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Laboratorium für Makromolekulare Chemie
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POLYCHAR 18 - Short CourseDYNAMIC-MECHANICAL PROPERTIES OF
POLYMERS
MECHANICAL PROPERTIES OF (POLYMERIC) MATERIALS UNDER THE INFLUENCE OF
DYNAMIC LOAD AND TEMPERATURE
mechanical modulus as a function of load, temperature, time
A project of the IUPAC Division IV (POLYMER DIVISION)
transitions, relaxations
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simple extension simple shear deformation(also in liquids possible)
simple deformations in a solid
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simple stress in a shear deformation
The total stress σij is a second rank tensor composed of normal andshear components
σ22
τ21
τ23τ32
σ33
τ31
σ11
τ12
τ13
1
2
3
ii ijnormal stress; shear stress! "= =
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(ideal) energy elasticity
•caused by deformation of bond angles and bond length at small deformations•the energy is stored and completely released after the load is removed•there is no (internal) friction
normal
0
0
0 0
1
FA
L L LE
L L
!
! " " #
=
$ %= & = = = $
in plane
0
FA
G
!
! "
=
= #
1J
E! " "= # = #
ε = strain; λ = uniaxial deformation ratio; γ = shear (angle); F = force [N]; A0 = initial areaE = Young modulus [Pa]; G = shear modulus [Pa]; J = compliance [Pa-1]
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Lo L
Cauchy orEngineering Strain
L-Lo = !L
Hencky or True Strain " = ln (!L/Lo)
" = !L/Lo
Kinetic Theoryof Rubber Strain " = 1/3{L/L o-(Lo/L)2}
Kirchhoff Strain
Murnaghan Strain
" = 1/2{ (L/L o)2-1}
" = 1/2{1-(Lo/L)2}
The different definitions of tensile strainbecome equivalent at very small deformations.
elongation
The stress [Pa = N/m2] refers to the initial cross section
Stress and strain are principally time-dependent stress can “relax” (at constant strain)
elongation can “creep” (at constant stress)
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The elastic limit: Hooke’s Law
=
Strain increaseswith increasing
Stress
!
"
slope = k
in the
Slope: elastic (Young-) modulus E
ideal stress-strain diagram
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extension → Young modulus Eshear → shear modulus Gcompression → bulk modulus B bending* → bending modulus Eb*three-point bending, 4 point bending
( ) ( )µµ 21312 !=+= BGE
!!"
#$$%
&'=
long
lat
((µ
The lateral strain εlat is the strain normal to the uniaxial deformation.
the major types of moduli
the different moduli can be converted into one another, see D. Ferry
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GE !"#" 35.0µThe volume change on deformation is for most elastomers negligible so thatµ=0.5 (isotropic, incompressible materials).
In a sample under small uniaxial deformation!!
The lateral strain εlat is the strain normal to the uniaxial deformation.
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!•
= shear rate
ideal liquid betweentwo parallel plates
an ideal liquid shows no elasticity
2
grad v
dvgrad v
dx
!
! " " " #•
= $ = $ = $
!σ
γ .
slope = η
dilatant
structuralviscous
η = dynamic viscosity [Pa s]; 1 centipoise = 1 mPa s
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plate-plate cone-plate Couette
constant shear ratealong the radius
torque
important rheometer types for viscous samples
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visco-elastic behaviour
James Clerk Maxwell, Phil. Trans. Roy. Soc. London 157 (1867) 52
single relaxation time τ spectrum of relaxation times τi
There is mater that shows elastic and viscous behaviour (e.g. pitch):fast deformation rather elastic, slow deformation rather viscous response
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major response types on deformational stress
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Young’s modulus is designed for elastic materials. Real materials consist of both elastic and viscous response.
E” – lost to friction and rearrangement - “the Loss Modulus”
E’ – stored and released –“the Storage Modulus” (conceptually like Young’s Modulus)
E”
E’
'' 20
' 20
12
rev
rev
Q E
w E
! "
"
=
=
dissipated energy
saved energy
''tan
'EE
! = damping (factor)
storage and loss of energy
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stress σ
strain ε
( )( )Ttf
Tf,,
==
!"#
'simple' static stress-strain experiment
yield strength
tensile strength
tensile strength
tensile strength
elongation at break
elongation at breakbrittleness B*):
1'b
BE!
