Experimental Observations Chapter 6fmorriso/cm4650/lectures/2020_Chapter6... · Experimental...
Transcript of Experimental Observations Chapter 6fmorriso/cm4650/lectures/2020_Chapter6... · Experimental...
Experimental Observations Chapter 6
1
© Faith A. Morrison, Michigan Tech U.
1
Done with Material Functions.
1. Intro2. Vectors/tensors3. Newtonian4. Standard Flows5. Material Functions6. Experimental
Behavior7. Inelastic effects8. Memory effects9. Advanced10. Rheometry
What now?
© Faith A. Morrison, Michigan Tech U.
3
Standard Velocity and pressure fields
HypotheticalConstitutive
equation
Material Functions (stress responses)
+
Standard Velocity and pressure fields
Ability to measure stresses
predictions of:
measurements of
=
=
+material +
What are material functions and why do we need them?
• We do not know the stress/deformation relationship
• We approach stress/deformation investigations from two directions (modeling, measuring) to reveal the physics;
• Material functions organize comparisons
For non-Newtonian
fluids:
1. Choose a material function2. Predict what Newtonian fluids would do3. See what non-Newtonian fluids do4. Hypothesize a 𝜏 𝑣5. Predict the material function6. Compare with what non-Newtonian fluids do7. Reflect, learn, revise model, repeat.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Investigating Stress/Deformation
Relationships
(Rheology)
Let’s focus more on material
observations
Experimental Observations Chapter 6
2
Chapter 6: Experimental Data
© Faith A. Morrison, Michigan Tech U.
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CM4650 Polymer Rheology
Michigan Tech
10
10000
10000000
10000000000
1E-13 1E-10 1E-07 0.0001 0.1 100 100000
G'
G''107
1010
104
modulus, Pa
oNG
10-4 10-1 102 10510-7
10-1010-13
frequency, rad/s
Small-Amplitude Oscillatory Shear - Storage and Loss Moduli
Figure 6.30, p. 192 Plazek and O’Rourke; PS
Terminal zone
Glassy zone
Rubbery zone
Imposed Kinematics:
𝑣 ≡𝜍 𝑡 𝑥
00
Steady Shear Flow Material Functions
Material Functions:
Viscosity
© Faith A. Morrison, Michigan Tech U.4
𝜍 𝑡 𝛾 constant
𝜂 𝛾 ≡�̃�𝛾
𝜏𝛾
Ψ 𝛾 ≡
Ψ 𝛾 ≡
Material Stress Response: �̃� 𝑡
�̃�
𝑡0
𝑡0
𝛾
𝜍 𝑡
𝑡0
𝛾
𝛾 0, 𝑡
𝑁 𝑡
𝑁 ,
𝑡0
First normal-stress coefficient
Second normal-stress coefficient
Experimental Observations Chapter 6
3
Experimental Data (Chapter 6)
Steady shear flow
•Linear Polymers
•Limits on measurability
•Material effects - MW, MWD, branching, mixtures, copolymers
•Temperature and pressure
Unsteady shear flow (SAOS, step strain, start up, cessation)Steady elongationUnsteady elongation
later ...
© Faith A. Morrison, Michigan Tech U.
5
Material Behavior Catalog (in terms of material functions)
First recipe card:
(the rest of the recipe cards)
What do we observe for
1.E+02
1.E+03
1.E+04
1.E+05
0.01 0.1 1 10 100
, Poise
1,
(dyn/cm2)s2
1, s
o1
o
1
Figure 6.1, p. 170 Menzes and Graessley conc. PB solution
Steady shear viscosity and first normal stress coefficient
© Faith A. Morrison, Michigan Tech U.
6
Material Behavior Catalog (in terms of material functions)
𝜂 𝛾
Ψ 𝛾
𝛾
Experimental Observations Chapter 6
4
Steady shear viscosity and first normal stress coefficient
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
0.01 0.1 1 10 100
, P
ois
e
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
1.E+09
1,
dyn
es/
cm2
1, s
1
Figure 6.2, p. 171 Menzes and Graessley conc. PB solution;
c=0.0676 g/cm3
813 kg/mol517 kg/mol350 kg/mol200 kg/mol
© Faith A. Morrison, Michigan Tech U.
