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Lecture 4 CM4655 Morrison 2016

1

1

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)

ProfessorFaithA.Morrison

DepartmentofChemicalEngineeringMichiganTechnologicalUniversity

2

log

log

o

Faith A. Morrison, Michigan Tech U.

Why do we need more than one method of measuring viscosity?

Torsional flows Capillary flowAt 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

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

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

4

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

3

5

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

Material Stress Response:

0

0 0

0,

,

0

First normal-stress coefficient

Second normal-stress coefficient

Lecture 4 CM4655 Morrison 2016

4

7

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

rHr

R

By carrying out a Rabinowitsch-like calculation, we can obtain the stress at the rim (r=R).

RRrz d

RTdRT ln

)2/ln(32/3

3

R

RrzR

)( Correction required

Torsional Shear Flow: Parallel-plate

8

Faith A. Morrison, Michigan Tech U.

0.1

1

10

0.1 1 10

slope is a function of R

32 RT

HR

R

RdRTdslope

log2/log 3

Parallel-Plate Shear Rate Correction

RRR d

RTdRT

ln)2/ln(32/)(

33

Torsional Shear Flow: Parallel-plate

Lecture 4 CM4655 Morrison 2016

5

9

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

constant 32

3 2

10

Faith A. Morrison, Michigan Tech U.

1st Normal Stress (C&P)

Measureables:Normal thrust F

r

R

(-planesection)

polymer melt

22

20

12)(

RF

The total upward thrust of the cone can be related directly to the first normal stress coefficient.

atm

R

pRrdrF 20 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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11

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( NRrNNp

(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

12

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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13

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

14

Start up of Steady Shear flow: obtain Steady State

Faith A. Morrison, Michigan Tech U.

Steady state must be

experimentally observed for each chosen

Torsional Shear Flow: Cone-and-plate

Obtained in the same run as More experimental noise

Lecture 4 CM4655 Morrison 2016

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15

Faith A. Morrison, Michigan Tech U.

Report Flow Curves ,

Torsional Shear Flow: Parallel-plate and Cone-and-plate

100

1000

10000

100000

0.01 0.1 1 10 100

Obtain widest range of possible within the ability of the instrument

Report on reproducibility

16

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

9

Small-Amplitude Oscillatory Shear Material Functions

SAOS stress

Faith A. Morrison, Michigan Tech U.17

00cos

, , sin sin cos cos sin

0

0,

0

0(linearviscoelasticregime)

Storage modulus

Loss modulus

0

cos sin

sin

phasedifferencebetweenstressandstrainwaves

Imposed Kinematics:

Material Functions:

Material Stress Response:

18

What is the strain in SAOS flow?

t

tdt

tdttt

t

sin

cos

)(),0(

0

00

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

10

19

h

1x

2x

h

1x

2x

thVttb o)(

ththtb

oo

sin

sin)(

Generating Shear

Faith A. Morrison, Michigan Tech U.

sin

Steady shear

Small-amplitude oscillatory shear

constant

periodic

Small-Amplitude Oscillatory Shear (SAOS)

20

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

11

21

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)

22

SAOS Material Functions

ttt

cossinsincos)(

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

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

Hookes Law for elastic solids

Similar to the linear spring law

What types of materials generate stress in proportion to the strain imposed? Answer: elastic soli

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