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EXTENSIONAL VISCOSITY OF DILUTE POLYMER SOLUTIONS - … · Extensional Viscosity of Dilute Polymer Solutions Jin Huang Master of Appikd Sckncc, 1999 Deputment of Mechrnicrrl and Industritl

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EXTENSIONAL VISCOSITY OF DILUTE POLYMER SOLUTIONS

Jin Huang

A thesis submitted in wdormity with the requirements for the degree of Master of Applied Science

Graduate Department of Mechanical and Industrial Engineering, In the University of Toronto

OCopynsht by Jin Huang, 1999

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Extensional Viscosity of Dilute Polymer Solutions Jin Huang

Master of Appikd Sckncc, 1999 Deputment of Mechrnicrrl and Industritl Engineering

University of Toronto

ABSTRACT

The purpose of this study was to obtain reliable extensionai viswsity

meanirements for dilute polymer solutions ushg a marnent-stretching rheometer at the

University of Toronto.

Initial extensional measurements of two Newtonian fluids gave data very close to

the expected values, which validated the experimentai technique.

Three dilute solutions of high-molecular-weight polystyrene in oligorneric styrene

and the Ml fluid, were then testeci. For the least viscous fluid, high strains were achieved

and a steady-state Trouton ratio of about 900 was obtained, at dl Deborah numbers. For

the more viscous Liquids, high strains were not achieved because the fluid filaments

detached from their holders. Reasomble agreement was obtained in the data cornparisons

with MIT and Monash University using similar test techniques and the same fluids.

The Fact that the steady-state value of SM-I was independent of the extensional

rate suggests that this value could be used as a m a t d property for SM-1.

ACKNOWLEDGMENT

1 wodd like to express my sincerest tfianks to Professor D. F. James for his

guidance and s u p e ~ s i o n of this work. His enthusiasm and patience have encourageci me

throughout the program. I wodd also like to thank Geoff M. Chandler for uimucting me

in the use of the filament stretching rheometer and for his continuous technical support.

His work has inspued me in many ways.

During the two yean, 1 received help of Mnous kinds from my wlleagues and

Wends. 1 wodd particularly like to thank Dorota Kiersnowsky, W e n g Liu, Navid

Mehdizadeh and Masoud Shams for their consistent assistance. Special tbanks are due to

Alison Collins, Angela Garabet and Vala Mehdinejad for kindly reviewing parts of the

manuscript.

Finally, I wodd like to express my love and appreciation to my parents: thank

you for bringing me to this wonderfl world and giving me the fieedom to explore it.

............................................................................. The ideal diameter history 3 5 ............................................................................ Velocity compensation 3 7

Experiwntil techniques ......l................-...H..........a~........................... 40 ........................................................................................ Loadiag technique -40

Disk cleaning ................................................................................................ 40 Effect of temperature and disk dimensions , .................................................. 4 1

.................................................................................. Effect of temperature 4 1 . . Initial aspect ratio .......................................................................................... 43

CEAPTER 6 CALIBRATIONS ................................................................................. 44

6.1 Force traasductr crilibration and noise rcduction -r------- ...HC.m...aH....-......... 44 ........................................................................................... 6.1.1 Force calriration 44

............................................................................................ 6.1.2 Noise reduction -47 6.2 Cdibmtion of diameter-mcuuring device and control of its position , ......... 51

.................................................................................... 6.2.1 Zumbach calibration 5 1 .............................................................. 6.2.2 Calibration of Zumbach positionhg 51

6.3 Vefocity dibmtion, ..,....l..................o......................................................... 54

7.1 Newtonirn fluids and thtir s b w propertics .....m............................................ 56 7.2 Non-Newtonian fluids propcrtia .,...m.....m......... ............................................. 57

................................................................ CaAPTER 8 EXTENSIONAL RESULTS 60

Newtoniao calibrations ................. .................................................................. 60 Extensional ruults for Noa-Newtonian Ruids ................................................. 67

.................................................................................................. Fluid SM-1 6 7

.................................................................................................. Fluid SM-2 7 4 Fluid SM-3 ................................................................................................... 78

......................................................................... ....................... Fluid Ml .,, 8 2 Cornparison with other Iaboratories ......e......................................................... 86

......................................................... ................................................ SM-1 .... 87 SM.2. ............................................................................................................ 89 Ml ................................................................................................................ 93

........................................... CakPTER 9 CONCLUSIONS AND FUTURE WORK 96 ..................................... 9.1 Conclusions ". ............................................................... 96 .................................................................. 9.2 Recommtndations for future work 98

LIST OF TABLES

Table 5.1 Temperature constants for SM fluids 41

Table 7.1 Properties of the Newtonian test fluids 57

Table 7.2 Composition and physical propenies of the wn-Newtonian test fluids 58

Table 7.3 Steady shear propertes of the non-Newtonian test fluids at 25 OC 58

LIST OF FIGURES

Conformation of a flexible polymer chah at rest

Conformation of a flexible polymer chah under shear

Steady shear flow

Molecule in shear flow using a dumbbell mode1

Deformation of a fiuid element in shear flow

Uniaxial extensional flow field

Molecule in extension using a dumbbell mode1 becornes aligned dong the

stretching direction

Tbree modes of extensional flow

Schematic diagram of uniaxial eldension of a cylhdrical element

Schematic diagram of the stagnation-point flow device

Spin-line devices

Contraction flow

Converging charnel rheometer

The filament-streching rheometer developed in Australia

Transient Trouton ratio of fluid Ml against strain

Schematic diagram of the filament stretching rheometer

Diameter measuring principle of Zumbach ODAC measuring head

P ~ c i p l e of the Zumbach positiorhg system

Force balance for the top halfof a filament

The diameter profile for SM-2 when an ideai extension imposed

Length and diameter data for SM-2 filament at an imposed extensionai rate of

2.8 s-', and the best-fit ames

Diameter of SM-2 &er velocity compensation

Calibration curve for the MOD405 force tratlsducer

Force measurement fiom the MOD404 10 g force transducer

Force measurement fiom the MOD405 1 g force transducer

Signal from the force traasducer MOD45 at i = 5 S-' during a blank nin

Force transducer noise for MOD405 during test period after noise reduction

Zumbach calt'bration for transparent objects

Zumbach calibration for opaque objects

Calibration of the Zumbach positioning system

Velocity calibration

Shear viscosity of Ml 6om the Brookfield viscameter

Diameter profile for Viscasil 12,500 at E = 5 S-'

