilities: MRSEC Program NSF DMR