3 A Demonstration of Polymer Viscoelasticity Poly(ethylene oxide) in water
4 “Memory” of Previous State Poly(styrene)Tg ~ 100 °C
5 Chapter 5. Viscoelasticity Is “silly putty” a solid or a liquid?Why do some injection molded parts warp?What is the source of the die swell phenomena that is often observed in extrusion processing?Expansion of a jetof an 8 wt% solution ofpolyisobutylene in decalinUnder what circumstances am I justified in ignoring viscoelastic effects?
6 Rheology is the science of flow and deformation of matter What is Rheology?Rheology is the science of flow and deformation of matterRheology Concepts, Methods, & Applications, A.Y. Malkin and A.I. Isayev; ChemTec Publishing, 2006
10 Time dependent processes: Viscoelasticity The response of polymeric liquids, such as melts and solutions, to an imposed stress may resemble the behavior of a solid or a liquid, depending on the situation.
13 Network of Entanglements There is a direct analogy between chemical crosslinks in rubbers and “physical” crosslinks that are created by the entanglements.The physical entanglements can support stress (for short periods up to a time tT), creating a “transient” network.
14 Entanglement Molecular Weights, Me, for Various Polymers Me (g/mole)Poly(ethylene) 1,250Poly(butadiene) 1,700Poly(vinyl acetate) 6,900Poly(dimethyl siloxane) 8,100Poly(styrene) 19,000
15 Pitch drop experimentStarted in 1927 by University of Queensland Professor Thomas Parnell.A drop of pitch falls every 9 yearsPitch drop experiment apparatusPitch can be shattered by a hammer
16 Viscoelasticity and Stress Relaxation Whereas steady-shear measurements probe material responses under a steady-state condition, creep and stress relaxation monitor material responses as a function of time.Stress relaxation studies the effect of a step-change in strain on stress.?
17 Physical Meaning of the Relaxation Time Constant strain appliedtimesStress relaxes over time as molecules re-arrangetimeStress relaxation:
18 Introduction to Viscoelasticity All viscous liquids deform continuously under the influence of an applied stress – They exhibit viscous behavior.Solids deform under an applied stress, but soon reach a position of equilibrium, in which further deformation ceases. If the stress is removed they recover their original shape – They exhibit elastic behavior.Viscoelastic fluids can exhibit both viscosity and elasticity, depending on the conditions.Viscous fluidViscoelastic fluidElastic solidPolymers display VISCOELASTIC properties
19 Static Testing of Rubber Vulcanizates Static tensile tests measure retractive stress at a constant elongation (strain) rate.Both strain rate and temperature influence the resultNote that at common static test conditions, vulcanized elastomers store energy efficiently, with little loss of inputted energy.
20 Dynamic Testing of Rubber Vulcanizates: Resilience Resilience tests reflect the ability of an elastomeric compound to store and return energy at a given frequency and temperature.Change of reboundresilience (h/ho) withtemperature T for:1. cis-poly(isoprene);2. poly(isobutylene);3. poly(chloroprene);4. poly(methyl methacrylate).
21 Hooke and NewtonIt is difficult to predict the creep and stress relaxation for polymeric materials.It is easier to predict the behaviour of polymeric materials with the assumption it behaves as linear viscoelastic behaviour.Deformation of polymeric materials can be divided to two components:Elastic component – Hooke’s lawViscous component – Newton’s lawDeformation of polymeric materials combination of Hooke’s law and Newton’s law.
22 Hooke’s law & Newton’s Law The behaviour of linear elastic were given by Hooke’s law:orThe behaviour of linear viscous were given by Newton’s Law:E= Elastic moduluss = Stresse = strainde/dt = strain rateds/dt = stress rate= viscosity** This equation only applicable at low strain
23 Viscoelasticity and Stress Relaxation Stress relaxation can be measured by shearing the polymer melt in a viscometer (for example cone-and-plate or parallel plate). If the rotation is suddenly stopped, ie. g=0, the measured stress will not fall to zero instantaneously, but will decay in an exponential manner..Relaxation is slower for Polymer B than for Polymer A, as a result of greater elasticity.These differences may arise from polymer microstructure (molecular weight, branching).
