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**What happens to Tg with increasing pressure?**

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Bar = 1 atm = 100 kPa Why?

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**A Demonstration of Polymer Viscoelasticity**

Poly(ethylene oxide) in water

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**“Memory” of Previous State**

Poly(styrene) Tg ~ 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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**Rheology is the science of flow and deformation of matter**

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**

The properties of polymers are strongly affected by temperature and 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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**increasing loading rate**

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 tT), creating a “transient” network.

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**Entanglement Molecular Weights, Me, for Various Polymers**

Me (g/mole) Poly(ethylene) 1,250 Poly(butadiene) 1,700 Poly(vinyl acetate) 6,900 Poly(dimethyl siloxane) 8,100 Poly(styrene) 19,000

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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 drop experiment apparatus Pitch can be shattered by a hammer

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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**

Constant strain applied time s Stress relaxes over time as molecules re-arrange time Stress relaxation:

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**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 fluid Viscoelastic fluid Elastic solid Polymers display VISCOELASTIC properties

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

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Hooke and Newton 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.

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**Hooke’s law & Newton’s Law**

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

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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. 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).

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STRESS RELAXATION CREEP 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**

Strain Elastic Stress Stress Viscoelastic Stress Viscous fluid Viscous fluid Viscous fluid Time, t

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**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 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**

perse Glassy behavior Transition Zone Terminal Zone (flow region) slope = -1 Plateau Zone Mc: critical molecular weight above which entanglements exist 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 tT), creating a “transient” network.

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**Relaxation Modulus for Polymer Melts**

Elastic tT = terminal relaxation time tT Viscous flow

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**Viscosity of Polymer Melts**

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

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

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**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: ho h ~ tTGP 3.4 ho ~ N3.4 N0 ~ N3.4 Universal behaviour for linear polymer melts Applies for higher N: N>NC Why? G.Strobl, The Physics of Polymers, p. 221

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**Application of Theory: Electrophoresis**

From Giant Molecules

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Mechanical Model 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: Elastic spring with modulus which follows Hooke’s law Viscous 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.

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**Dynamic Viscosity (dashpot)**

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

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stress stress input Strain in dashpot dashpot 27/06/46

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**Maxwell model In series Viscous strain remains after load removal.**

stress input Model Strain Response Maxwell model 27/06/46

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**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 Model Strain Response Voigt model 27/06/46

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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 Courtesy: TA Instruments

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**Shear stress Non Newtonian Fluids Shear rate Casson Plastic**

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

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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. 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 dashpot stress strain viscosity Gmodulus

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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 viscosity Gmodulus

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**Maxwell Model Let’s try to deform the Maxwell element stress**

strain viscosity Gmodulus

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**Maxwell model too primitive**

Maxwell: solid line Experiment: circles Maxwell model too primitive

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**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 time If the mechanical model is suddenly extended to a position and held there (g=const., g=0): . Exponential decay in stresses stress strain viscosity Gmodulus

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**Examples of Viscoelastic Materials**

Mattress, Pillow Tissue, skin 27/06/46

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

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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**

Purely Viscous Response (Newtonian Liquid) Purely Elastic Response (Hookean Solid) = 90° = 0° Stress Stress Strain Strain Courtesy: TA Instruments

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**Dynamic Mechanical Testing Viscoelastic Material Response**

Phase angle 0° < d < 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 * 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 d Complex Stress, * * = ' + i" Courtesy: TA Instruments Strain,

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**DMA Viscoelastic Parameters**

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

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**DMA Viscoelastic Parameters: Damping, tan **

Dynamic measurement represented as a vector It can be seen here that G* = (G’2 +G”2)1/2 G" 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

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**Frequency Sweep: Material Response**

Transition Region Rubbery Plateau Region Terminal Region Glassy Region log G'and G" 1 2 Storage Modulus (E' or G') Loss Modulus (E" or G") log Frequency (rad/s or Hz) 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 3.6x106.

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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 tand is on the order of 0.1

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**Dynamic Temperature Ramp or Step and Hold: Material Response**

Glassy Region Transition Region Rubbery Plateau Region Terminal Region Log G' and G" 1 Loss Modulus (E" or G") Storage Modulus (E' or G') 2 Temperature 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 d 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°<d<90° stress strain viscosity Gmodulus 3.29

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**One more time: Dynamic (Oscillatory) Testing**

By using trigonometry: (3-1) In-phase component of the stress, representing solid-like behavior Out-of-phase component of the stress, representing liquid-like behavior Let’s define: where: 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 modulus G’’loss modulus

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**Typical Oscillatory Data**

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

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**Typical Oscillatory Data**

G’ G’’ log G log Melt or solution G0 G’storage modulus G’’loss modulus 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 w-2 in the limit of low frequency (terminal region) G” decreases with w-1 in the limit of low frequency

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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 w-2 in the limit of low frequency (terminal region) G” decreases with w-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: Er(t) also a function of temperature Er(t) = (t)/e0

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