AMulti-FluidModel of MembraneFormation by Phase-Inversion

Douglas R. Tree1 and Glenn Fredrickson1,2

1Materials Research Laboratory2Departments of Chemical Engineering and Materials

University of California, Santa Barbara

Society of Rheology Annual MeetingFebruary 13, 2017

Acknowledgements

I Jan Garcia

I Dr. Kris T. Delaney

I Prof. Hector D. Ceniceros

I Lucas Francisco dos Santos

I Dr. Tatsuhiro Iwama (AsahiKasei)

I Dr. Jeffrey Weinhold (Dow)

2

Can we predict the microstructure of polymers?

I Microstructure dictates properties

I Microstructure depends on processhistory

A very generalproblem!

Polymer membranes

I clean water

I medical filters

Saedi et al. Can. J. Chem. Eng. (2014)

Polymer Blends

I commodityplastics (e.g.HIPS)

I block polymerthin films www.leica-microsystems.com

Polymer composites

I bulk hetero-junctions

I nano-compositesHoppe and Sariciftci J. Mater. Chem. (2006)

Biological patterning

I Eurasian jayfeathers

Parnell et al. Sci. Rep. (2015)

3

Can we predict the microstructure of polymers?

I Microstructure dictates properties

I Microstructure depends on processhistory

A very generalproblem!

Polymer membranes

I clean water

I medical filters

Saedi et al. Can. J. Chem. Eng. (2014)

Polymer Blends

I commodityplastics (e.g.HIPS)

I block polymerthin films www.leica-microsystems.com

Polymer composites

I bulk hetero-junctions

I nano-compositesHoppe and Sariciftci J. Mater. Chem. (2006)

Biological patterning

I Eurasian jayfeathers

Parnell et al. Sci. Rep. (2015)

3

Clean water is a present and growing concern

July 7, 2015

U.S. Drought Monitor

D0 Abnormally Dry

D1 Moderate Drought

D2 Severe Drought

D3 Extreme Drought

D4 Exceptional Drought

Intensity:

http://droughtmonitor.unl.edu/

Author:Brian FuchsNational Drought Mitigation Center

Why membranes?

I Water is projected tobecome increasinglyscarce.

I Filtration is a keytechnology for waterpurification.

http://www.kochmembrane.com/Learning-Center/Configurations/What-are-Hollow-Fiber-Membranes.aspx

4

Polymer membrane synthesis by immersion precipitation

Figure inspired by: www.synderfiltration.com/learning-center/articles/introduction-to-membranes

non-solvent bath

membranesubstrate

polymer solution

nonsolvent

solvent

polymer

solvent

non-solvent

H

L-LG L-G

5

Polymer membrane synthesis by immersion precipitation

Figure inspired by: www.synderfiltration.com/learning-center/articles/introduction-to-membranes

non-solvent bath

membranesubstrate

polymer solution

nonsolvent

solvent

polymer

solvent

non-solvent

H

L-LG L-G

5

Microstructural variety in membranes

Uniform “Sponge” Asymmetric“Sponge”

Fingers orMacro-voids

Skin Layer

Strathmann et al.Desalination. (1975)

6

ModelCharacterization

run

ModelDevelopment

NIPSquench process

ModelCharacterization

run

ModelDevelopment

NIPSquench process

How can we model microstructure formation?

A difficult challenge

I Complex thermodynamics out of equilibrium

I Spatially inhomogeneous (multi-phase)

I Multiple modes of transport (diffusion & convection)

I Large separation of length/time scales

Continuum fluid dynamics

Teran et al. Phys. Fluid. (2008)

Self-consistent field theory (SCFT)

Fredrickson. J. Chem. Phys. 6810 (2002)Hall et al. Phys. Rev. Lett. 114501 (2006)

Key idea – cheaper models

Classical density functional theory(CDFT)/“phase field” models

8

Multi-fluid models

Two-fluid modelI Momentum equation

for each species

I Large drag enforcescons. of momentum

de Gennes. J. Chem Phys. (1980)

The Rayleighian

A Lagrangian expression of “leastenergy dissipation” foroverdamped systems (Re = 0).

