0
Field-Based Simulations for the
Design of Polymer Nanostructures
                 Glenn H. Fredrickson

       Departments...
The Mitsubishi Chemical Center for
     Advanced Materials (MC-CAM)
MC-CAM was created in 2001 to
enable a research partne...
Polypropylene Block Copolymers
   A Mitsubishi Chemical—UCSB—Cornell
   collaboration




  sPP-EPR-sPP         iPP-EPR-iP...
Complex Fluids Design Consortium
The Complex Fluids Design Consortium (CFDC) is an academic-
industrial-national lab partn...
Postdocs:
                      Acknowledgements
   Dr.   Venkat Ganesan
                                   Funding:
   Dr...
The Problem—Design of Polymer
Formulations
Polymer formulations are
often inhomogeneous and
multi-component
   Multiphase ...
Nanoscale Morphology Control: Block
              Copolymers
• Microphase separation of block copolymers
                 ...
Enabling Chemistries to Create
Nanostructured Polymers
 The past decade has seen unprecedented advances
 in controlled (li...
Nanostructured Polymers via New Chemistry:
sPP-b-EPR Block Copolymers




                                                ...
Nanoparticles in Block Copolymers
                                              B.J. Kim et. al., Adv. Matl. 17, 2018 (200...
Scales and Approaches to Fluids Simulation
 Scale               DOF                      Method
Sub-atomic         Fields ...
From Particles to Fields
 Any classical “particle-based” model of an equilibrium fluid
 can be exactly converted to a stat...
Why Field-Based Simulations of Polymer
 Fluids?
Relevant spatial and time scales
cannot be accessed by atomistic
“particle...
Coarse-Grained “Particle-Based” Model:
Polymer Solution
Two-parameter “Edwards” model of homopolymers in an
implicit good ...
Edwards Field Theory (~1960)


 Energy functional




 Single-chain partition function




 Fokker-Planck equation for cha...
Generalizations
 Using similar methods, one can construct statistical
 field theories for a broad variety of polymer
 form...
Structure of the Field Theories
 The field theories have “saddle point”
 configurations w*(r) corresponding to stable and
...
Mean-Field Approximation: SCFT

•   SCFT is derived by a saddle point approximation to the
    field theory:




•   The a...
High-Resolution SCFT
By the above methods we can
compute saddle points using
~107 or more plane waves
  Unit cell calculat...
Unit Cell Calculation, Ia3d                  0.00
Symmetry specified initial guess               -0.05
              (E. C...
Mean-Field A-B Diblock Copolymer Melt Phase
Diagram    Matsen-Bates (1995), Cochran (2006)




                           ...
dark
ABA triblock + A homopolymer
Arkema (S. Sides)                              +
                                       ...
SW Sides and GHF,
Polymer 44, 5859 (2003)
                          +   light




106 plane waves
3000 field iterations


...
Simulation results (S. Sides)
Polydispersity: Acrylic BCs                                  <φA > ~ 0.65    <φΒ > =
       ...
Photolithography vs. Block Copolymer Lithography

                                              Basic steps


            ...
Defects in Laterally Confined Block
 Copolymer Thin Films
Large 2D arrays of spheres or cylinders
will exhibit defect popu...
SCFT studies of hexagonal confinement: “A wetting”




L = 14.75 Rg0        L = 16.25 Rg0        L = 18.00 Rg0

   Here we...
Tetragonal Ordering by Square Confinement
     AB block copolymers pack cylinders or monolayers of
     spheres in hexagon...
Multi-layer Films of Spherical AB Diblocks

              Gila Stein and Ed Kramer          Polymer – air
                ...
Stein-Kramer experiments reveal 3 structures:




       HCP spheres       Fm3m spheres – 100 plane
                      ...
1.16
BCC

                 1.12
       a1 / a2                                          bcc
                 1.08       hc...
A Simple Theory
• Assume that the surface excess free energy
  contributions are negligible beyond a single layer film,
  ...
Theory vs. Experiment
• The theory + SCFT explains the observation of a 1st
  order transition!
• The transition is predic...
Beyond Mean-Field Theory
In many situations, mean-field theory is inaccurate
   Polymer solutions
   Melts near a critical...
The “Sign Problem”
 When sampling a complex field theory, the
 statistical weight exp( – H[w]) is not positive-
 definite
...
Complex Langevin Sampling
               (Parisi, Klauder 1983)
A method to circumvent the sign problem in polymer
simulat...
Complex Langevin Dynamics
 A Langevin dynamics in the complex plane for generating
 Markov chains with stationary distribu...
Order-disorder transition of diblock copolymers
(E. Lennon)


f=0.396
χN = 14 ! 11
C=nRgd/V =60.0
L=17.8 Rg
483 lattice
IC...
Polyelectrolyte Complexation: Complex
Coacervates
   Aqueous mixtures of                  -
   polyanions and             ...
A Symmetric Model of Coacervation

