rheology introduction

TTTTooooppppiiiiccccssss ttttoooo bbbbeeee CCCCoooovvvveeeerrrreeeedddd

• Polymer Melt Rheology

• Introduction to Viscoelastic Behavior

• Time-Temperature Equivalence

Chapter 11 in CD (Polymer Science and Engineering)

PPPPoooollllyyyymmmmeeeerrrr MMMMeeeelllltttt RRRRhhhheeeeoooollllooooggggyyyy

ττττxy ∝∝∝∝ ˙ γγγγ xy

ττττxy ==== ηηηη ˙ γγγγ xy

˙ γγγγ xy ====˙ δδδδ z

==== v0

z

˙ γγγγ xy ====dvdz

====ddt

dxdz

v0

y

δ

z θ

τxy NNNNeeeewwwwttttoooonnnn’’’’ssss LLLLaaaawwww

tanθθθθ ====δδδδz ==== γγγγxy

The most convenient way to describe deformationunder shear is in terms of the angle θ through whichthe material is deformed;

Then if you shear a fluid at a constant rate

Just as stress is proportional to strain inHooke‛s law for a solid, the shear stress isproportional to the rate of strain for a fluid

NNNNeeeewwwwttttoooonnnniiiiaaaannnn aaaannnndddd NNNNoooonnnn ---- NNNNeeeewwwwttttoooonnnniiiiaaaannnnFFFFlllluuuuiiiiddddssss

Shear Thickening

ShearStress

Strain Rate

Shear Thinning

Newtonian Fluid(η = slope)

τxy = ηa (γ) γ. .

Slope = ηa

When the chips areWhen the chips aredown, what do you use?down, what do you use?

VVVVaaaarrrriiiiaaaattttiiiioooonnnn ooooffff MMMMeeeelllltttt VVVViiiissssccccoooossssiiiittttyyyywwwwiiiitttthhhh SSSSttttrrrraaaaiiiinnnn RRRRaaaatttteeee

CompressionMolding Calendering Extrusion

InjectionMolding

SpinDrawing

100

SHEAR RATES ENCOUNTERED IN PROCESSING

Zero Shear Rate Viscosity5

4

3

2

1

00 1 2 3 4-1-2-3

Log ηa

(Pa)

Log γ (sec-1).

101 102 103 104 105

Strain Rate (sec-1)

D ~ 1/M2

η0 ~ M3

HHHHoooowwww DDDDoooo CCCChhhhaaaaiiiinnnnssss MMMMoooovvvveeee ---- RRRReeeeppppttttaaaattttiiiioooonnnn

VVVVaaaarrrriiiiaaaattttiiiioooonnnn ooooffff MMMMeeeelllltttt VVVViiiissssccccoooossssiiiittttyyyywwwwiiiitttthhhh MMMMoooolllleeeeccccuuuullllaaaarrrr WWWWeeeeiiiigggghhhhtttt

Redrawn from the data of G. C.Berry and T. G. Fox, Adv. Polym.Sci., 5, 261 (1968)

ηm = KL(DP)1.0

ηm = KH(DP)3.4

54321

Poly(di-methylsiloxane)Poly(iso-butylene)Poly(ethylene)Poly(butadiene)Poly(tetra-methyl p-silphenyl siloxane)Poly(methyl methacrylate)Poly(ethylene glycol)Poly(vinyl acetate)Poly(styrene)

Log M + constant

Log

ηηηηm

+ c

onst

ant

EEEEnnnnttttaaaannnngggglllleeeemmmmeeeennnnttttssss

Short chains don‛t entangle but long ones do - think ofthe difference between a nice linguini and spaghettios,the little round things you can get out of a tin (we havesome value judgements concerning the relative merits ofthese two forms of pasta, but on the advice of ourlawyers we shall refrain from comment).