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Material Behavior Catalog (in terms of material functions)
𝛾
Steady shear viscosity and first normal stress coefficient
0.1
1
10
100
0.1 1 10 100
stress, Pa
1, sFigure 6.5, p. 173 Binnington and Boger; PIB soln
sPa ,
21, sPa
© Faith A. Morrison, Michigan Tech U.
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Material Behavior Catalog (in terms of material functions)
𝛾, 𝑠
Experimental Observations Chapter 6
5
0.1
1
10
100
1000
0.01 0.1 1 10 100
stea
dy
she
ar m
ater
ial f
un
ctio
ns
1 (Pa s2)
(Pa)
2 (Pa s2)
PS/TCP squaresPS/DOP circles
1, s
Steady shear viscosity and first and secondnormal stress coefficient
Figure 6.6, p. 174 Magda et al.; PS solns
© Faith A. Morrison, Michigan Tech U.
9
Material Behavior Catalog (in terms of material functions)
𝛾, 𝑠
Limits on Measurements: Flow instabilities in rheology
Figures 6.7 and 6.8, p. 175 Hutton; PDMS
Figure 6.9, p. 176 Pomar et al. LLDPE
cone and plate flow capillary flow
© Faith A. Morrison, Michigan Tech U.
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Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
6
10
100
1,000
10,000
1.E+05 1.E+06 1.E+07
LPE-1
LPE-2
LPE-3
LPE-4
(s -1 )3
4
R
Q
2/,2
cmdynesL
RP
Spurt instability
Figure 6.10, p. 177 Blyler and Hart; PE
2Rb
Q
B
A
Polymer meltreservoir
Piston
Barrel
F
Instroncapillary
rheometer
© Faith A. Morrison, Michigan Tech U.
11
Material Behavior Catalog (in terms of material functions)
Steady shear viscosity and first normal stress coefficient - Molecular weight effects
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
0.01 0.1 1 10 100
, P
ois
e
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
1.E+09
1, d
yne
s/cm
2
1, s
1
Figure 6.2, p. 171 Menzes and Graessley conc. PB solution;
various MW
813 kg/mol517 kg/mol350 kg/mol200 kg/mol
© Faith A. Morrison, Michigan Tech U.
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Material Behavior Catalog (in terms of material functions)
𝛾, 𝑠
Experimental Observations Chapter 6
7
log M + constant
1
constantlog 0
Variation of viscosity with molecular weight
Figure 6.12, p. 178 Berry and Fox, 1968; various polymers
slope=3.4-3.5
Mc
© Faith A. Morrison, Michigan Tech U.
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Material Behavior Catalog (in terms of material functions)
M<Mc M>Mc
Unentangledrelaxation is rapid
Entangledrelaxation is retarded
Entanglements strongly affect polymer relaxation
© Faith A. Morrison, Michigan Tech U.
14
Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
8
Effect of Distribution of Molecular Weight
Figure 6.14, p. 179 Berry and Fox; PVA solns in DEP
A - Mw/Mn=1.09
B - Mw/Mn= 2.0
C - branched
© Faith A. Morrison, Michigan Tech U.
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Material Behavior Catalog (in terms of material functions)
𝛾𝜏2
𝜂𝜂
Types of polymer architecture
short-chain branching
long-chain branching
star
hyperbranching
dendrimer
© Faith A. Morrison, Michigan Tech U.
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Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
9
branch points retard motion
chains move principally along
their contour
Entangled linear
polymer
Entangled branched polymer
once disentangled (high shear rate), branched
polymers flow more freely
Motion of Branched Polymers
© Faith A. Morrison, Michigan Tech U.
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Material Behavior Catalog (in terms of material functions)
Effect of Branching on 𝜼 𝜸
Figures 6.17, 6.18, pp. 181,3 Kraus & Gruver; PB
120 slow
linear3-arm star4-arm star
linear3-arm star4-arm star
© Faith A. Morrison, Michigan Tech U.
18
Material Behavior Catalog (in terms of material functions)
𝛾 low 𝛾 20𝑠
Experimental Observations Chapter 6
10
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
1.E+09
1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02
aT* shear rate, 1/s
vis
cosi
ty, p
ois
e
Gen 0Gen 1Gen 2Gen 3Gen 4Gen 5Gen 6
Steady shear rheology of PAMAM dendrimers
Figure 6.20 p. 183; from Uppuluri© Faith A. Morrison, Michigan Tech U.
