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What happens to Tg with increasing pressure?

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Why? Bar = 1 atm = 100 kPa

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Poly(ethylene oxide) in water A Demonstration of Polymer Viscoelasticity

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“Memory” of Previous State Poly(styrene) T g ~ 100 °C

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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 jet of an 8 wt% solution of polyisobutylene in decalin Under what circumstances am I justified in ignoring viscoelastic effects?

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What is Rheology? Rheology is the science of flow and deformation of matter Rheology Concepts, Methods, & Applications, A.Y. Malkin and A.I. Isayev; ChemTec Publishing, 2006

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Temperature & Strain Rate

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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.

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Stress Strain increasing loading rate

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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 T ), creating a “transient” network.

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Entanglement Molecular Weights, M e, for Various Polymers Poly(ethylene)1,250 Poly(butadiene)1,700 Poly(vinyl acetate)6,900 Poly(dimethyl siloxane)8,100 Poly(styrene)19,000 M e (g/mole)

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Pitch drop experiment Started in 1927 by University of Queensland Professor Thomas Parnell. A drop of pitch falls every 9 years Pitch can be shattered by a hammer Pitch drop experiment apparatus

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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. ?

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Physical Meaning of the Relaxation Time time Constant strain applied Stress relaxes over time as molecules re-arrange time Stress relaxation:

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Introduction to Viscoelasticity Polymers display VISCOELASTIC properties 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 fluid Viscoelastic fluid Elastic solid

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Static Testing of Rubber Vulcanizates Static tensile tests measure retractive stress at a constant elongation (strain) rate. –Both strain rate and temperature influence the result Note that at common static test conditions, vulcanized elastomers store energy efficiently, with little loss of inputted energy.

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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 rebound resilience (h/h o ) with temperature T for: 1. cis-poly(isoprene); 2. poly(isobutylene); 3. poly(chloroprene); 4. poly(methyl methacrylate).

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It 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 law Viscous component – Newton’s law Deformation of polymeric materials combination of Hooke’s law and Newton’s law. Hooke and Newton

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The behaviour of linear elastic were given by Hooke’s law: E = Elastic modulus = Stress e = strain de/dt = strain rate d /dt = stress rate = viscosity or The behaviour of linear viscous were given by Newton’s Law: ** This equation only applicable at low strain Hooke’s law & Newton’s Law

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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. =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).

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CREEP STRESS RELAXATION Constant strain is applied the stress relaxes as function of time Constant stress is applied the strain relaxes as function of time

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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

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metal elastomer Viscous liquid

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Static Modulus of Amorphous PS Glassy Leathery Rubbery Viscous Polystyrene Stress applied at x and removed at y

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Stress Relaxation Test Time, t Strain Stress Elastic Viscoelastic Viscous fluid 0 Stress Viscous fluid

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Stress relaxation Stress relaxation after a step strain o is the fundamental way in which we define the relaxation modulus: G o (or G N o ) is the “plateau modulus”: where M e is the average mol. weight between entanglements G(t) is defined for shear flow. We can also define a relaxation modulus for extension:

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Stress relaxation of an uncrosslinked melt M c : critical molecular weight above which entanglements exist perse Glassy behavior Transition Zone Terminal Zone (flow region) slope = -1 Plateau Zone 3.24

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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 T ), creating a “transient” network.

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Relaxation Modulus for Polymer Melts Viscous flow Elastic T = terminal relaxation time

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Viscosity of Polymer Melts Poly(butylene terephthalate) at 285 ºC For comparison: for water is 10 -3 Pa s at room temperature. Shear thinning behaviour Extrapolation to low shear rates gives us a value of the “zero-shear- rate viscosity”, o. oo

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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. oo Mn Slope = 1 Slope = 3.4 Entanglement molecular weight

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Scaling of Viscosity: ~ N 3.4 ~ T G P ~ N 3.4 N 0 ~ N 3.4 Universal behaviour for linear polymer melts Applies for higher N: N>N C Why? G.Strobl, The Physics of Polymers, p. 221 Data shifted for clarity! Viscosity is shear-strain rate dependent. Usually measure in the limit of a low shear rate: o 3.4

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Application of Theory: Electrophoresis From Giant Molecules

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Methods 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: 1.Elastic spring with modulus which follows Hooke’s law 2.Viscous dashpots with viscosity 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. Mechanical Model

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Dynamic Viscosity (dashpot) 1 centi-Poise = milli Pascal-second SI Unit: Pascal-second Shear stress Shear rate Lack of slipperiness Lack of slipperiness Resistance to flow Resistance to flow Interlayer friction Interlayer friction

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27/06/4642 stress input dashpot stress Strain in dashpot

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27/06/4643 Maxwell model In series Viscous strain remains after load removal. stress input ModelStrain Response Maxwell model

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27/06/4644 Kelvin or Voigt model In parallel Nonlinear increase in strain with time Strain decreases with time after load removal because of the action of the spring (and dashpot). stress input ModelStrain Response Voigt model

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Typical Viscosities (Pa. s) Asphalt Binder --------------- Polymer Melt ----------------- Molasses ---------------------- Liquid Honey ----------------- Glycerol ----------------------- Olive Oil ----------------------- Water -------------------------- Acetic Acid -------------------- 100,000 1,000 100 10 1 0.01 0.001 0.00001 Courtesy: TA Instruments

