Anna C. Balazs

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Multi-scale Modeling of Polymeric Mixtures
Anna C. Balazs Jae Youn Lee Gavin A. Buxton Olga
Kuksenok Kevin Good Valeriy V. Ginzburg Chemical
Engineering Department University of
Pittsburgh Pittsburgh, PA Dow Chemical
Company Midlland, MI
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Use Multi-scale Modeling to Examine

• Reactive A/B/C ternary mixture
• A and B form C at the
• A/B interface
• Critical step in many
• polymerization processes
• Challenges in modeling system
• Incorporate reaction
• Include hydrodynamics
• Capture structural evolutional and domain
  growth
• Predict macroscopic properties of mixture
• Results yield guidelines for controlling
  morphology and properties

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System

• A and B are immiscible
• Fluids undergo phase separation
• C forms at A/B interface
• Alters phase-separation
• Examples
• Interfacial polymerization
• Reactive compatibilization
• Two order parameters model
• here is
  density
• Challenges
• Modeling hydrodynamic interactions
• Predicting morphology formation of C

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Free Energy for Ternary Mixture FL

• Free energy F FL FNL
• Coefficients for FL yield 3 minima
• Three phase coexistence

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Free Energy Non-local part, FNL

• Free energy F FL FNL
• Reduction of A/B interfacial tension by C
• No formation of C in absence of chemical
  reactions ( )
• Cost of forming C interface .
• For (no reactions no C)
• Standard phase-separating binary fluid in
  two-phase coexistence region

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Evolution Equations

• Order parameters evolution
• Cahn Hilliard equations
• Mw(A) Mw(B) Mw(C) 2 Mw(A)
• Navier-Stokes equation
• C. Tong, H. Zhang, Y.Yang, J.Phys.Chem B, 2002

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Lattice-Boltzmann Method

• D2Q9 scheme
• 3 distribution functions
• Macroscopic variables
• Constrains Conservation Laws
• Governing equations in continuum limit are
• Cahn Hilliard equations

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• Single Interface
• System behavior vs ,
• Higher leads to wider C layer
• Choose parameters to have narrow interface
• Need other parameters and additional non-local
  terms to model wide interfaces.

• Steady-state distributions

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Diffusive Limit No Effects of Hydro

• Initial state
• 256 x 256 sites
• High visc.
• No reaction
• Domain growth
• Turn on reaction when R 4
• W/reactions steady-states
• Reactions arrest domain growth

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Viscous Limit Effect of Hydrodynamics

• No reaction
• Domain growth
• Evolution w/reactions ,
• Domain growth slows down but does not stop
• Velocities due to
• Advect interfaces and prevent equilibrium between
  reaction diffusion

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Compare Morphologies
Diffusive regime Steady-state
Viscous regime Early time
Viscous regime Late times
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Domain Growth vs. Reaction Rate Ratio,

• R(t) vs. time for different g
• Diffusive limit
• Freezing of R due to
• interfacial reactions
• Viscous limit
• Growth of R slows down,
• especially at early times
• R smaller for greater g

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Viscous Limit Evolution of Average
Interface Coverage, IC .

• Ic (L2 / LAB ) R
• IC saturates quickly value similar in diffusive
  and viscous limits
• Domain growth freezes in diffusive limit
• IC const for different R in viscous regimes

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Viscous Limit Evolution of

• Avg. Amount of C
• At early times, is the similar in both cases
  until IC saturates
• At late times, is smaller in viscous
  regime than in diffusive

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Dependence on Reaction Rates Diffusive
Regime,

• Reaction rates
• a) (
  )
• b) (
  )
• c) (
  )
• Steady-state
• The higher the reaction rates, the faster the
  interface saturates
• The lower the saturated value of R

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Dependence on Reaction Rates Viscous Regime,

• No saturation of R

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Dependence on Reaction Rates,

• Interface Coverage, IC
• Value of depends on values of reaction
  rates
• Higher reaction rates yield greater
• Sat. of IC faster for higher reaction rates
• Values similar for different cases

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Morphology Mechanical Properties

• Output from morphology study is input to Lattice
  Spring Model (LSM)
• LSM 3D network of springs
• Consider springs between nearest and
  next-nearest neighbors
• Model obeys elasticity theory
• Different domains incorporated via local
  variations in spring constants
• Apply deformation at boundaries
• Calculate local elastic fields

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Determine Mechanical Properties

• Use output from LB as input to LSM
• Morphology
• Strain Stress

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Symmetric Ternary Fluid

• Local in 3 phase coexistence
• Cost of A/C and B/C interfaces ( )
• Viscous limit,

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Conclusions

• Developed model for reactive ternary mixture
  with hydrodynamics
• Lattice Boltzmann model
• Compared viscous and diffusive regimes
• Freezing of domain growth in diffusive limit
  and slowing down in viscous limit
• R(t) depends on g and specific values of reaction
  rates
• Future work Symmetric
• ternary fluids
• A B C domain formation
• Determine mechanical properties