Anna C. BalazsJae Youn Lee

Gavin A. BuxtonOlga Kuksenok

Kevin GoodValeriy V. Ginzburg*

Chemical Engineering DepartmentUniversity of Pittsburgh

Pittsburgh, PA

*Dow Chemical CompanyMidlland, 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

CBA+

−

Γ

Γ⇔+

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

BA ρρϕ −=

cρψ =

CBAii ,,=ρ

• System CBA+

−

Γ

Γ⇔+

C

B

A

• Free Energy for Ternary Mixture: FL

Free energy F = FL + FNL

Coefficients for FL yield 3 minima

Three phase coexistence

∫⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+

+−+

+−

=22

22

404

303

202

440

220

ϕψ

ψψψ

ϕϕ

A

AAA

AA

rdFL

r

ϕ

ψ

A

B

C

4/120 =A

8/140 =A

4/102 =A103 =A

8/504 =A

122 =A

LF

• 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

∫⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

∇+

∇+∇−∇=

2

22

22

2

ψ

ϕϕψϕ

ψ

ϕ

r

rrrr

k

wskrdFNL

€

(kϕ − sψ 2) |r

∇ϕ |2

2ψψ ∇

rk

∫ ⎥⎦⎤

⎢⎣⎡ ∇++−=

24

812

41 ϕϕϕ ϕ

rrkrdF

0→ψ

0/ =δψδF

•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

μϕϕ 2∇=∇+∂

∂ vrrMu

t

δψδμδϕδμ ψ /,/ FF ==

ψϕψρϕψρ

μψψψψ

−+ Γ−−−+−Γ⋅

+∇=∇⋅+∂∂

))((2/1

2Mut

rr

[ ],2

1ψϕρρ −−=A [ ]ψϕρρ −+=

2

1B

CBA+

−

Γ

Γ⇔+

μϕμψηρρ ψ ∇−∇−∇+∇−=∇⋅+∂rrvrvrvv

uPuuut2

0)(

• 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

)()(),( ieq

ifiii ff,tftf −−=++ ϖδδ rer)()(),( i

eqigiii gg,tgtg −−=++ ϖδδ rer

iieq

ihiii Fhh,thth +−−=++ )()(),( ϖδδ rer

;∑ =i

eqif ρ

;fi

ieq

i ue∑ =ρ

;∑ =i

eqig ϕ ;∑ =

i

eqih ψ

βααββα ρ uuee∑ +=i

eqi Pf

βααβϕϕβα ϕδμ uuee∑ +=i

eqi Mg /

βααβψψβα ϕδμ uuee∑ +=i

eqi Mh /

;∑ =i

if ρ ;∑ =i

ig ϕ ;∑ =i

ih ψ ;fi

ii ue∑ =ρ

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.

•

−

+

ΓΓ

=γ )10( 4−−=Γ

CBA+

−

Γ

Γ⇔+

• Steady-state distributions:

x

y

γ

• 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

0=Γ=Γ +−3/1~ tR

)10( 4−−=Γ

1=γ 2=γ

€

ω f = 3⋅10−3

−

+

ΓΓ

=γ

• 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

1=γ

0=Γ=Γ +−

2/1~ tR

410−−=Γ

4103 −⋅=fω

μμψ ∇∇rr

,

4104⋅=t 5104⋅=t

• Compare Morphologies 2=γ

Diffusive regime: Steady-state )10( 5=t

Viscous regime:Early time )104( 4⋅=t

Viscous regime: Late times

)104( 5⋅=t )104.1( 6⋅=t

•Domain Growth vs. Reaction Rate Ratio,

R(t) vs. time for different γ

Diffusive limit: Freezing of R due to

interfacial reactions

Viscous limit: Growth of R slows down,

especially at early times R smaller for greater γ

γ

)(,1 diff=γ)(,2 diff=γ

)(,2 visc=γ

)(,1 visc=γ€

(visc)

•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

21

)(,1 diff=γ)(,1 visc=γ

)(,2 diff=γ)(,2 visc=γ

ψ ψ

€

•

•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

Velocity advects interfaces

larger for greater γ

∫= rr dL

)(1

2ψψ

)(,1 visc=γ

)(,2 visc=γ

)(,2 visc=γ

)(,1 visc=γ

)(,1 diff=γ

)(,2 diff=γ

ψ

ψ

€

ψ =Ic /R

ψ

ψ

• 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

2=γ

44 102;10 −+

−− ⋅=Γ=Γ

55 105;105.2 −+

−− ⋅=Γ⋅=Γ

44 108;104 −+

−− ⋅=Γ⋅=Γ

(b) (c)

• Dependence on Reaction Rates: Viscous Regime,

No saturation of R

R(t) smaller for higher reaction rates

2=γ

€

(Γ− = 2.5 ⋅10−5;Γ+ = 5 ⋅10−5)

€

(Γ− = 4 ⋅10−4;Γ+ = 8 ⋅10−4 )

4104⋅=t 5107⋅=t

• 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

2=γ

4/1=m 1=m 4=m

ψ

ψ

ψ

ψ

•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,

∫ ⎥⎦⎤

⎢⎣⎡ ∇+∇+=

22ψϕ ψϕ

rrrkkrdFF L

LF

CBA+

−

Γ

Γ⇔+

ψ∇r

20=γ410=t 4102⋅=t

510=t 5103⋅=t

• 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 γ and specific values of reaction rates

Future work: “Symmetric” ternary fluids

A & B & C domain formation

Determine mechanical properties

CBA ⇔+

:diffusive :viscous