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Shear stress Shear rate Newtonian Pseudoplastic (or Shear thinning)Dilatant (or Shear thickening) Bingham Plastic Casson Plastic Non Newtonian Fluids

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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. stress strain viscosityG modulus

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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: stress strain viscosityG modulus

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Maxwell Model Let’s try to deform the Maxwell element stress strain viscosityG modulus

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Maxwell: solid line Experiment: circles Maxwell model too primitive

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Maxwell Model The deformation rate of the Maxwell model is equal to the sum of the individual deformation rates: is the relaxation time If the mechanical model is suddenly extended to a position and held there ( =const., =0):. Exponential decay in stresses stress strain viscosityG modulus

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27/06/4652 Examples of Viscoelastic Materials Mattress, Pillow Tissue, skin

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The common mechanical model that use to explain the viscoelastic phenomena are: 1.Maxwell Spring and dashpot align in series 2.Voigt Spring and dashpot align in parallel 3.Standard linear solid One Maxwell model and one spring align in parallel. Elastic Viscous

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Measurements of Shear Viscosity Melt Flow Index Capillary Rheometer Coaxial Cylinder Viscometer (Couette) Cone and Plate Viscometer (Weissenberg rheogoniometer) Disk-Plate (or parallel plate) viscometer

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Weissenberg Effect

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Dough Climbing: Weissenberg Effect Other effects: Barus Kaye

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Courtesy: Dr. Osvaldo Campanella

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Dynamic Mechanical Testing Response for Classical Extremes Stress Strain = 0° = 90° Purely Elastic Response (Hookean Solid) Purely Viscous Response (Newtonian Liquid) Stress Strain Courtesy: TA Instruments

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Dynamic Mechanical Testing Viscoelastic Material Response Phase angle 0° < < 90° Strain Stress Courtesy: TA Instruments

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DMA Viscoelastic Parameters: The Complex, Elastic, & Viscous Stress The stress in a dynamic experiment is referred to as the complex stress * Phase angle Complex Stress, * Strain, * = ' + i " l 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. Courtesy: TA Instruments

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DMA Viscoelastic Parameters The Elastic (Storage) Modulus: Measure of elasticity of material. The ability of the material to store energy. G' = (stress*/strain)cos G" = (stress*/strain)sin The Viscous (loss) Modulus: The ability of the material to dissipate energy. Energy lost as heat. The Complex Modulus: Measure of materials overall resistance to deformation. G* = Stress*/Strain G* = G’ + iG” Tan = G"/G' Tan Delta: Measure of material damping - such as vibration or sound damping. Courtesy: TA Instruments

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DMA Viscoelastic Parameters: Damping, tan Phase angle G* G' G" Dynamic measurement represented as a vector It can be seen here that G* = (G’ 2 +G” 2 ) 1/2 l 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

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Frequency Sweep: Material Response Terminal Region Rubbery Plateau Region Transition Region Glassy Region 1 2 Storage Modulus (E' or G') Loss Modulus (E" or G") log Frequency (rad/s or Hz) log G'and G" Courtesy: TA Instruments

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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 3.6x10 6.

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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 conformations Plateau Zone: –Strain is accommodated by entropic changes to polymer segments between entanglements, providing good elastic response Transition 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 tan is on the order of 0.1

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Dynamic Temperature Ramp or Step and Hold: Material Response Temperature Terminal Region Rubbery Plateau Region Transition Region Glassy Region 1 2 Loss Modulus (E" or G") Storage Modulus (E' or G') Log G' and G" Courtesy: TA Instruments

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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 with respect to the strain wave. The phase angle will depend on the nature of the material (viscous, elastic or viscoelastic) Input Response where 0°< <90° 3.29 stress strain viscosityG modulus

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One more time: Dynamic (Oscillatory) Testing By using trigonometry: Let’s define: In-phase component of the stress, representing solid-like behavior Out-of-phase component of the stress, representing liquid-like behavior where: (3-1) 3.30

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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

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Typical Oscillatory Data Rubbers – Viscoelastic solid response: G’ > G” over the whole range of frequencies G’ G’’ log G log Rubber G’ storage modulus G’’ loss modulus

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Typical Oscillatory Data Polymeric liquids (solutions or melts) Viscoelastic liquid response: G” > G’ at low frequencies Response becomes solid-like at high frequencies G’ shows a plateau modulus and decreases with -2 in the limit of low frequency (terminal region) G” decreases with -1 in the limit of low frequency G’ G’’ log G log Melt or solution G0G0 G’ storage modulus G’’ loss modulus

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Typical Oscillatory Data For Rubbers – Viscoelastic solid response: G’ > G” over the whole range of frequencies For polymeric liquids (solutions or melts) – Viscoelastic liquid response: G”>G’ at low frequencies Response becomes solid-like at high frequencies G’ shows a plateau modulus and decreases with -2 in the limit of low frequency (terminal region) G” decreases with -1 in the limit of low frequency

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Sample is strained (pulled, ) rapidly to pre-determined strain ( ) Stress required to maintain this strain over time is measured at constant T Stress decreases with time due to molecular relaxation processes Relaxation modulus defined as: E r (t) also a function of temperature E r (t) = (t)/e 0

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