Chapter II Mechanical Characterizations

1. Chapter II Mechanical Characterizations *Most of figures appearing in this file are taken from the textbook “Dynamics of Polymeric Liquids” (Vol. 1). For more details, you are referred to the textbook and references cited therein.

2. Topics in Each Section • §2-1 Rheometry • Shear and Shearfree Flows • Flow Geometries & Viscometric Functions • §2-2 Basic Vector/Tensor Manipulations • Vector Operations • Tensor Operations • §2-3 Material Functions in Simple Shear Flows • Steady Flows • Unsteady Flows • §2-4 Material Functions in Elongational Flows

3. 2.1. Rheometry • Two standard kinds of flows, shear and shearfree, are used to characterize polymeric liquids (b) Shearfree (a) Shear Elongation rate FIG. 3.1-1. Steady simple shear flow Shear rate FIG. 3.1-2. Streamlines for elongational flow (b=0)

4. The Stress Tensor y x z Elongational Flow Shear Flow Total stress tensor* Stress tensor Hydrostatic pressure forces *See §2.2

5. (a) Shear Pressure Flow: • Classification of Flow Geometries Capillary Drag Flows: Concentric Cylinder Cone-and-Plate Parallel Plates (b) Elongation Moving Clamps

6. (a) Shear Concentrated Regime Dilute Regime • Typical Shear/Elongation Rate Range & Concentration Regimes for Each Geometries Homogeneous deformation:* Cone-and-Plate Concentric Cylinder Nonhomogeneous deformation: Parallel Plates Capillary (b) Elongation Moving clamps For Melts & High-Viscosity Solutions *Stress and strain are independent of position throughout the sample

7. Example: Concentric Cylinder • Viscometric Functions & Assumptions (From p.188 of ref 3) FIG. Concentric cylinder viscometer (homogeneous)

8. Flow Instability in a Concentric Cylinder Viscometer for a Newtonian Liquid Onset of Secondary Flow Ta (or Re)plays the central role! Laminar Secondary Turbulent Taylor vortices Turbulent

9. Rod Climbing is not a subtle effect, as demonstrated on the cover by Ph.D. student Sylvana Garcia-Rodrigues from Columbia. Ms. Garcia-Rodrigues is studying rheology in the Mechanical Engineering Department at U. of Wisconsin-Madison, USA. The Apparatus shown was created by UWMadison Professors Emeriti John L. Schrag and Arthur S. Lodge. The fluid shown is a 2% aqueous polyacrylamide solution, and the rotational speed is nominally 0.5 Hz. Photo by Carlos Arango Sabogal (2006)

10. Example 2.3-1: Interpretation of Free Surface Shapes in the Rod-Climbing Experiment (N) (P) I. Phenomenological Interpretation: (F): For the Newtonian fluids the surface near the rod is slightly depressed and acts as a sensitive manometer for the smaller pressure near the rod generated by centrifugal force (P):The Polymeric fluids exhibit an extra tension along the streamlines, that is, in the “θ” direction. In terms of chemical structure, this extra tension arises from the stretching and alignment of the polymer molecules along the streamlines. The thermal motions make the polymer molecules act as small “rubber bands” wanting to snap back

11. ‧ P (N) (P) II. Use of Equations of Change to Analyze the Distribution of the Normal Pressure The resultant formula derived in this example is:

12. Examples 1.3-4 & 10.2-1: Cone-and-Plate Instrument (From p.205 of ref 3) FIG. 1.3-4. Cone-and-plate geometry (homogeneous)

13. Example: p.530 Uniaxial Elongational Flow FIG. 10.3-1a. Device used to generate uniaxial elongational flows by separating Clamped ends of the sample Exptl. data see §2.4

14. Supplementary Examples • Capillary: • Example 10.2-3: Obtaining the Non-Newtonian Viscosity from the Capillary • Concentric Cylinders • Problem 10B.5: Viscous Heating in a Concentric Cylinder Viscometer • Parallel Plates: • Example 10.2-2: Measurement of the Viscometric Functions in the Parallel-Disk Instrument • Problem 1B.5: Parallel-Disk Viscometer • Problem 1D.2: Viscous Heating in Oscillatory Flow

15. 2.2. Basic Vector/Tensor Manipulations 3 • Vector Operations (Gibbs Notation) Vector: 2 1 Dot product:

16. Cross product:

17. 2 Tensor: • Tensor Operations Stresses acting on plane 1 1 3 FIG. The stress tensor The total momentum flux tensor for an incompressible fluid is: Normal stresses Stress tensor or Momentum flux tensor Hydrostatic pressure forces

18. Example:

19. Some Definitions & Frequently Used Operations: Cartesian coordinate Cartesian coordinate

20. 2.3. Material Functions in Simple Shear Flows • Remarks: • A variety of experiments performed on a polymeric liquid will yield a host of material functions that depend on shear rate, frequency, time, and so on • Representative fluid behavior will also be shown by means of sample experimental data • The description of the nature and diversity of material response to simple shearing and shearfree flow is given

21. FIG. 3.4-1. Various types of simple shear experiments used in rheology

22. Exp a: Steady Shear Flow • Steady Shear Flow Material Functions FIG. 3.3-1. Non-Newtonian viscosity η of a low-density polyethylene at several different temperatures The shear-rate dependent viscosity η is defined as: The first and second normal stress coefficients are defined as follows:

23. Relative Viscosity: FIG. 3.3-2. Master curves for the viscosity and first normal Stress coefficient as functions of shear rate for the Low-density polyethylene melt shown in previous figure Intrinsic Viscosity: FIG. 3.3-4. Intrinsic viscosity of polystyrene solutions, With various solvents, as a function of reduced shear rate β

24. Exp b: Small-Amplitude Oscillatory Shear Flow • Unsteady Shear Flow Material Functions FIG. 3.4-2. Oscillatory shear strain, shear rate, shear stress, and first normal stress difference in small-amplitude oscillatory shear flow

25. It is customary to rewrite the above eq to display the in-phase and out-of-phase parts of the shear stress Storage modulus Loss modulus FIG. 3.4-3. Storage and loss moduli, G’ and G”, as functions of frequency ω at a reference temperature of T0=423K for the low-density polyethylene melt shown in Fig. 3.3-1. The solid curves are calculated from the generalized Maxwell model, Eqs. 5.2-13 through 15

26. Exp c: Stress Growth upon Inception of Steady Shear Flow Transient Shear Stress:

28. Exp e: Stress Relaxation after a Sudden Shearing Displacement (Step-Strain Stress Relaxation) Relaxation Modulus:* For small shear strains The Lodge-Meissner Rule: *Example 5.3-2: Stress Relaxation after a Sudden Shearing Displacement

30. 2.4. Material Functions in Elongational Flows • Shearfree Flow Material Functions

31. The number average and weight average molecular weights of the samples: Monodisperse, but with a tail in high M.W. (GPC results)