Rheology-Processing / Chapter 2 1

Viscosity and

Polymer Melt Flow

Rheology-Processing / Chapter 2 2

Viscosity: a fluid property → resistance to flow(a more technical definition resistance to shearing)

Remember that:

dy

duμτ

shear stress F/A

viscosity

shear rate: ≈ U/H

Rheology-Processing / Chapter 2 3

POLYMER MELTS

THE VISCOSITY DECREASES AS THE SHEAR RATE (du/dy) INCREASES

(due to molecular alignments and disentanglements)

A characteristic value is the limiting viscosity for zero shear rate(frequently referred to as zero shear viscosity) usually denoted withηo.

Rheology-Processing / Chapter 2 4

ZERO SHEAR VISCOSITY is a function of POLYMER MOLECULAR WEIGHT

Critical molecular weight Mc ....... forentanglements to become effective.

Mc critical molecular weight for entanglements.Mc ≈ 4,000 for PE, Mc ≈ 36,000 for PS

Most (NOT all) commercial polymers have Mw=50,000 to 500,000

Rheology-Processing / Chapter 2 5

In characterization of polymers, the solution viscosity is often used.

For a polymer dissolved in a solvent:viscosity ↑ as concentration ↑

(solution viscosity definitions in pages 2.4 and 2.5)

For instance, the specific viscosity:

or the intrinsic viscosity:

The intrinsic viscosity is found by plotting ηsp/C against C andextrapolating to zero concentration

solvent

solventsolutionsp

η

ηηη

0

][

C

sp

C

ηη

Rheology-Processing / Chapter 2 6

The so called viscosity average molecular weight, introduced in chapter1, can be found from the Mark-Houwink equation:

[η]: intrinsic viscosityM: (viscosity) average molecular weightK, a: experimentally determined

constants (depend on polymer solventsystem)

αΚΜη ][

Rheology-Processing / Chapter 2 7

SHEAR-THINNING BEHAVIOR OF POLYMERS

From Newton’s law of viscosity

sPasrateshear

Pastressshear

dy

du

τμ

1

Viscosity is constant for Newtonian fluidsBUT

it DECREASES as SHEAR RATE INCREASES for polymer solutions and melts(non-Newtonian fluids)

This behavior is called: SHEAR-THINNING(due to molecular alignments and disentanglements )

Rheology-Processing / Chapter 2 8

We cannot talk about a constant viscosity μ in polymers, but rather about

srateshear

Pastressshearityvis

1cos

We usuallysymbolize itwith theGreek letterη

Rheology-Processing / Chapter 2 9

Most popular model to express the shear-thinning behavior: POWER-LAW

• m is called consistency index (the larger the m the more viscous the melt)

• n indicates the degree of Non-Newtonian behaviorn=1 means Newtoniann<1 for SHEAR THINNING POLYMERS

nγmτ (shear stress) = m (shear stress)n

γητ or or1 nγmη viscosity is a

function of shear rate

Rheology-Processing / Chapter 2 10

The power-law relation gives

γnmη log1loglog Note that consistency index m is equal to viscosity η atOn a log-log paper η vs is a straight line and the slope is equal to n-1

11 sγ

γ

• The power-law fit is OK for high shear rates, butfor low shear rates polymers exhibit aNewtonian plateau (e.g. at values say )

• The power-law model is good for most polymerprocessing operations because the shear rate isbetween 100 s-1 and 5000 s-1

13 sγ

Rheology-Processing / Chapter 2 11

THE MOST POPULAR MODELS:

Carreau–Yasuda: a

na

1

1

where ηo is the viscosity at zero shear and λ, α and n are fitted parameters

Cross model: n

11

UNFORTUNATELY NO ANALYTICAL SOLUTIONS ARE POSSIBLEWITH THESE VISCOSITY MODELS

USED IN COMPUTER SIMULATIONS FOR DETAILED LOCALFLOW ANALYSIS

WHAT ABOUT FITTING THE WHOLE VISCOSITY CURVE?

Rheology-Processing / Chapter 2 12

The consistency index m is sensitive to temperature. A commonrepresentation is:

oTTb

oemm

where mo the consistency index at a reference temperature To

Rheology-Processing / Chapter 2 13

Typical values:

For polymer melts at usual processing conditions m=1000 – 100000 Pa·sn

n=0.2 – 0.8b=0.01 – 0.1 oC-1

For example, for a commercially available polystyrene (PS) the followingparameters were obtained by curve fitting of viscosity data

mo=10800 Pa·sn, n=0.36, To=200oC, b=0.022 oC-1

The above value b corresponds to a viscosity reduction ~20% for 10oCtemperature rise. For isothermal flows we can use the power-law viscositymodel to solve problems of practical importance using analytical methodsof solution.