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Material Behavior Catalog (in terms of material functions)
𝛾𝑎 , 𝑠
Steady Shear Summary:
© Faith A. Morrison, Michigan Tech U.
20
1. General traits
2. Effect of MW on linear polymers
3. Effect of architecture
4. Measurement issues
Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
11
Mixtures of Polymers with other materials - Filler Effect
Figure 6.22, p. 184 Chapman and Lee; PP and filled PP
For more on filled systems, see Larson, The Structure and Rheology of Complex Fluids, Oxford, 1999.
(specific chemical interactions)
(no specific interactions)
© Faith A. Morrison, Michigan Tech U.
21
Material Behavior Catalog (in terms of material functions)
Mixtures of Polymers with other materials - Filler Effect
Figure 6.21, p. 184 Chapman and Lee; filled PP
For more on filled systems, see Larson, The Structure and Rheology of Complex Fluids, Oxford, 1999.
© Faith A. Morrison, Michigan Tech U.
22
(specific chemical interactions)
Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
12
0.001
0.01
0.1
1
10
1 10 100 1000 10000
47 42 38 27.2 12
1, s
)( sPa
vol % TiO2
Steady shear viscosity - shear thickening
Figure 6.27, p. 188 Metzner and Whitlock; TiO2/water
suspensions© Faith A. Morrison, Michigan Tech U.
23
For discussion of causes of shear thickening, see Wagner and Brady, Phys.
Today, October 2009
Material Behavior Catalog (in terms of material functions)
𝛾, 𝑠
24
Wagner and Brady, Phys. Today, Oct 2009
© Faith A. Morrison, Michigan Tech U.
Latex spheres in Newtonian fluid
Filler effect
Shear thickening
Material Behavior Catalog (in terms of material functions)
Steady shear viscosity - shear thickening
Experimental Observations Chapter 6
13
Steady shear viscosity - shear thickening
© Faith A. Morrison, Michigan Tech U.
25
Wagner and Brady, Phys. Today, Oct 2009
Hydrodynamic interaction causes shear thickening:
The pressure distribution in the Newtonian suspending fluid exerts forces on the particles and causes them to form structures in flow.
Particles bump into each other
Layers of particles slide
past each other
Clusters of particles bump into
each other
Material Behavior Catalog (in terms of material functions)
0.01
0.1
1
10
0.1 1 10 100
Poise
viscosity (Poise) 300 K
viscosity (Poise) 339 K
viscosity (Poise) 380 K
viscosity (Poise) 425 K
1, s
Steady shear viscosity - temperature dependence
Figure 6.28, p. 189 Gruver and Kraus; PB melt
© Faith A. Morrison, Michigan Tech U.
26
Material Behavior Catalog (in terms of material functions)
𝛾, 𝑠
Experimental Observations Chapter 6
14
10
100
1,000
10,000
100,000
0 50 100 150 200
hydrostatic pressure, MPa
app
aren
t vi
scos
ity,
Pa
s
PS, T=196CPE, T=149C
Steady shear viscosity - pressure dependence
Figure 6.29, p. 189 Maxwell and Jung; PS and PE
© Faith A. Morrison, Michigan Tech U.
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Material Behavior Catalog (in terms of material functions)
Steady Shear Summary:
© Faith A. Morrison, Michigan Tech U.
28
1. General traits
2. Effect of MW (linear polymers)
3. Effect of architecture
4. Measurement issues
5. Effect of chemical composition
6. Effect of temperature
7. Effect of pressure
Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
15
Unsteady shear flows (small and large strain) Steady elongationUnsteady elongation
Next:
© Faith A. Morrison, Michigan Tech U.
