Lecture 4 CM4655 Morrison 2016
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CM4655 Polymer Rheology Lab
© Faith A. Morrison, Michigan Tech U.
Torsional Shear Flow: Parallel-plate and Cone-and-plate
(Steady and SAOS)
r H
r
(-planesection)
(planesection)
Professor Faith A. Morrison
Department of Chemical EngineeringMichigan Technological University
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log
log
o
© Faith A. Morrison, Michigan Tech U.
Why do we need more than one method of measuring viscosity?
Torsional flows Capillary flow
•At low deformation rates, torques & pressures become low
•At deformation high rates, torques & pressures become high; flow instabilities set in
Torsional Shear Flow: Parallel-plate and Cone-and-plate
The choice is determined by experimental issues (signal, noise, instrument limitations.
Instabilities in torsional flows
Low signal in capillary flow
Lecture 4 CM4655 Morrison 2016
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x
y
x
y(z-planesection)
(z-planesection)
r H
r
(-planesection)
(planesection)
r
(z-planesection)
(-planesection)
(z-planesection)
(-planesection)
Experimental Shear Geometries
© Faith A. Morrison, Michigan Tech U.
rz
2R well-developed flow
exit region
entrance region
rz
2R well-developed flow
exit region
entrance region
rz
2R well-developed flow
exit region
entrance region
rz
2R well-developed flow
exit region
entrance region
Shear Flow
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© Faith A. Morrison, Michigan Tech U.
We will look at two flows measurable in torsional shear: steady and small-amplitude oscillatory shear (SAOS)
0 t
o
t
0 t
o
t21
0 t
o
t,021
a. Steady
.
t
tto cosf. SAOS
t
t21)sin( to
t,021
t
to sin
Material functions: G’(), G”() or ’(), ’’()
Material functions: 1(), 2(). . .
Torsional Shear Flow: Parallel-plate and Cone-and-plate
(Linear viscoelastic regime)
Lecture 4 CM4655 Morrison 2016
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© 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.
22
22* )(
GG
∗, ,
Imposed Kinematics:
≡ 00
Steady Shear Flow Material Functions
Material Functions:
Viscosity
© Faith A. Morrison, Michigan Tech U.6
constant
≡
Ψ ≡
Ψ ≡
Material Stress Response:
0
0 0
0,
,
0
First normal-stress coefficient
Second normal-stress coefficient
Lecture 4 CM4655 Morrison 2016
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© Faith A. Morrison, Michigan Tech U.
r
z
H
cross-sectionalview:
R
Torsional Parallel-Plate Flow - Viscosity
Measureables:Torque T to turn plateRate of angular rotation W
Note: shear rate experienced by fluid elements depends on their r position. R
r
H
rR
By carrying out a Rabinowitsch-like calculation, we can obtain the stress at the rim (r=R).
RRrz d
RTdRT
ln
)2/ln(32/
33
R
RrzR
)( Correction required
Torsional Shear Flow: Parallel-plate
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© Faith A. Morrison, Michigan Tech U.
0.1
1
10
0.1 1 10
slope is a function of R
32 R
T
HR
R
RdRTd
slopelog2/log 3
Parallel-Plate Shear Rate Correction
RRR d
RTdRT
ln
)2/ln(3
2/)(
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Torsional Shear Flow: Parallel-plate
Lecture 4 CM4655 Morrison 2016
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© Faith A. Morrison, Michigan Tech U.
Torsional Cone-and-Plate Flow Viscosity
Measureables:Torque T to turn coneRate of angular rotation
Since shear rate is constant everywhere, so is stress, and we can calculate stress from torque.
r
R
(-planesection)
polymer melt
Note: the introduction of the cone means that shear rate is independent of r.
No corrections needed in cone-and-plate
Torsional Shear Flow: Cone-and-plate
ΩΘ
constant32
3 Θ2 Ω
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© Faith A. Morrison, Michigan Tech U.
1st Normal Stress (C&P)
Measureables:Normal thrust F
r
R
(-planesection)
polymer melt
22
20
1
2)(
R
F
The total upward thrust of the cone can be related directly to the first normal stress coefficient.
atm
R
pRrdrF 2
0 2
2
(see text pp404-5; also DPL pp522-523)
Torsional Shear Flow: Cone-and-plate
Lecture 4 CM4655 Morrison 2016
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© Faith A. Morrison, Michigan Tech U.
