1. The Processing of Polymers Spring 2001

2. Module 3 Spring 2001Dr. Ken Lewis ISAT 4302 Introduction Three types of polymers of importance Thermoplastics Thermosets Elastomers As a group, polymers (plastics) possess Light weight Corrosion resistance Electrical insulating resistance Thermal insulating resistance

3. Module 3 Spring 2001Dr. Ken Lewis ISAT 4303 Introduction 2 Applications Automobile parts Packaging materials Electrical and electronic components Household articles Utensils Tubing Foamed products Fibers Films Paints, varnishes Fiber matrix composites

4. Module 3 Spring 2001Dr. Ken Lewis ISAT 4304 Introduction 3 Polymer usage is surpassing most other materials, at least in volume (low density!) Plastics are replacing Metals Glasses Woods

5. Module 3 Spring 2001Dr. Ken Lewis ISAT 4305 Introduction 4 – why plastics are important. Polymers are easily shaped into unlimited designs Many plastics are molded which is net shaping so little further processing is necessary Heating is needed, but far less than for most metal processes. Many times finishing or painting is not necessary.

6. Module 3 Spring 2001Dr. Ken Lewis ISAT 4306 Properties Used in Processing Enthalpy (Specific Heat) Thermal Conductivity Viscosity Most of these materials are all processed under similar constraints Both of these affect the initial plastification and the final cooling Affects the flow through the dies and spinnerets and mold cavities

7. Module 3 Spring 2001Dr. Ken Lewis ISAT 4307 Polymer Characteristics of importance High viscosity MaterialViscosity (Pascal Seconds) Water0.001 Oils0.1 – 1.0 Dough300 Molten Glass100 – 10,000 Polymers100 – 1500

8. Module 3 Spring 2001Dr. Ken Lewis ISAT 4308 In order to understand Polymer processing… We need some grounding in viscosity and viscous flow

9. Module 3 Spring 2001Dr. Ken Lewis ISAT 4309

10. Module 3 Spring 2001Dr. Ken Lewis ISAT 43010 Viscosity and Shear Rate Consider two large parallel plates separated by a fluid. At time t = 0 the upper plate is set in motion with a velocity v 0. As time proceeds, the fluid gains momentum and we arrive at the final steady state velocity distribution

11. Module 3 Spring 2001Dr. Ken Lewis ISAT 43011

12. Module 3 Spring 2001Dr. Ken Lewis ISAT 43012 Viscosity and Shear Rate Consider two very long and wide parallel plates. One is at rest and one is moving with velocity v 0.

13. Module 3 Spring 2001Dr. Ken Lewis ISAT 43013 Shear Rate The fluid adheres to both walls so the velocity of the fluid Is zero at the bottom plate Is v 0 at the top plate Is proportional to the distance from the bottom plate

14. Module 3 Spring 2001Dr. Ken Lewis ISAT 43014 Shear Rate We may rewrite this as: The proportionality constant is just the slope of the line, or:

15. Module 3 Spring 2001Dr. Ken Lewis ISAT 43015 Shear Rate The slope, or the rate of change of the x-velocity in the y direction is called the shear rate. Shear rates have unit of sec-1. The faster the plate moves, or the closer they are together… the more stress is imposed on the fluid

16. Module 3 Spring 2001Dr. Ken Lewis ISAT 43016 Shear Stress  yx & Viscosity  To support this motion, There must be a tangential force on the upper plate  v 0  1/Y F/A, the force per unit area  is called stress. We may rewrite this as:

17. Module 3 Spring 2001Dr. Ken Lewis ISAT 43017 Newton’s Law of Viscosity The shear force per unit area is proportional to the local velocity gradient. The constant of proportionality is called the viscosity

18. Module 3 Spring 2001Dr. Ken Lewis ISAT 43018 Newton’s Law of Viscosity In the neighborhood of the moving surface, the fluid acquires a certain amount of x-momentum. This fluid in turn, imparts some of its momentum to the adjacent ‘layer’ of liquid causing it to remain in motion in the x direction. Hence, x-momentum is transmitted through fluid in the y direction. Thus,  yx may be interpreted as the viscous flux of x- momentum in the y direction

19. Module 3 Spring 2001Dr. Ken Lewis ISAT 43019 Flux is the “rate of flow per unit area”

20. Module 3 Spring 2001Dr. Ken Lewis ISAT 43020 Shear Flow in a Cylinder Let’s go from plates to cylindrical flow Flow exhibited by fluid in pipes, capillaries, etc. The flow is purely axial No radial components

21. Module 3 Spring 2001Dr. Ken Lewis ISAT 43021 Shear Flow in a Cylinder Fluid velocity is zero at the wall. Fluid velocity remains constant on concentric cylindrical surfaces. The flow is purely axial The fluid velocity reaches a maximum at the center. This is called: Laminar Flow

22. Module 3 Spring 2001Dr. Ken Lewis ISAT 43022 Velocity Distribution in a Cylindrical Tube There is friction, both at the wall of the tube Within the fluid itself Thus, the fluid is: Accelerated by the pressure gradient Retarded by the frictional shearing stress Pressure gradient The fluid moves under the influence of a pressure gradient.

