Modeling Strong Extensional Flows of Polymer Solutions and Melts

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1 Modeling Strong Extensional Flows of Polymer Solutions and Melts Antony N. Beris CSCAMM Workshop Multiscale Modeling and Simulation of Complex Fluids April 19, 27 Acknowledgements: Dr. Joydeep Mukherjee, Department of Chemical Engineering, University of Michigan

Art Metzner, 1927-26 Commemorative Symposium:

2 Art Metzner, Commemorative Symposium: May 14, 27, Dept. of Chem. Engrg., Univ. of Delaware

3 Fiber Spinning of Semicrystalline Polymer Melts spinneret Disordered melt Die swell Necking Spinline Oriented semicrystalline morphology Take-up rolls

B1 41 (1967) 3 D.C. Prevorsek, P.J. Harget, R.K. Sharma and A.C. Reimschuessel, J. Macromol. Sci. Phys.

4 Semicrystalline Morphologies in Polymers Spherulite [1] Shish-kebab [2] Fibrillar[3] 1 A. Keller, G.R. Lester and L.B. Morgan, Phil. Trans. Roy. Soc. (London) A247 1 (1954) 2 A. Keller and M.J. Machin, J. Macromol. Sci. Phys. B1 41 (1967) 3 D.C. Prevorsek, P.J. Harget, R.K. Sharma and A.C. Reimschuessel, J. Macromol. Sci. Phys. B8 127 (1973) Most of the continuum models have a high degree of empiricism and lack capability to predict semicrystalline morphology.

Micro-Macro Hierarchical Approach [1] Processing conditions Temperature Take-up roll speed Macroscopic Model: Amorphous melt rheology and chain deformation Nonequilibrium thermodynamic potentials

5 Micro-Macro Hierarchical Approach [1] Processing conditions Temperature Take-up roll speed Macroscopic Model: Amorphous melt rheology and chain deformation Nonequilibrium thermodynamic potentials Microscopic Simulations Chain conformations and microscopic thermodynamic information under the influence of flow deformation for different semicrystalline morphologies using a microscopic model Predictions: Morphology 1 Mukherjee, Wilson, Beris, Flow-induced non-equilibrium thermodynamics of lamellar semicrystalline polymers, J. Non-Newt. Fluid Mech., 12 (1-3): (24).

6 Outline Objective: Examine/improve the internal consistency in micro-macro modeling following a hierarchical approach applied to a dense amorphous linear polymer system Microscopic modeling Lattice subdivision-based Monte Carlo method Macroscopic modeling Conformation tensor based models using Non- Equilibrium Thermodynamics Microscopic modeling under non-equilibrium conditions Conclusions

7 Microscopic Modeling: Lattice Models Originally developed by Flory and Yoon [1] Has been found to be useful in the study of dense semicrystalline polymers Various lattice based approaches: Random walk models Exact enumeration methods Monte Carlo methods Mean-field models 2 Layer 1. [1] Flory and Yoon, Polymer 18 (1977), pp

8 Lattice-based Metropolis Monte Carlo Scheme A modular lattice subdivision approach. At any Monte Carlo step perform any of Monte Carlo moves: Internal sublattice optimization Isomerization Accept the new state with a probability p exp( Eη N tight folds ) E η energetic penalty on tight folds nondimensionalized with respect to the Boltzmann factor k B T Tight folds Original state After a Monte Carlo move

9 Initial Macroscopic Modeling of the Dense Amorphous Polymer Phase: Phan-Thien Tanner Model [2] Based on a network model Can also be described from bracket formulation of non-equilibrium thermodynamics [3] The internal structure is described by the conformation tensor, c,the second moment of the end-to-end n-segment chain distance n-1 n c(n) = <R(n)R(n)> 1 2 R(n) [2] Phan-Thien N, Tanner R.I., J.Non-Newtonian Fluid Mech.,2(1977) [3]A.N. Beris, B.J. Edwards, Thermodynamics of Flowing Systems, Oxford University Press, New York, 1994

10 λ Phan-Thien Tanner (PTT)Model Constitutive equation 1 c = c 1 λ( c ) ( c ) σ = G 1 ( ) Internal structure dependent relaxation time 1 1 exp (tr( ) 3) ( c ) ( c ) = λ α Non-equilibrium thermodynamic information ( c ) ( c ) A = tr log det 2 M ( ) δa ( ( ) 1 ) M = I c 2 A - free energy of the system (non-dimensionalized by k B T) M- total number of entanglements in the system δc η - extensional viscosity λ - relaxation time G - elastic modulus λ - zero shear rate relaxation time α ~.5 for PE