 In the simplest case,
 assume symmetric
 polyacids & polybases          +
 mixed in eq...
Corresponding Field-Theory Model




w: fluctuating chemical pot.        lB =e2 /ε kBT: Bjerrum length
φ: fluctuating elec...
Complex Coacervation
CL Simulations of the Field Theory Model
 • 2D, 32x32 Rg02
 • C=2.0 B=1.0 E=64000
Future work
We believe that our CL simulation method can be used to
explore the phase behavior of a broad range of PE
comp...
Complexation of oppositely charged
            diblocks
           2D, 16x16 Rg02
           C=11.0 B=0.1
           E=64000
A Hybrid Particle-Field Simulation Approach
 S. W. Sides et. al. Phys. Rev. Lett. 96, 250601 (2006)
 Combine a field-based...
Block Copolymer Morphology Change Induced
by Nanoparticles (BJ Kim, et. al. PRL 96, 250601 (2006))
                       ...
Summary
  “Field-based” computer simulations are powerful tools for
  exploring equilibrium self-assembly in complex polym...
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Glenn Fredrickson of UCSB

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  1. 1. Field-Based Simulations for the Design of Polymer Nanostructures Glenn H. Fredrickson Departments of Chemical Engineering & Materials Mitsubishi Chemical Center for Advanced Materials (MC-CAM) Complex Fluids Design Consortium (CFDC) University of California, Santa Barbara The Equilibrium Theory of Inhomogeneous Polymers (Oxford, 2006)
  2. 2. The Mitsubishi Chemical Center for Advanced Materials (MC-CAM) MC-CAM was created in 2001 to enable a research partnership between Mitsubishi Chemical and UCSB Focus is new organic, inorganic, and hybrid materials for applications in Display technologies Specialty polymers Solid state lighting Energy devices, e.g. photovoltaics Funding has been ~$2.5M/yr ~50 patent disclosures to date
  3. 3. Polypropylene Block Copolymers A Mitsubishi Chemical—UCSB—Cornell collaboration sPP-EPR-sPP iPP-EPR-iPP Mn=300K, wPP=.24 Mn=100K, wPP=.24
  4. 4. Complex Fluids Design Consortium The Complex Fluids Design Consortium (CFDC) is an academic- industrial-national lab partnership aimed at developing computational tools for designing soft materials and analyzing multiphase complex fluids Academic partners: Fredrickson (Director), Banerjee, Ceniceros, Garcia-Cervera, Gusev (ETH), Cochran (Iowa St.) Industrial partners: Arkema, Mitsubishi Chemicals, Rhodia, General Electric, Dow Chemical, Kraton Polymers, Accelyrs, and Nestlé National lab partners: Los Alamos (Lookman, Redondo) Sandia (Curro, Grest, Frischknecht) http://www.mrl.ucsb.edu/cfdc/
  5. 5. Postdocs: Acknowledgements Dr. Venkat Ganesan Funding: Dr. Scott Sides NSF DMR-CMMT Dr. Eric Cochran NSF DMR-MRSEC Dr. Jonghoon Lee ACS-PRF Dr. Yuri Popov Dr. Kirill Katsov Complex Fluids Design Dr. Dominik Duechs Consortium: Students: Rhodia A. Alexander-Katz, S. Hur Mitsubishi Chemical E. Lennon, W. Lee, A. Bosse Arkema T. Chantawansri, M. Villet Dow Chemical Collaborators: GE CR&D Prof. Edward Kramer Nestlé Prof. Craig Hawker Kraton Polymers Prof. Hector Ceniceros Accelrys Prof. Carlos Garcia-Cervera SNL, LANL www.mrl.ucsb.edu/cfdc
  6. 6. The Problem—Design of Polymer Formulations Polymer formulations are often inhomogeneous and multi-component Multiphase plastics Solution formulations They exhibit complex phase behavior, including Nanostructured mesophases Coexistence of meso and macro phases (emulsions) Relationship between formulation, self-assembled structure, and properties difficult to establish Trial and error experimentation is norm Can Theory/Simulation help?
  7. 7. Nanoscale Morphology Control: Block Copolymers • Microphase separation of block copolymers ABA Triblock Thermoplastic Elastomer A B Holden & Legge A A (Shell – Kraton Polymers) f
  8. 8. Enabling Chemistries to Create Nanostructured Polymers The past decade has seen unprecedented advances in controlled (living or quasi-living) polymerization techniques: Controlled free radical methods Single site metallocene catalysts Improved ring-opening techniques “Change of mechanism” strategies Post-polymerization chemical modifications Living Ziegler-Natta methods These synthetic techniques enable the creation of block and graft architectures from a broad range of commodity-priced monomers