="
*) according to Brostow et al., J. Mater. Sci 21 (2006) 2422 not to be confused with the bulk modulus B (compression modulus)
b!
b!
b!
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frequency dependence of damping
temperature depending properties (elasticity, flow…)
long term prediction, fatigue…
TTS
time
temperature
superposition
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frequency range and applied technique
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the modern machines can rapidly change the measuring device so that solid and fluid samples can be measured
and many different modes can be applied
Thermomechanical AnalysisStress-Strain CurvesCreep RecoveryStress RelaxationDynamic Mechanical AnalysisSolvent Immersed testing
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molecular structure
processing conditions
product properties
MaterialBehavior
DMA relates:
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Free damping experiment
am
plitu
de
1
ln n
n
AA +
! "# = $ %& '
logarithmic decrement
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st
ress
st
rain
stress and strain are in phase in an ideal energy-elastic material, phase angle δ = 0°
stress and strain are out of phase in an ideal viscous material, phase angle δ = 90°
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( )0 0 0 0
0 0 0 0
' ''
Theory shows that the modulus is complex and can be split into
a real part ' and an imaginary part with '':
* cos sin cos sin
* ' ''
i
E E
E E
E e i i
E E iE
= = = + = +
! = +
!"#"$ !"#"$
"# # # ##" " " "
$ $ $ $ $
' 20
'' 20
12rev
rev
w E
Q E
=
=
!
" !
stored
dissipated (loss) Because Young’s Modulus isn’t enough…
Young’s modulus is designed for elastic materials. Real materials consist of both elastic and viscous response.
E” – lost to friction and rearrangement - “the Loss Modulus”
E’ – stored and released –“the Storage Modulus” (conceptually like Young’s Modulus)
E”
E’
The modulus of a visco-elastic material is a complex physical entity
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in the Gaussian plane of complex numbers thecomplex shear modulus G*, the Young modulus E*and the complex viscosity η* can be visualized as
the damping factor D is then given by the tan ofthe loss angle δ
tanD= !D shows a behaviour similar to Λ
* ; * ; *E G! ! "
#$ % %
•= = =
Correlation between moduli and phase angle (damping)
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Λ or tan δ
Λ or tan δ
Modulus, damping and their correlation with molecular motions
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There is not onlyone glass.The type of glassdepends on the thermal history.
slowly heating can causeannealing
a thermodynamic view at the 'glass transition'
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glassy visco-elastic rubbery T
G'
G''
tanδ
The glass transition in a dynamic experiment
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DMA and different molecular parameters
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0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
-100.0 0.0 100.0 200.0 300.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Sun Nov 26 21:02:11 1995
Mod
ulus
(Pa
x 10
9 )
# 2 Storage Modulus (Pa x 10 9)
Onset 83.29 Ctan
Temperature ( C)
# 1 pet film:demofilmtan
Onset 107.82 C
Onset 79.35 C
Curve 1: DMA Temp/Time Scan in ExtensionFile info: demofilm Wed Oct 11 17:06:48 1995Frequency: 1.00 Hz Amplitude: 21.949u
Tension: 110.000% pet film
PERKIN-ELMER7 Series Thermal Analysis System
TEMP1: -100.0 C TIME1: 0.0 min RATE1: 10.0 C/minTEMP2: 250.0 C
Tg are easily seen, as in PET Film
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Tg by DMA and DSCΔΗ
/(J/g)
Hea
t flo
w/m
WTemperature /C
Inflection Point ! ΔCp
Tf
Onset
Temperature/C
Mod
ulus
/Pa Tan δ
Onset E’ = 133.1 °C
Peak Tan δ = 140.5°C
Onset Tan δ = 130.0 °C
Onset E” = 127.3 °C
