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Lecture14Notes - Lecture 14 Viscoelasticity 1 Wednesday,...

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Unformatted text preview: Lecture 14 Viscoelasticity 1 Wednesday, November 17, 2010 Last time… • • • • • We showed the development of the affine, phantom and Mooney-Rivlin models of network deformation We derived an expression for modulus, G=nkT/V; this is (related) to the slope of stress strain curves The classical models fail to capture the effect of entanglements, which are particularly significant if the strand length between crosslinks is large; there is some dependence on preparation condition We discussed viscosity of polymers, introducing a time dependence We discussed how polymers entangle to form temporary crosslinks, and how the time scale of their movement (“reptation”) influences the viscous and elastic properties of the material 2 Wednesday, November 17, 2010 Entropy driven elasticity • • Flory constructed a simple method to separate energetic from entropic contribution to the elastic force Consider a typical temperature dependence of a retractive force, f, for a network of constant volume V at constant elongation L. ⎛ ∂f ⎞ slope = ⎜ ⎟ ⎝ ∂T ⎠V ,L ⎛ ∂E ⎞ fE = ⎜ ⎟ ⎝ ∂L ⎠T ,V ⎛ ∂f ⎞ ⎛ ∂S ⎞ f S = T⎜ ⎟ = −T⎜ ⎟ ⎝ ∂T ⎠V ,L ⎝ ∂L ⎠T ,V ⎛ ∂F ⎞ ⎛ ∂E ⎞ ⎛ ∂S ⎞ f = ⎜ ⎟ = ⎜ ⎟ − T⎜ ⎟ ⎝ ∂L ⎠V ,T ⎝ ∂L ⎠V ,T ⎝ ∂L ⎠V ,T (in ideal networks, this is approximately zero) (in crystalline materials, this is low or approximately zero) 3 Wednesday, November 17, 2010 This time • • • We’ll combine our discussions of viscous and elastic behavior We’ll discuss how to characterize, model and utilize viscoelastic behavior If there’s time we’ll talk about some other polymer properties and then about polymer processing. 4 Wednesday, November 17, 2010 Time dependent Deformation Deformation occurs via motion of polymer chains relative each other Sliding polymer chains past each other takes some time if we apply the load slowly, the chains will slide easily if the load is applied quickly, the chains do not have time to slide This is VISCOELASTIC behavior At high temperatures or low strain rates, polymers have more ductility At low temperatures or high strain rates, polymers have less ductility 5 Wednesday, November 17, 2010 Viscous versus elastic response to deformation 6 Wednesday, November 17, 2010 Viscoelasticity Creep: Permanent deformation to relieve stress. At constant stress, the strain increases with time. Stress Relaxation: Under (sudden) deformation to strain ε0, the stress decays with time, as the force required to maintain the deformation decreases. The amount of stress in the material as a result of a constant deformation decreases with time. Hystersis: non-conservative force; energy loss upon applied stress Creep Stress Relaxation Hysteresis deformation deformation strain spring (Hookean) creep (viscoelastic) stress stress relaxation (viscoelastic) force lost energy spring (Hookean) time Wednesday, November 17, 2010 time deformation 7 Creep • Creep: Permanent deformation to relieve stress. At constant stress, the strain increases with time. 8 Wednesday, November 17, 2010 Example: Creep in Sutures 9 Wednesday, November 17, 2010 Stress Relaxation • Stress Relaxation: Under (sudden) deformation to strain ε0, the stress decays with time, as the force required to maintain the deformation decreases. The amount of stress in the material as a result of a constant deformation decreases with time. 10 Wednesday, November 17, 2010 Example: Stress Relaxation in Bone Cement 11 Wednesday, November 17, 2010 Viscoelasticity In stress relaxation, the modulus is time dependent. The time dependent modulus is simply the time dependent stress divided by the (constant) strain: E (t) = σ (t ) ε0 12 Wednesday, November 17, 2010 Relaxation Modulus The viscoelastic relaxation modulus characterizes the stress relaxation behavior of a polymer Stress relaxation measures stress versus time at constant strain The relaxation modulus is G( t ) ≡ σ (t ) γ With time, the apparent stiffness of the material changes 13 Wednesday, November 17, 2010 How do we characterize this time-dependent behavior? 