24 STRESS RELAXATIONCREEPConstant strain is applied the stress relaxes as function of timeConstant stress is applied the strain relaxes as function of time
25 Time-dependent behavior of Polymers The response of polymeric liquids, such as melts and solutions, to an imposed stress may under certain conditions resemble the behavior of a solid or a liquid, depending on the situation.Reiner used the biblical expression that “mountains flowed in front of God” to define the DEBORAH number
27 Static Modulus of Amorphous PS GlassyLeatheryRubberyViscousPolystyreneStress applied at xand removed at y
28 Stress Relaxation Test StrainElasticStressStressViscoelasticStressViscous fluidViscous fluidViscous fluidTime, t
29 Stress relaxation Go (or GNo) is the “plateau modulus”: Stress relaxation after a step strain go is the fundamental way in which we define the relaxation modulus:Go (or GNo) is the “plateau modulus”:where Me is the average mol. weight between entanglementsG(t) is defined for shear flow. We can also define a relaxation modulus for extension:
30 Stress relaxation of an uncrosslinked melt perseGlassy behaviorTransition ZoneTerminal Zone(flow region)slope = -1Plateau ZoneMc: critical molecular weight above which entanglements exist3.24
32 Network of Entanglements There is a direct analogy between chemical crosslinks in rubbers and “physical” crosslinks that are created by the entanglements.The physical entanglements can support stress (for short periods up to a time tT), creating a “transient” network.
34 Viscosity of Polymer Melts hoExtrapolation to low shear rates gives us a value of the “zero-shear-rate viscosity”, ho.Shear thinning behaviourPoly(butylene terephthalate) at 285 ºCFor comparison: h for water is 10-3 Pa s at room temperature.
35 Rheology and Entanglements. The elastic properties of linear thermo-plastic polymers are due to chain entanglements. Entanglements will only occur above a critical molecular weight.When plotting melt viscosity o against molecular weight we see a change of slope from 1 to 3.45 at the critical entanglement molecular weight.oMnSlope = 1Slope = 3.4Entanglementmolecular weight
36 h ~ tTGP Scaling of Viscosity: ho ~ N3.4 ho ~ N3.4 N0 ~ N3.4 Why? 3.4 Data shifted for clarity!Viscosity is shear-strain rate dependent. Usually measure in the limit of a low shear rate: hoh ~ tTGP3.4ho ~ N3.4 N0 ~ N3.4Universal behaviour for linear polymer meltsApplies for higher N: N>NCWhy?G.Strobl, The Physics of Polymers, p. 221
40 Mechanical ModelMethods that used to predict the behaviour of visco-elasticity.They consist of a combination of between elastic behaviour and viscous behaviour.Two basic elements that been used in this model:Elastic spring with modulus which follows Hooke’s lawViscous dashpots with viscosity h which follows Newton’s law.The models are used to explain the phenomena creep and stress relaxation of polymers involved with different combination of this two basic elements.
41 Dynamic Viscosity (dashpot) Shear stressLack of slipperinessResistance to flowInterlayer frictionSI Unit: Pascal-secondShear rate1 centi-Poise = milli Pascal-second
42 stressstress inputStrain in dashpotdashpot27/06/46
43 Maxwell model In series Viscous strain remains after load removal. stress inputModelStrain ResponseMaxwell model27/06/46
44 Kelvin or Voigt model In parallel Nonlinear increase in strain with timeStrain decreases with time after load removal because of the action of the spring (and dashpot).stress inputModelStrain ResponseVoigt model27/06/46
47 The Theory of Viscoelasticity The liquid behavior can be simply represented by the Newtonian model. We can represent the Newtonian behavior by using a “dashpot” mechanical analog:The simplest elastic solid model is the Hookean model, which we can represent by the “spring” mechanical analog.The theory of linear viscoelasticity is phenomenological; no first principle prediction exists.The aim of the theory is to link properties in one circumstance to the measurable in the other.The basic components in the theory are the spring and the dashpotstressstrainviscosityGmodulus
48 Maxwell Model Let’s create a VISCOELASTIC material: At least two components are needed, one to characterize elastic and the other viscous behavior. One such model is the Maxwell model:stressstrainviscosityGmodulus
49 Maxwell Model Let’s try to deform the Maxwell element stress strainviscosityGmodulus
50 Maxwell model too primitive Maxwell: solid lineExperiment: circlesMaxwell model too primitive
51 Maxwell Model l is the relaxation time . Exponential decay in stresses The deformation rate of the Maxwell model is equal to the sum of the individual deformation rates:l is the relaxation timeIf the mechanical model is suddenly extended to a position and held there (g=const., g=0):.Exponential decay in stressesstressstrainviscosityGmodulus
52 Examples of Viscoelastic Materials Mattress, PillowTissue, skin27/06/46
53 ElasticViscousThe common mechanical model that use to explain the viscoelastic phenomena are:MaxwellSpring and dashpot align in seriesVoigtSpring and dashpot align in parallelStandard linear solidOne Maxwell model and one spring align in parallel.