  9. 9. Nanostructured Polymers via New Chemistry: sPP-b-EPR Block Copolymers sPP minority HPL phase P. Husted, J. Ruokolainen, R. Mezzenga, G. W. Coates, E. J. Kramer, GHF, Macrom. 38, 851 (2005)
  10. 10. Nanoparticles in Block Copolymers B.J. Kim et. al., Adv. Matl. 17, 2018 (2005) Central Interfacial 100nm 100nm Au Au 200 150 PVP PS PVP PS PVP PVP Number of Au Particles Number of Au Particles 150 100 100 50 50 0 0 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 Normalized Domain Size of PS-b-P2VP Normalized Domain Size of PS-b-P2VP
  11. 11. Scales and Approaches to Fluids Simulation Scale DOF Method Sub-atomic Fields Ab initio quantum < 1Å (wavefunctions, chemistry, electronic density functionals) structure Atomic to Particles Classical MD, MC, BD mesoscopic (positions, momenta) 1Å -- 1µm Continuum Fields PDEs of mass, > 1µm (densities, velocities, momentum, energy stresses) flow, elasticity Can we compute with fields in the atomic-mesoscopic regime?
  12. 12. From Particles to Fields Any classical “particle-based” model of an equilibrium fluid can be exactly converted to a statistical field theory E.g., monatomic fluid with invertable repulsive pair potential v(r) -- Hubbard-Stratonovich transformation microscopic density Particles are decoupled and rn coordinates can be traced out of the partition function Field theory is complex when repulsive interactions are present
  13. 13. Why Field-Based Simulations of Polymer Fluids? Relevant spatial and time scales cannot be accessed by atomistic “particle-based” simulations Use of fluctuating fields, rather than particle coordinates, has potential computational advantages: Copolymer nanocomposite BJ Kim `06 Simulations become easier at high density & high MW More seamless connection to continuum mechanics Systematic coarse-graining by numerical RG appears feasible Microemulsion, Bates ‘97
  14. 14. Coarse-Grained “Particle-Based” Model: Polymer Solution Two-parameter “Edwards” model of homopolymers in an implicit good solvent (v > 0): 2 s 0 v N R(s) v
  15. 15. Edwards Field Theory (~1960) Energy functional Single-chain partition function Fokker-Planck equation for chain propagator
  16. 16. Generalizations Using similar methods, one can construct statistical field theories for a broad variety of polymer formulations Models have been devised for: Block and graft copolymers of arbitrary architecture Molten polymer alloys Polyelectrolytes Liquid crystalline polymers (worm-like chains) Polymer brushes, thin films Supramolecular polymers Other ensembles, e.g. μVT, are straightforward
  17. 17. Structure of the Field Theories The field theories have “saddle point” configurations w*(r) corresponding to stable and metastable phases of the system Saddle points can be homogeneous (disordered phase) or inhomogeneous (ordered phase) Saddle points lie in the complex plane such that H[w*] is real
  18. 18. Mean-Field Approximation: SCFT • SCFT is derived by a saddle point approximation to the field theory: • The approximation is asymptotic for • We can simulate a field theory at two levels: • “Mean-field” approximation (SCFT): F ≈ H[w*] • Full stochastic sampling of the complex field theory: “Field-theoretic simulations” (FTS)
  19. 19. High-Resolution SCFT By the above methods we can compute saddle points using ~107 or more plane waves Unit cell calculations for high accuracy with variable cell shape to relax stress Initial condition has desired S. Sides, K. Katsov symmetry Large cell calculations for exploring self-assembly in new systems Initial condition is random Complex geometries can be addressed with a masking technique T. Chantawansri A. Bosse SPHEREPACK
  20. 20. Unit Cell Calculation, Ia3d 0.00 Symmetry specified initial guess -0.05 (E. Cochran) -0.10 Energy AB diblock melt, f = 0.39, χN = 20 -0.15 9.8 Rg -0.20 -0.25 0 10 20 30 40 Time 0 -1 Log Error -2 -3 -4 -5 0 10 20 30 40 Time
