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.  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: Newtonian Fluid Dynamics

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 MaterialGradeMI (g/10min) Density (g/cm 3 ) Thermal conductivity (Wm -1 K -1 ) MWPI LDPEExact 31397.50.9010.356,9501.99 HDPEPaxxon AB40003 0.30.9430.5105,2009.74

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, T in = T wall = 190 o C

25. Experimental Approach  Sample FEM calculation results Axial temperature profiles HDPE, T in = T wall = 190 o C

26. Experimental Approach  Sample FEM calculation results Radial temperature profiles HDPE, T in = T wall = 190 o C

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 T in = 190 o C 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, T in = 190 o C

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 T in = 190 o C 50 s -1 2 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 c zz is already known from the UCMM The derivative of the internal energy with respect to c zz 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, T in = 190 o C

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] Temperatureα0α0 α1α1 α2α2 α3α3 450K8.8427-77.9066521.951-2141.85 400K8.6677-30.5968-681.6816030.121 350K9.219-9.298-1573.3613064.89 300K11.9351-183.7562022.19-9320.51 450K ref. [1]9.1312-75.1865315.742-500.3518 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.

51. Equilibrium Simulations  The equilibrium mean-squared end-to-end distance Polynomial fits, all temperatures 1. Mavrantzas, V.G. and D.N. Theodorou, Macromolecules, 1998. 31(18): p. 6310-6332.

52. Equilibrium Simulations  The conformation tensor normalization factor μ Usually taken as a constant with respect to temperature (PEE assumption) Gupta and Metzner [1] proposed the following for the temperature dependence of μ This expression was used to fit our equilibrium simulation data with great success 1. Gupta, R.K. and A.B. Metzner, Journal of Rheology, 1982. 26(2): p. 181-198.

53. Equilibrium Simulations  Theoretical considerations for the behavior of μ with respect to temperature If B= - 1, μ is a constant and K(T) is a linear function of temperature  The configurational part of the internal energy density of a fluid particle given by the UCMM vanishes If B< - 1, μ increases with temperature and decreases with temperature  The configurational part of the internal energy density of a fluid particle given by the UCMM may become important at high degrees of orientation

54. Equilibrium Simulations  The temperature exponent B

55. Non-equilibrium Simulations  The applied “orienting field” α: The magnitude of α xx will uniquely describe the “strength” of the orienting field  Following the definition of α, the conformation tensor will also have a diagonal form Therefore, the trace of the conformation tensor may be used as a unique descriptor for the degree of orientation and extension developed in the simulations

56. Non-equilibrium Simulations  Molecular weight dependence of the degree of orientation All systems, T = 450K

57. Non-equilibrium Simulations  Temperature dependence of the degree of orientation C 36 system, all temperatures

58. Non-equilibrium Simulations  Energy balances for the oriented systems All systems, T = 450K

59. Non-equilibrium Simulations  Energy balances for the oriented systems C 36 system, all temperatures

60. Non-equilibrium Simulations  Internal energy broken down into individual components C 24 system, T = 400K

61. Non-equilibrium Simulations  The UCMM prediction for the change in Helmholtz free energy

62. Non-equilibrium Simulations  The conformational part of the heat capacity The MW dependence, T = 450K

63. Non-equilibrium Simulations  The conformational part of the heat capacity The temperature dependence, C 36 system

64. Part III: Summary  Equilibrium simulations Revealed a non-linear dependence of K(T) with respect to temperature Improved agreement with experiment in terms of the characteristic ratio C ∞ and temperature exponent B  Non-equilibrium simulations The changes in free energy and internal energy are of similar magnitude The examination of the individual components of the internal energy provided two useful insights  The elastic response of single chains is indeed purely entropic  The inter-molecular contribution to the internal energy of an ensemble of chains (missing in the isolated chain case) is very important and explains the trends observed during the experiments

65. Part IV: Published Research  “Structure Formation under Steady-State Isothermal Planar Elongational Flow of n-eicosane: A Comparison between Simulation and Experiment”[1] First, we examined the liquid structure predicted by simulation under equilibrium conditions  Simulation performed in the NVT ensemble (number of particles N, system volume V and temperature T are kept constant)  The state point was chosen the same as in the experiment case (T = 315K and ρ = 0.81 g/cc), and the experimental data were taken from literature (**) 1. Ionescu, T.C., et al.,. Physical Review Letters, 2006. 96(3). (*) A. Habenschuss and A.H. Narten, J. Chem. Phys., 92, 5692 (1990)

66. Simulated Elongated Structure  Next, we examined the structure when the flow field is turned on at steady-state in terms of the pair correlation function The applied velocity gradient is of the form: Results shown at a reduced elongation rate =1.0 The state point was the same as in the equilibrium case (T=315K and ρ = 0.81 g/cc)

67. Simulated Elongated Structure  Same structural data, in terms of the static structure factor s(k)

68. Comparison with Experiment  The structure factor s(k) determined via x-ray diffraction from the n-eicosane crystalline sample Identify two regions:  Inter-molecular region (k<6Å -1 ), where sharp Bragg peaks are present  Intra-molecular region (k>6Å -1 ), where the agreement with simulation is excellent (*) A. Habenschuss and A.H. Narten, J. Chem. Phys., 92, 5692 (1990).

69. Conclusions and Directions for Future Research  Conclusions We successfully combined experiment, theory and simulation to investigate the nature of the free energy stored by polymer melts subjected to deformation First, it was shown that the Theory of Purely Entropic Elasticity is applicable to polymer melts only at low deformation rates Second, molecular theory (the UCMM) was used to propose a recipe for eliminating the PEE assumption with great results In the end, the Molecular Simulation study helped us elucidate the trends observed in the experimental part  The simulated conformational dependent heat capacity was found in good qualitative agreement with experiment

70. Conclusions and Directions for Future Research  Directions for Future Research More polymers and processing conditions  Effects of molecular characteristics More accurate viscoelastic models Our work in Part IV already led to the development of a constant pressure version of the NEMD algorithm used  Longer chain systems and different flow situations also worth investigating

71. Acknowledgements  My advisors, Drs. Brian Edwards and David Keffer  Dr. Vlasis Mavrantzas  Dr. Simioan Petrovan  Doug Fielden  ORNL – Cheetah and UT SInRG Cluster  PRF, Grant No. 41000-AC7

72. Questions?

73. Experimental Approach  Same analysis performed for shear flow using a capillary die LDPE, T in = T wall = 170 o C, D = 1mm, L = 25mm

74. Experimental Approach  Shear flow temperature profile in the measurement device