A Multi-Fluid Model of Membrane Formation by Phase-Inversion Douglas R. Tree 1 and Glenn Fredrickson 1 , 2 1 Materials Research Laboratory 2 Departments of Chemical Engineering and Materials University of California, Santa Barbara Society of Rheology Annual Meeting February 13, 2017
Acknowledgements ◮ Jan Garcia ◮ Dr. Kris T. Delaney ◮ Prof. Hector D. Ceniceros ◮ Lucas Francisco dos Santos ◮ Dr. Tatsuhiro Iwama (Asahi Kasei) ◮ Dr. Jeffrey Weinhold (Dow) 2
Can we predict the microstructure of polymers? ◮ Microstructure dictates properties A very general ◮ Microstructure depends on process problem! history Polymer Blends Polymer membranes ◮ commodity ◮ clean water plastics (e.g. ◮ medical filters HIPS) ◮ block polymer Saedi et al. Can. J. Chem. Eng. (2014) thin films www.leica-microsystems.com Polymer composites Biological patterning ◮ bulk hetero- ◮ Eurasian jay junctions feathers ◮ nano- composites Parnell et al. Sci. Rep. (2015) Hoppe and Sariciftci J. Mater. Chem. (2006) 3
Can we predict the microstructure of polymers? ◮ Microstructure dictates properties A very general ◮ Microstructure depends on process problem! history Polymer Blends Polymer membranes ◮ commodity ◮ clean water plastics (e.g. ◮ medical filters HIPS) ◮ block polymer Saedi et al. Can. J. Chem. Eng. (2014) thin films www.leica-microsystems.com Polymer composites Biological patterning ◮ bulk hetero- ◮ Eurasian jay junctions feathers ◮ nano- composites Parnell et al. Sci. Rep. (2015) Hoppe and Sariciftci J. Mater. Chem. (2006) 3
Clean water is a present and growing concern Why membranes? U.S. Drought Monitor July 7, 2015 ◮ Water is projected to become increasingly Author: Brian Fuchs National Drought Mitigation Center scarce. Intensity: D0 Abnormally Dry D3 Extreme Drought ◮ Filtration is a key D1 Moderate Drought D4 Exceptional Drought D2 Severe Drought technology for water http://droughtmonitor.unl.edu/ purification. http://www.kochmembrane.com/Learning- Center/Configurations/What-are-Hollow-Fiber-Membranes.aspx 4
Polymer membrane synthesis by immersion precipitation polymer solution nonsolvent membrane solvent substrate non-solvent bath Figure inspired by: www.synder fi ltration.com/learning-center/articles/introduction-to-membranes 5
Polymer membrane synthesis by immersion precipitation polymer solution nonsolvent membrane solvent substrate non-solvent bath solvent H Figure inspired by: www.synder fi ltration.com/learning-center/articles/introduction-to-membranes G L-L L-G polymer non-solvent 5
Microstructural variety in membranes Uniform “Sponge” Asymmetric“Sponge” Skin Layer Fingers or Macro-voids Strathmann et al. Desalination. (1975) 6
Model Development Model Characterization run NIPS quench process
Model Development Model Characterization run NIPS quench process
How can we model microstructure formation? A difficult challenge ◮ Complex thermodynamics out of equilibrium ◮ Spatially inhomogeneous (multi-phase) ◮ Multiple modes of transport (diffusion & convection) ◮ Large separation of length/time scales Continuum fluid dynamics 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 Teran et al. Phys. Fluid. (2008) (CDFT)/“phase field” models 8
Multi-fluid models Two-fluid model The Rayleighian ◮ Momentum equation A Lagrangian expression of “least for each species energy dissipation” for overdamped systems ( Re = 0 ). ◮ Large drag enforces cons. of momentum ˙ free energy R [ { v i } ] = F [ { v i } ] dissipation + Φ[ { v i } ] − λG [ { v i } ] constraints δR ∂φ i & ∂t = −∇ · ( φ i v i ) δ v i Transport equations de Gennes. J. Chem Phys. (1980) Doi and Onuki. J Phys (Paris). 1992 9
Multi-fluid models Two-fluid model The Rayleighian ◮ Momentum equation A Lagrangian expression of “least for each species energy dissipation” for overdamped systems ( Re = 0 ). ◮ Large drag enforces cons. of momentum ˙ free energy R [ { v i } ] = F [ { v i } ] dissipation + Φ[ { v i } ] − λG [ { v i } ] constraints δR ∂φ i & ∂t = −∇ · ( φ i v i ) δ v i Transport equations de Gennes. J. Chem Phys. (1980) Doi and Onuki. J Phys (Paris). 1992 9
