Dynamics of Inhomogeneous Polymeric Fluids Douglas R. Tree - PowerPoint PPT Presentation

Dynamics of Inhomogeneous Polymeric Fluids Douglas R. Tree Materials Research Laboratory University of California, Santa Barbara CFDC Meeting February 3, 2016

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) 2

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 3

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 4

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 4

PFPD Software Phase-Field Polymer Dynamics Efficient, parallelized, object-oriented C++ program for simulating the flow and phase behavior of inhomogeous polymeric fluids. Scripts and plotting tools Operators Models Time Int. - Pseudospectral - Ternary FHG - Model B - Block polymers - Model H - Hybrid (FD) BCs Field vector/matrix operations Field library (KTD) 5

PFPD Software Phase-Field Polymer Dynamics Efficient, parallelized, object-oriented C++ program for simulating the flow and phase behavior of inhomogeous polymeric fluids. Scripts and plotting tools Operators Models Time Int. - Pseudospectral - Ternary FHG - Model B - Block polymers - Model H - Hybrid (FD) BCs Field vector/matrix operations Field library (KTD) 5

Integration of 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 6

Stable and efficient time integration Semi-implicit stabilization ◮ Unconditionally stable for practical use ◮ Inexpensive relative to fully implicit methods φ n +1 − φ n = ∇ · [ M ( φ ) ∇ µ n i ] + m ∇ 2 µ n +1 − m ∇ 2 µ n lin lin ∆ t Variable time-stepping ◮ Step-doubling (50% greater cost per step) ◮ Enables much larger step sizes for slow dynamics 7

State-of-the-art method for hydrodynamics Variable- η Stokes equation ◮ Fixed-point method ◮ Enhanced efficiency with 0 = −∇ p − ∇ · Π ∇ v + ∇ v T �� − Anderson mixing � � + ∇ · η ( φ ) − 1 st order continuation 0 = ∇ · v ◮ Solution for both PS and hybrid discretizations Doi and Edwards. (1986) ∇ 2 p = ∇∇ : (Θ n − Π) 1000000 t=500 (new code) v n +1 = 1 t=500 (old code) η ∗ ∇ · (Θ n − Π − I p ) ∇ 2 ˆ ) simulation time (sec model B(new code) 100000 where, 10000 Θ n = [ η ( φ ) − η ∗ ] ∇ v n + ( ∇ v n ) T � � 1000 (Figure courtesy of Tatsu Iwama) 1 100 10000 viscosity ratio (eta_P/eta_r) 8

PFPD Software Phase-Field Polymer Dynamics Efficient, parallelized, object-oriented C++ program for simulating the flow and phase behavior of inhomogeous polymeric fluids. Scripts and plotting tools Operators Models Time Int. - Pseudospectral - Ternary FHG - Model B - Hybrid - Block polymers - Model H (FD) BCs Field vector/matrix operations Field library (KTD) 9

Non-periodic Boundary Conditions Pseudo-spectral derivatives Finite differences ∂f ∂x ≈ f i +1 − f i − 1 ∂f ∂x ≈ FFT − 1 [ − ik x ˆ f ] 2∆ x ◮ periodic or homogeneous ◮ flexible BCs BCs only ◮ accuracy depends on ◮ very good accuracy order of FD x A hybrid method ◮ Periodic BCs in y (PS) y ◮ Arbitrary BCs in x (FD) 10

Spinodal decomposition example (diffusion only) Hybrid simulation ◮ Left and right BCs ∂ 3 φ p ∂φ p ∂x = 0 , ∂x 3 = 0 ∂ 3 φ n ∂φ n ∂x = 0 , ∂x 3 = 0 ◮ Top and bottom are periodic ◮ (Top) Polymer concentration ◮ (Bottom) Slice through y = 32 . Notice that the slope at x = 0 and x = 64 is zero. (Parameters: N = 5 , χ = 1 . 361 , κ = 4 ) 11

PFPD Software Phase-Field Polymer Dynamics Efficient, parallelized, object-oriented C++ program for simulating the flow and phase behavior of inhomogeous polymeric fluids. Scripts and plotting tools Operators Models Time Int. - Pseudospectral - Ternary FHG - Model B - Hybrid - Block polymers - Model H (FD) BCs Field vector/matrix operations Field library (KTD) 12