Force history for Viscasil 12,500 at E = 5 S-'

Diameter diagram for Viscasil 12,500 at & = 1 0 s-'

Force diagram for Viscasil 1 2,500 at d = 10 se'

Diameter diagram for Viscasil 30,000 at & = 5 s-'

Force diagram for Viscasil 30,000 at = 5 s"

Diameter d i a m for Viscasil 30,000 at I = 10 s-'

Force diagram for Viscasil 30,000 at i = 10 s-'

Transient Trouton ratio for Newtonian fluids

Cornparison of Newtonian data with two initial aspect ratios, for two nominal

extensionai rates

-viii-

Data for SM4 a$ De = 19.5. a diameter b. force history

Data for SM- 1 at De = 10.7. a diameter; b. force history

Comparison of Transient Trouton ratio for SM4 at ail Deborah numbers

Transient Trouton ratio meamernent at De = 10.7 for SM- 1

Data for SM-2 a De = 3 1.8. a diameter, bb. force history

Cornparison o f the Transient Trouton ratio for SM-2

Reproducibility of the Transient Trouton ratio measurement for SM-2

Data for SM-3 at De = 110. a. diameter; b. force history

Transient Trouton raito for SM-3 at De = 110.1 (t = 28.6 C)

Data for Ml fluid at i = 5.6 s". a. diameter, b. force

Data for Ml fluid at E = 10 s-'. a. diameter, b. force

Transient Trouton ratio for MI

Comparison of the transient Trouton ratio for SM-1, at tow De numbers

Comparison of the transient Trouton ratio at high De numbers, for SM-1

Comparison of the transient Trouton ratio for SM4

Comparison of the transient Trouton ratio for SM-2 with MIT

Comparison of transient Trouton ratio for SM-2 with Monash University

. Comparison of Ml witb Monash University

NOMENCLATURE

Tie-temperature superposition shifl factor

Coefficient in the shift factor

Coefficient in the shift factor

Filament mid-f ength diameter

initial filament diameter

Applied externa1 force on a filament (N)

Gravity coefficient @Lm/s2)

Filament length (mm)

Entrance length (mm)

Weight-average molecular weight @/mol)

Pressure (Pa)

Filament radius (mm)

time (s)

Ab solute temperature (K)

Force induced by surfkce tension (N)

Velocity of the bottom disk ( d s )

Weight (N)

r, e ,z Cylindricai co-ordinates

Greek symbols

Fining parameter in the Arrhenius equation (&as)

Exponent in the Arrhenius equation CK)

Extensionai deformation

Extensional rate tg1)

Shear rate (s-')

Shear viscosity (Pas)

Extensional viscosity (Pas)

ReIaxation time (s)

Rotating angle (O)

Density (kg/m3)

Surface tension c d c i e n t W/m)

Total stress

Characteristic time (s)

Stress associated with deformation

Subscript

O Fluid properties at zero shear rate;

Reference temperature;

Initial conditions

Dimensionless num bers

D e Deborah number

St Stokes number

Tr Trouton ratio

& Initiai aspect ratio

CHAPTER 1

INTRODUCTION

Polymers are found in a range of industrial fields, from plastics manufacturing to

food processing, with a variety of forms and fkctions. Not only are they used in a solid

form, such as plastics, but also in a liquid form like melts and solutions. An example of

the latter is that, a minute addition of a high molecular weight polymer to turbulent flow

can greatly reduce shear stress at a surface and consequently poIymers are used to reduce

wall friction in major oil and gasoline pipelines al1 over the world. In enhanced oil

recovery, dilute polymer solutions are used as pusher fluids to reduce fingering of cmde

oil in the reservoir and direct the oil toward production wells. In ftre fighting, polymers

are used as flow stabilizers because a srnail concentration can inhibit droplet formation at

the surface of the high-speed turbulent fiee jet [Glass 1986; Shalaby et al 1991; Schulz

and Glass 19911. Polymeric solutions also appear in coating, jet printing and many other

industrial applications.

Industrial applications such as these dernand understanding of the flow behaviour

of the fluids. In many of these applications, extensional deformation is dominant.

Consequently it is necessary to investigate the extensional flow behaviour of polymer

fluids. For polymer melts, extensional measurements have been made successflly at a

constant extensionai rate or under a constant stress. For mobile polymer solutions,

however, great diffimlty has been experienced in making extensional measurements. A

number of techniques, such as fibre-spinning and converging channel flow, have been

developed to mesu re the flow resistance of polymer solutions in an extensionally-

dominant flow field. However, creating a purely extensioaal flow with a constant

extensional rate is not easy. In fact, the flow fields created in most techniques are not well

defined. Extensional measurements were made by various techniques for a standard

polymer solution, the data were compared (the M 1 project), and large discrepancies were

found [J. Non-Newtonian Fluid Mech., 35, 199 11.

This situation prevailed until a filament stretching technique was developed by

Sridhar et al Cl99 11 in Australia. In this technique, a fluid sample is placed between two

vertical coaxial disks, which move apart in order to pull out a filament. A constant

extensional rate is achieved by programming the stretching of the filament. Extensional

measurements of several non-Newtonian fluids were made and a steady state was

achieved for some fluids. It was the first time that a 'me' measurement of extensional

viscosity of polymer solutions had been achieved.

A rheometer based on the filament stretching technique was constructeci at the

University of Toronto, similar to the original design but with improvernents in force

rneasurement and travel length. In this thesis, extensional measurements for Newtonian

and non-Newtonian fluids obtained tiom this rheometer will be presented and compared

with measurements fiom two other laboratories.

CHAPTER 2

BACKGROUND

The rheology related to this work is introduced in this chapter. This background

material includes the origin of fluid elasticity, types of deformation and an analysis of a

fluid filament under uniaxial extension.

2.1 Origin of elasticity

The polymers of interest in this study have a distinguishing feature: very long

flexible molecular chains. in a good solvent, a polymer chah swells and occupies a large

volume which is roughly spherical (Figure 2.1 a). M e n the solution is flowing, the

configuration of a polymer chain depends on the competition between Brownian motion

effects and hydrodynarnic effects. At low deformation rates, Brownian motion is strong

enough to offset the viscous pull of the solvent, and therefore polymer chains remain

close to their equilibrium configuration and the viscosity of the solution remains constant.