29
Experimental Data (continues)
Material Behavior Catalog (in terms of material functions)
10
10000
10000000
10000000000
1E-13 1E-10 1E-07 0.0001 0.1 100 100000
G'
G''107
1010
104
modulus, Pa
oNG
10-4 10-1 102 10510-7
10-1010-13
frequency, rad/s
Small-Amplitude Oscillatory Shear - Storage and Loss Moduli
Figure 6.30, p. 192 Plazek and O’Rourke; PS
© Faith A. Morrison, Michigan Tech U.
slope=2
slope=1
Rubbery plateau
30
Material Behavior Catalog (in terms of material functions)
𝜔, 𝑟𝑎𝑑/𝑠
Experimental Observations Chapter 6
16
10
10000
10000000
10000000000
1E-13 1E-10 1E-07 0.0001 0.1 100 100000
G'
G''107
1010
104
modulus, Pa
oNG
10-4 10-1 102 10510-7
10-1010-13
frequency, rad/s
Small-Amplitude Oscillatory Shear - Storage and Loss Moduli
Figure 6.30, p. 192 Plazek and O’Rourke; PS
© Faith A. Morrison, Michigan Tech U.
Terminal zone
Glassy zone
Rubbery zone
31
Material Behavior Catalog (in terms of material functions)
𝜔, 𝑟𝑎𝑑/𝑠
1.E+01
1.E+04
1.E+07
1.E+10
0.01 10 10000 10000000 10000000000
1E+13104 107 1010 1013
time, s
107
1010
104
G(t) Pa
104
Step Shear Strain - Relaxation Modulus
Figure 6.31, p. 192 Plazek and O’Rourke; PS
© Faith A. Morrison, Michigan Tech U.
Glassy zone
Terminal zone
Rubbery zone
Rubbery plateau
32
Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
17
Ferry’s Summary of Viscoelastic properties of several classes of polymers
Figure2-3 from Ferry, Viscoelastic Properties of
Polymers, Wiley, 1980© Faith A. Morrison, Michigan Tech U.
Storage modulus
33
Material Behavior Catalog (in terms of material functions)
𝜔, 𝑟𝑎𝑑/𝑠
Key to Ferry’s plots
I. Dilute polymer solutions: atactic polystyrene, 0.015 g/ml in Aroclor 1248, a chlorinated diphenyl with viscosity 2.47 poise at 25oC. Mw=86,000, Mw/Mn near 1.
II. Amorphous polymer of low molecular weight: poly(vinyl acetate), M=10,500, fractionated.
III. Amorphous polymer of high molecular weight: atactic polystyrene, narrow MW distribution, Mw=600,000.
IV. Amorphous polymer of high molecular weight with long side groups: fractionated poly(n-octyl methacrylate), Mw=3.62 X 106.
V. Amorphous polymer of high molecular weight below its glass transition temperature: poly(methyl methacrylate).
VI. Lightly cross-linked amorphous polymer: lightly vulcanized Hevea rubber.
VII. Very lightly cross-linked amorphous polymer: styrene butadiene random copolymer, 23.5% styrene by weight.
VII. Highly crystalline polymer: linear polyethylene.
34
Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
18
© Faith A. Morrison, Michigan Tech U.
Ferry’s Summary of Viscoelastic properties of several classes of polymers
Figure2-4 from Ferry, Viscoelastic Properties of
Polymers, Wiley, 1980
Loss modulus
35
Material Behavior Catalog (in terms of material functions)
© Faith A. Morrison, Michigan Tech U.
Ferry’s Summary of Viscoelastic properties of several classes of polymers
Figure2-2 from Ferry, Viscoelastic Properties of
Polymers, Wiley, 1980
Relaxation modulus
36
Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
19
© Faith A. Morrison, Michigan Tech U.
Ferry’s Summary of Viscoelastic properties of several classes of polymers
Figure2-8 from Ferry, Viscoelastic Properties of
Polymers, Wiley, 1980
Loss tangent
37
Material Behavior Catalog (in terms of material functions)
© Faith A. Morrison, Michigan Tech U.
Cox-Merz Rule
)()( *
Figure 6.32, p. 193 Venkataraman et al.; LDPE
An empirical way to infer steady shear data from SAOS
data.
38
Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
20
© Faith A. Morrison, Michigan Tech U.
Figure 6.34, p. 195 Onogi et al; narrow MWD PS
Small-Amplitude Oscillatory Shear - G’molecular weight dependence
Breadth of plateau depends on MW
When MW is below Mc, there is no plateau.All materials
show terminal behavior
39
Level of plateau GN
0
related to Me
Material Behavior Catalog (in terms of material functions)
log 𝑎 𝜔, 𝑟𝑎𝑑/𝑠
Entanglements form a temporary network
© Faith A. Morrison, Michigan Tech U.
