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) What is rheology? ρειν (Greek) = to flow τα παντα ρει = every thing flows rheology = study of flow?, i.e. fluid mechanics? honey and mayonnaise honey and mayo stress= viscosity = f/area stress/rate rate of deformation rate of deformation What is rheology? ρειν (Greek) = to flow τα παντα ρει = every thing flows rheology = study of flow?, i.e. fluid mechanics? honey and mayo rubber band and silly putty viscosity modulus = f/area rate of deformation time of deformation 4 key rheological phenomena rheology = study of deformation of complex materials fluid mechanician: simple fluids complex flows materials chemist: complex fluids complex flows rheologist: complex fluids simple flows rheologist fits data to constitutive equations which - can be solved by fluid mechanician for complex flows - have a microstructural basis from: Rheology: Principles, Measurement and Applications, VCH/Wiley (1994). ad majorem Dei gloriam Goal: Understand Principles of Rheology: (stress, strain, constitutive equations) stress = f (deformation, time) Simplest constitutive relations: Newton’s Law: Hooke’s Law: dγ τ = η = ηγ τ = Gγ dt Key Rheological Phenomena • shear thinning (thickening) h ( g ) • time dependent modulus G(t) • normal stresses in shear N1 η η • extensional > shear stress u > k k 1 ELASTIC SOLID 1 2 The power of any spring L´ is in the same proportion with the tension thereof. Robert Hooke (1678) k1 k f ∝ ∆L 2 f f = k∆L L´ Young (1805) modulus τ = Gγ stress τ 1-8 strain Uniaxial Extension Natural rubber G=3.9x105Pa f T = 11 a L extension α = L′ L - L or strain e = = a -1 L a = area natural rubber G = 400 kPa 1-9 Shear gives different stress response γ = 0 γ = -0.4 γ = 0.4 Goal: explain different results in extension and shear Silicone rubber G = 160 kPa obtain from Hooke’s Law in 3D If use stress and deformation te nsors 1-10 Stress Tensor - Notation T = xˆxˆTxx + xˆyˆTxy + xˆzˆTxz T T T xx xy xz T = T T T +yˆxˆTyx + yˆyˆTyy + yˆzˆTyz yx yy yz Tzx Tzy Tzz +zˆxˆTzx + zˆyˆTzy + zˆzˆTzz dyad xˆ ,yˆ ,zˆ = xˆ 1 ,xˆ 2 ,xˆ 3 T = xˆt x + yˆt y + zˆt z T11 T12 T13 = Tij T21 T22 T23 T31 T32 T33 xyˆ ˆT xy 3 3 direction of stress on plane T = xˆ ixˆ jTij = Tij plane stress acts on i=1 j=1 σ Π Other notation be sides Tij: ij or ij 1-11 Rheologists use very simple T 1. Uniaxial Extension T 0 0 11 T = 0 0 0 T22 = T33 = 0 0 0 0 or T 22 0 0 0 T = 0 −T 0 22 T11 = 0 − T33 0 0 T33 T22 = T33 1-12 T11 -T22 causes deformation Consider only normal stress components T11 0 0 = Tij 0 T22 0 0 0 T33 Hydrostatic Pressure T11 = T22 = T33 = -p -p 0 0 1 0 0 Tij = 0 - p 0 = -p 0 1 0 0 0 - p 0 0 1 1 0 0 T = - pI I = 0 1 0 Then only τ the extra or viscous stresses 0 0 1 cause deformation T I = T T = -pI + τ If a liquid is and only the normal stress differences incompressible cause deformation 1-13 G ≠ f(p) η ≠ f(p) τ τ T11 - T22 = 11 - 22 ≡ N1 (shear) Rheologists use very simple T 2. Simple Shear T xˆ2 12 xˆ2 T21 xˆ1 xˆ1 T21 xˆ3 xˆ3 T 0 0 0 0 T12 0 12 = = Tij T21 0 0 Tij 0 0 0 0 0 0 0 0 0 But to balance angular momentum T21 = T12 T T in general T T = (x x T ) = x x T T = T i j i j ij i j i j ji T tn = nˆ gT = T gnˆ Stress tensor for simple shear Only 3 components: T11 T12 0 T12 Tij = T12 T22 0 τ τ T11 – T22 = 11 – 22 ≡ N1 0 0 T 1-14 33 τ τ T22 – T33 = 22 – 33 ≡ N2 Stress Tensor Summary 1. stress at point n T T on any plane 11 11 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 + τ. 5. τ causes deformation τ τ 6. normal stress differences cause deformation, 11- 22 = T11-T22 T 7. symmetric T = T i.e. T12=T21 Hooke → Young → Cauchy → Gibbs Einstein (1678) (~1801) (1830’s) (1880,~1905) ˆ x2 Q Deformation Gradient Tensor Q s Motion s’ is a vector