Block Copolymer Simulations using Self Consistent Field Theory for the Surface of a Sphere

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Block Copolymer Simulations using Self Consistent
Field Theory for the Surface of a Sphere

• Tanya L. Chantawansri
• Glenn H. Fredrickson, Hector D. Ceniceros, and
  Carlos J. García-Cervera
• November 18,2006
• CSE IGERT Review

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Collaborators

• August Bosse Physics
• Alexander Hexemer Materials
• Glenn Fredrickson and Ed Kramer Chemical
  Engineering, Materials
• Hector Ceniceros and Carlos García-Cervera
  Mathematics

IGERT Advisors
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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
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Polymers

• Formed from chemically coupling a large quantity
  of small reactive molecules, monomers
• Homopolymer linking together N monomers of only
  one chemically distinct type
• Copolymer linking together N monomers of two or
  more chemically distinct types

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Bulk Phases of Diblock Copolymers

• In 3D we can see 6 bulk phases
• L Lamellar
• C Cylindrical
• S Spherical
• G Gyroid
• PL Perforated-lamellar
• D Double Diamond
• In 2D we can only see 2 of these phases
• L Lamellar
• C Cylindrical

M.W. Matsen, J.Physics Condens. Matter. 2002,
14, R21-R47
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Block Copolymers on Curved Surfaces

• Substrates with regions of both negative and
  positive Gaussian curvature

Alexander Hexemer (LBL and Kramer Group)
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Diblock Copolymers on a Spherical Surface

• Thin but finite film
• Composition only varies parallel to the film
  surface
• ? Colatitude ? 0,p
• F Longitude ? 0,2p)

Thin Block Copolymer Film
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Diblock Copolymers in Flat Space

• Field Theory Representation of the Hamiltonian
  HW,W-
• W Pressure Field
• W- Exchange Chemical Potential Field
• f Fraction of A monomers in the polymer chain
• ?N AB Flory Parameter, Index of Polymerization
• Qwa,wb Partition function

Matsen, (2002)

• M.W. Matsen and M. Schick, PRL 72, 2660 (1994).

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Diblock Copolymers in Curved Space

• Effective Hamiltonian
• Mean field (saddle point) solution (SCFT)
• Solving for Qwa,wb most numerically expensive
  step

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

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Cylindrical Phase
12 (5-fold disclinations)
69 (5-fold) , 350 (6-fold) , and 57 (7-fold)
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SCFT Model Cylindrical Phase ?N25.0, f 0.8
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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, 1716-1718.
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SCFT Model Grain Boundary Scars R20 Rg,
?N25.0, f0.8
Total of 446 domains
69 (5-fold) , 350 (6-fold) , and 57
(7-fold)
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Lamellar phase
Spiral
Hedgehog
Quasibaseball
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SCFT Model Lamellar Phase ?N12.5, f 0.5
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SCFT Model Lamellar Phase ?N12.5, f 0.5
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Acknowledgements

• Kirrill Katsov, Richard Elliot, David R. Nelson,
  Vincenzo Vitelli, Erin M Lennon, Won Bo Lee.
• Funding NSF IGERT grant DGE02-21715
• MRL Central Facilities MRSEC Program NSF
  DMR05-20415