How can we model the free energy? Field theory Analytical Numerical simulations approximations approximations (SCFT/CL) to a field theory to a field theory 10 1 Γ - 1 0.1 0.01 0.01 0.1 1 10 kR g / 2 1 / 2 13

How can we model the free energy? Field theory Analytical Numerical simulations approximations approximations (SCFT/CL) to a field theory to a field theory 10 1 Γ - 1 0.1 0.01 0.01 0.1 1 10 kR g / 2 1 / 2 13

Deriving free energy functionals Exact DFT � � � F [ φ ] = − k B T ln Dw Dφ exp( − βH [ φ, w ]) − J ( r ) φ ( r ) + Mean-Field Approximation & Weak-Inhomogeneity* ↓ Random Phase Approximation (RPA) F [ φ ] = F 0 [ φ ] + 1 � � d r ′ Γ( r − r ′ ) δφ ( r ) δφ ( r ′ ) + O ( δφ 3 ) d r 2 * Other approximations are possible, e.g. slow gradient expansion G.H. Fredrickson. The Equilibrium Theory of Inhomogeneous Polymers. Oxford (2006). 14

Square-gradient (Cahn–Hilliard) models For a simple mixture the RPA (or gradient expansion) simplifies to: � � � f 0 ( φ ) + 1 2 κ ( φ ) |∇ φ | 2 F [ φ ] = d r Flory–Huggins–de Gennes Ginzburg–Landau f 0 ( φ ) = a ( φ ) 2 + bφ 4 2 φ i � f 0 ( φ ) = ln φ i + χ 12 φ 1 φ 2 N i κ ( φ ) = κ i =1 � 1 � κ ( φ ) = b 2 − χ 18 φ 1 φ 2 P.G. de Gennes. J. Chem. Phys. (1980). Cahn and Hilliard. J. Chem. Phys. (1958). 15

Stability is a challenge at strong segregation solvent An unstable code is bad H ◮ The parameter space is very limited G L-L ◮ The quench depth can L-G vary with time polymer non-solvent Why is it hard? Accuracy Small w and small ∆ φ means we need a fine grid (small ∆ x ) and accurate time integration (small ∆ t ). 16

Interaction between small w and small ∆ φ binary polymer solution N = 30 , χ = 0 . 979 The key challenge Resolve the curvature of the asymmetric interfacial profile within the order of accuracy of ∆ φ . 17

Regularizing the free energy Modified Flory–Huggins f ( φ ) = φ N ln φ + (1 − φ ) ln(1 − φ ) + χφ (1 − φ ) + A exp( − φ/δ ) 0.12 0.5 0.10 0.4 0.08 0.3 0.06 0.2 0.04 0.1 0.02 0.2 0.4 0.6 0.8 1.0 - 0.01 0.01 0.02 0.03 0.04 0.05 N = 40 , χ = 3 . 5 A = 10 − 2 , δ = 5 × 10 − 3 18

Effect of regularization on the phase diagram No regularization Regularized ( A = δ = 5 e − 3 ) 19

Effect of regularization on the dynamics No regularization Regularized ( A = δ = 5 e − 3 ) 20

What about inhomogeneous polymer models? Limitations of the RPA 10 ◮ Γ( k ) is a complex 1 function Γ - 1 ◮ Limited to O ( δφ 2 ) , i.e. 0.1 “weak segregation” Leibler. J. Chem. Phys. (1980) 0.01 0.01 0.1 1 10 kR g / 2 1 / 2 Ohta-Kawasaki procedure 10 ◮ Get non-local terms from the small k and large k 1 limits of the RPA Γ - 1 ◮ Local approximation 0.1 beyond O ( δφ 2 ) 0.01 0.01 0.1 1 10 Ohta & Kawasaki. Macromol. (1986) kR g / 2 1 / 2 21

Ohta-Kawasaki proof of principle � � κ i � 2 |∇ φ i | 2 F [ φ ] = d r f ( { φ i } ) + i � +1 � � d r ′ G ( r , r ′ ) δφ i ( r ) δφ i ( r ′ ) � ξ i d r 2 i “OK Model” – Lam “OK Model” – Hex 22

How can we model the free energy? Field theory Analytical Numerical simulations approximations approximations (SCFT/CL) to a field theory to a field theory 10 1 Γ - 1 0.1 0.01 0.01 0.1 1 10 kR g / 2 1 / 2 23