What happens to Tg with increasing pressure?

Why?

Bar = 1 atm = 100 kPa

Poly(ethylene oxide) in water

A Demonstration of Polymer Viscoelasticity

“Memory” of Previous State

Poly(styrene)

Tg ~ 100 °C

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 of polyisobutylene in decalin

Under what circumstances am I justified in ignoring viscoelastic effects?

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

Temperature & Strain Rate

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.

S

C

t

sticcharacteriDe

λ=≡

ndeformatio theof scale time timematerial

Str

ess

Strain

increasing loading rate

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.

Entanglement Molecular Weights, Me, 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

Me (g/mole)

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

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.

γ (strain)

time

τ (stress)

timeto=0 to=0

γo

?

Physical Meaning of the Relaxation Time

time

γ

Constant strain applied

Stress relaxes over time as molecules re-arrange

timeγ

teGt =)(Stress relaxation:

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

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.

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).

• 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

• The behaviour of linear elastic were given by Hooke’s law:

Ee=

E= Elastic modulus

= Stress

e = strain

de/dt = strain rate

d/dt = stress rate

= viscosity

ordt

deE

dt

d=

• The behaviour of linear viscous were given by Newton’s Law:

dt

de =

** This equation only applicable at low strain

Hooke’s law & Newton’s Law

Viscoelasticity and Stress RelaxationStress 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. γ=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).

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

Time-dependent behavior of PolymersThe 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

S

C

tndeformatio theof scale time

timematerial sticcharacteriDe

λ=≡

metal

elastomerViscous liquid

Static Modulus of Amorphous PS

Glassy

Leathery

Rubbery

Viscous

Polystyrene

Stress applied at x and removed at y

Stress Relaxation Test

Time, t

Strain

Stress

Elastic

Viscoelastic

Viscous fluid

0

StressStress

Viscous fluidViscous fluid

Stress relaxationStress relaxation after a step strain γo is the fundamental way in which we define the

relaxation modulus:

o

)t()t(G

γ

=

Go (or GNo) is the

“plateau modulus”:

e

oN M

RTG

ρ=

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: o

)t()t(E

ε

=

Stress relaxation of an uncrosslinked melt

Mc: critical molecular weight above which entanglements exist

perse

Glassy behavior

Transition Zone

Terminal Zone (flow region) slope = -1

Plateau Zone

3.24

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.

Relaxation Modulus for Polymer Melts

Viscous flow

Elastic T = terminal relaxation time

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.

γ&

o

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

Scaling of Viscosity: ~ N3.4

~ TGP

~ 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

Data shifted for clarity!

Viscosity is shear-strain rate dependent. Usually measure in the limit of a low shear rate: o

3.4

Application of Theory: Electrophoresis

From Giant Molecules

• 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.

1. 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

μγ

=&

Dynamic Viscosity (dashpot)

1 centi-Poise = milli Pascal-second

SI Unit: Pascal-second

Shear stress

Shear rate

Slope of lineμ =

• Lack of slipperinessLack of slipperiness• Resistance to flowResistance to flow• Interlayer frictionInterlayer friction

27/06/46 42

stress input

dashpot

stress

Strain in dashpot

27/06/46 43

Maxwell model In series Viscous strain remains after load removal.

stress input

Model Strain Response

Maxwell model

27/06/46 44

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

Typical Viscosities (Pa.s)

Asphalt Binder ---------------Polymer Melt -----------------Molasses ----------------------Liquid Honey -----------------Glycerol -----------------------Olive Oil -----------------------Water --------------------------Acetic Acid --------------------

100,0001,0001001010.010.0010.00001

Courtesy: TA Instruments

Shear stress

Shear rate

NewtonianPse

udoplastic (

or Shear

thinning)

Dilata

nt (or S

hear thick

ening)

Bingham PlasticCass

on Plastic

Non Newtonian Fluids

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:

γ= &

The simplest elastic solid model is the Hookean model, which we can

represent by the “spring” mechanical analog.

γ= G

stress γstrain viscosity G modulus

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 γstrain viscosity G modulus

Maxwell Model

Let’s try to deform the Maxwell element

stress γstrain viscosity G modulus

Maxwell: solid lineExperiment: circles

Maxwell model too primitive

Maxwell ModelThe deformation rate of the Maxwell model is equal to the sum of the

individual deformation rates:

γ=λ+

γ=

+

+

=γ

γ+γ=γ

&&

&&

&&

&&&

G

G

solidfluid

λ is the relaxation time

If the mechanical model is suddenly extended to a position and held

there (γ=const., γ=0):.

λ−= /toe Exponential decay in stresses

stress γstrain viscosity G modulus

27/06/46 52

Examples of Viscoelastic Materials

Mattress, Pillow

Tissue, skin

• 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

– Standard linear solid• One Maxwell model and one spring align in

parallel.

