IΞΩΔΟΕΛΑΣΤΙΚΟΤΗΤΑ VISCOELASTICITY · 2020. 4. 30. · “Rheology, Principles,...

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Rheology-Processing / Chapter 3 1 IΞΩΔΟΕΛΑΣΤΙΚΟΤΗΤΑ VISCOELASTICITY

Transcript of IΞΩΔΟΕΛΑΣΤΙΚΟΤΗΤΑ VISCOELASTICITY · 2020. 4. 30. · “Rheology, Principles,...

  • Rheology-Processing / Chapter 3 1

    IΞΩΔΟΕΛΑΣΤΙΚΟΤΗΤΑVISCOELASTICITY

  • Rheology-Processing / Chapter 3 2

    • Liquids with complex structure, such as polymer solutions,

    polymer melts, suspensions of particles, soap solutions,

    whole human blood, slurries, pastes etc behave in unusual

    ways.

    • The flow behavior of these liquids is the object of rheology

    (science of deformation and flow of materials).

    παντα ρει (panta rei, all things flow)

  • Rheology-Processing / Chapter 3 3

    Macromolecular (polymeric) solutions and melts are

    quite weird!!!

    They exhibit many unexpected flow phenomena

    beyond their shear thinning behavior and they are

    perhaps the most interesting from the rheological

    point of view.

  • Rheology-Processing / Chapter 3 4

    Die or Extrudate Swell (Διόγκωση Εκβαλόμενου)

    Bird et al. (1987).

  • Rheology-Processing / Chapter 3 5

    Entry Flow (sudden contraction)

    Bird et al. (1987).

  • Rheology-Processing / Chapter 3 6

    Recoil of polymeric liquid when pumping stops

    From Bird et al. (1987).

  • Rheology-Processing / Chapter 3 7

    Pressure difference during annular flow

    From Darby (1976).

  • Rheology-Processing / Chapter 3 8

    • The response of polymeric liquids to an imposed stress may,

    under certain conditions, resemble the behavior of a solid, in

    addition to the non–linear dependence of stress on shear rate.

    • These liquids are composed of very long molecular chains of

    molecular weight usually in the range of 10,000 to 10,000,000 with

    many commercial products being in the range of 50,000 to 500,000.

  • Rheology-Processing / Chapter 3 9

    When these liquids are at rest, the molecular chains are

    randomly distributed. When an external stress is applied, the

    intermolecular bonds are stretched, the chains commence to

    flow past another, to disentangle and to align in the direction of

    the flow.

    For these processes to occur

    certain time is required.

  • Rheology-Processing / Chapter 3 10

    • Response time of water:

    10-12 seconds (INSTANTANEOUS)

    • Response time of polymeric liquids:

    10-3 to 103 seconds(the lower values for solutions and the higher for melts).

  • Rheology-Processing / Chapter 3 11

    A constant is necessary to describe this behavior:

    Deborah Number

    𝐷𝑒 =𝜆

    𝜃=𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 𝑡𝑖𝑚𝑒

    𝑝𝑟𝑜𝑐𝑒𝑠𝑠 𝑡𝑖𝑚𝑒

    Let us choose a typical polymer melt with a characteristic time λ=1 s. • If the process time is very large (θ →∞ and De →0) the material will behave like a

    fluid.

    • when the process time is very short (θ →0 and De →∞) the polymer melt will behave like a solid.

  • Rheology-Processing / Chapter 3 12

    We need to develop constitutive equations to study the behavior

    of viscoelastic materials

    The simplest of them involves a simple combination of:

    a Newtonian fluid and an elastic (Hookean) solid.

    dashpot

    spring

  • Rheology-Processing / Chapter 3 13

    For the Newtonian fluid we have a linear relation between stress (τ)and rate of strain ( ), where η is the viscosity.

    For the elastic (Hookean) solid we have a linear relation between stress (τ) and the strain (γs) where G is the modulus of elasticity.