EEEEnnnnttttaaaannnngggglllleeeemmmmeeeennnnttttssss

At a critical chain length chains start to becometangled up with one another, however

Viscosity - a measure of the frictional forcesacting on a molecule

Small molecules - the viscosity varies directlywith size

Then

ηm = KL(DP)1.0

ηm = KH(DP)3.4

EEEEnnnnttttaaaannnngggglllleeeemmmmeeeennnnttttssss aaaannnndddd tttthhhheeee EEEEllllaaaassssttttiiiicccc PPPPrrrrooooppppeeeerrrrttttiiiieeeessss ooooffff PPPPoooollllyyyymmmmeeeerrrr MMMMeeeellllttttssss

Depending upon the rate atwhich chains disentangle relativeto the rate at which they stretchout, there is an elastic componentto the behavior of polymer melts.There are various consequencesas a result of this.

Jet Swelling

1.0

1 .2

1.4

1.6

1.8

2.0160°C

180°C

200°C

d/d0

10210-210-1 100 101 103

Strain Rate at Wall (sec-1)

dd0

MMMMeeeelllltttt FFFFrrrraaaaccccttttuuuurrrreeee

Reproduced with permission from J. J. Benbow, R. N. Browne and E. R. Howells, Coll. Intern.Rheol., Paris, June-July 1960.

VVVViiiissssccccooooeeeellllaaaassssttttiiiicccciiiittttyyyy

If we stretch a crystalline solid,The energy is stored in theChemical bonds

If we apply a shear stress toA fluid,energy is dissipated In flow

Ideally elasticbehaviour

Ideally viscousbehaviour

VISCOELASTIC

Homer knew that the firstthing To do on getting yourchariot out in the morningwas to put the wheels backin.(Telemachus,in The Odyssey,would tip his chariot againsta wall)

Robin hood knew never toleave his bow strung

VVVViiiissssccccooooeeeellllaaaassssttttiiiicccciiiittttyyyy

EEEExxxxppppeeeerrrriiiimmmmeeeennnnttttaaaallll OOOObbbbsssseeeerrrrvvvvaaaattttiiiioooonnnnssss

•Creep

•Stress Relaxation

•Dynamic Mechanical Analysis

CCCCrrrreeeeeeeepppp aaaannnndddd SSSSttttrrrreeeessssssss RRRReeeellllaaaaxxxxaaaattttiiiioooonnnn

Creep - deformation under a constant load as a function of time

TIME Stress Relaxation - constant deformation experiment

5lb 4lb 3lb 2lb

TIME

5lb5lb

5lb5lb

CCCCrrrreeeeeeeepppp aaaannnnddddRRRReeeeccccoooovvvveeeerrrryyyy

800060004000200000

10

20

30

40

50

Time (hours)

Cre

ep(%

)

1018 psi1320 psi

1690 psi

2008 psi2305 psi

2505 psi

2695psi

Redrawn from the data of W. N. Findley, ModernPlastics, 19, 71 (August 1942)

} Permanentdeformation

Creep

Recovery

Time

Stra

in

Stressapplied

Stressremoved

(c) VISCOELASTIC RESPONSE


Creep

Recovery

Time

Strain

Stressapplied

Stressremoved


SSSSttttrrrraaaaiiiinnnn vvvvssss.... TTTTiiiimmmmeeee PPPPlllloooottttssss ---- CCCCrrrreeeeeeeepppp

TimeStressapplied

Stressremoved

(a) PURELY ELASTIC RESPONSE

Strain

TimeShearStressapplied

ShearStress

removed

PermanentDeformation

(b) PURELY VISCOUS RESPONSE

Strain

10

9

8

7

0.001 0.01 0.1 1 10 100 1000

Stress relaxationof PMMA

Time (hours)

400C

Log

E(t

), (

dyne

s/cm

2 )

600C

800C920C

1000C

1100C

1120C

1150C1200C

1250C

1350C

SSSSttttrrrreeeessssssss RRRReeeellllaaaaxxxxaaaattttiiiioooonnnn

The data are not usually reported as astress/time plot, but as amodulus/time plot. This timedependent modulus, called therelaxation modulus, is simply the timedependent stress divided by the(constant) strain

E(t ) ==== σσσσ (t )εεεε0

Redrawn from the data of J.R. McLoughlin and A.V.Tobolsky. J. Colloid Sci., 7, 555 (1952).

