Thermodynamics of Nonisothermal Polymer Flows: Experiment, Theory, and Simulation

1. Thermodynamics of Nonisothermal Polymer Flows:Experiment, Theory, and Simulation Brian J. Edwards Department of Chemical and Biomolecular Engineering University of Tennessee-Knoxville University of Kentucky Lexington, Kentucky February 18, 2009

2. Collaborators and Funding • Tudor Ionescu: Graduate student, UTK • Vlasis Mavrantzas, Professor, University of Patras • Grant #41000-AC7, The Petroleum Research Fund, American Chemical Society

3. Outline • Part I: Introduction and Background • Introduction to Viscoelastic Fluids • Definition of the concept of Purely Entropic Elasticity • Objective • Part II: Experiment and Theory • Experimental Approach • Theoretical Approach • Part III: Molecular Simulations • Equilibrium Simulations • Nonequilibrium Simulations • Conclusions

4. Part I: Introduction and Background • The phenomenon described in this presentation is one manifestation of viscoelastic fluid mechanics • Viscoelastic fluids display complex non-Newtonian flow properties under the application of an external force: • Pressure gradient • Shear stress • Extensional strain (stretching)

5. Paint (&) Crude oil Asphalt Cosmetics Biological fluids Blood Protein solutions Pulp and coal slurries Toothpaste Grease Foodstuffs Ketchup Dough Salad dressing Plastics Polymer melts Rubbers Polymer solutions Examples of Viscoelastic Fluids

6. Newtonian Fluid Dynamics • The dynamics of an incompressible Newtonian fluid can be described completely with three equations: • The Cauchy momentum equation: • The divergence-free condition: • The Newtonian constitutive equation:

7. Newtonian Flow Equations Are Remarkably Robust: • Simple, low-molecular-weight, structureless fluids are well described in three dimensions: • Laminar shear and extensional flows • Turbulent pipe and channel flows • Free-surface flows • The simple, structureless fluid:

8. Viscoelastic Fluid Dynamics • A viscoelastic fluid has a complex internal microstructure • Today’s topic: Polymer melts • A high-molecular-weight polymer is dissolved in a simple Newtonian fluid • At equilibrium, the polymer molecules assume their statistically most probable conformations, random coils: Polymer solution

9. Viscoelastic Flow Behavior • These conformational rearrangements produce very bizarre “non-Newtonian” flow phenomena! • Viscoelastic fluids have very long relaxation times: Viscoelastic fluid Newtonian fluid t Flow off

10. Viscoelastic Flow Behavior • Viscoelastic fluids typically display shear-rate dependent viscosities: Shear-thinning fluid Newtonian fluid

11. Viscoelastic Flow Behavior • Viscoelastic fluids develop very large normal stresses: • Example: Paint Viscoelastic fluid Newtonian fluid

12. Nonisothermal Flows of VEs • Nonisothermal flow problems defined by a set of four PDE’s: • 1) Equation of motion: • 2) Equation of continuity: • Incompressible fluid: • 3) Internal energy equation: • 4) An appropriate constitutive equation: • Upper-Convected Maxwell Model (UCMM)

13. The concept of Purely Entropic Elasticity • For simplicity, the internal energy of a viscoelastic liquid is considered as a unique function of temperature (i.e. not a function of deformation) [1,2]: • This let us define the constant volume heat capacity as: • For an incompressible fluid with PEE, the heat equation becomes: • PEE is always assumed in flow calculations!!! 1. Sarti, G.C. and N. Esposito, Journal of Non-Newtonian Fluid Mechanics, 1977. 3(1): p. 65-76. 2. Astarita, G. and G.C. Sarti, Journal of Non-Newtonian Fluid Mechanics, 1976. 1(1): p. 39-50.