40
crosslinked elastic network –constant modulus G at all 𝜔
Entangled polymers – over a range of w when unable to disentangle, exhibit a constant modulus 𝐺
M>Mc
Me
Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
21
© Faith A. Morrison, Michigan Tech U.
Small-Amplitude Oscillatory Shear - G’molecular weight dependence
41
Level of plateau 𝐺𝑁0 is related to 𝑀𝑒
(molecular theory for temporary networks)
TkG BN 5
40 𝑛 density of effective cross links
𝑛 cross links/volume
e
BAN
A
volumecrosslinks
molecrosslinks
volumemass
e
M
TkNG
NM
5
40
ec MM 2Larger the MW between entanglements, the softer the network
See Larson, The Structure and Rheology of Complex Fluids, Oxford, 1999
Material Behavior Catalog (in terms of material functions)
© Faith A. Morrison, Michigan Tech U.
Figure 6.36, p. 196 Onogi et al; narrow MWD PS
Small-Amplitude Oscillatory Shear - G”molecular weight dependence
Breadth of minimum depends on MW; structure is more complex
When MW is below Mc, there is no minimum.All materials
show terminal behavior
42
Material Behavior Catalog (in terms of material functions)
log 𝑎 𝜔, 𝑟𝑎𝑑/𝑠
Experimental Observations Chapter 6
22
Small-Strain Unsteady Shear Summary:
© Faith A. Morrison, Michigan Tech U.
43
1. General traits
2. Effect of MW (linear polymers)
3. Effect of architecture
4. Relationship to steady flow material functions
5. Measurement issues
6. Effect of chemical composition
Material Behavior Catalog (in terms of material functions)
Small-Amplitude Oscillatory Shear - G’ as a function of temperature for copolymers
© Faith A. Morrison, Michigan Tech U.
Figure 6.39, p. 198 Cooper and Tobolsky; SIS block and SBS random
Block copolymers Random copolymers
44
Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
23
1.E-10
1.E-09
1.E-08
1.E-07
1.E-06
1.E-05
10 100 1,000 10,000 (hz)
J' (
cm2 /d
yne
)
-14.3 -9.6
-5 -0.1
4 9.9 15.1 19.8
25.3 30
34.2 38.8
44.4 50.2 54.4 59.8
65.8 70.9
80.2 89.4
99.8 109.4 120.3 129.5
T (oC)
Figure 6.43, p. 202 Dannhauser et al.; P-OctylMethacrylate
© Faith A. Morrison, Michigan Tech U.
Small-Amplitude Oscillatory Shear -temperature dependence
45
Material Behavior Catalog (in terms of material functions)
Time-Temperature Superposition
Material functions depend on gi, li
),,(
),,(
ii
ii
gGG
gGG
relaxation times
relaxation moduli
gi, li are in turn functions of temperature and material properties
Theoretical result: in the linear-viscoelastic regime, material functions are a function of wli
rather than of w and li individually.
© Faith A. Morrison, Michigan Tech U.
46
Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
24
Example: plot a simple function )sin(),(f
0.01
0.1
1
10
0.01 0.1 1 10 or
sin()
+
0.2
0.3
0.4
0.5
0.6
1.0
=
© Faith A. Morrison, Michigan Tech U.
.vs)sin(
47
Material Behavior Catalog (in terms of material functions)
© Faith A. Morrison, Michigan Tech U.
In general,
)),(),(),(( 321 TTTGG
Suppose that the temperature-dependence of li could be factored out. Let aTi(T) be the temperature-dependence of li.
iTii TaT ~)()(
Then we could group the temperature-dependence function with the frequency.
),~,~,~( 332211 TTT aaaGG
not a function of temperature
48
Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
25
Time-Temperature Superposition
•Relaxation times decrease strongly as temperature increases
•Moduli associated with relaxations are proportional to absolute temperature; depend on density
Empirical observation: for many materials, all the relaxation times and moduli have the same
functional dependence on temperature
)(~
)( TaT Tii temperature dependence of
all relaxation times
)(~)( TTgTg ii temperature dependence of
all moduli
© Faith A. Morrison, Michigan Tech U.