AAAAmmmmoooorrrrpppphhhhoooouuuussss PPPPoooollllyyyymmmmeeeerrrrssss ---- RRRRaaaannnnggggeeee ooooffff VVVViiiissssccccooooeeeellllaaaassssttttiiiicccc BBBBeeeehhhhaaaavvvviiiioooouuuurrrr

Vs

T

gT

Hard Glass

SoftGlass

LeatherLike

Rubbery

Semi-Solid

LiquidLike

VVVViiiissssccccooooeeeellllaaaassssttttiiiicccc PPPPrrrrooooppppeeeerrrrttttiiiieeeessss 0000ffff AAAAmmmmoooorrrrpppphhhhoooouuuussss PPPPoooollllyyyymmmmeeeerrrrssss

Measured overSome arbitrary Time period - say10 secs

4

3

5

6

7

8

10

9

GlassyRegion

RubberyPlateau

Cross-linkedElastomers

MeltLow

MolecularWeight

Log E(Pa)

Temperature

- 10

GlassyRegion

RubberyPlateau

Transition

LowMolecularWeight

- 8 - 4- 6 0- 2 + 2Log Time (Sec)

Log

E(t

) (

dyne

s/cm

2 )

HighMolecularWeight

4

3

5

6

7

8

10

9

Glass

Stretch sample anarbitrary amount,measure the stressrequired to maintainthis strain.Then E(t) = σ(t)/ε

VVVViiiissssccccooooeeeellllaaaassssttttiiiicccc PPPPrrrrooooppppeeeerrrrttttiiiieeeessss 0000ffff AAAAmmmmoooorrrrpppphhhhoooouuuussss PPPPoooollllyyyymmmmeeeerrrrssss

- 10

GlassyRegion

RubberyPlateau

Transition

LowMolecularWeight

- 8 - 4- 6 0- 2 + 2Log Time (Sec)

Log

E(t

) (

dyne

s/cm

2 )

HighMolecularWeight

4

3

5

6

78

109

Glass

TTTTiiiimmmmeeeeTTTTeeeemmmmppppeeeerrrraaaattttuuuurrrreeeeEEEEqqqquuuuiiiivvvvaaaalllleeeennnncccceeee

4

3

5

6

78

109

GlassyRegion

RubberyPlateau


MeltLow

MolecularWeight

Log E(Pa)

Temperature

RRRReeeellllaaaaxxxxaaaattttiiiioooonnnn iiiinnnn PPPPoooollllyyyymmmmeeeerrrrssss

First consider a hypothetical isolatedchain in space,then imagine stretching this chain instantaneously so that thereis a new end - to - end distance.Thedistribution of bond angles (trans,gauche, etc) changes to accommodatethe conformations that are allowedby the new constraints on the ends.Because it takes time for bond rotationsto occur, particularly when we also addin the viscous forces due to neighbors,we say the chain RELAXES to the newstate and the relaxation is describedby a characteristic time τ.

How quickly can I doHow quickly can I dothese things?these things?

AAAAmmmmoooorrrrpppphhhhoooouuuussss PPPPoooollllyyyymmmmeeeerrrrssss ---- tttthhhheeee FFFFoooouuuurrrr RRRReeeeggggiiiioooonnnnssss ooooffff VVVViiiissssccccooooeeeellllaaaassssttttiiiicccc BBBBeeeehhhhaaaavvvviiiioooorrrr

GLASSY STATE - conformationalchanges severely inhibited.

Tg REGION - cooperative motionsof segments now occur,but themotions are sluggish ( a maximum intan δ curves are observed in DMAexperiments)

RUBBERY PLATEAU - τ t becomesshorter,but still longer than thetime scale for disentanglement

TERMINAL FLOW - the time scale for disentanglement becomesshorter and the melt becomes more fluid like in its behavior

4

3

5

6

7

8

10

9

PolymerMelt

LowMolecular

Weight

Log E(Pa)

Temperature

SSSSeeeemmmmiiiiccccrrrryyyyssssttttaaaalllllllliiiinnnneeee PPPPoooollllyyyymmmmeeeerrrrssss

•Motion in the amorphous domains constrained by crystallites

•Motions above Tg are often more complex,often involving coupled processes in the crystalline and amorphous domains

•Less easy to generalize - polymers often have to be considered individually - see DMA data opposite

Redrawn from the data of H. A. Flocke, Kolloid–Z. Z. Polym., 180, 188 (1962).