R[{vi}] = F [{vi}] free energy

+ Φ[{vi}] dissipation

− λG[{vi}] constraints

δR

δvi&

∂φi∂t

= −∇ · (φivi)

Transport equations

Doi and Onuki. J Phys (Paris). 19929

Multi-fluid models

Two-fluid modelI Momentum equation

for each species

I Large drag enforcescons. of momentum

de Gennes. J. Chem Phys. (1980)

The Rayleighian

A Lagrangian expression of “leastenergy dissipation” foroverdamped systems (Re = 0).

R[{vi}] = F [{vi}] free energy

+ Φ[{vi}] dissipation

− λG[{vi}] constraints

δR

δvi&

∂φi∂t

= −∇ · (φivi)

Transport equations

Doi and Onuki. J Phys (Paris). 19929

A ternary solution model

F [{vi}] free energy

Φ[{vi}] dissipation

λG[{vi}] constraints

Transport Equations

I Diffusion & Momentum

I Coupled, Non-lin. PDEs

Solve numerically

I Pseudo-spectral on GPUs

I Semi-implicit stabilization

Ternary polymer solution(Flory–Huggins–de Gennes)

F =

∫dr

[f({φi}) +

1

2

∑i

κi |∇φi|2]

Newtonian fluid withφ-dependent viscosity

Φ =1

2

∫dr

[∑i

ζi(vi − v)2

+2η({φi})D : D

]Incompressibility

λG = p∇ · v10

Transport equations

Model B

Model H

∂φi∂t

+ v · ∇φi = ∇ ·

∑j

Mij({φ})∇µj

Convection-Diffusion

µi =δF [{φi}]δφi Chemical Potential

0 = −∇p+∇ ·[η({φ})(∇v +∇vT )

]−N−1∑i=0

φi∇µi Momentum

0 = ∇ · vIncompressibility

11

ModelCharacterization

run

ModelDevelopment

NIPSquench process

12

Characterization of model thermodynamics

0 32 64 96 128

x/R

0.0

0.2

0.4

0.6

0.8

1.0φp

(a)

0 32 64 96 128

x/R

0.0

0.2

0.4

0.6

0.8

1.0

φn

(b)

0 32 64 96 128

x/R

0.0

0.2

0.4

0.6

0.8

1.0

φs

(c)

φp φn

φs(d)

1

N = 50κ = 12χ = 0.912

13

Calculated interface width for many parameters

100

101

102

10−2 10−1 100

l/l ∞

χ∗

21

N = 1N = 2N = 5N = 10

N = 20N = 50N = 80N = 100

l∞ =1

2

(κ

χ

)1/2

χ∗ = (χ− χc)/χc

14

What explains the interface width data?

100

101

102

10−2 10−1 100

l/l ∞

χ∗

21

N = 1N = 2N = 5N = 10

N = 20N = 50N = 80N = 100

We are near the critical point, χcWe recover the mean-field critical exponent,

l = l∞

(χ− χcχc

)−1/215

Characterization of phase separation dynamics

16

There are two dynamic regimes

100

101

102

100 101 102 103 104 105

dom

ain

size

simulation time

17

Early-time regime — initiation of spinodal decomposition

0.5 1.0 1.5 2.0 2.5 3.0k

- 20

- 15

- 10

- 5

5

10λ λ+

λ-

Two key parameters

I qm – fastest growingmode

I λm – rate of spinodaldecomposition

Linear stability analysis

Exponential growth of thefastest growing mode,

δψ = exp[λ+(q)t]