At high deformation rates, on the other hand, hydrodynamic forces overcome the

randomising effect of Brownian motion, and the coils are deforrned and oriented in the

Figure 2.1 Conformation of a flemile poiymer chah. (a) at rest; @) under shear

flow direction, as shown in Figure 2.1 b. niese oriented ellipsoidai-shaped mils impede

the flow less than the original sphericai coils and a decrease in the viscosity is ofien

observed; Le., the fluid is shear thinning. As defoxmation rates continue to increase, the

coils become virtually aligneci with the flow. The viscosity of the solution decreases and

remains at a lower value thereafter. For dilute polymer solutions, where the distance

between the molecules is large that hydrodynamic interactions between them are

negligible, the shear-ihinning effect is very weak.

One of the main probIems in investigating the elasticity of polymers is to

distinguish between elasticity and shear-thinning of these fluids. For moa non-

Newtonian fluids like melts and concentrated solutions, elastic effects are accompanied

with significant shear-thinning. In order to dinerentiate between elastic and viscous

effect, a higldy elastic fluid with a constant viscosity was developed in 1977 poger,

19771. A 'Boger fluid' is a dilute solution of a high molecular weight polymer in a highly

viscous solvent. Aithough the solution does display shear-thinning, the f d in its viscosity

is very smaii compared with the zero-shear value and for practicai purposes, the viscosity

appears to be constant Barnes a al, 19891.

As describeci earliq Brownian motion is always trying to bring the chains back to

the equilibrium random configuration, regardless of the deformation rate. When the

molecules are stretched and then the apptied stress is removed, the molecular chains relax

to their quiiibrium configuration The relaxation is not instantaneous, as it is for

Newtonian fluids, and thus the elasticity of polymer fluids becomes evident. The time

constant associateci with relaxation is a property of the fluid, called the relaxation time 1,

The relaxation time for polymer solutions can Vary from a fiaction of a second to a few

minutes. For fluids with high relaxation times, elastic effects are observed easiiy because

the stresses relax so slowly; for materiais with low k, the elasticity can not usually be

observed but it can be generated provided the defoxmation rate is high enough. In other

words, both the juid characteristic time (the relaxation time) and the f70w characteristic

time (such as the inverse of the deformation rate) are important in determining the

amount of elasticity in a fluid or the elastic response of the fluid. The ratio of these two

times is an important dirnensionless group called the Deborah number p i r d et ai, 19871:

where t is a characteristic time of the flow. in extensional flow, this ratio becomes L k ,

where & is the extensional rate. The Deborah number a n also be interpreted as the ratio

of the elastic forces to the viscous forces. At small Deborah numbers, the polymer

molecules are close to their equilibrium configurations and the polymer fluid shows only

minor differences from a Newtonian fluid. When the Deborah number is large, polymer

moiecules are distorted and cannot relax during the defornion, and thus the fiuid

behaves more like an elastic soiid, that is, elastic effects become significant when the

Deborah number is above unity (J3ird et al, 1987).

2.2 Sbear and extensional flows

2.2. f Defulltion of two types of flow

Flows may be classified into weak and strong flows [Giesekus, 1962; Tanner,

19761. A flow is considered strong if neighbouring fluid particles separate at an

exponential rate in the flow field. The flow is coasidered weak otherwise.

Steady simple shear is an example of a weak flow. This flow is illustrateci in

Figure 2.2 a, with the fluid confineci between two parallel plates. The top plate is moving

at a constant velocity V. The velocity components of the fluid element in Cartesian co-

ordinates are:

v, =y,,x,; v , =O; v , = O (2-2)

where y?, is the velocity gradient and its abdu t e value is calted the shear rate. The

distance L between two neighbourng fluid particles in the x2-direction, initially b apart,

increases Iinemlly with time t as:

In shear flow, molecular chains experience both extensional and rotational motion. In

Figure 2.2 b, where a polymer molemle is depicted as a dumbbeil of two beads joined by

a spring, the molecule rotates and stretches in the flow. The extension and rotation in

shear flow can be iilustrated more clearly by a fluid element. As shown in Figure 2.3, the

shear main can be considered as the sum of extension dong diagonals plus rotation.

Figure 2.2 a Steady shear flow

Figure 2.2 b Moleaile in shear flow using a dumbbell mode1 [Tirtaatmadja, 19931.

Figure 2.3

extension rotation

Deformation of a fluid eiement in shear flow

Extensional flow is a strong flow and the streamlines for one type of extensional

flow, namely uniaxial extensional flow with a constant rate of extension, are shown in

Figure 2.4 a. The velocity cornponents for this flow, in Cartesian cc~ordinates, are

1 v, = &x,; v , = -- 1 Lx,; v , = --x,

2 2 (2-4)

where ti is the extensional rate. Fluid is stretched in the XI-direction and contracts in the

other two. Two fluid particles, which are initiay a distance Lo apart in the xpdirection,

are a distance of L apart d e r time t according to

L = L , exp (kt) . (2-5)

In this flow field, molecules become aligned and stretched dong the xi-direction, as

shown in Figure 2.4 b. In a purely extensional deformation, there is no rotation or

shearing of the fluid element; the element experiences only extension.

Figure 2.4 a Uniaxial extensional flow field

Figure 2.4 b Molecule in extensional flow using a dumbbell model bewrnes aligned

dong the stretching direction [Tirtaatmadja, 19931.

2.2.2 Types of extensional flow

There are three major types of extensionai flow: uniaxial, biaxial, and planar.

Figure 2.5 shows the three types of extension for a cubicai element of material. The

dotted lines represent the shape of the element &er deformation. In uniaxial extension,

the material is stretched in one direction and compressed equally in the other two; in

biaxiai extension, the material is stretched equally in two directions and compressed in

the third; and in planar extension the material is stretched in one direction ody,

compressed in the second direction, and held to the same dimension in the third.

2.3 Uniaxial extensional flow of a fluid filament

Of the three modes, uniaxial extension is of particular interest because of its

industrial applications, such as in coating and printing- Moreover, it is easier to make

extensional measurements under this mode in a laboratory. Therefore most extensionaI

rheometers are designed to create a predominantly uniaxial extensional flow and to

rneasure the fluid response.

This mode of extension can be visualised as stretching a rod of material along its

length. The general flow field is given by Eqn. 2.4 in Cartesian CO-ordinates. In this

section, a material element of a particular shape is considered for convenience, a

cylindncal one with initial diameter Do and initial length Lo. As shown in Figure 2.6, the

elernent is stretched homogeneously at a constant extensional rate dong the axial

direction z and the radius of the element decreases uniformly dong the length.