  21. 21. Mean-Field A-B Diblock Copolymer Melt Phase Diagram Matsen-Bates (1995), Cochran (2006) f χ : strength of A-B monomer Repulsion N: degree of polymerization
  22. 22. dark ABA triblock + A homopolymer Arkema (S. Sides) + light volume fraction of homopolymer Nt /Nh = 2 fraction of A monomers on each triblock fA χNh = 16.0
  23. 23. SW Sides and GHF, Polymer 44, 5859 (2003) + light 106 plane waves 3000 field iterations 256 Rg ∼2.5 μm
  24. 24. Simulation results (S. Sides) Polydispersity: Acrylic BCs <φA > ~ 0.65 <φΒ > = 0.35 (dark) PBA pdi = PMMA (light) 1.00 Experimental data (Arkema/ESPCI) <φPMMA > ~ 0.65<φPBA > ~ 0.35 <φA > ~ 0.65 <φΒ > ~ 0.35 PDI ~ pdi= 2.0- 1.225 3.0 TEM data courtesy of A.-V. Ruzette 200 nm 12 Rg ~214 nm
  25. 25. Photolithography vs. Block Copolymer Lithography Basic steps 1. Coating polymers 2. Alignment of Expensive microdomains 3. Removal of one component Low Features Cost 5 -20 nm Chuanbing Tang Materials Research Laboratory University of California Santa Barbara
  26. 26. Defects in Laterally Confined Block Copolymer Thin Films Large 2D arrays of spheres or cylinders will exhibit defect populations, even at equilibrium However, lateral confinement can be used to induce order in smaller 2D systems—”graphoepitaxy” (Kramer, Segalman, Stein) Top-down lithography for creating μm scale “wells”, e.g. stripes, squares, or hexagons Bottom-up self-assembly to achieve perfect long-range registry of nm scale microdomains Segalman et al. Macromolecules 36, 6831 (2003)
  27. 27. SCFT studies of hexagonal confinement: “A wetting” L = 14.75 Rg0 L = 16.25 Rg0 L = 18.00 Rg0 Here we examine f = 0.7, χ = 17, and χw = 17 (majority A-monomer is attracted to the wall) We have identified “commensurability windows” of side length L, for which various annealing conditions always produced a defect free configuration
  28. 28. Tetragonal Ordering by Square Confinement AB block copolymers pack cylinders or monolayers of spheres in hexagonal lattices SCFT simulations show we can use graphoepitaxy with square wells to force tetragonal (square) packing Limitations: Need to add majority block A homopolymer (φA=0.23, Nh/N =1.75) Surface/bulk competition restricts method to small lattices Total A A homopolymer segment segment Support: FENA- concentration concentration MARCO, UCLA
  29. 29. Multi-layer Films of Spherical AB Diblocks Gila Stein and Ed Kramer Polymer – air interaction Polymer – substrate interaction 1 layer many layers HCP spheres – 111 plane (p6m 2D symmetry) BCC spheres – 110 plane
  30. 30. Stein-Kramer experiments reveal 3 structures: HCP spheres Fm3m spheres – 100 plane BCC spheres – 110 plane (p6m 2D symmetry) Face-centered orthorhombic a2 a2 a2 a1 a1 a1 a1 / a2 = 1 1 < a1 / a2 < 2 /√3 a1 / a2 = 2 /√3 = 1.155
  31. 31. 1.16 BCC 1.12 a1 / a2 bcc 1.08 hcp Fmmm Fmmm (bulk behavior) 1.04 Experiment HCP 1.00 0 5 10 15 20 # Layers a1 a2
  32. 32. A Simple Theory • Assume that the surface excess free energy contributions are negligible beyond a single layer film, n=1 • The free energy per chain as a function of the order parameter η =a1/a2 is: • The model can be parameterized by SCFT simulations of a 1-layer system (d1,f1) and a unit cell calculation of a bulk system (d1b, fb)
  33. 33. Theory vs. Experiment • The theory + SCFT explains the observation of a 1st order transition! • The transition is predicted at n=7 (χN=60) vs. n=4 (expt.) G. E. Stein et. al. Phys. Rev. Lett. 98, 158302 (2007)
  34. 34. Beyond Mean-Field Theory In many situations, mean-field theory is inaccurate Polymer solutions Melts near a critical point or ODT In such cases, the field theory is dominated by w configurations far from any saddle point w* w plane X Ia3d Physical X Lam path X DIS How do we statistically sample the full field theory?
  35. 35. The “Sign Problem” When sampling a complex field theory, the statistical weight exp( – H[w]) is not positive- definite Phase oscillations associated with the factor exp(-i HI[w]) dramatically slows the convergence of stochastic sampling methods, e.g. MC techniques This sign problem is encountered in other branches of chemistry and physics: QCD, lattice gauge theory, correlated electrons, quantum rate processes
