Multi-scale Modeling of Polymeric Mixtures and Ternary Fluid Reactions

1. 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

2. 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

3. B A C 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

4. A C B Free Energy for Ternary Mixture: FL • Free energy F = FL + FNL • Coefficients for FL yield 3 minima • Three phase coexistence

5. 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

6. 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

7. 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

8. 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:

9. 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

10. 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

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

12. 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

13. 1 2 Viscous Limit: Evolution of Average Interface Coverage, IC . • Ic = (L2 / LAB ) = R • ICsaturates quickly; value similar in diffusive and viscous limits • Domain growth freezes in diffusive limit • IC= const for different R in viscous regimes

14. Viscous Limit: Evolution of • Avg. Amount of C: • At early times, is the similar in both cases until ICsaturates • At late times, is smaller in viscous regime than in diffusive • Velocity advects interfaces • larger for greater g

15. (c) (b) 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

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

17. 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

18. 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

19. Determine Mechanical Properties • Use output from LB as input to LSM • Morphology • Strain Stress

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

21. 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