Chris Macosko Department of Chemical Engineering and Materials Science NSF- MRSEC

1. IMA Annual Program Year Tutorial An Introduction to Funny (Complex) Fluids: Rheology, Modeling and Theorems September 12-13, 2009 Understanding silly putty, snail slime and other funny fluids Chris Macosko Department of Chemical Engineering and Materials Science NSF- MRSEC (National Science Foundation sponsored Materials Research Science and Engineering Center) IPRIME (Industrial Partnership for Research in Interfacial and Materials Engineering)

2. What is rheology? rein(Greek)= ta panta rei= rheology = to flow every thing flows study of flow?, i.e. fluid mechanics? honey and mayonnaise honey and mayo stress= f/area viscosity = stress/rate rate of deformation rate of deformation

3. What is rheology? rein(Greek)= ta panta rei= rheology = honey and mayo viscosity rate of deformation to flow every thing flows study of flow?, i.e. fluid mechanics? rubber band and silly putty modulus = f/area time of deformation

4. 4 key rheological phenomena

5. fluid mechanician: simple fluids complex flows materials chemist: complex fluids complex flows rheologist: complex fluids simple flows rheology = study of deformation of complex materials rheologist fits data to constitutive equations which - can be solved by fluid mechanician for complex flows - have a microstructural basis

6. from: Rheology: Principles, Measurement and Applications, VCH/Wiley (1994). ad majorem Dei gloriam

7. Goal: Understand Principles of Rheology: (stress, strain, constitutive equations) stress = f (deformation, time) • Simplest constitutive relations: • Newton’s Law: Hooke’s Law: Key Rheological Phenomena • shear thinning (thickening) • time dependent modulus G(t) • normal stresses in shear N1 • extensional > shear stress hu> h

8. 1 k1 k2 k1 k2 L´ f L´ Young (1805) modulus stress t strain ELASTIC SOLID The power of any spring is in the same proportion with the tension thereof. Robert Hooke (1678) f µ DL f = kDL 1-8

9. Natural rubber G=3.9x105Pa a = area natural rubber G = 400 kPa Uniaxial Extension 1-9

10. Shear gives different stress response g = 0 g = -0.4 g = 0.4 Silicone rubber G = 160 kPa Goal: explain different results in extension and shear obtain from Hooke’s Law in 3DIf use stress and deformation tensors 1-10

11. direction of stress on plane plane stress acts on Stress Tensor - Notation dyad Other notation besides Tij: sijor Pij 1-11

12. Rheologists use very simple T or T22 T11 = 0 T33 T22 = T33 T11 -T22causes deformation 1. Uniaxial Extension T22 = T33 = 0 1-12

13. Hydrostatic Pressure T11 = T22 = T33 = -p Consider only normal stress components Then only t the extra or viscous stresses cause deformation T = -pI + t and only the normal stress differences cause deformation T11 - T22 = t11 - t22 ≡ N1 (shear) If a liquid is incompressible G ≠ f(p) h ≠ f(p) 1-13

14. Rheologists use very simple T T21 T21 T12 T12 in general Stress tensor for simple shear Only 3 components: T12 T11 –T22 = t11 –t22 ≡ N1 T22 –T33 = t22 –t33 ≡ N2 2. Simple Shear But to balance angular momentum 1-14

15. Stress Tensor Summary n T11 T11 • stress at point on any plane • 2. in general T = f( time or rate, strain) • 3. simple T for rheologically complex materials: • - extension and shear • 4. T= pressure + extra stress = -pI+t. • 5. τ causes deformation • 6. normal stress differences cause deformation,t11-t22= T11-T22 • 7. symmetric T = TT i.e. T12=T21

16. Deformation Gradient Tensor Q w s P w′ y s′ y′ a new tensor ! s′ = w′ - y′ w’ = y’ + s’ s’ is a vector connecting two very close points in the material, P and Q Q P s = w – y w = y + s x = displacement functiondescribes how material points move 1-16

17. Apply F to Uniaxial Extension Displacement functions describe how coordinates of P in undeformed state, xi‘ have been displaced to coordinates of P in deformed state, xi. 1-17

