anna c. balazs jae youn lee gavin a. buxton olga kuksenok kevin good valeriy v. ginzburg* chemical...
TRANSCRIPT
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