Rheology-Processing / Chapter 2 14

To solve general flow problems we must set a momentum balance. It turnsout that the momentum balance can be written verbally

Rheology-Processing / Chapter 2 15

• Polymers are characterized by extremely high viscosities (about amillion times more viscous than water) in molten state.

Rheology-Processing / Chapter 2 16

Therefore:

• Pressure p is a scalar

• Velocity is vector V i.e. Vx, Vy, Vz having components in the

x, y and z directions

• Stress is defined as the ratio Force/Area and can be normal or

tangential

• Stress is a tensor having nine components:

(τxx, τyy, τzz) → normal stresses

The rest components → shear stresses

τp 0

Rheology-Processing / Chapter 2 17

For planar unidirectional flows the equation

is simplified to

y

τ

x

p yx

0

A good way to remember it, is symbolically:

flowtonormal

stressshear

flowofdirection

p

)(0

τp 0

Rheology-Processing / Chapter 2 18

PRESSURE DRIVEN FLOW OF A POWER-LAW FLUID BETWEEN TWO FLAT PLATES (book page 2.15)

The absolute value || is needed because sometimes is negative and n-1<0 for polymers.yVx

Rheology-Processing / Chapter 2 19

←boundary condition

Delicate point: the right-hand side is negative and thusmust be negative.yVx

Rheology-Processing / Chapter 2 20

Apply NO-SLIP boundary condition: Vx=0 at y=b

This is the velocity profile!

Rheology-Processing / Chapter 2 21

The maximum velocity Vmax is at y=0

The average velocity Vavg is calculated by

Rheology-Processing / Chapter 2 22

The volume flow rate per unit width W (plate width) is given by

The pressure drop may then be easily calculated

Rheology-Processing / Chapter 2 23

From a previous relation we saw that

which means that the stress varies linearly in the gap.

The maximum value is at the wall (i.e. at y=b)

the w subscript refers to…. “wall”

The negative sign simply indicates that when this quantity is multiplied by thearea, it gives a force (i.e. Fw=τw·A) that is exerted by the plate on the fluid which is,of course, in the negative x direction. The force exerted by the fluid on the wettedplate should therefore be positive.

Rheology-Processing / Chapter 2 24

Another important quantity that we need to calculate is the shear rate

We may then calculate the shear rate at the wall (absolute value)

The maximum shear stress at the wall can also be expressed as

Rheology-Processing / Chapter 2 25

PRESSURE DRIVEN FLOW OF A POWER-LAW FLUID IN A TUBE (book page 2.19)

governing equation

Rheology-Processing / Chapter 2 26

Boundary conditions

The solution gives the following

velocity profile

maximum velocity

average velocity

volume flow rate

pressure drop

Rheology-Processing / Chapter 2 27

• In the previous equations if we set n=1 we end up with the well-known Hagen-Poiseuille formula for Newtonian fluids.

• Pressure driven flows are also referred to as Poiseuille flows.

For SHEAR-THINNING fluids, the velocity profiles are more flat than the parabolic profiles of Newtonian fluids.

Rheology-Processing / Chapter 2 28

From the previous equations it is easy to see that the stress varies linearly

and the maximum value is at the wall (i.e. at r=R)

The linearity of the stress is often expressed as

τw is taken as positive

Rheology-Processing / Chapter 2 29

The shear rate is:

And the shear rate at the wall:

The maximum shear stress at the wall can be then calculated from

Rheology-Processing / Chapter 2 30

CAPILLARY VISCOMETER ANALYSIS (book page 2.23)

For Newtonian fluids, the Hagen-Poiseuille formula

can be used for direct determination of viscosity μ, from the measurement of pressuredrop ΔP at flow rate Q, through a tube of length L and radius R.