Peak E” = 136.7 °C
(a) (b)
differential scanning calorimetry
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Operating Range by DMA
-100.0 0.0 100.0 200.00.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Sun Nov 26 20:13:53 1995
Modu
lus (P
a x 10
10)
PE DMA7 R&D LABTemperature ( C)
# 1 EPOXY PC BOARD AT 7 Hz:gamma_1Storage Modulus (Pa x 10 10)
Curve 1: DMA Temp/Time Scan in 3 Point BendingFile info: gamma_1 Thu Jun 30 02:17:24 1988Frequency: 7.00 Hz Dynamic Stress: 1.86e+06Pa
Static Stress: 1.86e+06Pa EPOXY PC BOARD AT 7 Hz
PERKIN-ELMER7 Series Thermal Analysis System
TEMP1: -180.0 C TIME1: 0.0 min RATE1: 10.0 C/minTEMP2: 300.0 C
tan
(x 10
-1)
-> # 2 tan (x 10 -1 )
BetaTg
Operatingrange
(b)
-150.0 -100.0 -50.0 0.0 50.0 100.0
0.5
1.0
1.5
2.0
2.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Sun Nov 26 20:58:36 1995
Modu
lus (P
a x 10
9 )
AMP Flame Retardant Polypropylene KPMTemperature ( C)
# 1 LeBrun samples:AMPfrPP.1Storage Modulus (Pa x 10 9)
Curve 1: DMA Temp/Time Scan in 3 Point BendingFile info: AMPfrPP.1 Wed Oct 27 13:49:06 1993Frequency: 1.00 Hz Dynamic Stress: 950.0mN
Static Stress: 1000.0mN LeBrun samples
PERKIN-ELMER7 Series Thermal Analysis System
TEMP1: -160.0 C TIME1: 0.0 min RATE1: 5.0 C/minTEMP2: 300.0 C
tan
(x 10
-1)
# 2 tan (x 10 -1 )
(c)Operating
range
(a)Operating
range
leather-like state
toughness is okmodulus is ok
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amorphous semicrystalline thermoplasts
temperature of use'leather' vacuum
forming
blowforming extrusion
injection mouldingrigid
rubberviscous
temperature of usecold forming
vacuum forming
blowforming
extrusioninjection moulding
shea
r mod
ulus
shea
r mod
ulus
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Cold Crystallization in PET seen by DMA and DSC
Tg
Cold Crystallization
Tm
DSC
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0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
-150.0 -100.0 -50.0 0.0 50.0 100.0 150.00.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
Sat Oct 15 14:32:54 1994
LeBrun samples
tan
(x 10
-1)
Modu
lus (P
a x 10
9 )
AMP good part 20% glass filled Nylon 6/6 KPMTemperature ( C)
Curve 1: DMA Temp/Time Scan in 3 Point BendingFile info: AMP66gp.1 Tue Oct 26 16:05:29 1993Frequency: 1.00 Hz Dynamic Stress: 190.0mN
Static Stress: 200.0mN LeBrun samples
PERKIN-ELMER7 Series Thermal Analysis System
TEMP1: -160.0 C TIME1: 0.0 min RATE1: 5.0 C/minTEMP2: 300.0 C
β Transitions
Tg
Good Impact Strength
Poor
Higher Order Transitions affect toughness
Impact was good if Tg/Tβ was 3 or less.
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Tg
Cold Crystallization
DMTA
DSC
Left: silicon rubber with a glass transition at –117°C and a melting transition at –40°C. Beyond the meltingtemperature this crosslinked (vulcanised) material shows rubber-elasticity with modulus that increases with thetemperature.Right: also a silicone rubber that contains silicone oil as diluent, as plasticizer. The oil causes a stress-relaxation atthe beginning of the melting transition around –47°C.
stress-relaxation in a silicone rubber
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Temperature/K Temperature/K
Polymer A
Temperature/K
E’E’
Polymer B+ =
E’
Temperature/K
Both Tgs
Block CopolymersGraft CopolymersImmiscible Blends
Random Copolymers &
Miscible Blends
E’
Single Tg
Exact T depends on concentration of A and B
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Temperature dependency of E' and tanδ of PVC at different frequencies, afterBecker, Kolloid-Z.140 (1955) 1
The frequency-dependence of dynamic experiments
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Time-Temperature-Superposition Principle
experimentalwindow
experimentalwindow
NBS-poly(isobutylene, after A. Tobolski
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( )( )
( ) ( )20,4
ln ln ; 50102
s sT s g
s
T T Ta T T K
T T T!
= = = ++ !