11 Hamiltonian Functional Formalism Beris and Edwards, Thermodynamics of Flowing Systems, Oxford UP, 1994 For any arbitrary functional F, its time evolution can be described as the sum of two contributions: a reversible one, represented by a Poisson bracket: {F,H} an irreversible one, represented by a dissipative bracket: [F,H] The final dynamic equations are recovered through a direct comparison with the expression derived by differentiation by parts: df { FH, } [ FH, ] F d dv dt = + = δ x δ x dt

12 Advantages of Hamiltonian Formalism (1) It only requires knowledge of the following: A set of macroscopic variables, taken uniformly as volume densities. The include, in addition to the equilibrium thermodynamic ones (the component mass density, ρ i, for every component i, the entropy density si), the momentum density, ρv, and any additional structural parameter, again expressed as a density The total energy of the system or any suitable Lagrange transform of it, typically the total Helmholtz free energy, expressed as a functional of all other densities with the temperature substituting for the entropy density The Poisson bracket, {F,H} The dissipation bracket, [F,H]

13 A set of macroscopic variables can easily be assumed depending on the physics that we want to incorporate to the problem The total Helmholtz free energy can also easily be constructed as the sum of kinetic energy plus an extended thermodynamic free energy that typically includes an easily derived expression (in terms of the structural parameters) in addition to a standard equilibrium expression The Poisson bracket, {F,H} is rarely needed by itself: only when an equation is put together for the first time characteristic of the variables involved in this system; otherwise, its effect is probably already known from previous work: it corresponds to a standard reversible dynamics. For viscoelastic flows, this corresponds to the terms defining an upper convected derivative The dissipation bracket, [F,H] is the only one to contain major new information and is typically where our maximum ignorance lies. Barren any other information (say, by comparison against a microscopic theory) the main information that we can use is a linear irreversible thermodynamics expression: according to that, the dissipation bracket becomes a bilinear functional in terms of all the nonequilibrium Hamiltonian gradients with an additional nonlinear (in H) correction with respect to δf/δs (entropy correction) that can be easily calculated so that the conservation of the total energy is satisfied: [H,H] =. Advantages of Hamiltonian Formalism (2)

14 Final Equations for Single Conformation Viscoelasticity For an isothermal system, we get the standard dynamics for a viscoelastic medium (together with the divergencefree velocity constraint): D ρ Dt v = p + η v+ T D ξ Dt 2 T c v c c v= ( γ c+ c γ) Λ: c s T a T T a = 2(1 ξ ) c c

15 Key Question: Consistency with the Underlying Microscopic Physics Approach: Solve the Continuum Macroscopic Model for a well defined flow to obtain: Structural information: Conformation Tensor, c Thermodynamic information: (Non-equilibrium) Free Energy, A and Free Energy potentials, δa/δc For the given flow, and the given potentials, solve directly for the corresponding microscopic structure Use a specially developed Non-Equilibrium Monte-Carlo approach applied to a 3d Lattice model with full periodicity Attest on the consistency through a direct comparison of the microscopic/macroscopic results Interpret results and suggest improvements to the macroscopic model repeating the procedure if necessary

16 Notice that at equilibrium (We ~ ), these potentials go to zero! Uniaxial Extensional flow for PTT Solve the equations for uniaxial extensional flow at steady state Diagonal Conformation tensor cxx c = cxx = cyy c zz 1 1 cxx = and c zz =, 1+ We 1 2We where, We <.5 ε 2 z x y We is the Weissenberg number λ( c ) = ε 1 ε Non-equilibrium thermodynamic potentials δ A δ A M δ A δc δc 2 δc = = We ; = MWe xx yy zz Macro-micro bridges

17 Non-equilibrium Thermodynamic Bias Need to formulate the proper non-equilibrium thermodynamic guidance to drive the microscopic simulations Ansatz: Accept the new microstate with a probability δa p exp : δc < RR > Non-equilibrium forcing term R(n)

18 Evaluation of the Non-equilibrium Bias 1. Evaluate the non-equilibrium thermodynamic potential from Macroscopic model (PTT) : δ A δ A M δ A δc δc 2 δc = = We ; = MWe xx yy zz R(n) x y 2. Use it to evaluate the Non-equilibrium forcing term δ A < RR > : < RR > where < RR > =, δc ( n / ) N e : entanglement length (typically 1 lattice chain segments for polyethylene). For N e segment chain, n = N e, we have z δ < > e A 1 2 : RR = We 2 Rz c N e all possible multisegment selections of length N δ R + R 2 2 x y 2