59 Dynamic Mechanical Testing Viscoelastic Material Response Phase angle 0° < d < 90°StrainStressCourtesy: TA Instruments
60 DMA Viscoelastic Parameters: The Complex, Elastic, & Viscous Stress The stress in a dynamic experiment is referred to as the complex stress *The complex stress can be separated into two components:1) An elastic stress in phase with the strain. ' = *cos' is the degree to which material behaves like an elastic solid.2) A viscous stress in phase with the strain rate. " = *sin" is the degree to which material behaves like an ideal liquid.Phase angle dComplex Stress, ** = ' + i"Courtesy: TA InstrumentsStrain,
61 DMA Viscoelastic Parameters The Complex Modulus: Measure of materials overall resistance to deformation.G* = Stress*/StrainG* = G’ + iG”The Elastic (Storage) Modulus: Measure of elasticity of material. The ability of the material to store energy.G' = (stress*/strain)cosThe Viscous (loss) Modulus:The ability of the material to dissipate energy. Energy lost as heat.G" = (stress*/strain)sinTan Delta:Measure of material damping - such as vibration or sound damping.Tan = G"/G'Courtesy: TA Instruments
62 DMA Viscoelastic Parameters: Damping, tan Dynamic measurement represented as a vectorIt can be seen here that G* = (G’2 +G”2)1/2G"Phase angle G'The tangent of the phase angle is the ratio of the loss modulus to the storage modulus.tan = G"/G'"TAN DELTA" (tan )is a measure of the damping ability of the material.Courtesy: TA Instruments
63 Frequency Sweep: Material Response TransitionRegionRubbery PlateauRegionTerminal RegionGlassy Regionlog G'and G"12Storage Modulus (E' or G')Loss Modulus (E" or G")log Frequency (rad/s or Hz)Courtesy: TA Instruments
64 Viscoelasticity in Uncrosslinked, Amorphous Polymers Logarithmic plots of G’ and G” against angular frequency for uncrosslinked poly(n-octyl methacrylate) at 100°C (above Tg), molecular weight 3.6x106.
65 Dynamic Characteristics of Rubber Compounds Why do E’ and E” vary with frequency and temperature?The extent to which a polymer chains can store/dissipate energy depends on the rate at which the chain can alter its conformation and its entanglements relative to the frequency of the load.Terminal Zone:Period of oscillation is so long that chains can snake through their entanglement constraints and completely rearrange their conformationsPlateau Zone:Strain is accommodated by entropic changes to polymer segments between entanglements, providing good elastic responseTransition Zone:The period of oscillation is becoming too short to allow for complete rearrangement of chain conformation. Enough mobility is present for substantial friction between chain segments.Glassy Zone:No configurational rearrangements occur within the period of oscillation. Stress response to a given strain is high (glass-like solid) and tand is on the order of 0.1
66 Dynamic Temperature Ramp or Step and Hold: Material Response Glassy RegionTransitionRegionRubbery PlateauRegionTerminal RegionLog G' and G"1Loss Modulus (E" or G")Storage Modulus (E' or G')2TemperatureCourtesy: TA Instruments
67 One more time: Dynamic (Oscillatory) Testing In the general case when the sample is deformed sinusoidally, as a response the stress will also oscillate sinusoidally at the same frequency, but in general will be shifted by a phase angle d with respect to the strain wave. The phase angle will depend on the nature of the material (viscous, elastic or viscoelastic)InputResponsewhere 0°<d<90°stressstrainviscosityGmodulus3.29
68 One more time: Dynamic (Oscillatory) Testing By using trigonometry:(3-1)In-phase component of the stress, representing solid-like behaviorOut-of-phase component of the stress, representing liquid-like behaviorLet’s define:where:3.30
69 Physical Meaning of G’, G” Equation (3-1) becomes:We can also define the loss tangent:For solid-like response:For liquid-like response:G’storage modulusG’’loss modulus
70 Typical Oscillatory Data G’G’’log Glog RubberG’storage modulusG’’loss modulusRubbers – Viscoelastic solid response:G’ > G” over the whole range of frequencies
71 Typical Oscillatory Data G’G’’log Glog Melt or solutionG0G’storage modulusG’’loss modulusPolymeric liquids (solutions or melts) Viscoelastic liquid response:G” > G’ at low frequenciesResponse becomes solid-like at high frequenciesG’ shows a plateau modulus and decreases with w-2 in the limit of low frequency (terminal region)G” decreases with w-1 in the limit of low frequency
72 Typical Oscillatory Data For Rubbers – Viscoelastic solid response:G’ > G” over the whole range of frequenciesFor polymeric liquids (solutions or melts) – Viscoelastic liquid response:G”>G’ at low frequenciesResponse becomes solid-like at high frequenciesG’ shows a plateau modulus and decreases with w-2 in the limit of low frequency (terminal region)G” decreases with w-1 in the limit of low frequency
74 Sample is strained (pulled, ) rapidly to pre-determined strain () Stress required to maintain this strain over time is measured at constant TStress decreases with time due to molecular relaxation processesRelaxation modulus defined as:Er(t) also a function of temperatureEr(t) = (t)/e0
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