what happens to tg with increasing pressure?
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What happens to Tg with increasing pressure?. Bar = 1 atm = 100 kPa. Why?. A Demonstration of Polymer Viscoelasticity. Poly(ethylene oxide) in water. “Memory” of Previous State. Poly(styrene) T g ~ 100 °C. Chapter 5. Viscoelasticity. Is “silly putty” a solid or a liquid? - PowerPoint PPT PresentationTRANSCRIPT
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. ViscoelasticityIs “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
tsticcharacteriDe λ
=≡ndeformatio theof scaλe time
timemateriaλ
Stre
ss
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 tT), 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
s Stress relaxes over time as molecules re-arrange
timetγs
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=s
E= Elastic moduluss = Stresse = strainde/dt = strain rateds/dt = stress rateh = viscosity
ordtdeE
dtd
=s
• The behaviour of linear viscous were given by Newton’s Law:
dtdehs =
** 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(Gγt
=
Go (or GNo) is the
“plateau modulus”:
e
oN M
RTG r=
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(Ees
=
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 tT), creating a “transient” network.
Relaxation Modulus for Polymer Melts
Viscous flow
t
Elastic tT = terminal relaxation time
Viscosity of Polymer Melts
Poly(butylene terephthalate) at 285 ºC
For comparison: h 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”, ho.
γ&
ho
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 ho against molecular weight we see a change of slope from 1 to 3.45 at the critical entanglement molecular weight.
ho
Mn
Slope = 1
Slope = 3.4Entanglement molecular weight
Scaling of Viscosity: ho ~ N3.4
h ~ tTGP
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
Data shifted for clarity!
Viscosity is shear-strain rate dependent. Usually measure in the limit of a low shear rate: ho
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 h 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
tmγ
= &
Dynamic Viscosity (dashpot)
1 centi-Poise = milli Pascal-second
SI Unit: Pascal-second
Shear stress
Shear rate
Slope of linem =
• 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
NewtonianPseudoplastic
(or Shear
thinning)
Dilatant (or Shear th
ickening)
Bingham PlasticCasson 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:
γh=t &
The simplest elastic solid model is the Hookean model, which we can represent by the “spring” mechanical analog.
γ=t G
tstress γstrain hviscosity 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:
tstress γstrain hviscosity G modulus
Maxwell ModelLet’s try to deform the Maxwell element
tstress γstrain hviscosity 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:
γh=tλ+t
γh=th
+t
t+
ht
=γ
γ+γ=γ
&&&&
&&
&&&
G
G
solidfluid
λ is the relaxation timeIf the mechanical model is suddenly extended to a position and held there (γ=const., γ=0):
.
λ−t=t /toe Exponential decay in stresses
tstress γstrain hviscosity 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
d=AnglePhase
)(')(''tan
ωωδ
GG=
Loss Tangent
LiquidViscousMaterialicViscoelastSolidElasticHookean
o
o
90900
0
=<<
=
δδ
δ
Viscoelastic MeasurementsTorque bar
SampleCup
Bob
Strain γStress σ
oσ
oγ
OscillatorPhase Angle δ
0
cos)('γ
δσω oG =
Storage Modulus
0
sin)(''γ
δσω oG =
L o s s M o d u l u s
Courtesy: Dr. Osvaldo Campanella
Dynamic Mechanical TestingResponse for Classical Extremes
Stress
Strain
d = 0° d = 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° < d < 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 s*
Phase angle d
Complex Stress, s*
Strain, e
s* = s' + is"
The complex stress can be separated into two components: 1) An elastic stress in phase with the strain. s' = s*cosd s' is the degree to which material behaves like an elastic
solid.2) A viscous stress in phase with the strain rate. s" = s*sind s" 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)cosd
G" = (stress*/strain)sind
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 d = G"/G'
Tan Delta: Measure of material damping - such as vibration or sound damping.
Courtesy: TA Instruments
DMA Viscoelastic Parameters: Damping, tan d
Phase angle d
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 d = G"/G'"TAN DELTA" (tan d)is a measure of the damping ability of the material.
Courtesy: TA Instruments
Frequency Sweep: Material Response
Terminal Region
Rubbery PlateauRegion
TransitionRegion
Glassy Region
1 2Storage 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 tand is on the order of 0.1
Dynamic Temperature Ramp or Step and Hold: Material Response
Temperature
Terminal RegionRubbery PlateauRegion
TransitionRegion
Glassy Region
12Loss Modulus (E" or G")
Storage Modulus (E' or G')Log
G' a
nd G
"
Courtesy: TA Instruments
One more time: Dynamic (Oscillatory) TestingIn 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)
)tsin(o ωγ=γ
Input
Response
)tsin(o d+ωt=twhere 0°<d<90°
3.29tstress γstrain hviscosity G modulus
One more time: Dynamic (Oscillatory) TestingBy using trigonometry:
)tcos()tsin()tsin( ooo ωt′′+ωt′=d+ωt=t
Let’s define: oooo G and G γ′′=t ′′γ′=t′
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 maximumstress phasein)(G
o
o
o
o
γt′′
=−−
=ω′′
γt′
=−
=ω′where:
(3-1)
3.30
Physical Meaning of G’, G”
[ ])tcos()("G)tsin()(Go ωω+ωω′γ=τEquation (3-1) becomes:
GGtan′′′
=dWe can also define the loss tangent:
)tsin(GG ospring ωγ=γ=tFor solid-like response:
°=d=d=′′=′∴ 0 0, tan0,G ,GGFor liquid-like response:
)tcos(odashpot ωωhγ=γh=t &
°=d∞=dhω=′′=′∴ 09 , tan,G ,0GG’ 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 frequenciesResponse becomes solid-like at high frequenciesG’ 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, e) rapidly to pre-determined strain (s)•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) = s(t)/e0