19 Consistency at low We between Microscopic and PTT Models

20 Serious Qualitative Inconsistency at higher We

21 Inconsistencies in C zz at high We between PTT and MC C zz We Maximum limit PTT MC-data

22 Modified FENE-P/PTT Model Constitutive equation 2 1 ( ) L 3 = f ; f = 2 c c 1 λ( c ) L trc ( c ) σ = G f 1 L - extensibility parameter λ - relaxation time G - elastic modulus Internal structure dependent relaxation time λ 1 1 exp (tr( ) 3) ( c ) ( c ) = λ α λ - zero shear rate relaxation time α ~.5 for PE Non-equilibrium thermodynamic information M 2 δa M ( ) ( ( )) = c c f = f 2 δc 2 c I c A (L 3)log(f ) log det Ma A - free energy of the system (non-dimensionalized by k B T) M- total number of entanglements in the system ( ( ) ( ) 1 )

23 Uniaxial Extensional flow for FENE-P/PTT This model is also compatible to the same microscopic simulations! Solve the equations for uniaxial extensional flow at steady state Diagonal Conformation tensor cxx c = cxx = cyy c zz 1 1 cxx = and c zz =, f + We f 2We where, We is unrestricted! ε 2 z x y We is the Weissenberg number λ( c ) = ε 1 ε Same Non-equilibrium thermodynamic potentials as before! δ A δ A M δ A δc δc 2 δc = = We ; = MWe xx yy zz Macro-micro bridges

24 Comparison between FENE-P/PTT, PTT and MC C zz We Maximum limit PTT MC-data FENE-P/PTT

25 Inconsistencies in the free energy 14 FENE-P/PTT PTT 12 1 Free Energy The total free energy difference between fully extended and fully random state cannot be more than logw L We

26 Bounded Free Energy FENE-PB/PTT Model Constitutive equation c c 1 λ( c ) L trc 2 B ( ) L 3 = f ; f = 2 2 ( ) L σ = GBf c 1; B= 2 af + L Internal structure dependent relaxation time λ 1 1 exp (tr( ) 3) ( c ) ( c ) = λ α 2 L - extensibility parameter λ - relaxation time G - elastic modulus λ - zero shear rate relaxation time α ~.5 for PE Non-equilibrium thermodynamic information 2 δa M (L a f ) A = M = Bf 2 δc 2 c I c (L + a f ) A - free energy of the system (non-dimensionalized by k B T) M- total number of entanglements in the system ( ( ) ( ) 1 )

27 Uniaxial Extensional flow for FENE-PB/PTT The microscopic simulations are also compatible to this model! Solve the equations for uniaxial extensional flow at steady state Diagonal Conformation tensor cxx c = cxx = cyy c zz B B cxx = and c zz =, Bf + We Bf 2We where, We is (asbefore) unrestricted! ε 2 z x y We is the Weissenberg number λ( c ) = ε 1 ε Same Non-equilibrium thermodynamic potentials as before! δ A δ A M δ A δc δc 2 δc = = We ; = MWe xx yy zz Macro-micro bridges

28 FENE-PB/PTT has always a finite free energy 14 FENE-P/PTT PTT FENE-PB/PTT 12 1 Maximum limit for FENE-PB/PTT Free Energy We

29 There is good agreement between FENE-PB/PTT and MC

30 There is also consistency in C zz between FENE-PB/PTT and MC C zz We Maximum limit PTT MC-data FENE-P/PTT FENE-PB/PTT

31 Extensional viscosity properties of the modified models are also realistic FENE-P/PTT FENE-PB/PTT 2 15 Trouton ratio We

32 Conclusions Efficient 3-D lattice-subdivision based Monte Carlo simulations have enabled us to check for microscopic consistency various continuum viscoelastic models for dense linear polymer models. Previous models have shown to be defective at high levels of molecular extension. A new phenomenological model has been developed (FENE-PB/PTT) that is consistent to finite levels on both the molecular extension and the total free energy. Preliminary results show that the new model captures well the sudden changes accompanying the coil-to-stretch transition at high extensions.

33 Acknowledgements Dr. Joydeep Mukherjee Dr. Stevan Wilson Randy Oswald Brian Snyder Funding: NSF GOALI (Materials processing and manufacturing division). Dupont-Nylon

34 Evaluating Microscopic Thermodynamic Information A (E η,we ) can be evaluated incrementally close to static equilibrium as: A(E,We) A(E,We) A(E,We ) dwe We η η = η = + We Equilibrium A A Now we calculate from the relationship of A with A expanded = A : c and the We c fact that in the MC we calculate A A : c We expanded = δa log Z = log exp : < R R > i microstate δc A A ==> = A expanded : We We + < RR > = c A A A : < > + : < > + : < > = We RR We RR RR c c c We < RR > i

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