topics to be covered - zeus.plmsc.psu.eduzeus.plmsc.psu.edu/~manias/matse259/lecture11-12.pdf ·...
Post on 31-Jan-2018
225 Views
Preview:
TRANSCRIPT
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
} Permanentdeformation
Creep
Recovery
Time
Strain
Stressapplied
Stressremoved
(c) VISCOELASTIC RESPONSE
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
Cross-linkedElastomers
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
} Permanentdeformation
Creep
Recovery
Time
Strain
Stressapplied
Stressremoved
(c) VISCOELASTIC RESPONSE
SSSSiiiimmmmpppplllleeee MMMMooooddddeeeellllssss ooooffff tttthhhheeee VVVViiiissssccccooooeeeellllaaaassssttttiiiicccc BBBBeeeehhhhaaaavvvviiiioooorrrr ooooffff AAAAmmmmoooorrrrpppphhhhoooouuuussss PPPPoooollllyyyymmmmeeeerrrrssss
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
SSSSiiiimmmmpppplllleeee MMMMooooddddeeeellllssss ooooffff tttthhhheeee VVVViiiissssccccooooeeeellllaaaassssttttiiiicccc BBBBeeeehhhhaaaavvvviiiioooorrrr ooooffff AAAAmmmmoooorrrrpppphhhhoooouuuussss PPPPoooollllyyyymmmmeeeerrrrssss
Hooke's law
Extension
σ = Eε ∆l
l0
σ
I represent linearI represent linearelastic behaviorelastic behavior
σσσσ ==== Eεεεε
Newtonian fluid
Viscous flow
τ = ηγxy.
SSSSiiiimmmmpppplllleeee MMMMooooddddeeeellllssss ooooffff tttthhhheeee VVVViiiissssccccooooeeeellllaaaassssttttiiiicccc BBBBeeeehhhhaaaavvvviiiioooorrrr ooooffff AAAAmmmmoooorrrrpppphhhhoooouuuussss PPPPoooollllyyyymmmmeeeerrrrssss
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
} Permanentdeformation
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
0σ = σ exp[-t/τ ]t22
0σ = σ exp[-t/τ ]t33
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
Cross-linkedElastomers
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
top related