A ternary solution model Ternary polymer solution ˙ free energy (Flory–Huggins–de Gennes) F [ { v i } ] � � dissipation � f ( { φ i } ) + 1 Φ[ { v i } ] � κ i |∇ φ i | 2 F = d r 2 i λG [ { v i } ] constraints Newtonian fluid with φ -dependent viscosity Transport Equations �� Φ = 1 � ◮ Diffusion & Momentum ζ i ( v i − v ) 2 d r 2 ◮ Coupled, Non-lin. PDEs i � +2 η ( { φ i } ) D : D Solve numerically Incompressibility ◮ Pseudo-spectral on GPUs ◮ Semi-implicit stabilization λG = p ∇ · v 10
Transport equations Model H Model B ∂φ i � Convection-Diffusion ∂t + v · ∇ φ i = ∇ · M ij ( { φ } ) ∇ µ j j µ i = δF [ { φ i } ] Chemical Potential δφ i N − 1 η ( { φ } )( ∇ v + ∇ v T ) � � � 0 = −∇ p + ∇ · − φ i ∇ µ i Momentum i =0 0 = ∇ · v Incompressibility 11
Model Development Model Characterization run NIPS quench process 12
Characterization of model thermodynamics (a) (b) 1.0 1.0 N = 50 0.8 0.8 κ = 12 χ = 0 . 912 0.6 0.6 φ n φ p 0.4 0.4 0.2 0.2 0.0 0.0 0 32 64 96 128 0 32 64 96 128 x/R x/R φ s (c) (d) 1.0 0.8 0.6 φ s 0.4 0.2 0.0 0 32 64 96 128 φ p φ n x/R 13
Calculated interface width for many parameters 10 2 � 1 / 2 � κ l ∞ = 1 2 χ χ ∗ = ( χ − χ c ) /χ c 1 l/l ∞ 10 1 2 N = 1 N = 20 N = 2 N = 50 N = 5 N = 80 N = 10 N = 100 10 0 10 − 2 10 − 1 10 0 χ ∗ 14
What explains the interface width data? 10 2 1 l/l ∞ 10 1 2 N = 1 N = 20 N = 2 N = 50 N = 5 N = 80 N = 10 N = 100 10 0 10 − 2 10 − 1 10 0 χ ∗ We are near the critical point, χ c We recover the mean-field critical exponent, � − 1 / 2 � χ − χ c l = l ∞ χ c 15
Characterization of phase separation dynamics 16
There are two dynamic regimes 10 2 domain size 10 1 10 0 10 0 10 1 10 2 10 3 10 4 10 5 simulation time 17
Early-time regime — initiation of spinodal decomposition λ λ + Linear stability analysis 10 λ - 5 Exponential growth of the 3.0 k fastest growing mode, 0.5 1.0 1.5 2.0 2.5 - 5 - 10 δψ = exp[ λ + ( q ) t ] - 15 - 20 0.72 φ s 0.64 Two key parameters 0.56 ◮ q m – fastest growing 0.48 mode 0.40 q m ◮ λ m – rate of spinodal 0.32 decomposition 0.24 0.16 0.08 φ p φ n 0.00 18
Long-time regime — coarsening hydrodynamics surface diffusion bulk diffusion slope=1/4 3 domain size domain size domain size / 1 = slope=1 e p o l s time time time 19
Comparing simulations to the LSA 10 2 domain size, 2 π/ � q � φ s 10 1 φ p φ n 10 0 10 0 10 1 10 2 10 3 10 4 10 5 simulation time, t 20
Comparing simulations to the LSA 10 1 scaled domain size, q m / � q � 1 4 10 0 10 − 1 10 − 3 10 − 2 10 − 1 10 0 10 1 10 2 10 3 10 4 scaled simulation time, λ m t 20
Model Development Model Characterization run NIPS quench process 21
How does a quench happen by mass transfer? Qualitative features of NIPS (mass-transfer) v. TIPS (temp.) 1. Inherent anisotropy and inhomogeneities 2. Driving force (solvent exchange) and phase separation inseparably linked by mass transfer Important questions 1. What is the effect of the initial fi lm bath/film concentration? 2. What role does film thickness play? 3. How does mass transfer path affect bath microstructure? 22
Anisotropic quench The bath interface gives rise to: ◮ Surface-directed spinodal decomposition ◮ Surface hydrodynamic instabilities Ball and Essery. J. Phys.-Condens. Mat. 2, 10303 (1990) 23
Early-time behavior – the infinite film limit Key concept – time scales ◮ Phase separation is faster than solvent exchange ◮ At short times we can neglect the role of film thickness. Pego. P. Roy. Soc. A-Math. Phy. 422, 261 (1989) Simple diffusion from a initial step 24
Early-time behavior – the infinite film limit Key concept – time scales ◮ Phase separation is faster than solvent exchange ◮ At short times we can neglect the role of film thickness. Pego. P. Roy. Soc. A-Math. Phy. 422, 261 (1989) Three possible cases 1. No phase separation, just diffusion (steady) Simple diffusion from a initial step 24
Early-time behavior – the infinite film limit Key concept – time scales ◮ Phase separation is faster than solvent exchange ◮ At short times we can neglect the role of film thickness. Pego. P. Roy. Soc. A-Math. Phy. 422, 261 (1989) Three possible cases 1. No phase separation, just diffusion (steady) 2. Phase separation, single domain film (steady) Simple diffusion from a initial step 24
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