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 Multi-scale Modeling of Polymeric Mixtures

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

 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 System C B A

Free Energy for Ternary Mixture: F L  Free energy F = F L + F NL   Coefficients for F L yield 3 minima  Three phase coexistence A B C

Free Energy : Non-local part, F NL  Free energy F= F L + F NL  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

Evolution Equations *  Order parameters evolution:  Cahn –Hilliard equations   M w (A) = M w (B); M w (C) = 2* M w (A)  Navier-Stokes equation * C. Tong, H. Zhang, Y.Yang, J.Phys.Chem B, 2002

Lattice-Boltzmann Method  D2Q9 scheme  3 distribution functions  Macroscopic variables:  Constrains & Conservation Laws  Governing equations in continuum limit are:  Cahn –Hilliard equations  Navier-Stokes equation

 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:

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

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

Compare Morphologies Diffusive regime: Steady-state Viscous regime: Early time Viscous regime: Late times

Domain Growth vs. Reaction Rate Ratio,  R(t) vs. time for different   Diffusive limit:  Freezing of R due to interfacial reactions  Viscous limit:  Growth of R slows down, especially at early times  R smaller for greater 

Viscous Limit: Evolution of Average Interface Coverage, I C.  I c = (L 2 / L AB ) = R  I C saturates quickly; value similar in diffusive and viscous limits  Domain growth freezes in diffusive limit  I C = const for different R in viscous regimes 21

Viscous Limit: Evolution of  Avg. Amount of C:  At early times, is the similar in both cases until I C saturates  At late times, is smaller in viscous regime than in diffusive  Velocity advects interfaces  larger for greater 

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 (b) (c)

Dependence on Reaction Rates: Viscous Regime,  No saturation of R  R(t) smaller for higher reaction rates

Dependence on Reaction Rates,  & Interface Coverage, I C  Value of depends on values of reaction rates  Higher reaction rates yield greater  Sat. of I C faster for higher reaction rates  Values similar for different cases

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

Determine Mechanical Properties  Use output from LB as input to LSM  Morphology  Strain Stress

Symmetric Ternary Fluid;   Local in 3 phase coexistence;  Cost of A/C and B/C interfaces ( )  Viscous limit,

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  and specific values of reaction rates  Future work: “Symmetric” ternary fluids  A & B & C domain formation  Determine mechanical properties