0.00

0.05

0.10

0.15

0.20

0.25

0.30

-200 -150 -100 -50 0 50 100 150

LDPELPE

αααααααα

ββββββββ

γγγγγγγγ

TTTTooooppppiiiiccccssss ttttoooo bbbbeeee CCCCoooovvvveeeerrrreeeedddd

• Simple models of Viscoelastic Behavior

• Time-Temperature Superposition Principle

Chapter 11 in CD (Polymer Science and Engineering)

MMMMeeeecccchhhhaaaannnniiiiccccaaaallll aaaannnnddddTTTThhhheeeeoooorrrreeeettttiiiiccccaaaallllMMMMooooddddeeeellllssss ooooffff

VVVViiiissssccccooooeeeellllaaaassssttttiiiiccccBBBBeeeehhhhaaaavvvviiiioooorrrr

GOAL - relate G(t) and J(t) torelaxation behavior.We will only consider LINEARMODELS. (i.e. if we doubleG(t) [or σ(t)], then γ(t) [orε(t)] also increases by a factorof 2 (small loads and strains)).

D(t) ==== εεεε(t)σσσσ0

J(t) ==== γγγγ(t)ττττ 0

E(t ) ==== σσσσ (t)εεεε0

G(t) ==== ττττ (t)γγγγ 0

E ==== 1D

G ==== 1J

E(t ) ≠≠≠≠ 1D(t)

G(t) ≠≠≠≠ 1J(t)

TENSILETENSILEEXPERIMENTEXPERIMENT

SHEARSHEAREXPERIMENTEXPERIMENT

Stress RelaxationStress Relaxation

CreepCreep

Linear Time Independent BehaviorLinear Time Independent Behavior

Time Dependent BehaviorTime Dependent Behavior

SSSSiiiimmmmpppplllleeee MMMMooooddddeeeellllssss ooooffff tttthhhheeee VVVViiiissssccccooooeeeellllaaaassssttttiiiicccc BBBBeeeehhhhaaaavvvviiiioooorrrr ooooffff AAAAmmmmoooorrrrpppphhhhoooouuuussss PPPPoooollllyyyymmmmeeeerrrrssss

Keep in mind that simple creep and recovery data for viscoelastic materials looks something like this


Creep

Recovery

Time

Strain

Stressapplied

Stressremoved



While stress relaxationdata look something likethis;

- 10

GlassyRegion

RubberyPlateau

Transition

LowMolecularWeight

- 8 - 4- 6 0- 2 + 2Log Time (Sec)

Log

E(t

) (

dyne

s/cm

2 )

HighMolecularWeight

4

3

5

6

7

8

10

9

Glass


Hooke's law

Extension

σ = Eε ∆l

l0

σ

I represent linearI represent linearelastic behaviorelastic behavior

σσσσ ==== Eεεεε

Newtonian fluid

Viscous flow

τ = ηγxy.


vy

v0

σσσσ ==== ηηηη dεεεεdt

PURELY VISCOUS RESPONSEPURELY VISCOUS RESPONSE

TimeShearStressapplied

ShearStress

removed

PermanentDeformation

Strain

σσσσ ==== ηηηη dεεεεdt

SSSSttttrrrraaaaiiiinnnn vvvvssss.... TTTTiiiimmmmeeee ffffoooorrrr SSSSiiiimmmmpppplllleeeeMMMMooooddddeeeellllssss

PURELY ELASTIC RESPONSEPURELY ELASTIC RESPONSE

Time

Stressapplied

Stressremoved

Strain

σσσσ ==== Eεεεε

=_dεdt

_ση

_dσdt

_1Ε

+

MMMMaaaaxxxxwwwweeeellllllll MMMMooooddddeeeellll

σ = Eε Maxwell started with Hooke‛s law

Then allowed σ to vary with time = _dεdt

_dσdt Ε

Writing for a Newtonian fluid σ η = _dεdt

Then assuming that the rate of strainis simply a sum of these two contributions

Maxwell was interested in creep and stress relaxation anddeveloped a differential equation to describe these properties