φp φn

φs

0.00

0.08

0.16

0.24

0.32

0.40

0.48

0.56

0.64

0.72

q m

18

Long-time regime — coarsening

dom

ain si

ze

time

slope=1/4

dom

ain si

ze

time

slope

=1/3

dom

ain si

ze

time

slope

=1

surface diffusion bulk diffusion hydrodynamics

19

Comparing simulations to the LSA

100

101

102

100 101 102 103 104 105

dom

ain

size

,2π/〈q〉

simulation time, t

φp φn

φs

1

20

Comparing simulations to the LSA

10−1

100

101

10−3 10−2 10−1 100 101 102 103 104

41

scal

eddo

mai

nsize

,qm/〈q〉

scaled simulation time, λmt20

ModelCharacterization

run

ModelDevelopment

NIPSquench process

21

How does a quench happen by mass transfer?

film

bath

Qualitative features of NIPS(mass-transfer) v. TIPS (temp.)

1. Inherent anisotropy andinhomogeneities

2. Driving force (solvent exchange) andphase separation inseparably linkedby mass transfer

Important questions

1. What is the effect of the initialbath/film concentration?

2. What role does film thickness play?

3. How does mass transfer path affectmicrostructure?

22

Anisotropic quench

The bath interface gives rise to:

I Surface-directed spinodal decomposition

I Surface hydrodynamic instabilitiesBall and Essery. J. Phys.-Condens. Mat. 2, 10303 (1990)

23

Early-time behavior – the infinite film limit

Key concept – time scales

I Phase separation is fasterthan solvent exchange

I At short times we canneglect the role of filmthickness.

Pego. P. Roy. Soc. A-Math. Phy. 422, 261 (1989)

Simple diffusion from a initial step

Three possible cases

1. No phase separation, justdiffusion (steady)

2. Phase separation, singledomain film (steady)

3. Phase separation, multipledomain film (unsteady)

24

Early-time behavior – the infinite film limit

Key concept – time scales

I Phase separation is fasterthan solvent exchange

I At short times we canneglect the role of filmthickness.

Pego. P. Roy. Soc. A-Math. Phy. 422, 261 (1989)

Simple diffusion from a initial step

Three possible cases

1. No phase separation, justdiffusion (steady)

2. Phase separation, singledomain film (steady)

3. Phase separation, multipledomain film (unsteady)

24

Early-time behavior – the infinite film limit

Key concept – time scales

I Phase separation is fasterthan solvent exchange

I At short times we canneglect the role of filmthickness.

Pego. P. Roy. Soc. A-Math. Phy. 422, 261 (1989)

Simple diffusion from a initial step

Three possible cases

1. No phase separation, justdiffusion (steady)

2. Phase separation, singledomain film (steady)

3. Phase separation, multipledomain film (unsteady)

24

Early-time behavior – the infinite film limit

Key concept – time scales

I Phase separation is fasterthan solvent exchange

I At short times we canneglect the role of filmthickness.

Pego. P. Roy. Soc. A-Math. Phy. 422, 261 (1989)

Simple diffusion from a initial step

Three possible cases

1. No phase separation, justdiffusion (steady)

2. Phase separation, singledomain film (steady)

3. Phase separation, multipledomain film (unsteady)

24

Immediate spinodal decomposition into multi-domain films

25

A finite-sized film can exhibit delayed phase-separation

φp φn

φs

polymer film

nonsolventbath

Depending on parameters andinitial conditions, a delayed phaseseparation produces either

I single domain films (shown)

I multiple domain films

26

Finite-film data collapse with a similarity variable

0 5 10 15 20 25

ξ

0.0

0.2

0.4

0.6

0.8

1.0φn

f= 0. 05

f= 0. 025

φp φn

φs

1

27

Conclusions (1D)

1. Inherent anisotropy?

− SDSD-like wave

2. Film thickness?

− Short v. long-time− Scales with xt−1/2

3. Initial conditions?

− No PS, single/multiple domains− Instantaneous v. delayed PS

Saedi et al. Can. J. Chem. Eng. (2014)

Future: microstructure (2D)

I Pore gradients

φp φn

φs

0.00

0.08

0.16

0.24

0.32

0.40

0.48

0.56

0.64

0.72

q m

I Macrovoids

Sternling and Scriven. AICHE J. (1959)

28