""
empirical WLF equation≈Tg+50K
Arrhenius-type
Williams-Landel-Ferry (WLF) equation
Temperature-dependence of the viscosityof PMMA (M=63.000 g/mol) after Bueche
lg η
The glass transition temperature seen by viscosity
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( )( )
338.0
lg
lglglg338.0
338.0lg
lim
!!"
#$$%
&
'=(
!!"
#$$%
&=!!"
#$$%
&==''
'=')
**
**
g
g
gT
T
ttATgT
TgTTA
TgT
TTS gives the frequency-dependence of the glass transition temperature:
An increase of the measuring frequency (heating rate) by a factor 10 (or a decrease of the time frame by a factor of 10) near Tg the glass-transition temperature is found about 3 K higher.
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Master Curves*) extend the range
• We can collect data from 0.01 to 100 Hz.• If we do this at many temperatures, we can
“superposition” the data.
• After TTS, our range is 1e-7 to 1e9 Hertz (1/sec)• Then x scale (frequency) can then be inverted to get time
TTS
*) modulus or compliance; compliance = (modulus)-1
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BUT...
TTS assumes that:“all relaxation times are equally affected by
temperature.”
THIS IS KNOWN TO OFTEN BEINVALID.
J. Dealy
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Failure of TTSLo
g J*
compliance J = 1/E
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Analysis of a Cure by DMA
50.0 70.0 90.0 110.0 130.0 150.010 0
10 1
10 2
10 3
10 4
10 5
10 6
10 7
10 8
E’
E”
Mod
ulus
!
"#
E’-E” Crossover ~ gelation point
vitrification point
Minimum Viscosity (time, length,temperature )
106 Pa ~ Solidity
Melting
Curing
time-temperature-transition diagramafter Gillham
experiment at a constant heating rate
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Activation Energy tells us about the molecule
• For example, are these 2 Tgs or a Tg and a Tβ?
• Because we can calculate the Eact for the peaks, we candetermine if both are glass transitions.
ElastomerSample
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351.7 kJ/mol
146.5 kJ/mol
.1log constTR
Ef a +!=
Determination of the apparent energy of activation
How can we do this experimentally??
MULTIPLEXING
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Multiplexing…
Instead of just the Tg
Film
Sheet
Fiber
multiple frequencies in one run
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Or you can use the Synthetic Oscillation Mode
Take five frequencies
Sum together
And applythecomplexwave formto thesample
Temperature in C
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Gelation Point by Multiplexing
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PMMA (0.01~100Hz)
Master curve
Activation energy
We can then do further analysis
Why?…
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To Review, DMA ties together...
molecular structure
processing conditions
product propertiesMolecular weightMW DistributionChain BranchingCross linkingEntanglementsPhasesCrystallinityFree VolumeLocalized motionRelaxation Mechanisms Stress
StrainTemperatureHeat HistoryFrequencyPressureHeat set
MaterialBehavior
Dimensional StabilityImpact properties
Long term behaviorEnvironmental resistance
Temperature performanceAdhesion
TackPeel
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Kevin P. Menard, Dynamic-Mechanical Analysis, CRC-Press (1999) Boca Raton W. Brostow, Performance of Plastics, Carl Hanser Verlag (2000) Munich I. M. Ward, Mechanical Properties of Solid Polymers, Wiley (1983) New YorkJ. J. Aklonis, W. J. McKnight, Introduction to Polymer Viscoelasticity, Wiley-Interscience (1983) New YorkN. W. Tschloegl, The Theory of Viscoelastic Behaviour, Acad. Press (1981) New YorkD. Ferry, Viscoelastic Properties of Polymers, Wiley (1980) New YorkB. E. Read, G. D. Dean, The Determination of Dynamic Properties of Polymers and Composites, Hilger (1978) BristolL. E. Nielsen, Polymer Rheology, Dekker (1977) New YorkL. E. Nielsen, Mechanical Properties of Polymers, Dekker (1974) New YorkL. E. Nielsen, Mechanical Properties of Polymers and Composites Vol. I & II, Dekker (1974) New YorkA. V. Tobolsky, Properties and Structure of Polymers, Wiley (1960) New York
Further Readíng
Many examples by courtesy of Kevin Menard(University of North Texas, Department of Materials Scienceand Perkin Elmer Corp.)