MMMMAAAAXXXXWWWWEEEELLLLLLLL MMMMOOOODDDDEEEELLLL ---- Creep aaaannnndddd RRRReeeeccccoooovvvveeeerrrryyyy

Strain

Time0 t

Strain

Time0 t

Creep andrecovery

=_dεdt

_ση

A picture representation of Maxwell‛s equation

Recall that real viscoelasticbehaviour looks somethinglike this

MMMMaaaaxxxxwwwweeeellllllll MMMMooooddddeeeellll ----SSSSttttrrrreeeessssssss RRRReeeellllaaaaxxxxaaaattttiiiioooonnnn

=_dεdt

_ση

_dσdt

_1Ε +

=_dεdt 0

In a stress relaxation experiment

_σ =dσ _σ

ηdt

0σ = σ exp[-t/τ ]t

_ηΕtτ =

Hence

Where

Relaxation time

- 5

- 6

- 4

- 3

- 2

- 1

0

- 2 - 1 0 1

Log (Er /E0 )

Log (t/τ )

MMMMaaaaxxxxwwwweeeellllllll MMMMooooddddeeeellll ----SSSSttttrrrreeeessssssss RRRReeeellllaaaaxxxxaaaattttiiiioooonnnn

4

3

5

6

7

8

10

9

Log E(t)(Pa)

Time

Real data looks like thisReal data looks like this

The MaxwellThe Maxwellmodel givesmodel gives

curves like this,curves like this, OR like this OR like this

MAXWELL MODELMAXWELL MODEL

E(t ) ====σσσσ (t)

εεεε0

====σσσσ0

εεεε0

exp −−−−t

ττττ t

VVVVooooiiiiggggtttt MMMMooooddddeeeellllMaxwell model essentially assumes a uniform distribution Of stress.Now assume uniform distribution of strain - VOIGT MODEL

Picture representation

Equation

__dε(t)dt

σ(t) = Eε(t) + η

(Strain in both elements of themodel is the same and the totalstress is the sum of the twocontributions)

σσσσ ==== σσσσ1111 ++++ σσσσ2222

σσσσ1111 σσσσ2222

VVVVooooiiiiggggtttt MMMMooooddddeeeellll ---- CCCCrrrreeeeeeeepppp aaaannnndddd SSSSttttrrrreeeessssssss RRRReeeellllaaaaxxxxaaaattttiiiioooonnnn

Gives a retarded elastic response but does not allowfor “ideal” stress relaxation,in that the modelcannot be “instantaneously” deformed to a givenstrain.But in CREEP σ = constant, σ0

σ(t) = σ 0__dε(t)dt

= Eε(t) + η

σΕ

0_ε(t) = [1- exp (-t/τ ‘ )]t

τ ‘t - retardation time (η/E)

Strain

σσσσ ==== σσσσ1111 ++++ σσσσ2222

σσσσ1111 σσσσ2222

SSttrraaiinn

TimeTime

t1 t2

RETARDED ELASTIC RESPONSERETARDED ELASTIC RESPONSE

=

__dε(t)dt

__ ε(t) τ ‘

ση

0_+t

SSSSuuuummmmmmmmaaaarrrryyyy

What do the strain/time plots look like?