The strain in the z-direction is

Uniaxial extensional flow

Biaxial cxtensiona l flo w

Phar cxttnsianal flow

Figure 2.5 Three modes of extensional flow. (a) uniaxial (b) biaxial (c) planar

and the deformation rate is

where v, is the axial velocity . Hence

For an incompressible flui4 fiom the continuity equation,

from which the radial (inward) velocity is found to be

Figure 2.i 6 Schematic diagram of uniaxial extension of a cyiindricai element

with the other velocity component ve king zero. For this flow field, strearnlines are

given by

r ' z = contant . (2-1 1) By integraing Eqn. (2.8), it is found that the length of the specirnen L(t) increases

exponentially fiom its initiai length La, as given by Eqn. 2.5. From Eqn (2.10), the

filament diarneter De) at any time t decreases fiom its initial value Do as

In this work, E is referred to Hencky strain, which is defined as hm), where

is the original length and L is the length of the sample. In a constant extension, the

Hencky strain can be alo wntten as Et , where is the effective extensional rate.

Now consider the uniaxial extension to start at t = 0, with the fluid initially in a

stress fiee state. The fiuid resists this motion and the resistance to extensional

deformation is expressed as exfemiomI viscosity, analogous to the term shear viscosity,

which expresses resistance to shearhg motion. Extensional viscosity is wrtten as q ~ , in

acwrdance with nomenclature suggested by the Socim of RheoIogy.

In the same way that shear viscosity is defined as stress divided by shear rate,

extensional viscosity is defined as stress in the fiow direction, a,, divided by extensional

rate E . The total stress o, can be measured experimentally but it includes the dmown

pressure p. This pressure p is eliminated by subtracting the n o r d stress in the r-

direction um fiorn G. The difference is quivalent to 'tn - T~~ the difference between the

stresses associatecl with deformation in the two directions. Thus, the definition for q~ is

For Newtonian fluids with a shear viscosity of qo,

and

so that

q E = 3 % -

Hence q ~ : is mt an independent fluid property for Newtonian fluids.

For mobile viscoelastic fluids, QE depends on both duration t and main rate & . In

most cases, E is not constant during the residence time t, and thus q~ is a tnction of

strain history. For extensional flow, the preferred strain history is the one in which the

fluid is initially stress-fiee and then subjected to a constant- 6 deformation (a step change

in strain rate). The extensional viscosity is then a fiinction of extensional rate and time:

T ~ E =%(kt)- (2.17)

Hence, extensional viscosity is a fr-ent matend property.

It is common in rheoiogy to report extensional viscosity data in terms of the

Trouton ratio Tr, dehed as

where r\ is the shear viscosity. For al1 inelastic fluids, including Newtonian fluids, the

Trouton ratio gives a constant value of 3. This fact will be used in the calibration of the

extensional rheometer.

CHAPTER 3

TECHNIQUES FOR MEASURING EXTENSIONAL VISOCSITY

In non-Newtonian flows, the extensional viscosity may be at least as important as

the shear viscosity. In applications such as fibre spinning, jet break-up and turbulent drag

reduction, it is the more dominant property. The industrial importance of this property

has motivated the development o f rheorneters to measure it and thereby characterise a

fluid's resistance to extensional motion-

While analogous to shear viscosity, it is very difficult to create experimental

conditions to measure extensionai viscosit y properly . In the same way that shear viscosity

is measured in a simple shear flow with a uniform shear rate throughout, the extensional

viscosity should be measured when the deformation is purely extensional and the

extensional rate is constant throughout. The t k s t challenge in extensional rheometry is to

generate a well-defined extensional field, fiee of signifiant shear. As discussed

previously in Section 2.3, uniaxial flow is ideai for experimental investigation. However,

most measuring techniques do not create the requud shear-free flow field. It is also far

fiom easy to achieve the preferred strain history, Le., to generate an extensional flow

instantaneously fiom a state of rest so that the liquid is stress-& before experiencing

extensional motions, and to provide a constant extensional rate, even for a short penod of

tirne.

While the filament-stretching technique of Sridhar et al meets these conditions, it

is useful to review other techniques which have been deveIoped for measuring

extensional viscosity. In this chapter, they will be describeci, and their advantages and

drawbacks discussed. Then the objectives of this thesis are addressed.

3.1 Stagnation-point flow techniques

Fuller et al 119871, Laun and Hingmann Cl9951 and others pioneered a technique

based on stagnation-point flow. The principle is illustrated in Figure 3.1 : fluid is drawn

I

Pivot 1

Hinged Tube

Figure 3.1 Schematic diagram of the stagnation-point flow device [James and

Walters, 19931

into opposing tubes to create a stagnation-point Bow about the mid-plane. One inlet is

stationary and the other is part of a balance ami to mesure the force exerted by the

entering liquid. By measuring the force requued to keep the tube entrances a fixed

distance apart, an apparent extensional viscosity can be determineci.

With this technique, viscosities can range fiom 1 mPa.s to 10 Pas, and a wide

range of strain rates can be achieved. Easy operation is another advantage of this

technique.

The technique, however, has severai disadvantages. One is that the overall strain

cannot be changed systematically. Secondly, the strain history is not the same for al1

particles in the flow field even though the extensional rate in the stagnation zone is

constant. The closer a particle is to the stagnation point, the longer is its residence time in

the flow. In an analysis carried out by Schunk and Scriven Cl9901 using a finite element

method, the velocity and strain rate fields around the opposed entry tubes were

determined for a Newtonian fluid. It was found that the fluid enters the tubes in a state

which consists much more of shear than extension. The flow close to the stagnation point

is purely extensional, but the region is much smaller than expected. Overall, the fluid is

not subjected to pure extensional motion except near the stagnation point.

Another disadvantage of this technique is that inertia is a factor in testing low

viscosity fluids vermansky and Boger, 19951, so that inertial effects have to be

separated out.

3.2 Fibre-spinning Devices

The first method developed for measuring q~ was one based on fibre spinning. As

shown in Figure 3.2, fluid is drawn down fiom an orifice or a tube, using either a rotating

drum, as in Ferguson's original design [1976], or a suction device of the sort descnbed by

Gupta and Sridhar [1984).

For an elastic Iiquid, a steady downward flow can usually be generated. Tension

in the filament is measured either kom the torque on the drurn or fiom the force on the

exit tube. The diameter and length of the filament can be determined with reasonable

accuracy using a video carnera. The stress is therefore readily determined fiom the

measured tension, flow rate and filament dimensions.