18. Assume: • constant volume V′ = V 2) symmetric about the x1 axis Can we write Hooke’s Law as ? 1-18

19. Can we write Hooke’s Law as ? Solid Body Rotation – expect no stresses For solid body rotation, expect F = I t = 0 But F ≠ I F ≠ FT Need to get rid of rotation create a new tensor! 1-19

20. Finger Tensor Solid Body Rotation Bij gives relative local change in area within the sample. 1-20

21. Neo-Hookean Solid 1. Uniaxial Extension since T22 = 0 1-21

22. 2. Simple Shear agrees with experiment Silicone rubber G = 160 kPa 1-22

23. Finger Deformation Tensor Summary • area change around a point on any plane 2. symmetric 3. eliminates rotation 4. gives Hooke’s Law in 3D fits rubber data fairly well predicts N1, shear normal stresses

24. Course Goal: Understand Principles of Rheology: (constitutive equations) stress = f (deformation, time) • Simplest constitutive relations: • Newton’s Law: Hooke’s Law: Key Rheological Phenomena • shear thinning (thickening) • time dependent modulus G(t) • normal stresses in shear N1 • extensional > shear stress hu> h

25. 2 VISCOUS LIQUID The resistance which arises From the lack of slipperiness Originating in a fluid, other Things being equal, is Proportional to the velocity by which the parts of the fluids are being separated from each other. Isaac S. Newton (1687)

26. Adapted from Barnes et al. (1989). Familiar materials have a wide range in viscosity Bernoulli • measured h in shear • 1856 capillary (Poiseuille) • 1880’s concentric cylinders (Perry, Mallock, Couette, Schwedoff) Newton, 1687 Stokes-Navier, 1845

27. Bernoulli • measured h in shear • 1856 capillary (Poiseuille) • 1880’s concentric cylinders (Perry, Mallock, Couette, Schwedoff) • measured in extension • Troutonhu= 3h Newton, 1687 Stokes-Navier, 1845 “A variety of pitch which gave by the traction method l = 4.3 x 1010 (poise) was found by the torsion method to have a viscosity m = 1.4 x 1010 (poise).” F.T. Trouton (1906) To hold his viscous pitch samples, Trouton forced a thickened end into a small metal box. A hook was attached to the box from which weights were hung.

28. polystyrene 160°C Münstedt (1980) • Goal • Put Newton’s Law in 3 dimensions • rate of strain tensor 2D • show hu = 3h

29. recall Deformation Gradient Tensor, F Q w s Q P w′ y s′ P y′ s′ = w′ - y′ Separation and displacement of point Q from P s = w - y

30. Alternate notation: Viscosity is “proportional to the velocity by which the parts of the fluids are being separated from each other.” —Newton Velocity Gradient Tensor

31. Rate of Deformation Tensor D Other notation: Vorticity Tensor W Can we write Newton’s Law for viscosity as t = hL? solid body rotation t12 ≠ t21

32. Show that 2D = 0 for solid body rotation Example 2.2.4 Rate of Deformation Tensor is a Time Derivative of B.

33. Time derivatives of the displacement functions for simple, shear Newtonian Liquid t = h2DorT = -pI + h2D Steady simple shear Here planes of fluid slide over each other like cards in a deck.

34. Newtonian Liquid Steady Uniaxial Extension

35. Newtonian Liquid Apply to Uniaxial Extensiont = h2D From definition of extensional viscosity • Newton’s Law in 3 Dimensions • predicts h0 low shear rate • predicts hu0 = 3h0 • but many materials show large deviation

36. Summary of Fundamentals n T11 T11 2. area change around a point on plane symmetric, eliminates rotation gives Hooke’s Law in 3D, E=3G 3. rate of separation of particles symmetric, eliminates rotation gives Newton’s Law in 3D, • stress at point on plane simple T - extension and shear T= pressure + extra stress = -pI+t. symmetric T = TT i.e. T12=T21

37. Course Goal: Understand Principles of Rheology: stress = f (deformation, time) Key Rheological Phenomena • shear thinning (thickening) • time dependent modulus G(t) • normal stresses in shear N1 • extensional > shear stress hu> h • NeoHookean: Newtonian: t = h2D