PPPPrrrroooobbbblllleeeemmmmssss wwwwiiiitttthhhh SSSSiiiimmmmpppplllleeee MMMMooooddddeeeellllssss

•The maxwell model cannot account for a retarded elastic response

•The voigt model does not describe stress relaxation

•Both models are characterized by single relaxation times - a spectrum ofrelaxation times would provide a better description

NEXT - CONSIDER THE FIRST TWOPROBLEMSTHEN -THE PROBLEM OF A SPECTRUM OF RELAXATION TIMES

I canI can’’t do creept do creep

And I canAnd I can’’t do stress relaxationt do stress relaxation

Elastic + viscous flow + retarded elastic

FFFFoooouuuurrrr ---- PPPPaaaarrrraaaammmmeeeetttteeeerrrrMMMMooooddddeeeellll

ε [1- exp (-t/τ )]tσΕ

0_M

__ση M

0 t= +

σΕ

0_M

+

Eg CREEP


ElasticResponse

Retarded orAnelastic Response

Time

Strain

Stressapplied

Stressremoved

EEMM

EEVV ηηηηηηηηVV

ηηηηηηηηMM

DDDDiiiissssttttrrrriiiibbbbuuuuttttiiiioooonnnnssss ooooffff RRRReeeellllaaaaxxxxaaaattttiiiioooonnnn aaaannnndddd RRRReeeettttaaaarrrrddddaaaattttiiiioooonnnn TTTTiiiimmmmeeeessss

0

E(t ) ==== E(ττττ t )exp(−−−− t / ττττ t )dττττ t

∞∞∞∞

∫∫∫∫

D(t) ==== D( ′′′′ ττττ t ) 1 −−−− exp(−−−− t / ′′′′ ττττ t )[[[[ ]]]] d ′′′′ ττττ t0

∞∞∞∞

∫∫∫∫D(t) ==== D0 1 −−−− exp −−−−t / ′′′′ ττττ t(((( ))))[[[[ ]]]]

E(t )==== expE0 −−−−( t )ττττ t/

Stress RelaxationStress Relaxation

CreepCreep

We have mentioned that although theMaxwell and Voigt models are seriouslyflawed, the equations have the rightform.

What we mean by that is shownopposite, where equations describing theMaxwell model for stress relaxation andthe Voigt model for creep are comparedto equations that account for acontinuous range of relaxation times.

These equations can be obtained by assuming that relaxation occurs at a rate thatis linearly proportional to the distance from equilibrium and use of the Boltzmannsuperposition principle. We will show how the same equations are obtained frommodels.

TTTThhhheeee MMMMaaaaxxxxwwwweeeellllllll ---- WWWWiiiieeeecccchhhheeeerrrrtttt MMMMooooddddeeeellll

=_dεdt

_ση

_dσdt

_1Ε +1

1 11

= _ση

_dσdt

_1Ε +2

2 22

= _ση

_dσdt

_1Ε +3

3 33

Consider stress relaxation_dεdt = 0

0σ = σ exp[-t/τ ]t11



EE11 EE22 EE33

ηηηηηηηη11111111 ηηηηηηηη22222222 ηηηηηηηη33333333

DDDDiiiissssttttrrrriiiibbbbuuuuttttiiiioooonnnnssss ooooffff RRRReeeellllaaaaxxxxaaaattttiiiioooonnnn aaaannnndddd RRRReeeettttaaaarrrrddddaaaattttiiiioooonnnn TTTTiiiimmmmeeeessss

Stress relaxation modulus

E(t) = σ(t)/ε 0

σ(t) = σ + σ + σ1 2 3

E(t) = exp (-t/τ )σ__ε

010

01 + exp (-t/τ )σ__ε

020

02 + exp (-t/τ )σ__ε

030

03

Or, in generalσ__ε

0n0

E(t) = Σ E exp (-t/τ )tnn E =nwhere

Similarly, for creep compliance combine voigt elements to obtain

D(t) =ΣD [1- exp (-t/τ ‘ )]tnn

43210-1-2

0

2

4

6

8

10

Log time (min)

Log

E(t

) (

Pa)

DDDDiiiissssttttrrrriiiibbbbuuuuttttiiiioooonnnnssss ooooffff RRRReeeellllaaaaxxxxaaaattttiiiioooonnnn aaaannnnddddRRRReeeettttaaaarrrrddddaaaattttiiiioooonnnn TTTTiiiimmmmeeeessss

EEEExxxxaaaammmmpppplllleeee ---- tttthhhheeee MMMMaaaaxxxxwwwweeeellllllll ---- WWWWiiiieeeecccchhhheeeerrrrtttt MMMMooooddddeeeellll WWWWiiiitttthhhhnnnn ==== 2222