This technique provides an approximate value of the extensionai viscosity, and an

accurate value is not possible because of several problems. First of dl, varying the strain

Figure 3.2 Spin-line devices [James and Walters, 19931

rate in the thread is not easy and hence the extensional strain rate range with this

technique is quite limited. Secondly, the extensional strain rate is generally not constant

along the filament and, thirdly, significant shear can be present.

The possibility of a sig~ficant and largely unexpected shearing component brings

non-ideality in the flow field. The main disadvantage of the fibre-spiming methods is the

shearing in the upsueam tube or orifice. Since q~ is a function of strain history for mobile

fluids, the preshear bas a significant influence on the stress field.

3.3 Contraction-flow devices

This technique was pioaeered by Cogswell [1972] for melts and advanced by

Binding and CO-workers [1988,1990] for mobile fluids. As shown schematically in Figure

3.3. a device based on sudden-contraction flow has the advantage of easy operation-

Figure 3.3 Contraction flow [Cogswell, 19721

Vortex enhancement generally occurs on the upstream side of the contraction, and flow in

the core is primarily extensional. The relevant dynamic measurement is provided by a

pressure transducer which detects the pressure drop across the sudden contraction or

across an orifice plate.

The extensional rate is not constant in a contraction flow. When the flow pattern

is known, the rneasured flow rate yields a simple measure of overall strain rate. However,

converging flow is complex and therefore assumptions are required for data

interpretation.

3.4 Converging cbannel rheometer

An instrument based on converging-channel flow has been developed by James et

al [1990]. Its primary component is a converging channel of a shape such that the

extensional rate is constant in the central region. A schematic diagram is shown in Figure

3.4. An important aspect of this design is that the flow is at high Reynolds numbers so

that shear effects are restricted to the wall region. To determine q~ for elastic liquids,

knowledge of the behaviour of the inelastic component of the fluid is required and this is

provided by using the known viscous behaviour of the test fluid and assuming the flow

field. The boundary Iayer is taken into account in the analysis and a major assumption

(which is yet to be verified) is that the boundary layer characteristics are not unduly

modified by viscoelasticity. If this assumption stands, flow in the converging-channel

rheometer will provide an approximation to the Heaviside strain rate assumption (that the

strain rate E is irnposed at the entrance of the channel) and the technique will be useful in

determining transient extensional viscosity function [James and Walters, 19931.

Figure 3.4 Converging channel rheometer [James et al, 19903

This technique is suitable for low-viscosity fluids due to the high Reynolds

numbers required. Ln addition, high strain rates can be achieved. Drawbacks are the large

inenia effects at high Reynolds nurnbers and shear effects in the boundary layer. Also,

this device cannot be calibrated using Newtonian fluids.

3.5 Filament-stretching rheometer

A filament-stretching technique for extensiond viscosity measurement was

initiated by Matta and Tytus [1990J and fiilly developed by Sridhar and CO-workers

[I99 1, 19931. A liquid sample is initially held between two closely-spaced coaxial disks,

forming a stress-fie cylindrical filament. As show in Figure 3.5, the sample is then

stretched into a thin filament by moving both disks @) apart at an increasing velocity.

Figure 3.5 The filament stretching rheometer devekoped in Australia [Srid har and

Tirtaatrnadja 19931

For a viscoelastic fluid, a long filament is usualfy obtained before it breaks. The disk

motions are controlled by a DC motor such that the extensional rate at the mid-length of

the stretched filament is constant. The tension in the filament is measured by a force

transducer connecteci to one disk, and a laser measuring device is used to measure the

midpoint diameter dunng the stretching process.

Using this technique, Tirtaatmadja and Sridhar [1993] obtained constant

extensional rates and deterrnined values of the transient Trouton ratio for a few fluids.

Figure 3 -6 is a plot of the transient Trouton ratio for a special international fluid, obtained

with their original filament-stretching rheometer [Ti~atmadja, 1993, p. 1951. Steady state

values at high strains were achieved for this fluid.

Sridhar's method is the only accurate one to date of measuring the extensional

viscosity of a fluid. This technique provides the possibility of generating an extensional

motion fiom a state of rest, Le., there is no prehistoiy of deformation. Constant

extensional rates can be obtained by controlling the stretching velocity of the disks.

Furthemore, this technique prevents the presence of significant shear component. For the

tirst tirne, proper measurements of extensional viscosity were made for polymer

solutions.

Figure 3.6 Transient Trouton ratio of fluid Ml against strain [Tirtaatmadja, 19931

3.6 Thesis objectives

As discussed previously, the filament stretching technique is a breakthrough for

the nieasurement of extensional viscosity of mobile polymer fiuds. After Tirtaatmadja

and Sridhar, severai groups amund the world, including Spiegelberg a al [1996], Kroger

[1992] and Berg et al [1994]. Solomon and Muller [ 19961, van Niaiwkoop and Muller

von Czernicki [19%] and Verhoef et al [1999], constnicted filament-stretching devices of

various designs. In most of the devices, high strains could aot be achieved because of

limitations of length and motor speed-

Another filment stretching rheometer was built in the Wwology Labratory at

the University of Toronto in 1998. In this design, a travel length of ZOO0 mm is possible

to overmne the length constraint and to achieve high strains. The measurement of force

was irnproved by fixing one disk and attaching the force transducer to it, so that the

vibration-induced noise was reduced.

To evduate extensiona1 measurements fiom this rheometer, dslta fkom it should be

compared with data fkom other instruments. However, as discussed previously, the

extensional viscosity of dilute polymer fluids is a fnction of strain history. Therefore TE

data fiom different instruments can be properly compared only when the strain histow is

the same or when differences in strain histories are s d l and can be accounted for. To

ensure proper cornparisons, a collaborative project was undertaken with other

laboratories. The objective of this project was to compare extensional viocosity

meawements of some specially-made polymer soiutions using the filament stretching

rheometers at MIT, Momh University, and the new oae et the University of Toronto.

In this thesis, the fmt objective was to achieve accurate extensional

measurements for Newtonian fluids using the new rheometer, to validate the instrument.

The second objective was to obtain reproducible extensionai viscosity measurements of

three standard polymer solutions and to compare these data with results from MIT and

Monash University.

CHAPTER 4

EXPERIMENTAL SETUP

The filament-stretching technique has been reaiised in different designs, and in

this chapter the details of the filament stretching rheometer at the University of Toronto

will be presented.