49
(for the 𝑖𝑡ℎ relaxation
mode)
Material Behavior Catalog (in terms of material functions)
Second theoretical result: the 𝑔𝑖 enter into the functions for 𝐺’, 𝐺’’ such that 𝑇𝜌 can be
factored out of the function
)~
,(~
)~
,(~
iT
iT
ahT
G
afT
G
Therefore if we plot reduced variables, we can suppress all of the temperature dependence of the moduli.
)()~
,()(
)()~
,()(
refrefrefiTrefref
r
refrefrefiTrefref
r
TGTahT
TTGG
TGTafT
TTGG
Plots of 𝐺 ,𝐺 versus 𝑎 𝜔 will therefore be independent of temperature.
© Faith A. Morrison, Michigan Tech U.
(will still depend on the material through the 𝜆 )
50
Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
26
1.E-10
1.E-09
1.E-08
1.E-07
1.E-06
1.E-05
1.E-07 1.E-05 1.E-03 1.E-01 1.E+01 1.E+03 1.E+05
aT (hz)
J' (
cm
2 /dyn
e)
Figure 6.44, p. 202 Dannhauser et al.; P-OctylMethacrylate
© Faith A. Morrison, Michigan Tech U.
Small-Amplitude Oscillatory Shear -temperature dependence
51
Material Behavior Catalog (in terms of material functions)
-50 0 50 100 150
T (oC)
aT
experimental WLF
10-8
10-6
10-4
10-2
100
Small-Amplitude Oscillatory Shear -temperature dependence
Figure 6.45, p. 203 Dannhauser et al.; P-OctylMethacrylate
© Faith A. Morrison, Michigan Tech U.
52
Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
27
Shift Factors
refT TTR
Ha
11exp
Arrhenius equation
ref
refT
TTc
TTca
0
2
01log
Williams-Landel-Ferry (WLF) equation
found to be valid for 𝑇 𝑇 100 𝐶
found to be valid w/in
100𝑜𝐶 of 𝑇𝑔
© Faith A. Morrison, Michigan Tech U.
53
Material Behavior Catalog (in terms of material functions)
12
2
tan1
/1tan1
/1
tan
)(
)(
GJ
GJ
G
G
TG
TG
Shifting other Material Functions
Other linear viscoelasticmaterial functions:
© Faith A. Morrison, Michigan Tech U.
54
)()~
,()(
)()~
,()(
refrefrefiTrefref
r
refrefrefiTrefref
r
TGTahT
TTGG
TGTafT
TTGG
Independent of temperature
Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
28
refrefr
refrefr
T
refref
T
refrefr
T
refref
T
refrefr
T
TTJJ
T
TTJJ
Ta
T
Ta
TTG
Ta
T
Ta
TTG
)(
)(
)(
)(
Ta
TTa
T
refrefTr
)()(
Shifting other Material Functions
linear viscoelastic steady shear
© Faith A. Morrison, Michigan Tech U.
55
G
G
tan = independent of temperature when
plotted versus reduced frequency)
Material Behavior Catalog (in terms of material functions)
0.01
0.1
1
10
0.01 0.1 1 10 100
r,Poise
1, saT
0.01
0.1
1
2 2.5 3 3.5
1000/T (K)
aT
0.01
0.1
1
10
0.1 1 10 100
Poise
viscosity (Poise) 300 K
viscosity (Poise) 339 K
viscosity (Poise) 380 K
viscosity (Poise) 425 K
1, s
Steady shear viscosity -Temperature dependence
Figure 6.46, p. 204 Gruver and Kraus; PB melt
© Faith A. Morrison, Michigan Tech U.
56
Material Behavior Catalog (in terms of material functions)
𝑎 𝛾, 𝑠𝛾, 𝑠
Experimental Observations Chapter 6
29
10
10000
10000000
10000000000
1E-13 1E-10 1E-07 0.0001 0.1 100 100000
G'
G''107
1010
104
modulus, Pa
oNG
10-4 10-1 102 10510-7
10-1010-13
frequency, rad/s
Another consequence of 𝜆 𝑇 𝜆 𝑎 𝑇 is the similarity
between log 𝐺’ 𝜔 and log 𝐺’ 𝑇 .