For non-Newtonian fluids, the viscosity η is a function of the shear rate (i.e. )and special treatment is necessary. We will determine the viscosity from its basicdefinition

The wall shear stress can be calculated from

γ γη

γ

τγη

Rheology-Processing / Chapter 2 31

The shear rate requires special manipulations. Again for Newtonian fluids (n=1) theshear rate can be obtained by differentiating the velocity profile

and using the Hagen-Poiseuille formula we get

For non-Newtonian fluids we will develop a general expression for the shear rate at thewall by starting from the definition of the volume rate of flow through a tube

(see subsequent steps followed in pages 2.24 and 2.25. Next slide goes directly to the result)

Rheology-Processing / Chapter 2 32

This equation is usually referred to as the Rabinowitsch equation. It gives the shearrate in terms of Q, R and τw. The term in the parentheses may be considered as a“correction” to the Newtonian expression which is simply 4Q/πR3. To obtain wemust plot Q versus τw on logarithmic coordinates to evaluate the derivative dlnQ/dlnτw

for each point of the curve.

wγ

Rheology-Processing / Chapter 2 33

The previous method can be simplified if POWER-LAW fluid is assumednγmτ

We may then write an empirical expressionn

Rπ

Qm΄τ

3

4

in which n is the slope of the logτw versus log(4Q/πR3) plot, that is

3

4log

log

Rπ

Qd

τdn w

It turns out that for the derivative dlnQ/dlnτw we have dlnQ/dlnτw=1/n, therefore

n

n

Rπ

Q

nRπ

Qγw

4

134

4

1

4

3433

n

n

nm΄m

13

4

This means that the relation between the apparent m΄ and the true m is

Rheology-Processing / Chapter 2 34

STEPS FOR DETERMINATION OF m and n

1. Determine τw from pressure drop Δp data

2. Determine

3. Plot logτw versus log(4Q/πR3) to get n (slope)and m΄ (intercept at )

4. Correct m΄ using

3

4

Rπ

Qγapp

11 sγapp

n

n

nm΄m

13

4

Rheology-Processing / Chapter 2 35

Tapered dies (slits or truncated cones) are used very often in polymer meltprocessing

(a) Flow through a tapered slit

Start from the pressure drop expression derived for flow between parallel plates

Rheology-Processing / Chapter 2 36

For an infinitesimal slit of length dz

Where for a tapered slit

Integration between z=0 and z=L gives

Po PL

note that ΔP=Po-PL

Rheology-Processing / Chapter 2 37

Further noting that

We get

The derivation of the above expression was based on the implicit assumption of nearlyparallel flow. It is a very good approximation for half angles θ of up to 15o.

Rheology-Processing / Chapter 2 38

(b) Flow through a truncated cone

We start from

It gives the pressure for flow through a tube of length dz. We integrate from z=0 to z=L,noting that

or

cotθ=1/tanθ

This expression, which is based on the nearly parallel flow assumption, is a very goodapproximation for half angle θ up to 15o.

Rheology-Processing / Chapter 2 39

DRAG FLOW BETWEEN FLAT PLATES

0

So the velocity profile will be

Velocity profile:

The average velocity is: Vavg=(1/2)Vo

The volumetric flow rate, for a channelof width W will be Q=(1/2)VoBWThis gives the amount of fluid that isdragged by the moving plate

Governing equation:

Rheology-Processing / Chapter 2 40

COMBINED PRESSURE AND DRAG FLOW BETWEEN FLAT PLATES

The pressure gradient may be aiding or opposing to the drag flow:

V

V

Rheology-Processing / Chapter 2 41

Governing equation

The power-law expression

must be introduced into the equation. Unfortunately, the resulting equation DOES NOThave a simple closed-form solution. Since this type of flow is of considerable practicalimportance, we present the Newtonian solution, i.e. n=1. We have

where η=μ is a constant Newtonian viscosity.

Rheology-Processing / Chapter 2 42

B.C.s

The pressure gradient is in general

Solving the governing equation of theprevious slide we obtain thevelocity profile

V

V

Rheology-Processing / Chapter 2 43

The volume flow rate for width W

integrationgives

or(in terms ofpressure gradient)

This may be integrated to give thepressure as a linear function of x:

Rheology-Processing / Chapter 2 44

INTRODUCTION OF AN EQUIVALENT NEWTONIAN VISCOSITY

• Combined pressure and drag flow→ no closed-form solution• For problem solution: apply numerical methods or literature for analytical (series

???) solution.• It is also possible to come up with a Newtonian approximation (for back-of-the-

envelope calculations)

Viscosity:

We can determine an equivalent Newtonian viscosity by choosing a characteristic shearrate

Then and further we may carry out calculations using the ηref in theNewtonian solution

Rheology-Processing / Chapter 2 45

For a reference shear rate it might be a good idea to use the apparent viscosity for flowin tubes with for some problems

For slit dies of gap H and width W, the reference (apparent) shear rate can easily beobtained from

Calculations involving equivalent Newtonian viscosity should be carried out withcaution for (rough) approximation purposes only.