The instrument was designed and built by G. M. Chandler in 1998. It consists of

four basic parts: the force transducer system, the diarneter measuring system, the actuator

and the control system. As shown in the schematic diagram (Figure 4. l), a test sample is

placed in the initial 2 mm gap between two 3 mm-diameter coaxial Teflon disks (B) and

stretched downward. The top disk is attached to a force transducer (C) to measure the

force exerted by the stretched filament. The bottom disk is attached to an actuator (D)

that is driven downward by a motor at a controlled speed. A laser device (E) is used to

measure the instantaneous diameter at the mid-point of the filament. To measure the

diarneter at the midpoint, the motion of the device is controlled so that its displacement is

half of the displacement of the actuator. A computer is used to generate controlling

signals and to process signals from the transducer and diameter measuring systems.

Laser beam .--------------

Force Transducer C

Disk B D = 3.0

Control cornputer

F A

w Diameter

Measuring , , , , , ,

Actuator

Figure 4.1 Schematic diagram of the filament-str&ching rheometer. The diameter-

measuring device moves half the distance L of the actuator.

4.1 Force transducer system

Since tensile force is a crucial element in determining the extensional viscosity,

accuracy in the force measurement is essential. In this experiment, the tension in the

stretched thin filament is smaii, the magnitude being of order 104 N. Also, a test run

takes only a few seconds or less. Therefore the transducer rnust be accurate at small

forces and be able to respond quickly.

Because tensile forces in the filaments Vary widely, in order to obtain accurate

force rneasurements, two transducer systems were selected, MOD405 and MOD404,

supplied by Aurora Scientifk Inc., Canada. Each transducer is a variable displacement

capacitor whose plates are formed by vacuum metalization on the surface of cantilevered

fsed silica barns. A matched reference device of identical construction is placed beside

the capacitor to compensate for thermal effects, mechanical vibration and extemai noise.

For the MOD404 transducer, the fiII scale force is *10 g force, its resolution is

200 pg force and its resonant fiequency is 2.0 kHz. For MOD405, the fi11 scale is *l g

force, with a resolution of 15 force and a resonant fiequency of 0.6 kHz. The latter

transducer was used for the lower-viscosity fluids.

Initially, a force transducer was mounted on the support stand for the actuator.

Since this set-up produced high levels of noise, the transducer was then mounted on a

nearby wall. Translation stages were part of the wall mounting. The upper disk was

comected to the transducer by a g las rod and the weight of both the disk and the rod

were compensated for in the design of the transducer.

4.2 Diameter measuring device

Accuracy on a srnail s d e and fast response were also required for the diameter

measuring system The systern, fiom ZUMBACH Electronics, consists of a measuring

head Zumbach ODAC 16J and a processor unit Zumbach USYS 10. They are normally

used to obtain ou-Iine diameter measurement of extnided fibres (both transparent and

opaque) for quality fontml at manufacturing plants. As shown in Figure 4.2, the

measuring head scans the filament with a laser beam. The size of the shadow cast by the

filament is detected eIaPoncaiIy by a receiving window and sent to the processor unit,

where it is converteci into a digital signal and sent to the control computer.

This device can measure diameters in the range of 0.05 mm to 8 mm with a

Figure 4.2 Diameter measuring principle of Zumbach ODAC meamring head.

a. laser generator; b. laser sheet; c. fluid filament; d. laser detedor

resolution of 0.00 I mm, at a scan rate of 240 scandsec.

Since it is essential to measure the response for the same portion o f fluid

throughout the process, and because only the mid-point fiuid in the filament experiences

pure extension at al1 tirnes, that point was chosen for extensional measurernent. White

total stress at the mid-point can be obtained by measuring the force at the top disk, as

analysed in Section 5 - 1, the diameter must be measured at the mid-point itself. Hence, the

Zumbach measunng head must be at the midpoint initially and during the extension. This

can be achieved using another actuator, a separate one, to give the required displacement

!hL(t), where L(t) is the displacement of the bottom disk. This idea has been applied to

some filament stretching rheometers. In this rheorneter, only one actuator is used and the

Zumbach position is controlled by a simple mechanism.

The initial midpoint position of the Zumbach was set as follows. The translation

stage on which the transducer was mounted could be adjusted vertically with an accuracy

of 0.01 mm. M e r the disks were set to the initiai distance, = 2 mm, the translation

stage was adjusted to lower the top disk by 1 mm, the bottom one remaining at its

position. Then the Zumbach position was shifted so that the laser beam pointed along the

bottom edge of the top disk. At this stage, the measuring laser sheet was 1 mm away fiom

the bottom disk. The top disk was raised 1 mm, back to its original position, and so the

laser beam was set at the midpoint of the two disks.

Controlling the Zumbach to follow the midpoint of the filament during extension

was achieved by a lever system. The prnciple is s h o w in Figure 4.3: one end of the

lever (O) is fixed while the other end (C) follows the movement of the bottom disk (the

top disk is stationary). The mid-length of the lever (B) is the support for the Zumbach

measuring head. Thus, if the displacement in the r-direction is negligible, the

displacement of the Zumbach measuriag head is half of that of the bottom disk. In

Figure 4.3 Principle of the Zumbach positioning system

practise, in order to avoid a significant r-direction displacement, the lever rotation was

Iimited to s d l angles. Therefore, the Zumbach measuring head did not foliow the entire

travel of the bottom disk. However, the Zumbach measuring head need not follow the

whole length because photograpbs by Sridhar et al [1991] show that the filament

diameter is uniforrn over alrnost its entire length after a main of 1.8 has been reached. If

the Zumbach measuring head follows the midpoint position until the critical strain is

reached, the diameter can be measured anywhere dong the length afterward.

The positioning lever used in the rheorneter is 200 mm long and the total

Zumbach displacement is 48 mm. D u ~ g the process, the maximum [email protected] of the lever

arrn is 14 O, producing an error of l e s than 3% in the Zumbach position.

4 3 Actuator

The actuator moves the bottom disk according to a velocity signal sent fiorn the

wmputer. A carriage holding the bottom disk moves down a vertical track by a

transmission belt driven by a DC motor. The motor is controlled by a computer

programmeci to produce the desired speed history. The position of the actuator carriage,

and therefore the position of the bottom disk, is sent as a feedback signal to the control

system for verification.

The total length of travel is 2000 mm, the longest arnong al1 filament-stretching

rheometers. The maximum speed of the motor is 3000 mdsec with a resolution of 3

mrn/sec.

