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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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**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 Polymers have both Viscous (liquid) and elastic (solid) characteristics

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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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**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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**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. •Liquid favored by longer time scales & higher temperatures • Solid favored by short time and lower temperature De is large, solid behavior, small-liquid behavior.

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**increasing loading rate**

Stress Strain increasing loading rate

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Time and temperature

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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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**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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**Mathematical models: 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: stress strain viscosity Gmodulus

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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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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 stress strain viscosity Gmodulus

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**Ideal Liquid h= viscosity de/dt = strain rate**

The viscous response is generally time- and rate-dependent.

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Ideal Liquid

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**Ideal (elastic) Solid Hooks Law**

response is independent of time and the deformation is dependent on the spring constant.

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Ideal Solid

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**Polymer is called visco- elastic because:**

Showing both behaviour elastic & viscous behaviour Instantaneously elastic strain followed by viscous time dependent strain Load released elastic Load added viscous viscous elastic

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stress strain viscosity Gmodulus

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Maxwell Model

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Kelvin Voigt Model

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Burger Model

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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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**Dynamic Mechanical Analysis**

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**Spring Model g = g0⋅sin (ω⋅t) g0 = maximum strain w = angular velocity**

Since stress, t, is t = Gg t = Gg0sin(wt) And t and g are in phase

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Whenever the strain in a dashpot is at its maximum, the rate of change of the strain is zero ( g = 0). Whenever the strain changes from positive values to negative ones and then passes through zero, the rate of strain change is highest and this leads to the maximum resulting stress. Dashpot Model

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Kelvin-Voigt Model

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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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**Real Visco-Elastic Samples**

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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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Blend

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Epoxy

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**Nylon-6 as a function of humidity**

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E’storage modulus Polylactic acid E’’loss modulus

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Tg 87 °C

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Tg -123 °C (-190 F) Tm 135 °C (275 F)

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G’storage modulus G’’loss modulus These data show the difference between the behaviour of un-aged and aged samples of rubber, and were collected in shear mode on the DMTA at 1 Hz. The aged sample has a lower modulus than the un-aged, and is weaker. The loss peak is also much smaller for the aged sample.

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**Tan d of paint as it dries**

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**Epoxy and epoxy with clay filler**

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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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