14. Implications of PEE • What happens to the energy equation if one does not assume PEE? • First, the internal energy is taken as a function of temperature and an appropriate internal structural variable (conformation tensor): • Next, the heat capacity is defined as: • Then, the substantial time derivative of the internal energy becomes: • The complete form of the heat equation becomes:

15. Objective • Test the validity of PEE under a wide range of processing conditions using experimental measurements, theory and molecular simulation • Experimental approach • Solve the temperature equation numerically using a finite element modeling method (FEM) • Measure the temperature increase due to viscous heating, and compare the results to the FEM predictions • Theoretical approach • Identify all possible causes for the deviations from the FEM predictions observed in the experimental measurements • Use a theoretical model to propose a more accurate form of the temperature equation and test it through the FEM analysis • Molecular simulation approach • Use a molecular simulation technique to evaluate the energy balances under non-equilibrium conditions for compounds chemically similar to the ones used in the experiments

16. Part II: Experiment and Theory • Experimental Approach • Identify a flow situation in which high degrees of orientation are developed • Uniaxial elongational flow generated using the semi-hyperbolically converging dies (Hencky dies) • The analysis is not possible in capillary shear flow • Find numerical solutions to the temperature equation at steady state using the PEE assumption for this particular flow situation • The solution to this equation will yield the spatial temperature distribution profiles inside the die channel • Compute the average temperature value for the exit axial cross-section of the die • Under the same conditions used in the FEM calculations, measure the temperature increase due to viscous heating

17. Experimental Approach • The semi-hyperbolically converging die (Hencky die) • Proven to generate a uniaxial elongational flow field under special conditions Hencky 6 Die:

18. Experimental Approach • Materials used in this study

19. Experimental Approach • Calculation of the steady-state spatial temperature distribution profiles • Used a FEM method to find numerical solutions to the temperature equation • First, elongational viscosity measurements are needed in order to evaluate the viscous heating term: • The elongational viscosity is identifiable with the “effective elongational viscosity” [1] which can be measured using the Hencky dies and the Advanced Capillary Extrusion Rheometer (ACER) 1. Feigl, K., F. Tanner, B.J. Edwards, and J.R. Collier, Journal of Non-Newtonian Fluid Mechanics, 2003. 115(2-3): p. 191-215.

20. Experimental Approach • Advanced Capillary Extrusion Rheometer (ACER 2000)

21. Experimental Approach • Effective elongational viscosity results • HDPE

22. Experimental Approach • Effective elongational viscosity results • LDPE

23. Experimental Approach • FEM calculations • The heat capacity is considered a function of temperature • the tabulated values for generic polyethylene are used from [1] • The thermal conductivity is considered isotropic, and taken as a constant with respect to temperature and position [1] • The input velocity field corresponds to a uniaxial elongational flow field in cylindrical coordinates [2] • The effective elongational viscosity is taken as a function of temperature [3], according to our own experimental measurements 1. Polymer Handbook. 1999, New York: Wiley Interscience. 2. Feigl, K., F. Tanner, B.J. Edwards, and J.R. Collier, Journal of Non-Newtonian Fluid Mechanics, 2003. 115(2-3): p. 191-215. 3. Dressler, M., B.J. Edwards, and H.C. Ottinger, Rheologica Acta, 1999. 38(2): p. 117-136.

24. Experimental Approach • Sample FEM calculation results • HDPE, Tin = Twall = 190oC

25. Experimental Approach • Sample FEM calculation results • Axial temperature profiles • HDPE, Tin = Twall = 190oC

26. Experimental Approach • Sample FEM calculation results • Radial temperature profiles • HDPE, Tin = Twall = 190oC

27. Experimental Approach • Complete FEM calculation results • Average exit cross-section temperature increases with respect to the inlet • HDPE

28. Experimental Approach • Complete FEM calculation results • Average exit cross-section temperature increases with respect to the inlet • LDPE

29. Experimental Approach • Experimental design for the temperature measurements

30. Experimental Approach • Complete temperature measurement results • HDPE

31. Theoretical Approach • Identify all the factors that may be responsible for the deviations observed at high strain rates • Key assumptions made for the derivation of the temperature equation used in the FEM analysis • Started with the general heat equation • Assumption 1: Incompressible fluid • Assumption 2: Flow is steady and • Assumption 3: Fluid is Purely Entropic and • Obtained the temperature equation solved using FEM

32. Theoretical Approach • Furthermore • As a consequence of Assumption 3, the heat capacity is a function of temperature only • Assumption 4: the thermal conductivity is isotropic • Assumption 5: the velocity flow field corresponds to uniaxial elongational stretching (with full-slip boundary conditions) • Identified Assumptions 3, 4, and 5 as possible candidates responsible for the deviations mentioned earlier

33. Theoretical Approach • Elimination of Assumptions 4 and 5 • Considered anisotropy into the thermal conductivity • Increased k|| by 20% • Decreased k┴ by 10% • Axial temperature profile calculated for HDPE at Tin = 190oC and a strain rate of 34s-1

34. Theoretical Approach • Clearly, the PEE assumption seems to be the only remaining factor that is potentially responsible for the deviations observed at high strain rates • How do we eliminate it? • Start with the complete form of the temperature equation for an incompressible fluid defined earlier (*) • First correction: introduce conformation information into the heat capacity [1,2] • Second correction: introduce the second term on the left side of equation (*) 1. Dressler, M., B.J. Edwards, and H.C. Ottinger, Rheologica Acta, 1999. 38(2): p. 117-136. 2. Dressler, M., The Dynamical Theory of Non-Isothermal Polymeric Materials. 2000, ETH: Zurich.