4.4 Control system

The rheorneter is controlled by a computer using a Visual Basic program d e n

by G. M. Chandler of Aurora Scientific Inc. and revised by the author. Control

parameters are calculateci by the program according to required experimental conditions

and then converted to executive signals by a data acquisition processor. Thus signals to

the actuator create the desired constant extensional motion in the filament. The

measurements of force and mid-length diameter, as fnctions of time, are sent back to the

computer for processing.

CHAPTER 5

EXPERIMENTAL METHODS

As mentioned in previous chapters, the three filament stretching rheometers in the

joint research project have different mechanical designs. in this chapter, a detailed

discussion of the experimental method and data reduction procedure of the instrument at

the University of Toronto is presented. The operation and data processing of the other

two are described in the literature rirtaatmadja and Sndhar, 19933 [Spiegelberg et al,

19961. Data cornparison between the three labs will be shown and discussed in a later

chapter.

5.1 Force balance

To find the stress difference r, - T, fiom the measured force, which is required

for t l ~ , it is necessary to derive an equation based on force balances in the axial and radial

directions. The balance in the axial direction is made on a deforming fluid filament in the

region O < z I W2, as shown in Figure 5.1. In addition to the viscous stresses fn and T, in

the fluid, surface tension, weight, and inertia contribute to the tension in the filament.

Figure 5.1 Force balance for the top half of a stmching filament, O S z < U2.

Because sample sizes are small and the polymer solutions used in this work are highly

viscous, inestia is not a factor until very high mains are achieved or high strain rates are

imposed.

The filament starts as a cylindrical shape, with an initiai diameter do and initial

Iength b. WhiIe the filament is stretched a distance L, a force balance in the axial

direction z, between the top disk and the mid-point, yields

where o, is the total stress in the z-direction at the mid-point, d is the mid-point

diameter, T is the force due to sunace tension, W is the weight of the liquid and F is the

applied force. The surface-tension induced force T is at the interface of the filament and

the surrouding air, expressed as zda, where a is the surface tension coeffecient for the

fluid. Since half of the filament is wnsidered in the balance, the weight in Eqn. 5.1 is half

of the liquid loaded, ~4~g7c&~/4), where p is the density of the fluid,

where p is the isotropie pressure in the deforming filament at z = U2.

A force balace in the radial direction on the free surface at z = W2 gives

- 2 0 6, - 2, -Po-. d (5.3)

Combining (5.2) and (5.3) gives the tensile stress difference

Since the tensile force and the mid-point filament diameter are measued, the

stress ciifference can be calculated f?om Eqn 5.4. Eqns. 2.17 and 2.23 are then used to

obtain the extensional viscosity and the Trouton ratio.

5.2 Generating a constant extensional rate

5.2.1 The ideal diameter history

As shown in Chapter 2, for uniaxial stretching, the ideal diameter and length as

hnctions of time are

D(t) = D ,exp(- W2) , ( 5 - 5 )

L(t) = L , exp( lt) . (5-6) Ideally, the stcain rate experienced by every fluid element is identicai to the strain rate

i m posed b y the extensional apparatus. However, non-homogeneous kinematics is induced

by the disks, Le., because of them, the test sample is exposed to shear as well as extension

prior to the predominantly extensional motion. For polymer melts, a long test sarnple is

used to localise the influence the end regions and thus to decrease the errors. For mobile

polymer solutions, their low viscosity requires a short initial sample length in order to

stablize the filament, that is, to prevent it from sagging under gravity and fkom breaking

up because of surface tension. Therefore, non-homogeneity in the filament under

elongation is not negligible. Figure 5.2 is a typical mid-length diameter profile on a semi-

log plot for non-Newtonian fluids when the controiied length L is hexp($ t). In this case,

instead of a straight tine, two distinct regions for D(t) are obtained, where the filament

diarneter decreases exponentialty with time. In the first region, the filament undergoes

significant necking, and the mid-point diameter decreases faster than that for

homogeneous deformation. Therefore, the actud main rate is higher than the imposed

1 1.5

Time, s

Figure 5.2 The diameter profile for SM-2 when an ideal extension imposed.

suain rate. In the second region, the strai-n-hardeneci central portion of the filament pulls

out some of the end Iiquid (not strain-hardened) at each disk, making the volume of the

filament larger. Thus the amial extensional rate is lower than what is imposed. For

Newtonian fluids, the second strain-hardening region is not observed and the effective

strain rate is always greater than the imposed value.

Because of these non-ideal kinematics, the midpoint diameter profie is not an

indication of the deformation experienced by the whole filament. However, a numericai

analysis carried out by Kolte et al [1997] indicates that, if a fluid element at the midpoint

of the filament contracts in the same rnanner as a cylindricai column undergoing ideal

uniaxial extension, then the measured rheological response will be virtually identical to

that experienced in homogeneous shear-fiee flow. So, the desired fluid property can still

be obtained if mid-point quantities are measured. The challenge now is to create a

velocity control such that the mid-point diameter follows D(t) = D,exp(- W2) .

5.2.2 Velocity compensation

A procedure for getting the desked velocity fnction was developed, which is as

foilows:

1. Run tests at a nominal extensional rate, with the bottom disk velocity given by

V(t) = EL,exp( t) . (5-7)

2. Plot the measured diameter and length as D n ( t ) versus L(t)/Lo plot, an

example of which is shown in Figure 5.3.

3. Curve fit the data DdD(t) versus L(t)/Lo, to describe

= f[~,/D(t)]. L(t)k 0 -

More than one curve is usually rquired because a single n w e cannot f i t the data well.

The solid line in Figure 5.3 represents the curves fitted for SM-2 at an imposed

extensionai rate of 2.8 S.' .

4. Substitute the ideal diameter

D(t) = Do exp(- d2) (5-9)

into the ftnction for L(t), so that

L(t) = L , f [exp (a )] - (5.10)

DiEerentiate Eqn. (5.10) with time t to yield the correct velocity fnction V'(t):

Figure 5.3 Length and diameter data for SM-2 filament at an imposed extensionai

rate of 2.8 s-', and the bestfit curves.

df v '(t) = L , - . dt

This fiinction applies to a specific fluid at a specific elongation rate.

5. At the sarne temperature, remn the test with V'(t) to obtain a new diameter plot.

Figure 5.4 shows the resulting diameter for SM-2 &er the velocity compensation. It is a

straight line on a semi-log plot and the actuai extensional rate is found fiom the slope of

the straight line.

