(c) G.H. McKinley 2019 8/21/19 DO NOT DUPLICATE Linear Viscoelasticity (its all a matter of time) Gareth H. McKinley Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, Massachusetts 02139 Including material from: Randy Ewoldt, University of Illinois Gerry Fuller, Stanford University Tim Lodge, U. Minnesota Introduction to Viscoelasticity | Göteborg, Sweden | Aug. 21 2019LVE-1 Key References • Rheology: Principles, Measurements and Applications – C.W. Macosko (Wiley, 1994) • Viscoelastic Properties of Polymers – J.D. Ferry (3rd Edition, 1980) LVE-2 NRC Rheology Short Course 2019 1 (c) G.H. McKinley 2019 8/21/19 DO NOT DUPLICATE The Goal of Today… (in honor of 20th anniversary) VISCOELASTICITY • “I’m going to learn Jiu-Jitsu VISCOELASTICITY? Neo(Hookean) How to Learn Viscoelasticity Before… After… want some more??? LVE-3 So What is a Complex Fluid? Isaac Newton, FRS 1672 • Complex fluids possess an underlying microstructure that can be affected by (and in turn then affect) a flow field – Gives rise to Visco-Elasticity • Examples include: – Polymer solutions, polymer melts, liquid crystals – Foams, gels, bubbly-liquids, – Suspensions, emulsions, slurries, mud.. – Food stuffs, paints, adhesives and other consumer products Non- • Basically everything except air, oil, water! Newtonian • These fluids violate Newton’s law of viscosity : Robert Hooke, FRS 1663 ∂vx t τ yx = µ τ = µ{∇v + ∇v } Neo-Hookean!!! ∂y •Rheology: study of the material properties of complex fluids in specified/known flow fields •Non-Newtonian Fluid Dynamics: self-consistent solutions of conservation of mass, momentum PLUS a constitutive model (rheological equation of state) For example: “Neo-Hookean Dumbbell Model” *Posthumous Portrait by Rita Greer (Wikipedia) NRC Rheology Short Course 2019 2 (c) G.H. McKinley 2019 8/21/19 DO NOT DUPLICATE Four Key Rheological Phenomena shear Impact time force G(t) : viscoelasticity η(γ!) shear thinning torque extension shear N ( ) 1 γ! shear normal stresses ηE (ε!) strain hardening Image from Chris Macosko’s book 5 Rheology = study of deformation of complex materials EXAMPLES: Dilute Polymer Solution (oil additive) Entangled Polymers (polyethylene melt) Surfactant Solution (body wash) Emulsion Suspension Gel (mayonnaise) (latex paint) (gelatin) 6 NRC Rheology Short Course 2019 3 (c) G.H. McKinley 2019 8/21/19 DO NOT DUPLICATE Probing the Viscoelastic Response of Polymeric Materials Natural Time Scale of Complex Fluids • Natural time scale λ ≡ τ material ≈ 10 sec. • The Deborah number is a dimensionless measure compared with the time scale of the deformation... http://www.sillyputty.com Images courtesy of Cambridge Polymer Group De < 1 Viscous Liquid De ~ 1 Viscoplastic ‘Solid’ De ~ 10 Elastic Solid (Relaxation time of the material) τ material De ~ = (The timescale of the process) T process Brittle Solid De ~1000! ©MIT; Harold.EdGerton Strobe Lab. NRC Rheology Short Course 2019 4 (c) G.H. McKinley 2019 8/21/19 DO NOT DUPLICATE Origin of the Deborah number see M. Reiner, Physics Today, January 1964, p. 62 Judges 5:5 9 A creep test: A creep test (constant load) on silly putty Another creep test (silicone gum!!) From Youtube: LVE-10 NRC Rheology Short Course 2019 5 (c) G.H. McKinley 2019 8/21/19 DO NOT DUPLICATE USGS Image Library “Byrd Glacier Flow”. Photo: Dr Bill Servais (http://www.rosssea.info/glaciers.html) 11 10mm 0˚ 30˚ 60˚ 90˚ Elastic increasing amplitude (of applied stress and/or deformation) Viscous “Linear viscoelasticity” is the limit of small loading amplitude (does not disrupt microstructure) ideal elastic : ideal