14 Wednesday, November 17, 2010 Dynamic Mechanical Testing Response for Classical Extremes Purely Elastic Response (Hookean Solid) Purely Viscous Response (Newtonian Liquid) δ = 90° δ = 0° Stress Stress Strain Strain γ ( t ) = γ 0 sin 2πft = γ 0 sin ωt τ ( t ) = Gγ 0 sin ωt d (γ 0 sinωt ) dt τ ( t ) = ηγ 0 cosωt . τ (t) = η γ = η Courtesy of TA Instruments Wednesday, November 17, 2010 15 Dynamic Mechanical Testing: Viscoelastic Material Response γ ( t ) = τ 0 sin(ωt + δ ) τ ( t ) = (τ 0 cosδ ) sinωt + (τ 0 sinδ ) cosωt Phase angle 0° < δ < 90° Strain Stress Courtesy of TA Instruments Wednesday, November 17, 2010 16 DMA Viscoelastic Parameters The Complex, Elastic, & Viscous Stress The stress in a dynamic experiment is referred to as the complex stress τ* The complex stress can be separated into two components: 1) An elastic stress in phase with the strain. τ' = τ*cosδ τ' is the degree to which material behaves like an elastic solid. 2) A viscous stress in phase with the strain rate. τ" = τ*sinδ τ" is the degree to which material behaves like an ideal liquid. Phase angle δ τ* = τ' + i τ" Complex Stress, τ* Strain, ε Wednesday, November 17, 2010 Courtesy: TA Instruments 17 DMA • • τ ( t ) = (τ 0 cosδ ) sin ωt + (τ 0 sin δ ) cosωt Clearly, there is an in-phase and out-of-phase component to this shear stress Since we know modulus is stress divided by strain, we can rewrite the equation: ⎡⎛ τ ⎤ ⎞ ⎛τ0 ⎞ 0 τ ( t ) = γ 0 ⎢⎜ cosδ ⎟ sin ωt + ⎜ sin δ ⎟ cosωt⎥ ⎠ ⎝ γ0 ⎠ ⎣⎝ γ 0 ⎦ • This yields expressions for the in-phase and out-of- phase modulus (moduli): τ0 cos δ γ0 τ G" (ω ) = 0 sin δ γ0 G' (ω ) = τ ( t ) = γ 0 [G' (ω ) sinωt + G" (ω ) cosωt ] Courtesy of TA Instruments Wednesday, November 17, 2010 18 DMA Viscoelastic Parameters τ ( t ) = γ 0 [G' (ω ) sinωt + G" (ω ) cosωt ] The Complex Modulus: Measure of materials overall resistance to deformation. The Elastic (Storage) Modulus: Measure of elasticity of material. The ability of the material to store energy. The Viscous (Loss) Modulus: The ability of the material to dissipate energy. Energy lost as heat. Tan Delta: Measure of material damping - such as vibration or sound damping. G* = Stress*/Strain G* = G’ + iG” G' (ω ) = τ0 cos δ γ0 τ0 G" (ω ) = sin δ γ0 tan δ = G"/G' Courtesy of TA Instruments Wednesday, November 17, 2010 19 Complex and Dynamic Viscosity The viscosity measured in an oscillatory experiment is a Complex Viscosity much the way the modulus can be expressed as the complex modulus. The complex viscosity contains an elastic component and a term similar to the steady state viscosity. The Complex viscosity is defined as: η* = η’ - iη” or η* = G*/ω 20 Wednesday, November 17, 2010 DMA 21 Wednesday, November 17, 2010 Frequency Sweep: Material Response Transition Region log G'and G" Terminal Region Rubbery Plateau Region Glassy Region Storage Modulus (E' or G') Loss Modulus (E" or G") log Frequency (rad/s or Hz) Wednesday, November 17, 2010 Courtesy: TA Instruments 22 Dynamic Temperature Ramp or Step and Hold: Material Response Log G' and G" Glassy Region Transition Region Rubbery Plateau Region Terminal Region