© Faith A. Morrison, Michigan Tech U.
Figure 6.39, p. 198 Cooper and Tobolsky; SIS block and SBS random
Figure 6.30, p. 192 Plazek and O’Rourke; PS
rad/s,CT o, CT o,
57
Material Behavior Catalog (in terms of material functions)
)~
,()(
)~
,()(
iTrefref
r
iTrefref
r
ahT
TTGG
afT
TTGG
© Faith A. Morrison, Michigan Tech U.
Take data for G’, G’’ at a fixed w for a variety of T.
but, what is aT(T)?
We do not know.
But since log aT is approximately a linear function of T,
-50 0 50 100 150
T (oC)
aT
experimental WLF
10-8
10-6
10-4
10-2
100
curves of log G’ versus T (not log T ) at constant wresemble slightly skewed plots of log G’ versus log aTw.
(mirror image)58
Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
30
Using G’(T) in research on pressure-sensitive
adhesives
© Faith A. Morrison, Michigan Tech U.
Figure 6.48, p. 207 Kim et al.; SIS block copolymer with tackifier
59
0 wt% 30 wt% 50 wt% 70 wt% tackifier
Application of SAOS
Material Behavior Catalog (in terms of material functions)
M>Mc
Entanglements form a temporary network
© Faith A. Morrison, Michigan Tech U.
60
crosslinked elastic network – constant modulus G at all
w
Entangled polymers – over a range of w when unable to disentangle, exhibit a constant modulus GN
0
Me
Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
31
© Faith A. Morrison, Michigan Tech U.
Small-Amplitude Oscillatory Shear - G’molecular weight dependence
61
Level of plateau GN0
is related to Me
(molecular theory for temporary networks) e
BAN M
TkNG
5
40
Larger the MW between entanglements, the softer the network
Material Behavior Catalog (in terms of material functions)
Using G’(T) in research on pressure-sensitive
adhesives
© Faith A. Morrison, Michigan Tech U.
Figure 6.48, p. 207 Kim et al.; SIS block copolymer with tackifier
Height of plateau modulus and temperature of glass
transition are key performance factors for
PSAs.0NG
gT62
Tack – if not tacky, will not produce bond
Shear holding – if too fluid, will slide under shear
tack increasing
Shear holding decreasing
Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
32
Small-Strain Unsteady Shear Summary:
© Faith A. Morrison, Michigan Tech U.
63
1. General traits
2. Effect of MW (linear polymers)
3. Effect of architecture
4. Relationship to steady flow material functions
5. Measurement issues
6. Effect of chemical composition
7. Effect of temperature
Material Behavior Catalog (in terms of material functions)
Experimental Data (continued)
Unsteady shear flow
•Small strain - SAOS, step strain
•Large strain - start-up, cessation, creep, large-amplitude step strain
Steady elongationUnsteady elongation
lastly . ..
linear polymers, material effects, temperature effects
© Faith A. Morrison, Michigan Tech U.
64
Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
33
Startup of Steady Shearing
© Faith A. Morrison, Michigan Tech U.
Figures 6.49, 6.50, p. 208 Menezes and Graessley, PB soln 65
Material Behavior Catalog (in terms of material functions)
𝜂 ≡𝜏 𝑡𝛾
Ψ ≡𝜏 𝜏
𝛾
𝑣 ≡𝜍 𝑡 𝑥
00
𝜍 𝑡0 𝑡 0𝛾 𝑡 0
Cessation of Steady Shearing
© Faith A. Morrison, Michigan Tech U.
Figures 6.51, 6.52, p. 209 Menezes and Graessley, PB soln
66
Material Behavior Catalog (in terms of material functions)
𝑣 ≡𝜍 𝑡 𝑥
00
𝜍 𝑡 𝛾 𝑡 00 𝑡 0
𝜂 ≡𝜏 𝑡𝛾
Ψ ≡𝜏 𝜏
𝛾
Experimental Observations Chapter 6
34
Shear Creep
© Faith A. Morrison, Michigan Tech U.
0
210
),0(),(
t
tJrefref
p T
TTJJ
)(
Data have been corrected for vertical shift.