viscous : viscoelastic dγ σ = Gγ ! σ = ⌘ = ⌘γ˙ dt NRC Rheology Short Course 2019 6 (c) G.H. McKinley 2019 8/21/19 DO NOT DUPLICATE ideal elastic : ideal viscous : dγ σ = Gγ viscoelastic σ = ⌘ = ⌘γ˙ stress strain dt (old literature: elastico-viscous)! strain-rate “shear rate” Basic tests: apply measure calculate γ Step strain: γ 0 s G modulus t t t s s0 γ J Creep: compliance t t t γ s G’ Dynamic: G” complex wt wt moduli w Stress Relaxation Modulus, G(t) Mathematically: g Impose: step strain g 0 = constant Heaviside Step Function (How? Use a rheometer) time σ(t, γ0) Observe : stress σ(t, γ0) Material function : G(t, γ0)= γ0 log G st Linear response: g small enough so that G(t) is independent of g t LVElog-14 t NRC Rheology Short Course 2019 7 (c) G.H. McKinley 2019 8/21/19 DO NOT DUPLICATE Stress Relaxation Modulus, G(t), Limiting Cases τs Hookean σ(t, γ0) G(t, γ0)= γ0 Experimental difficulty: True “step” displacement impossible (instrument inertia at short times) time τ sτ s Newtonian viscoelastic solid viscoelastic liquid time time LVE-15 Maxwell Viscoelastic Model • Let’s motivate these linear tests using a simple model • Elastic solid: σ e = G γ spring • Viscous liquid: σv = ⌘γ˙ dashpot • The Maxwell model is a superposition of these two: total strain : γ = γe + γv Go ηo stress is equal : σ = σe = σv ⌘0 dσ dγ dσ dγ σ + = ⌘0 σ + λ = ⌘0 G0 dt dt ! dt dt λ = ⌘ /G : relaxation time 0 0 16 NRC Rheology Short Course 2019 8 (c) G.H. McKinley 2019 8/21/19 DO NOT DUPLICATE Example: Maxwell Model in Step Strain • In a step-strain experiment, the material is subjected to an instantaneous strain of magnitude go • Following the step, the strain is held fixed at this constant value and the stress is monitored in time. • The initial stress is then G0go • Solving the Maxwell model, we have 0 dσ dγ σ + λ = ⌘ dt 0 dt t/λ t/λ σ = σ(t = 0)e− = G0γ0e− • Dividing the stress by the strain, we get the “relaxation modulus” t/λ • No strain dependence G(t)=G0e− • Single relaxation time t/λ G e− γ γ0 G(t) 0 t t 17 Comparing with Measurements of G(t) 1. Impose: step strain g 0 = constant g 2. Observe : stress σ(t, γ0) σ(t, γ0) time 3. Material function : G(t, γ0)= γ0 t/λ 4. Choose a model and fit to data; e.g. Maxwell G(t)=G0e− Modulus vs. time: AP Example: Wormlike micellar solution (e.g. hair Modulusshampoo) vs. time: AP G 0 G G0 0 G0 NRC Rheology Short Course 2019 9 (c) G.H. McKinley 2019 8/21/19 DO NOT DUPLICATE There are many other possible models… Relaxation and creep function of (A) Maxwell, (B) Voigt, and (C) Kelvin model (standard linear solid). Eiji Tanaka, and Theo van Eijden CROBM 2003;14:138-150 Copyright © by International & American Associations for Dental Research Generalized Maxwell Model G(t) G exp( t / ) = ∑ k − λk k A spectrum of relaxation times {λk} “Prony Series” Extract spectrum by inversion (tricky) or Fit to model e.g. for polymers, Rouse, Zimm, reptation… G(t) usually follows a kind of “equipartition”: ρ N G(t) av k T exp( t / ) = B ∑ − λk M k other models: G(t) = At −n power-law spectra, near gel point LVE-20 NRC Rheology Short Course 2019 10 (c) G.H. McKinley 2019 8/21/19 DO NOT DUPLICATE Fitting to a sum of exponentials A spectrum of relaxation modes {λk} “Prony Series” “Mechanical Spectroscopy” Gi k =1 k =1 G(t) k =2 k =3 k =4 k =N ... lk t LVE-21 Creep Compliance, J(t) s s0 Impose : step stress σ0 = constant time γ(t, σ0) Observe : strain γ(t, σ0) Material function : J(t, σ0)= σ0 J viscoelastic liquid ϕ = 1/η γ˙ (t, σ0) dJ Fluidity : φ(t, σ0)= = σ0 dt viscoelastic solid time LVE-22 NRC Rheology Short Course 2019 11.