IΞΩΔΟΕΛΑΣΤΙΚΟΤΗΤΑ VISCOELASTICITY · 2020. 4. 30. · “Rheology, Principles,...
Transcript of IΞΩΔΟΕΛΑΣΤΙΚΟΤΗΤΑ VISCOELASTICITY · 2020. 4. 30. · “Rheology, Principles,...
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Rheology-Processing / Chapter 3 1
IΞΩΔΟΕΛΑΣΤΙΚΟΤΗΤΑVISCOELASTICITY
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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)
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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.
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Rheology-Processing / Chapter 3 4
Die or Extrudate Swell (Διόγκωση Εκβαλόμενου)
Bird et al. (1987).
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Rheology-Processing / Chapter 3 5
Entry Flow (sudden contraction)
Bird et al. (1987).
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Rheology-Processing / Chapter 3 6
Recoil of polymeric liquid when pumping stops
From Bird et al. (1987).
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Rheology-Processing / Chapter 3 7
Pressure difference during annular flow
From Darby (1976).
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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.
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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.
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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).
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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.
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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
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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
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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
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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
0γ
0 τλτ
0dt
τdλτ
λ
dt
τ
τd
λteCτ 1
Let τ = το at t=0
λt
ο
eτ
τ
We see that for t=λ
37.011
ee
τ
τ
ο
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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.
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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.
0γ
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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.
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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.
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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.
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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).
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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)
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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
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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
Rπ
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.
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Rheology-Processing / Chapter 2 25
Cone-and-plate instrument
(also known as Weissenberg rheogoniometer)
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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)
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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)
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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.
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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 …
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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
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Rheology-Processing / Chapter 2 31
𝜂 = 𝑚 ሶ𝛾𝑛−1
𝜂𝑒 = 𝐿 ሶ𝜀𝑞
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Rheology-Processing / Chapter 2 32
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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.
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Rheology-Processing / Chapter 2 34
Sentmanat extensional rheometer (SER)
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Rheology-Processing / Chapter 2 35
a b
c
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Rheology-Processing / Chapter 2 36
Flow in a Sudden Contraction
Solve numerically the equations
below to obtain the flow field
(FEM, FVM, FD)
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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.
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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.
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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
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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
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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.
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Rheology-Processing / Chapter 2 42
Original Bagley experiments (1957)
𝜏𝑤 =𝛥𝑝𝑡𝑜𝑡.
2𝐿𝑅+ 𝑒
The zero length die method and the
Bagley method are equivalent