    We assume that the combined material will have a shear rate equal to the sum of the two shear rates

    fγfγητ

    sγGτ

    sf γγγ

  • Rheology-Processing / Chapter 3 14

    G

    τ

    η

    τγ

    γητG

    ητ

    The ratio η/G has dimensions of time and is usually denoted by λ

    This mathematical model is referred to as a Maxwell fluid.

    γητλτ

  • Rheology-Processing / Chapter 3 15

    The mechanical model is extended to a position and

    held there. This means that we impose a constant

    extension (strain γ=const.) and therefore .

    Equation on the previous slide becomes

    or

    0 τλτ

    0dt

    τdλτ

    λ

    dt

    τ

    τd

    λteCτ 1

    Let τ = το at t=0

    λt

    ο

    τ

    We see that for t=λ

    37.011

    ee

    τ

    τ

    ο

  • Rheology-Processing / Chapter 3 16

    Thus, λ represents the time for the stress to decay by a factor1/e=0.37 and is called the RELAXATION TIME. The physicalmeaning of this quantity can be better understood by referringto the mechanical model again: If we impose a sudden extension, the spring will respond

    instantaneously. However, the stress will be relaxed gradually

    (exponentially) as the dashpot will start and keep moving.If enough time is given, the stress will become zeroeventually.

  • Rheology-Processing / Chapter 3 17

    More real fluid flow experiment:

    If the rotation is suddenly stopped, i.e. , the measured stress will notbecome instantaneously zero (as for Newtonian Fluids) but it will decay inan exponential manner.

  • Rheology-Processing / Chapter 3 18

    The relaxation behavior is not the only unusual time response

    for polymeric liquids. If we start suddenly shearing from rest, a

    Newtonian fluid will respond instantaneously, while a polymer

    solution or melt will exhibit an overshoot.

  • Rheology-Processing / Chapter 3 19

    Obviously it is not sufficient to characterize polymers in term of theirviscosity but also in terms of their relaxation times.

    If a material has long relaxation times, it is possible during processing tosolidify the material before the stresses are completely relaxed. So wemay end up producing a product with a considerable amount ofFROZEN-IN stresses.

    These stresses may eventually be released and may lead to (undesirable)shrinkage and warpage phenomena or premature cracking or aging.

    The relaxation characteristics are influenced by the size and flexibility ofthe polymer.

  • Rheology-Processing / Chapter 3 20

    Why rod-climbing, why die swell ?

    Under shearing the long molecular chains can be

    thought of as acting as springs or rubber bands. By

    shearing, the springs are stretched around a rotating

    shaft and exert a contraction force towards the

    axis of rotation like a “strangulation” which

    forces the fluid towards the axis. This results in

    the rod-climbing, or Weissenberg effect.

    Similarly, when a polymeric liquid exits from a

    tube, the “springs” which are extended inside

    the tube, contract and this causes the

    phenomenon of extrudate swell.

  • Rheology-Processing / Chapter 3 21

    NORMAL STRESSES (ΟΡΘΕΣ ΤΑΣΕΙΣ)

    “A fluid that’s macromolecular

    is really quite weird – in particular

    the big normal stresses

    the fluid possesses

    give rise to effects quite spectacular”

    Bird R.B., Armstrong R.C., Hassager O., Dynamics of Polymeric Liquids:

    Volume 1 Fluid Mechanics, 2nd Edition, John Wiley & Sons, USA (1987).

  • Rheology-Processing / Chapter 3 22

    SHEAR

    ONLY

    σ11=-p+τ11σ22=-p+τ22σ33=-p+τ33

    Whenever a polymeric liquid is sheared, normal stresses are developed

    because shearing results also in extension in the x-direction

    (think of deck of ELASTIC cards sliding)

  • Rheology-Processing / Chapter 2 23

    Ν1 = σ11 – σ22 = τ11 – τ22 (first NORMAL stress Diff.)