23. Module 3 Spring 2001Dr. Ken Lewis ISAT 43023 Shear Rate The Driving Force is: The Resisting Force is: At equilibrium, they must balance:

24. Module 3 Spring 2001Dr. Ken Lewis ISAT 43024 Shear Rate 2 At equilibrium, they must balance: Solving for the shear stress: So, Stress is greatest At the wall And zero at the center

25. Module 3 Spring 2001Dr. Ken Lewis ISAT 43025 Shear Rate 3 If we insert Newton’s Law:

26. Module 3 Spring 2001Dr. Ken Lewis ISAT 43026 Shear Rates 4 Shear rate 0 at the center (r = 0) Max at the wall (r = R) Shear rate is an indication of the stress being seen by the fluid, and how fast it sees it! The shear rate at the wall for a Newtonian fluid is: Q = volumetric flow rate D = diameter

27. Module 3 Spring 2001Dr. Ken Lewis ISAT 43027 Viscosities Material Viscosity  (Pa s) Water 20°C0.001 Water 100°C0.00028 Air 20°C1.8 x 10 -5 Air 100°C2.1 x 10 -5 Mercury 20°C0.0016 Machine Oil 20°C0.1 Pancake Syrup 20°C50 Polymer A 150°C225 Polymer A 250°C25 Glass (SiO 2 ) 540°C10 12 Glass (SiO 2 ) 1095°C10 3 Glass (SiO 2 ) 1370°C15

28. Module 3 Spring 2001Dr. Ken Lewis ISAT 43028 Volumetric Newtonian Flow in a Tube The laminar flow of a Newtonian fluid in a pipe or tube may be expressed: Where: Q = the volumetric flow rate [=] m 3 /s or gal/min  P = the pressure drop or driving force [=] kg/m 2 or Pa R = the radius of the tube [=] m or cm L = the length of the pipe [=] m or cm  = the Newtonian viscosity [=] Pa s

29. Module 3 Spring 2001Dr. Ken Lewis ISAT 43029 The effect of viscosity on Pressure Drop The Pressure drop across a pipe is a measure of the energy necessary to drive a fluid through the pipe. Assume a Newtonian Fluid Two cases: A viscosity of 0.001 Pa s (like water) A viscosity of 500 Pa s (like many polymers)

30. Module 3 Spring 2001Dr. Ken Lewis ISAT 43030 The effect of viscosity on Pressure Drop Let: Then:

31. Module 3 Spring 2001Dr. Ken Lewis ISAT 43031 So the effect of viscosity on fluid transport can be IMPORTANT

32. Module 3 Spring 2001Dr. Ken Lewis ISAT 43032 Viscosity For a Newtonian fluid, the viscosity  is constant. This holds for simple fluids like water, all gases. However For almost all polymeric fluids, the viscosity is NOT constant. Many times it is a function of the shear rate!

33. Module 3 Spring 2001Dr. Ken Lewis ISAT 43033 Newton’s Law of Viscosity or

34. Module 3 Spring 2001Dr. Ken Lewis ISAT 43034 Power Law Fluids The Ostwald-de Waele Model Known as the Power Law Model Note that for n=1, this reduces to Newton’s Law of viscosity with m = 

35. Module 3 Spring 2001Dr. Ken Lewis ISAT 43035 Power Law Fluids The deviation of n from unity indicates the degree of Non-Newtonian behavior. If n < 1, material behavior is pseudoplastic If n> 1, material behavior is dilatant.

36. Module 3 Spring 2001Dr. Ken Lewis ISAT 43036 Power Law Viscosity For most polymers, the isothermal viscosity decreases with increasing shear rate. Effect of shear on the entangled polymer chains Usually, in the literature, the viscosity is not shown as “  ”, but rather “  ” So:

38. Module 3 Spring 2001Dr. Ken Lewis ISAT 43038 Viscosity Newtonian Fluid Viscosity (slope) constant Non-Newtonian Fluid Viscosity is not constant Profound affect on processing

39. Module 3 Spring 2001Dr. Ken Lewis ISAT 43039 Power Law Viscosity For a power law fluid: The effect of shear rate on viscosity can be enormous! Remember for a Newtonian fluid n=1  is constant