Storage Modulus (E' or G') Loss Modulus (E" or G") Temperature Wednesday, November 17, 2010 Courtesy: TA Instruments 23 Viscoelastic Behavior of Amorphous Materials • Polymer properties are time and temperature dependent 24 Wednesday, November 17, 2010 Viscoelastic Behavior of Amorphous Materials • Polymer properties are time and temperature dependent 25 Wednesday, November 17, 2010 Modulus-Temperature Curve from: Dynamic Mechanical Analysis: A Practical Introduction 2nd Edition by Kevin Menard 26 Wednesday, November 17, 2010 How do we describe viscoelastic behaviour? 27 Wednesday, November 17, 2010 Viscoelasticity Material has a viscous component and elastic component; typical of polymers Models of viscoelasticity separate the viscosity component from the elasticity component ε Parallel: Voigt model ∂ε( t ) σ ( t ) = Eε( t ) + η ∂(t) η 1 ∂σ σ ∂ε( t ) += E ∂t η ∂t Serial: Maxwell model η ε (η is viscosity) 28 Wednesday, November 17, 2010 Higher Order Models • • Real materials are characterized by a spectrum of relaxation times More complicated models have been developed to try to describe this behaviour 29 Wednesday, November 17, 2010 Can we predict properties over a broad time range? 30 Wednesday, November 17, 2010 Temperature Dependence on Deformation Increasing T decreases E decreases yield strength increases ductility 31 Wednesday, November 17, 2010 Williams, Landel and Ferry (WLF) model and the Time Temperature Super-Position Principle (TTS) Used for predicting polymer behavior over time, when measurements would otherwise take years. Can only be used for materials above their Tg; below Tg, Arrhenius-type dependence. log aτ = C1 (T − Tg ) 17.44 (T − Tg ) log τ (T ) =− =− log τ (T0 ) 51.6 + T − Tg C2 + (T − Tg ) 10 4 T1 (-90 C) log a( !) T2 (-85 C) T Tg T3 (-75 C) 3 10 E(t) T 4 (- 70 C) T 5 (- 65 C) T6 (-60 C) 2 10 1 10 !ref ! 10 T7 (-50 C) T8 (-45 C) T9 (-35 C) 0 10 log t -5 -4 10 10 -3 10 -2 -1 10 10 0 10 1 2 10 3 10 10 4 5 10 6 10 10 7 time Free volume justification: Wednesday, November 17, 2010 f = f g + (T − Tg )α f aτ = τ (T ) η(T ) =− τ (Tg ) η(Tg ) 32 Other Properties to Consider 33 Wednesday, November 17, 2010 Other Physical Properties Brittleness - materials that are brittle experience little or no plastic deformation upon fracture (~5% or less fracture strain) Ductility- a measure of the degree/extent of plastic deformation upon fracture Resilience - the capacity of a material to absorb energy on elastic deformation and then allow this energy to be recovered on unloading (energy storage) Hardness - resistance to localized plastic deformation. Rockwell, Shore (Durometer) tests Toughness - energy to break a unit volume of material, approximately the area under the stress-strain curve 34 Wednesday, November 17, 2010 Ductility • Amount of plastic deformation prior to failure – % elongation brittle – % reduction in area ⎛ A0 − A f % RA = ⎜ ⎜A 0 ⎝ ductile Stress ⎛ l f − l0 ⎞ % EL = ⎜ ⎜ l ⎟ ×100 ⎟ ⎝0⎠ ⎞ ⎟ ×100 ⎟ ⎠ strain Temperature dependent! Wednesday, November 17, 2010 l0 35 Resilience • • Capacity to absorb and release energy during elastic deformation Modulus of resilience Ur σy = strain energy per unit volume εy U r = ∫ σ dε for linear elastic materials 1 U r = σ yε y 2 Stress 0 0.002 εy strain 36 Wednesday, November 17, 2010 Toughness • Capacity to absorb and release energy prior to fracture Low strain rate: toughness = ∫ εf 0 σ dε Stress • σy 0.002 εf strain 37 Wednesday, November 17, 2010 Hardness test simple (easy prep) – need smooth, flat surface – stay away from edge & from adjacent indentations inexpensive less-destructive can estimate Tensile Strength from hardness – both measure resistance to plastic deformation TS [MPa ]= 3.45 × HB TS [psi ]= 500 × HB 38 Wednesday, November 17, 2010 Polymer Processing 39 Wednesday, November 17, 2010 Polymer Additives Improve mechanical properties, processability, durability, etc. • Fillers – Added to improve tensile strength & abrasion resistance, toughness & decrease cost – ex: carbon black, silica gel, wood flour, glass, limestone, talc, etc. • Plasticizers – Added to reduce the glass transition temperature Tg – commonly added to PVC - otherwise it is brittle 40 Wednesday, November 17, 2010 Polymer Additives • Stabilizers – Antioxidants – UV protectants • Lubricants – Added to allow easier processing – “slides” through dies easier – ex: Na stearate • Colorants – Dyes or pigments • Flame Retardants – Cl/F & B 41 Wednesday, November 17, 2010 Bioactive Contaminants Leach from Disposable Laboratory Plasticware G. Reid McDonald, Alan L. Hudson, Susan M. J. Dunn, Haitao You, Glen B. Baker, Randy M. Whittal, Jonathan W. Martin, Amitabh Jha, Dale E. Edmondson, Andrew Holt Science 2008, 322, 917 42 Wednesday, November 17, 2010 Bisphenol-A as endocrine disruptor • BPA is used as a monomer in some polycarbonates, as well as a plasticizer for many plastics • BPA can mimic effect of estrogen? 43 Wednesday, November 17, 2010 Processing of Plastics • Thermoplastic – – can be reversibly cooled & reheated, i.e. recycled – heat till soft, shape as desired, then cool – ex: polyethylene, polypropylene, polystyrene, etc. • Thermoset – when heated forms a network – degrades (not melts) when heated – mold the prepolymer then allow further reaction – ex: urethane, epoxy 44 Wednesday, November 17, 2010 Processing Plastics - Molding • Compression and transfer molding – thermoplastic or thermoset Adapted from Fig. 15.23, Callister 7e. (Fig. 15.23 is from F.W. Billmeyer, Jr., Textbook of Polymer Science, 3rd ed., John Wiley & Sons, 1984. ) 45 Wednesday, November 17, 2010 Processing Plastics - Molding • Injection molding – thermoplastic & some thermosets Adapted from Fig. 15.24, Callister 7e. (Fig. 15.24 is from F.W. Billmeyer, Jr., Textbook of Polymer Science, 2nd edition, John Wiley & Sons, 1971. ) 46 Wednesday, November 17, 2010 Processing Plastics – Extrusion Adapted from Fig. 15.25, Callister 7e. (Fig. 15.25 is from Encyclopædia Britannica, 1997.) 47 Wednesday, November 17, 2010 Polymer Types: Fibers Fibers - length/diameter >100 • Textiles are main use – Must have high tensile strength – Usually highly crystalline & highly polar • Formed by spinning – ex: extrude polymer through a spinnerette • Pt plate with 1000’s of holes for nylon • ex: rayon – dissolved in solvent then pumped through die head to make fibers – the fibers are drawn – leads to highly aligned chains- fibrillar structure 48 Wednesday, November 17, 2010 Polymer Types • Coatings – thin film on surface – i.e. paint, varnish – To protect item – Improve appearance – Electrical insulation • Adhesives – produce bond between two adherands – Usually bonded by: 1. Secondary bonds 2. Mechanical bonding • Films – blown film extrusion • Foams – gas bubbles in plastic 49 Wednesday, November 17, 2010 Blown-Film Extrusion Adapted from Fig. 15.26, Callister 7e. (Fig. 15.26 is from Encyclopædia Britannica, 1997.) 50 Wednesday, November 17, 2010 Next time... • An important class of viscoelastic polymeric materials: hydrogels! 51 Wednesday, November 17, 2010 ...
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This note was uploaded on 11/29/2010 for the course BME 104 taught by Professor Kasko during the Fall '10 term at UCLA.

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