connecting s' ˆ two very close points in the x1 P P material, P and Q Rest or past state at t' Deformed or present state at t xˆ 3 Q Q w′ w s′ s y′ P y P s′ = w′ - y′ s = w – y w’ = y’ + s’ w = y + s a new tensor ! x = displacement function x x describes how material points move F= or F = i x ij x w = x(w ) = x(y + s ) j x = x ( y ) + s + Os 2 x y s = w - y = F s = x(y ) + F s 1-16 = y + F s 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. Fij = xi x j = x1 x1 = a1 x1 x2 = 0 x1 x3 = 0 x2 x1 = 0 x2 x2 = a2 x2 x3 = 0 x3 x1 = 0 x3 x2 = 0 x3 x3 = a3 a1 0 0 Fij = 0 a2 0 1-17 0 0 a3 a1 0 0 Fij = xi x j = 0 a2 0 0 0 a3 Assume: 2) constant volume V′ = V 2) symmetric about the x1 axis a1 0 0 -1 2 Fij = 0 a1 0 -1 2 0 0 a1 Can we write Hooke’s Law as τ = G ( F - I ) ? 1-18 Can we write Hooke’s Law as τ = G ( F - I ) ? Solid Body Rotation – expect no stresses For solid body rotation, expect F = I τ = 0 But F ≠ I F ≠ FT Need to get rid of rota1t-io19n create a new tensor! Finger Tensor T F·FT = ( V·R)·( V·R) F = VgR = V·R·RT ·VT V = stretch = V·I·VT R = rotation = V 2 T xi x j B = FgF or Bij = Fik Fjk = xk xk Solid Body Rotation cosq -sinq 0 cosq sinq 0 1 0 0 Bij = sinq cosq 0 -sinq cosq 0 = 0 1 0 0 0 1 0 0 1 0 0 1 Bij gives relative local change in are a within the sample. 1-20 Neo-Hookean Solid τ = G( B - I) or T = − pI + GB 1. Uniaxial Extension since T 22 = 0 1-21 2. Simple Shear τ = G ( B - I ) t = Gg s 21 2 x1 = x1 + x2 = x1 + g x2 t11 -t 22 = Gg Dx '2 x x 2 = 2 agrees with experiment x3 = x3 1 g 0 Fij = 0 1 0 0 0 1 2 1 g 0 1 0 0 1+ g g 0 Bij = 0 1 0 g 1 0 = g 1 0 0 0 1 0 0 1 0 0 1 1-22 Silicone rubber G = 160 kPa Finger Deformation Tensor Summary 1. B = F g F T 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 Course Goal: Understand Principles of Rheology: (constitutive equations) stress = f (deformation, time) Simplest constitutive relations: Newton’s Law: Hooke’s Law: dγ τ = η = ηγ τ = Gγ dt τ = G ( B - I ) Key Rheological Phenomena • shear thinning (thickening) h ( g ) • time dependent modulus G(t) • normal stresses in shear N1 η η • extensional > shear stress u > 2 VISCOUS LIQUID The resistance which arises From the lack of slipperiness du x t yx=h Originating in a fluid, other dy 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) dv dvx t = h t yx = h dy dy Bernoulli measured η in shear 1856 capillary (Poiseuille) Newton, 1687 Stokes-Navier, 1845 1880’s concentric cylinders (Perry, Mallock, Couette, Schwedoff) Material Approximate Viscosity (Pa - s) Glass 1040 Molten Glass (500°C) 1012 Asphalt 108 Molten polymers 103 Heavy syrup 102 Honey 101 Glycerin 100 Olive oil 10-1 Light oil 10-2 Water 10-3 -5 Air 10 Adapted from Barnes et al. (1989). Familiar materials have a wide range in viscosity dv dvx t = h t yx = h dy dy Bernoulli measured η in shear 1856 capillary (Poiseuille) Newton, 1687 Stokes-Navier, 1845 1880’s concentric cylinders (Perry, Mallock, Couette, Schwedoff) measured in extension 7 Trouton η η u = 3 “A variety of pitch which gave by the traction method λ = 4.3 x 1010 (poise) was found by the torsion method to have a viscosity µ = 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. polystyrene 160°C Münstedt (1980) Goal 2.Put Newton’s Law in 3 dimensions • rate of strain tensor 2D η η • show u = 3 recall Deformation Gradient Tensor, F ˆ x2 Q Q s Motion s' ˆ x1 P P Rest or past state at t' Deformed or present state at t ˆ x3 Separation and displacement of point Q from P Q Q w = x(y , t) + F s w′ w s′ s P y′ y P s = w - y = F s s′ = w′ - y′ s = w - y Viscosity is “proportional to the velocity by which the parts of the fluids are being separated from each other.” —Newton Velocity Gradient Tensor L is the velocity gradient tensor.