  36. 36. Complex Langevin Sampling (Parisi, Klauder 1983) A method to circumvent the sign problem in polymer simulations (V. Ganesan) Extend the field w(r) to the complex plane Compute averages by: The CL method is a stochastic dynamics that serves to Verify the existence of the real, positive weight P[wR,wI] To importance sample the distribution
  37. 37. Complex Langevin Dynamics A Langevin dynamics in the complex plane for generating Markov chains with stationary distribution P[wR, wI] Thermal noise is asymmetrically placed and is Gaussian and white satisfying usual fluctuation-dissipation relation:
  38. 38. Order-disorder transition of diblock copolymers (E. Lennon) f=0.396 χN = 14 ! 11 C=nRgd/V =60.0 L=17.8 Rg 483 lattice IC: 23 unit cells of stress-free gyroid from SCFT
  39. 39. Polyelectrolyte Complexation: Complex Coacervates Aqueous mixtures of - polyanions and - - polycations complex to form dense liquid aggregates + + + Fluctuation-dominated: SCFT fails! Applications include: Cooper et al (2005) Curr Opin Coll. & Interf. Sci. Food/drug encapsulation 10, 52-78. Drug/gene delivery vehicles Purification/separations Bio-inspired adhesives H. Waite (UCSB) “Sandcastle worms”
  40. 40. A Symmetric Model of Coacervation In the simplest case, assume symmetric polyacids & polybases + mixed in equal + + proportions (no counterions) + Polymers are flexible and carry total charge - Z§ =§ σN - Implicit solvent - - Interactions: Coulomb and excluded volume Uniform dielectric medium: ε
  41. 41. Corresponding Field-Theory Model w: fluctuating chemical pot. lB =e2 /ε kBT: Bjerrum length φ: fluctuating electrostatic pot. v: Excluded volume parameter
  42. 42. Complex Coacervation CL Simulations of the Field Theory Model • 2D, 32x32 Rg02 • C=2.0 B=1.0 E=64000
  43. 43. Future work We believe that our CL simulation method can be used to explore the phase behavior of a broad range of PE complexation phenomena: Block copolyampholytes Block copolymers with charged blocks and uncharged blocks (hydrophobic or hydrophilic) Charged graft, star, and branched polymers Polymer-surfactant complexes Delivery vehicles: Enzymes Drugs Genes
  44. 44. Complexation of oppositely charged diblocks 2D, 16x16 Rg02 C=11.0 B=0.1 E=64000
  45. 45. A Hybrid Particle-Field Simulation Approach S. W. Sides et. al. Phys. Rev. Lett. 96, 250601 (2006) Combine a field-based description of a polymeric fluid with a particle-based description of the nanoparticles The particles are described as cavities in the fluid. They can: Be of arbitrary size, shape, and aspect ratio Have a surface treatment to attract or repel any fluid component Have grafted polymers of any architecture on their surfaces The fluid field equations are solved inside the cavities for computational efficiency The forces on the particles can computed in a single sweep of the fluid field A variety of MC and BD update schemes can be applied
  46. 46. Block Copolymer Morphology Change Induced by Nanoparticles (BJ Kim, et. al. PRL 96, 250601 (2006)) Hybrid FTS PS-b-P2VP 58k-57k/ Au-PS Low particle conc. Low particle conc. Lamellar By adding PS coated nanoparticles High particle conc. High particle conc. Hexagonal from S. W. Sides, G. H. Fredrickson
  47. 47. Summary “Field-based” computer simulations are powerful tools for exploring equilibrium self-assembly in complex polymer formulations Good numerical methods are essential! Free energy evaluation, multiscale methods, and numerical RG remain to be explored Emerging areas are Hybrid simulations with nanoparticles and colloids Polyelectrolyte complexes Supramolecular polymers Nonequilibrium extensions to coupled flow and structure This is an exciting frontier research area that brings together topics from Theoretical physics and applied math Numerical and computational sciences Materials science Real-world applications! The Equilibrium Theory of Inhomogeneous Polymers (Oxford, 2006) G. H. Fredrickson et. al., Macromolecules 35, 16 (2002)
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