•Cone and Plate:
•MEMS used to manufacture sensors at different radial positions
S. G. Baek and J. J. Magda, J. Rheology, 47(5), 1249-1260 (2003)
J. Magda et al. Proc. XIV International Congress on Rheology, Seoul, 2004.
221022 ln)2( NR
rNNp
(see Bird et al., DPL)
Need normal force as a function of r / R
2nd Normal Stress (C&P)
RheoSense Incorporated (www.rheosense.com)
Torsional Shear Flow: Cone-and-plate
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© Faith A. Morrison, Michigan Tech U.
Comparison with other instruments
S. G. Baek and J. J. Magda, J. Rheology, 47(5), 1249-1260 (2003)
RheoSense Incorporated
Torsional Shear Flow: Cone-and-plate
Lecture 4 CM4655 Morrison 2016
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Start up of Steady Shear flow: obtain Steady State
© Faith A. Morrison, Michigan Tech U.
Torsional Shear Flow: Cone-and-plate
Choose Take frequent data points
Steady state must be experimentally observed for
each chosen
••
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Start up of Steady Shear flow: obtain Steady State
© Faith A. Morrison, Michigan Tech U.
Ψ
Steady state must be
experimentally observed for each chosen
Torsional Shear Flow: Cone-and-plate
Obtained in the same run as More experimental noise
Ψ
••
Lecture 4 CM4655 Morrison 2016
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© Faith A. Morrison, Michigan Tech U.
Report Flow Curves , Ψ
Torsional Shear Flow: Parallel-plate and Cone-and-plate
100
1000
10000
100000
0.01 0.1 1 10 100
Ψ
• Obtain widest range of possible within the ability of the instrument
• Report on reproducibility
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Limits on Measurements: Flow instabilities in rheology
Figures 6.7 and 6.8, p. 175 Hutton; PDMS
cone and plate flow
© Faith A. Morrison, Michigan Tech U.
Torsional Shear Flow: Cone-and-plate
High (steady shear) or (SAOS) cause these
instabilities to be observed.
Lecture 4 CM4655 Morrison 2016
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Small-Amplitude Oscillatory Shear Material Functions
SAOS stress
© Faith A. Morrison, Michigan Tech U.17
≡ 00
cos
, , sin sin cos
′ ≡cos ′′ ≡
sin
0
0,
0
0(linear viscoelastic regime)
Storage modulus
Loss modulus
0
cos sin
sin
phase difference between stress and strain waves
Imposed Kinematics:
Material Functions:
Material Stress Response:
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What is the strain in SAOS flow?
t
tdt
tdtt
t
t
sin
cos
)(),0(
0
0
0
0 2121
The strain imposed is sinusoidal.
© Faith A. Morrison, Michigan Tech U.
The strain amplitude is:
Small-Amplitude Oscillatory Shear (SAOS)
Lecture 4 CM4655 Morrison 2016
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h
1x
2x
h
1x
2x
thVttb o)(
th
thtb
o
o
sin
sin)(
Generating Shear
© Faith A. Morrison, Michigan Tech U.
sin
Steady shear
Small-amplitude oscillatory shear
constant
periodic
Small-Amplitude Oscillatory Shear (SAOS)
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In SAOS the strain amplitude is small, and a sinusoidal imposed strain induces a sinusoidal measured stress (in the linear regime).
)sin()( 021 tt
tttt
tt
cossinsincossincoscossin
)sin()(
00
00
021
portion in-phase with strain
portion in-phase with strain-rate
© Faith A. Morrison, Michigan Tech U.
Small-Amplitude Oscillatory Shear (SAOS)
Lecture 4 CM4655 Morrison 2016
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© Faith A. Morrison, Michigan Tech U.
-3
-2
-1
0
1
2
3
0 2 4 6 8 10
is the phase difference between the stress wave and the strain wave
)(21 t
)(),0( 2121 tt
Small-Amplitude Oscillatory Shear (SAOS)
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SAOS Material Functions
ttt
cossin
sincos)(
0
0
0
0
0
21
portion in-phase with strain
portion in-phase with strain-rate
G G
For Newtonian fluids, stress is proportional to strain rate: 2121
G” is thus known as the viscous loss modulus. It characterizes the viscous contribution to the stress response.