35. Theoretical Approach • Both corrections mentioned above require knowledge of the conformation tensor • We can use the UCMM to evaluate the conformation tensor components inside the die channel • In Cartesian coordinates, the diagonal components of the normalized conformation tensor work out to be:

36. Theoretical Approach • Relaxation time measurements • Complete results for HDPE and LDPE

37. Theoretical Approach • Conformation tensor predictions using the UCMM • HDPE, Tin = 190oC

38. Theoretical Approach • Conformation tensor predictions using the UCMM • HDPE, all temperatures

39. Theoretical Approach • Correlation between the conformation at the exit cross-section and the difference between the measured and calculated ΔT

40. Theoretical Approach • First correction: the conformation dependent heat capacity • For example, the total heat capacity evaluated at the die axis for HDPE at Tin = 190oC 2 s-1 50 s-1

41. Theoretical Approach • Second correction • Rearranging the complete form of the heat equation and making the appropriate simplifications, we get: • The axial gradient of czz is already known from the UCMM • The derivative of the internal energy with respect to czz can also be evaluated using the UCMM [1]: 1. Dressler, M., The Dynamical Theory of Non-Isothermal Polymeric Materials. 2000, ETH: Zurich.

42. Theoretical Approach • Examining the effect of introducing corrections 1 and 2 detailed above • HDPE, Tin = 190oC

43. Part II: Summary • Provided experimental evidence that PEE is not universally valid • Verified a new form for the temperature equation by essentially eliminating the PEE assumption • Using the UCMM, two corrections have been made to the traditional temperature equation • 1) The conformational dependent heat capacity • Was found to have a significant decrease with increasing orientation • Had a negligible effect on the calculated temperature profiles • 2) The extra heat generation term • Quantified the temperature profiles in agreement with the experimental values

44. Part III: Molecular Simulations • Simulation Details • NEMC scheme developed by Mavrantzas and coworkers was used • Polydisperse linear alkane systems with average lengths of 24, 36, 50 and 78 carbon atoms were investigated • Temperature effects were also investigated (300K, 350K, 400K and 450K) • A uniaxial orienting field was applied • Simulations were run at constant temperature and constant pressure P=1atm

45. Molecular Simulations • Background • The conformation tensor is defined as the second moment of the end-to-end vector R • The normalized conformation tensor is: • The overall chain spring constant is then defined as: • The “orienting field” α:

46. Molecular Simulations • Thermodynamic Considerations • How do we test the validity of PEE under this framework? • The steps involved in accomplishing this task include: • Evaluate ΔA via thermodynamic integration • Evaluate ΔU directly from simulation

47. Molecular Simulations • Potential Model Details • Siepmann-Karaborni-Smit (SKS) force field σ ε

48. Equilibrium Simulations • The equilibrium mean-squared end-to-end distance • Used in the evaluation of the conformation tensor normalization factor and the chain spring constant • Can be evaluated for the entire molecular weight distribution interval • Its molecular weight dependence can be fitted to a polynomial function proposed by Mavrantzas and Theodorou [1] 1. Mavrantzas, V.G. and D.N. Theodorou, Macromolecules, 1998. 31(18): p. 6310-6332.

49. Equilibrium Simulations • The equilibrium mean-squared end-to-end distance • All systems at T = 450K

50. Equilibrium Simulations • The equilibrium mean-squared end-to-end distance • The polynomial fitting constants • For polyethylene, the measured characteristic ratio at T = 413K [2] 1. Mavrantzas, V.G. and D.N. Theodorou, Macromolecules, 1998. 31(18): p. 6310-6332. 2. Fetters, L.J., W.W. Graessley, R. Krishnamoorti, and D.J. Lohse, Macromolecules, 1997. 30(17): p. 4973-4977.