40. The Effect of Shear Rate on Viscosity

41. Module 3 Spring 2001Dr. Ken Lewis ISAT 43041 The Effect of Shear Rate on Viscosity The effect can be enormous In this case the zero shear viscosity is about 1000 Pa s. At a shear rate of 1000 sec- 1, the viscosity has dropped to about 5 Pa s

42. Module 3 Spring 2001Dr. Ken Lewis ISAT 43042 The Effect of Shear Rate on Viscosity The effect can be enormous Polymer B 90°C Shear Rate (sec -1 ) Viscosity (Pa s) 01000 10020 10003

43. Module 3 Spring 2001Dr. Ken Lewis ISAT 43043 Power Law Shear Rates. It can be shown that for the flow of a power law fluid through a cylindrical pipe, the maximum shear rate is; Note If n = 1, This reduces to The Newtonian Shear rate

44. Module 3 Spring 2001Dr. Ken Lewis ISAT 43044 Power Law Shear Rates 2 And the volumetric flow rate Q for a Power Law fluid through a pipe can be shown to be: Note If n = 1, This reduces to The Newtonian Flow rate

45. Module 3 Spring 2001Dr. Ken Lewis ISAT 43045 Properties Polymer TgTg TmTm TpTp  n Polyethylene LDPE-100120160-240650.35 HDPE-115130200-2822400.5 Polyvinyl chloride80212160-210800.3 Polystyrene100240180-2602200.3 Nylon 6,65526260-2901000.75 Polycarbonate150230280-3102250.7 Polyester (ABS)115180-2402100.25

46. Module 3 Spring 2001Dr. Ken Lewis ISAT 43046 The effect of shear rate on viscosity which affects pressure drop. Remember the problem of finding the pressure drop necessary to push a fluid through a pipe at a desired flow rate. Two cases The Newtonian fluid (water) with a viscosity of 0.001 Pa s. The polymer with a zero shear viscosity of 500 Pa s. Let the power law exponent n = 0.55 And remember the conditions:

47. Module 3 Spring 2001Dr. Ken Lewis ISAT 43047 The effect of shear rate on viscosity which affects pressure drop. Remember the problem of finding the pressure drop necessary to push a fluid through a pipe at a desired flow rate. In the first case, the results are the same since the fluid is Newtonian and the viscosity is constant….

48. Module 3 Spring 2001Dr. Ken Lewis ISAT 43048 The effect of shear rate on viscosity which affects pressure drop. Remember the problem of finding the pressure drop necessary to push a fluid through a pipe at a desired flow rate. In the second case The fluid is non-Newtonian This means that the apparent viscosity will be a function of the shear rate Thus, we must first find the shear rate at the above conditions, Then using our power law relationships find the apparent viscosity at that shear rate Finally using the power law equation, calculate the pressure drop that will occur.

49. Module 3 Spring 2001Dr. Ken Lewis ISAT 43049 The effect of shear rate on viscosity which affects pressure drop. We know:

50. Module 3 Spring 2001Dr. Ken Lewis ISAT 43050 The effect of shear rate on viscosity which affects pressure drop And from the equation for a power law viscosity Remember: m is the zero shear viscosity 500 Pa s And n = 0.55 Look at the Difference!

51. Module 3 Spring 2001Dr. Ken Lewis ISAT 43051 The effect of shear rate on viscosity which affects pressure drop We know the power law equation for the volumetric flow rate, Q Rearranging and solving for  P

52. Module 3 Spring 2001Dr. Ken Lewis ISAT 43052 The effect of shear rate on viscosity which affects pressure drop Rearranging and solving for  P Inserting and solving:

53. Module 3 Spring 2001Dr. Ken Lewis ISAT 43053 The effect of shear rate on viscosity which affects pressure drop Compare this to the non shear thinned value of:

54. Module 3 Spring 2001Dr. Ken Lewis ISAT 43054 Velocity profiles of Newtonian and Non- Newtonian Fluids Note the difference. The Newtonian profile is parabolic The Power Law fluid is blunted. Why? Remember the viscosity for the PL fluid is a function of shear rate. Shear rate is highest at / near the wall The shear gets dissipated and the central part of the PL flow is called “plug flow”. Ramifications – the fluid there is stagnant

55. Module 3 Spring 2001Dr. Ken Lewis ISAT 43055 Velocity profiles of Newtonian and Non- Newtonian Fluids Note that the PL flow rate for the same pressure drop is higher The shear rates are higher and the viscosity becomes lower.

56. Module 3 Spring 2001Dr. Ken Lewis ISAT 43056 Effect of Temperature on Viscosity Usually models using a form of the Arrhenius equation Shear Rate (sec -1 ) Activation energy E  (kcal/mole) 012.8 10 -1 11.4 1010.3 10 1 8.5 10 2 7.2 10 3 6.1