© Faith A. Morrison, Michigan Tech U.
Small-Amplitude Oscillatory Shear (SAOS)
Lecture 4 CM4655 Morrison 2016
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2
121 x
uG
1u 21 xv
2x
initial state,no flow,no forces
deformed state,
Hooke's law forelastic solids
spring restoring force
11 xkf
initial state,no forceinitial state,no force
deformed state,
Hooke's law forlinear springs
f
1x1x
Hooke’s Law for elastic solids
Similar to the linear spring law
What types of materials generate stress in proportion to the strain imposed? Answer: elastic solids
2121 G
© Faith A. Morrison, Michigan Tech U.
Small-Amplitude Oscillatory Shear (SAOS)
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SAOS Material Functions
ttt
cossin
sincos)(
0
0
0
0
0
21
portion in-phase with strain
portion in-phase with strain-rate
G G
For Hookean solids, stress is proportional to strain : 2121 G
G’ is thus known as the elastic storage modulus. It characterizes the elastic contribution to the stress response.
(note: SAOS material functions may also be expressed in complex notation. See pp. 156-159 of Morrison, 2001)
© Faith A. Morrison, Michigan Tech U.
Small-Amplitude Oscillatory Shear (SAOS)
Lecture 4 CM4655 Morrison 2016
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Linear-Viscoelastic Regime
© Faith A. Morrison, Michigan Tech U.
Small-Amplitude Oscillatory Shear (SAOS)
sin cos
Impose: sin
Measure: sin
Report:
The response must be independent of the strain amplitude
cos ∗ cos
sin ∗ sin
Choose to obtain good, strong signal, but within the linear regime
∝
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Linear-Viscoelastic Regime: Strain Sweep
© Faith A. Morrison, Michigan Tech U.
Small-Amplitude Oscillatory Shear (SAOS)
LVE Limit
The linear regime must be experimentally determined for your material by doing strain
sweeps.
Lecture 4 CM4655 Morrison 2016
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Linear-Viscoelastic Regime: Strain Sweep
© Faith A. Morrison, Michigan Tech U.
Small-Amplitude Oscillatory Shear (SAOS)
LVE Limit
will be a function of frequency
(check at low and high ).
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Linear-Viscoelastic Regime: Frequency Sweeps
© Faith A. Morrison, Michigan Tech U.
Small-Amplitude Oscillatory Shear (SAOS)
0.00001
0.0001
0.001
0.01
0.1
1
10
0.001 0.01 0.1 1 10 100 1000
1
1'
G
1
1"
G
10-5
10-4
• Frequency sweeps give the
and curves
′
′′
• Should be independent of strain amplitude
• Collect as a function of temperature to expand the dynamic range (tTshifting)
Lecture 4 CM4655 Morrison 2016
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© Faith A. Morrison, Michigan Tech U.
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07
aT, rad/s
G' (Pa)
G'' (Pa)
k k (s) gk(kPa) 1 2.3E-3 16 2 3.0E-4 140 3 3.0E-5 90 4 3.0E-6 400 5 3.0E-7 4000
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Figure 8.8, p. 284 data from Vinogradov, polystyrene melt
Linear-Viscoelastic Regime: Time-Temperature Superposition
Data taken at different temperatures may be combined to make master curves of ′ and versus
(see earlier lecture)
and a Generalized Maxwell model may be fit (see later lecture)
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SAOS Material Functions
© Faith A. Morrison, Michigan Tech U.
Small-Amplitude Oscillatory Shear (SAOS)
∗
tan′′′
′′
∗ ∗
∗ 1∗
1/ ′1 tan
1/1 tan
Lecture 4 CM4655 Morrison 2016
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Assignment:
© Faith A. Morrison, Michigan Tech U.
For the PDMS polymer in the lab
•Measure and report on the true steady shear viscosity at room temperature and other assigned temperatures, as a function of shear rate, as measured with the torsional cone-plate rheometer
•Report ′ and at room temperature and other assigned temperatures. You must determine the appropriate strains to stay in the linear-viscoelastic regime
•Check to see if the Cox-Merz rule holds for PDMS.
•See memo for additional objectives (time-temperature superposition)
Cox-Merz Rule
)()( *