E(t) = Σ E exp (-t/τ )tnn

n = 2

Log Time (Sec)- 10

GlassyRegion

RubberyPlateau

Transition

- 8 - 4- 6 0- 2 + 2

Log

E(t

) (

dyne

s/cm

2 )

Melt4

3

5

6

7

8

10

9

Glass

TTTTiiiimmmmeeee ---- TTTTeeeemmmmppppeeeerrrraaaattttuuuurrrreeee SSSSuuuuppppeeeerrrrppppoooossssiiiittttiiiioooonnnn PPPPrrrriiiinnnncccciiiipppplllleeee

Recall that we have seen thatthere is a time - temperatureequivalence in behavior

This can be expressedformally in terms of asuperposition principle

- 10

GlassyRegion

RubberyPlateau

Transition

LowMolecularWeight

- 8 - 4- 6 0- 2 + 2Log Time (Sec)

Log

E(t

) (

dyne

s/cm

2 )

HighMolecularWeight

4

3

5

6

7

8

109

Glass

4

3

5

6

78

109

GlassyRegion

RubberyPlateau


MeltLow

MolecularWeight

Log E(Pa)

Temperature

TTTTiiiimmmmeeee TTTTeeeemmmmppppeeeerrrraaaattttuuuurrrreeee SSSSuuuuppppeeeerrrrppppoooossssiiiittttiiiioooonnnnPPPPrrrriiiinnnncccciiiipppplllleeee ---- CCCCrrrreeeeeeeepppp

G'

T3

T3 > T2 > T1

T2 T1

log t

ε(t) Tσ0

0

T0

T[log t - log aT]

log aTε(t) T

σ0

Master Curve at T0 0C

-76.7°C

-80.8°C

-74.1°C-70.6°C

-65.4°C

-58.8°C-49.6°C-40.1°C-0.0°C

+25°C+50°C

-80° -60°

10-14 10-12 10-10 10-8 10-6 10-4 10-2 10-0 10+2

105

10210010-2

106

Stre

ss R

elax

atio

n M

odul

usdy

nes/

cm2

1011

1010

109

108

107

Time (hours)

-40° -20° -10° 0° 25°

Master Curve at 250C

StressRelaxation

Data

-40 -4

+4

+8

-80 0 40 80 Temperature 0C

TEMPERATURE SHIFT FACTOR vs

0

Log

Shi

ft F

acto

r

TTTTiiiimmmmeeee TTTTeeeemmmmppppeeeerrrraaaattttuuuurrrreeeeSSSSuuuuppppeeeerrrrppppoooossssiiiittttiiiioooonnnn

PPPPrrrriiiinnnncccciiiipppplllleeee ---- SSSSttttrrrreeeessssssssRRRReeeellllaaaaxxxxaaaattttiiiioooonnnn

RRRReeeellllaaaaxxxxaaaattttiiiioooonnnn PPPPrrrroooocccceeeesssssssseeeessss aaaabbbboooovvvveeee TTTTgggg ---- tttthhhheeee WWWWLLLLFFFF EEEEqqqquuuuaaaattttiiiioooonnnn

From empirical observation

Originally thought that C1 and C2 were universal constants,= 17.44 and 51.6, respectively,when Ts = Tg. Now known that these vary from polymer to polymer.Homework problem - show how the WLF equation can beobtained from the relationship of viscosity to freevolume as expressed in the Doolittle equation

Log a = _________-C (T - T )s

C + (T - T )s2T For Tg > T < ~(Tg + 1000C)

1

SSSSeeeemmmmiiii ---- CCCCrrrryyyyssssttttaaaalllllllliiiinnnneeee PPPPoooollllyyyymmmmeeeerrrrssss

Temperature

Tm

Tg

Amorphous melt

Rigid crystalline domainsRubbery amorphous domains

Rigid crystalline domainsGlassy amorphous domains

NON - LINEAR RESPONSE TO STRESS.SIMPLEMODELS AND THE TIME - TEMPERATURESUPERPOSITION PRINCIPLE DO NOT APPLY

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rheology introduction