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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

2

log

log

o


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


The choice is determined by experimental issues (signal, noise, instrument limitations.

Instabilities in torsional flows

Low signal in capillary flow


2

3

x

y

x

y(z-planesection)

(z-planesection)

r H

r

(-planesection)

(planesection)

r

(z-planesection)

(-planesection)

(z-planesection)

(-planesection)

Experimental Shear Geometries


rz

2R well-developed flow

exit region

entrance region

rz


exit region

entrance region

rz


exit region

entrance region

rz


exit region

entrance region

Shear Flow

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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(). . .


(Linear viscoelastic regime)


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5


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


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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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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/)(

33

Torsional Shear Flow: Parallel-plate


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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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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)



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•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)


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Comparison with other instruments

S. G. Baek and J. J. Magda, J. Rheology, 47(5), 1249-1260 (2003)

RheoSense Incorporated



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Start up of Steady Shear flow: obtain Steady State



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


Ψ

Steady state must be

experimentally observed for each chosen


Obtained in the same run as More experimental noise

Ψ

••


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15


Report Flow Curves , Ψ


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



High (steady shear) or (SAOS) cause these

instabilities to be observed.


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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.


The strain amplitude is:

Small-Amplitude Oscillatory Shear (SAOS)


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19

h

1x

2x

h

1x

2x

thVttb o)(

th

thtb

o

o

sin

sin)(

Generating Shear


sin

Steady shear

Small-amplitude oscillatory shear

constant

periodic


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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




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21


-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


22

SAOS Material Functions

ttt

cossin

sincos)(

0

0

0

0

0

21



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.




12

23

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



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ttt

cossin

sincos)(

0

0

0

0

0

21



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)




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Linear-Viscoelastic Regime



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



LVE Limit

The linear regime must be experimentally determined for your material by doing strain

sweeps.


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Linear-Viscoelastic Regime: Strain Sweep



LVE Limit

will be a function of frequency

(check at low and high ).

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Linear-Viscoelastic Regime: Frequency Sweeps



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)


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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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∗

tan′′′

′′

∗ ∗

∗ 1∗

1/ ′1 tan

1/1 tan


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Assignment:


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

)()( *