    Ν2 = σ22 – σ33 = τ22 – τ33 (second NORMAL stress Diff.)

    SHEAR

    ONLYMeasurements of σ11, σ22 and σ33will not be useful in assessing the

    elasticity level of the fluid

    because the pressure p can be set

    arbitrarily from an external source

    (e.g. pump). To eliminate the

    contribution of pressure we take

    the differences

  • Rheology-Processing / Chapter 2 24

    The first normal stress difference can be measured

    directly with a cone-and-plate instrument. As the cone

    turns the tendency to climb up the rotating shaft is

    converted in a normal force NF which can be measured by

    a suitable mechanical or electronic device. From flow

    analysis of the cone-and-plate instrument, it turns out that

    the first normal stress difference is

    222111

    2

    ΝττN F

    The second normal stress difference is much more

    difficult to measure. Up to the mid 1960’s it was though

    that N2=0. More recent measurements showed that N2 is

    negative and approximately 10-20% of the magnitude

    of N1.

  • Rheology-Processing / Chapter 2 25

    Cone-and-plate instrument

    (also known as Weissenberg rheogoniometer)

  • Rheology-Processing / Chapter 2 26

    MEASUREMENT OF FIRST AND SECOND NORMAL

    STRESS DIFFERENCES N1 AND N2.

    The normal separating force

    generated when the cone-and-

    plate rheometer rotates gives

    N1

    The normal separating force

    generated when the parallel

    plate rheometer rotates gives

    N1–N2

    By making measurements with both instruments you get N1 and N2. See C.W. Macosko

    “Rheology, Principles, Measurements and Applications”, VCH Publishers, New York (1994)

  • Rheology-Processing / Chapter 2 27

    For molten polymers the first normal stress difference

    obeys expressions in the form

    bτAN 121

    For molten polystyrenes a rough approximation

    might be A=0.00347 and b=1.66.

    Under usual processing conditions for the fabrication

    of plastic parts by extruding a molten polymer through a die,

    the shear stress is likely to be τ12=105 Pa. Using the above

    equation, we get approximately N1=7×105 Pa i.e. under

    customary processing conditions the first normal stress

    difference is much larger than the shear stress (roughly

    10 times larger at a die exit for film of fiber extrusion)

  • Rheology-Processing / Chapter 2 28

    ELONGATIONAL (EXTENSIONAL) VISCOSITYΕΚΤΑΤΙΚΟ ΙΞΩΔΕΣ

    When we talk about flow we usually mean shear.Extensional or elongational flow involves stretching.While it is difficult to visualize stretching of a lowviscosity liquid like water, molten polymers aresubjected to a lot of extensional deformations duringprocessing.

  • Rheology-Processing / Chapter 2 29

    EXTENSIONAL FLOW

    A

    Fσ 11tensile stress:

    stretch rate:dt

    dL

    LL

    U

    z

    Vε z

    1

    EXTENSIONAL (or ELONGATIONAL) VISCOSITY

    is defined as:

    ε

    AF

    ε

    σηe

    11

    This is a “new” fluid property. It is important whenever

    polymers are stretched…e.g. in fiber spinning, film blowing, blow

    molding, thermoforming, squeezing flows, converging flows …

  • Rheology-Processing / Chapter 2 30

    Q. How is ELONGATIONAL VISCOSITY (ηe),related to the ordinary (SHEAR) VISCOSITY (η) ?

    A. For Newtonian fluids it can be proven rigorouslythat :

    Q. What about polymers in molten or semi-moltenstate?

    A. ηe varies from ~3η to ~100η or more,depending on molecular structure and processing

    conditions!!!!

    relationTrouton3ηηe

  • Rheology-Processing / Chapter 2 31

    𝜂 = 𝑚 ሶ𝛾𝑛−1

    𝜂𝑒 = 𝐿 ሶ𝜀𝑞

  • Rheology-Processing / Chapter 2 32

  • Rheology-Processing / Chapter 2 33

    Measuring devices for elongational viscosity

    Schematic of the Meissner (BASF) rheometer.