67
Material Behavior Catalog (in terms of material functions)
Figure 6.53, p. 210 Plazek; PS melt
𝑣 ≡𝛾 𝑡 𝑥
00
𝜏 𝑡0 𝑡 0𝜏 𝑡 0
© Faith A. Morrison, Michigan Tech U.
Figures 6.54, 6.55, p. 211 Plazek; PS melt
Shear Creep - Recoverable Compliance
0
)()(t
tRtJ
ttt r )()(
total strain
recoverable strain
non-recoverable
strain
68
Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
35
0
1
10
100
1,000
10,000
1 10 100 1000 10000
time, s
G(t), Pa
<1.87
3.34
5.22
6.68
10
13.4
18.7
25.4
Step shear strain - strain dependence
Figure 6.57, p. 212 Einaga et al.; PS soln
© Faith A. Morrison, Michigan Tech U.
69
Material Behavior Catalog (in terms of material functions)
)(log)(log),(log
)()(),(
00
00
htGtG
htGtG
Shear Damping Function
Observation: step-strain moduli curves have similar shapes and
appear to be shifted down with strain.
Damping function
The damping function gives the strain-dependence of the step-strain relaxation
modulus.
When the behavior is called time-strain
separable.
)()(),( 00 htGtG This behavior is predicted by some advanced constitutive
equations.
70
Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
36
1
10
100
1000
10000
100000
1 10 100 1000 10000
time, t
shif
ted
G(t
), P
a
<1.87
3.345.22
6.68
10
13.418.7
25.4
Step shear strain -Damping Function
Figure 6.58, p. 213 Einaga et al.; PS soln
0.01
0.1
1
10
0.1 1 10 100
strain
dam
pin
g f
un
ctio
n,
h
© Faith A. Morrison, Michigan Tech U.
71
Material Behavior Catalog (in terms of material functions)
Large-Strain Unsteady Shear Summary:
© Faith A. Morrison, Michigan Tech U.
72
1. General traits
2. Measurement issues
Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
37
Experimental Data (continued)
Unsteady shear flow
•Small strain - SAOS, step strain
•Large strain - start-up, cessation, creep, large-amplitude step strain
Steady elongationUnsteady elongation
lastly . ..
linear polymers, material effects, temperature effects
© Faith A. Morrison, Michigan Tech U.
73
Material Behavior Catalog (in terms of material functions)
Steady State Elongation Viscosity
© Faith A. Morrison, Michigan Tech U.
Figure 6.60, p. 215 Munstedt.; PS melt
Both tension thinning and
thickening are observed.
0
TrTrouton ratio:
74
Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
38
© Faith A. Morrison, Michigan Tech U.
Figure 6.63, p. 217 Inkson et al.; LDPE
Figure 6.64, p. 218 Kurzbeck et al.; PP
Start-up of Steady
Elongation
Strain-hardening
Fit to an advanced constitutive equation (12 mode pom-pom
model)
75
Material Behavior Catalog (in terms of material functions)
© Faith A. Morrison, Michigan Tech U.
Elongation Step-Strain
h(t) R(t)
R(to)
h(to) x1
x3
thin, lubricatinglayer on eachplate
Figures 6.68, 6.69, pp. 220-1 Soskey and Winter; LDPE
Damping function for step biaxial elongation
76
Material Behavior Catalog (in terms of material functions)
Relaxation function for step biaxial elongation
Experimental Observations Chapter 6
39
Elongational Flow Summary:
© Faith A. Morrison, Michigan Tech U.
77
1. General traits
2. Measurement issues
Material Behavior Catalog (in terms of material functions)
© Faith A. Morrison, Michigan Tech U.
More on Material Behavior
Polymer Behavior
Larson, Ron, The Structure and Rheology of Complex Fluids (Oxford, 1999)Ferry, John, Viscoelastic Properties of Polymers (Wiley, 1980)
Suspension Behavior
Mewis, Jan and Norm Wagner, Colloidal Suspension (Cambridge, 2012)
78
Journals
Journal of RheologyRheologica ActaMacromolecules
Material Behavior Catalog (in terms of material functions)
Experimental Observations Chapter 6
40
© Faith A. Morrison, Michigan Tech U.79
Done with Experimental Data.
Let’s move on to Generalized
Newtonian Fluids
© Faith A. Morrison, Michigan Tech U.80