Elastic Viscous

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

Weissenberg Effect

Dough Climbing: Weissenberg Effect

Other effects: Barus Kaye

δ=AnglePhase

)('

)(''tan

ω

ωδ

G

G=

Loss Tangent

LiquidViscous

MaterialicViscoelast

SolidElasticHookean

o

o

90

900

0

=

<<

=

δ

δ

δ

Viscoelastic MeasurementsTorque bar

Sample

Cup

Bob

Strain γStress σ

oσ

oγ

Oscillator

Phase Angle δ

0

cos)('

γδσ

ω oG =

Storage Modulus

0

sin)(''

γδσ

ω oG =

L o s s M o d u l u s

Courtesy: Dr. Osvaldo Campanella

Dynamic Mechanical TestingResponse for Classical Extremes

Stress

Strain

δ = 0° δ = 90°

Purely Elastic Response(Hookean Solid)

Purely Viscous Response

(Newtonian Liquid)

Stress

Strain

Courtesy: TA Instruments

Dynamic Mechanical Testing Viscoelastic Material Response

Phase angle 0° < δ < 90°

Strain

Stress

Courtesy: TA Instruments

DMA Viscoelastic Parameters:The Complex, Elastic, & Viscous Stress

The stress in a dynamic experiment is referred to as the complex stress *

Phase angle δ

Complex Stress, *

Strain, ε

* = ' + i"

The complex stress can be separated into two components: 1) An elastic stress in phase with the strain. ' = *cosδ ' is the degree to which material behaves like an elastic

solid.2) A viscous stress in phase with the strain rate. " = *sinδ " is the degree to which material behaves like an ideal liquid.

Courtesy: TA Instruments

DMA Viscoelastic Parameters

The Elastic (Storage) Modulus: Measure of elasticity of material. The ability of the material to store energy.

G' = (stress*/strain)cosδ

G" = (stress*/strain)sinδ

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*/StrainG* = G’ + iG”

Tan δ = G"/G'

Tan Delta: Measure of material damping - such as vibration or sound damping.

Courtesy: TA Instruments

DMA Viscoelastic Parameters: Damping, tan δ

Phase angle δ

G*

G'

G"

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

The tangent of the phase angle is the ratio of the loss modulus to the storage modulus.

tan δ = G"/G'"TAN DELTA" (tan δ)is a measure of the damping ability of the material.

Courtesy: TA Instruments

Frequency Sweep: Material Response

Terminal Region

Rubbery PlateauRegion

TransitionRegion

Glassy Region

12

Storage Modulus (E' or G')

Loss Modulus (E" or G")

log Frequency (rad/s or Hz)

log

G'a

nd G

"

Courtesy: TA Instruments

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.

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 tanδ is on the order of 0.1

Dynamic Temperature Ramp or Step and Hold: Material Response

Temperature

Terminal Region

Rubbery PlateauRegion

TransitionRegion

Glassy Region

1

2Loss Modulus (E" or G")

Storage Modulus (E' or G')Log

G' a

nd G

"

Courtesy: TA Instruments

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 δ with respect to the strain wave. The phase angle will depend on the nature of the material (viscous, elastic or viscoelastic)

)tsin(o ωγ=γ

Input

Response

)tsin(o δ+ω=where 0°<δ<90°

3.29stress γstrain viscosity G modulus

One more time: Dynamic (Oscillatory) Testing

By using trigonometry:

)tcos()tsin()tsin( ooo ω′′+ω′=δ+ω=

Let’s define: oooo G and G γ′′= ′′γ′=′

In-phase component of the stress, representing solid-like behavior

Out-of-phase component of the stress, representing liquid-like behavior

Modulusor Loss Viscous ,strain maximum

stress phaseofout)(G

ModulusStorageor Elastic, strain maximum

stress phasein)(G

o

o

o

o

γ′′

=−−

=ω′′

γ′

=−

=ω′where:

(3-1)

3.30

Physical Meaning of G’, G”

[ ])tcos()("G)tsin()(Go ωω+ωω′γ=τEquation (3-1) becomes:

G

Gtan

′′′

=δWe can also define the loss tangent:

)tsin(GG ospring ωγ=γ=For solid-like response:

°=δ=δ=′′=′∴ 0 0, tan0,G ,GG

For liquid-like response:

)tcos(odashpot ωωγ=γ= &

°=δ∞=δω=′′=′∴ 09 , tan,G ,0G

G’ storage modulus G’’ loss modulus

Typical Oscillatory Data

Rubbers – Viscoelastic solid response:

G’ > G” over the whole range of frequencies

G’

G’’

log G

log ω

Rubber

G’ storage modulus

G’’ loss modulus

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 ω-2 in the limit of low frequency (terminal region)

G” decreases with ω-1 in the limit of low frequency

G’G’’

log G

log ω

Melt or solution

G0

G’ storage modulus

G’’ loss modulus

Typical Oscillatory DataFor Rubbers – Viscoelastic solid response:

G’ > G” over the whole range of frequenciesFor 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 ω-2 in the limit of low frequency (terminal region)

G” decreases with ω-1 in the limit of low frequency

•Sample is strained (pulled, ε) 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