    • some accurate useful measurements of elongational viscosity

    • not suitable for routine measurements.

  • Rheology-Processing / Chapter 2 34

    Sentmanat extensional rheometer (SER)

  • Rheology-Processing / Chapter 2 35

    a b

    c

  • Rheology-Processing / Chapter 2 36

    Flow in a Sudden Contraction

    Solve numerically the equations

    below to obtain the flow field

    (FEM, FVM, FD)

  • Rheology-Processing / Chapter 2 37

    The simulations will show that there is an EXCESS PRESSURE DROP at the

    capillary ENTRY

    • The large excess pressure drop at the

    entrance for polymeric liquids is

    apparently due to large elongational

    viscosities exhibited by these substances.

    • Entry flow is mainly elongational in

    character.

  • Rheology-Processing / Chapter 2 38

    EXCESS PRESSURE DROP also referred to as PRESSURE LOSSES

    Expressed usually in dimensional form

    𝑛𝐵 =𝛥𝑝𝑒2𝜏𝑤

    known as the Bagley correction in capillary viscometry

    For Newtonian fluids

    𝑛𝐵 =𝛥𝑝𝑒2𝜏𝑤

    = 0.587

    For polymer melts measurements usually range from the Newtonian value at low shear rates to about nB=10.

  • Rheology-Processing / Chapter 2 39

    Cogswell’s Method for Elongational Viscosity Determination

    ሶ𝜀 =4𝜂 ሶ𝛾𝛼

    2

    3 𝑛 + 1 𝛥𝑝𝑒

    𝜂𝑒 =9

    32

    𝑛 + 1 𝛥𝑝𝑒2

    𝜂 ሶ𝛾𝛼2

  • Rheology-Processing / Chapter 2 40

    COGSWELL EXAMPLE

    Assume m=10000 Pa·sn and n=0.35. The calculated pressure drop through a L=16 mm and D=1 mm die is

    ΔP=6.69 MPa for shear rate 818/s and a measured total pressure drop is ΔPtotal=8.69, thus ΔPe=(ΔPtotal –

    ΔPcapillary)=2 MPa, determine the elongational viscosity.

    Paτw 1045848181000035.0

    sPa 12881810000 135.0

    sPaηe

    23883

    10458489.1

    102135.0128

    26

    1

    6135.0

    2

    421023

    8181284

    s

    NOTE THAT at shear rate η= 1000x42(0.35-1)=881 Pa·s while

    at stretch rate , ηe=23883 Pa·s142 sε 27

    881

    23883Ratio

    e

    142 s

  • Rheology-Processing / Chapter 2 41

    The Bagley Correction of Capillary Viscometry

    𝛥𝑝𝑐𝑎𝑝𝑖𝑙𝑙𝑎𝑟𝑦 = 𝛥𝑝𝑡𝑜𝑡. − 𝛥𝑝𝑒

    • Modern capillary viscometers (two bore) have two pistons and two dies.

    • The long die will be typically of L/D=16–20 and the short die of practically

    zero length (say L=0.3 mm).

    • The pressure drop caused by the zero length die is simply subtracted from the

    total pressure, to get the Δpcapillary

    𝜏𝑤 =𝛥𝑝𝑐𝑎𝑝𝑖𝑙𝑙𝑎𝑟𝑦

    2𝐿𝑅

    The true viscosity is then calculated by dividing the above value of wall

    shear stress by the Rabinowitsch corrected wall shear rate.

  • Rheology-Processing / Chapter 2 42

    Original Bagley experiments (1957)

    𝜏𝑤 =𝛥𝑝𝑡𝑜𝑡.

    2𝐿𝑅+ 𝑒

    The zero length die method and the

    Bagley method are equivalent