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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)

    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

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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

    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

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    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

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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

    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

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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

    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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    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

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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.

    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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    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

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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

    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:

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    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

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    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

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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

    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

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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

    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