6. Verie the resuithg diarneter by plotthg @-d - Dideri) / Didd as a hnction

of time for the test period.

1 O Measured diameter

Figure 5.4 Diameter of SM-2 after velocity compensation.

7. Ifthe diameter deviation is greater than 5%, repeat Steps 5 - 7 until the error is

smaller than 5%.

In this procedure, temperature measurements are essential because the test fluids

are highly sensitive to temperature. It is necessary to perform extensional measurements

at the same temperature in order to obtain a satisfactory diameter profile.

Under the nominal extension, as descnbed by Eqn. 5.5, the two slopes in the

diameter plot Vary acwrding to the fluid and the extensional rate. Since a fluid's response

is generally di fferent at each extensional rate, the velocity compensation needs to be

performed at each rate.

5.3 Experimental techniques

5.3.1 Loading technique

The loading consists of setting the two disks apart at the desired initial distance of

2 mm and loading a test sample between them by a syringe. The amount injected is

careflly controlled to obtain a cylindrical test sample. M e r the injection, the sample is

allowed to relax before a test is run. Since the initial distance Lo is 2 mm, the injecting

needle is required to be smaller than 2 mm in diameter. It was observed that a needle with

a diameter smaller than 1 mm caused crystailisation of the polymer in the injected

sample. Hence B-D16G surgical needles with an outside diameter of 1.65 mm were used.

5.3.2 Disk cleaning

In addition to carenil filling, other procedures are necessary before deformation

commences. A wet edge on the bottom disk can cause the fluid to drip over the edge, and

so the disks should be cleaned before a sample is loaded. For hydrocarbon liquids like the

test fluids, cleaning with water or alcohol is not sufficient, and hence acetone was used as

the cleaning agent and solvent. To d u c e the Wear of the Teflon disks, Q-tips were used

to apply the acetone.

5.4 Effect of temperature and disk dimensions

5.4.1 Effect of temperature

Lie most viscoelastic fluids the test fluids are very sensitive to temperature- For

the specially-made test fluids, the shear viscosity at temperature T is given by

where qo is the shear viscosity at the reference temperature To and a~ is the shift factor.

The factor a~ is given by

where T is the absoiute temperature [KI, To is a reference temperature w] and a and b are constants. Values of the two constants are listed in Table 5.1. The temperature sensitivity

- --

Table 5.1 Temperature constants for SM fluids [Anna and McKinley, 19991.

Fluid

SM- 1

SM02

SM-3

a

16.9

22.4

36.9

b

75 -4

99.6

160

wi be illustrateci by the fct that a shifl of 0.5 OC in the experimental temperature can

cause a change as high as 12 % in the shiA factor a,.

As for Ml , the shear viscosity of this fluid was measured by many researcbers

[eg. Binding et al 19901 and its zero shear viscosity obeys the Arrhenius relationship:

7 = ae ( 5 - 14)

where a=3.7x 1 o - ~ P a s and P=6000 K pinding et ai, 1 9901.

Temperature dso affects the relaxation time and therefore the Deborah number.

The shifi fiictors used for shear viscosity, as given by Eqn. 5.13 and 5.14, were also used

to correct the De number and the relaxation tirne.

It is therefore important to know and maintain the temperature in the experiments.

Since the rheometer cannot control the temperature, the foilowing steps were taken to

reduce temperature fluctuations and errors in the temperature measurement:

1. A uniform and stable temperature was achieved in the laboratory by using

heating and cooling devices.

2. A test sample was exposed to the ambient temperature of the laboratory for

some time.

3.The temperature of the test sample was measured by a thermocouple

irnmediately before being loaded between the disks. This temperature was taken as the

experi mental temperature.

The temperature error was estimated to be Hl. 1 OC, corresponding to a Si% error

in the shifi -or.

5.4.2 Initial aspect ratio

The initial aspect ratio of the filament is defined as

-Ao = LJR, - (5.15)

where Lo and & are the initial length and radius of the filament. This ratio was first

introduced in the measurements for polymer melts in order to deal with the non-

homogeneous kinematics generated by endplate fixtures. For meits, I\o is usuaily larger

than 20, which minimizes the error in the measured tende force [Anna et al, 19991. For

polymer solutions, however, to insure an initial cylindncal sample, I\o has to be less than

about one. In this work, because of the loading procedure and the small disk size, AO was

4/3.

Tests with Newtonian fluids were camed out with two initial aspect ratios to

analyse the effect of the initial ratio. The results are reported in Chapter 8.

CHAPTER 6

CALIBRATIONS

Calibrations are crucial steps in an experimental investigation. Accuracy and fast

response are requued in the present experiment and this chapter describes the calibration

procedure for the force tansducers, the diameter-measuring device and the actuator

system-

6.1 Force transducer calibration and noise reduction

6.1.1 Force calibration

As described earlier in Section 4.1, two force transducers were used to rneasure

tensile force over a wide range. For the MOD404 transducer with a fll scale of 10 g

force, standard weights were used for the calibration- The standard weights were hung on

the upper disk by fine nylon thread. For the MOD405 transducer with a full scale of 1 g

force, custorn-made calibration weights were used. A piece of thin soit metal we was

shaped into a hook and hung on the disk to hold the weights. The centre of mass was

along the tine of the glass rod attsched to the transducer. The weight of the hook was

added to the calibration weights, which were d m made of thin wire. The caiibration

weights were measured by a microbalance, to an accuracy of 0.001 g.

The force transducers were calibrated several times, at different temperatures, and

linear calibration curves were obtained. Figure 6.1 gives a typical calibration m e for

the MOD405 1 g force transducer. The symbols are the output voltages fiom the

transducer and a calibration coefficient of 1 . 3 2 ~ 1 ~ ' NNolt was obtained fiom the

straight line fitted to the data. The coefficient for the MOD404 transducer was found to

be 1 . 3 0 ~ 1 O" NNolt.

Force measurements afier calibration fiom the two transducers are given in

Figures 6.2-6.3. The force data, represented by the symbols, agree with the known

Figure 6.1 Calibration curve for the MOD405 force transducer. and O are the

output voltage data.

i d e a l force

transducer nading

O 0.02 0.04 0.06 0.08 O. 1

Weight. N

Figure 6.2 Force measurements f?om the MOD404 10 g transducer.

i d e a l brce

transducer reading

O 0.002 0.004 0.006 0.008 0.01

Weight, N

Figure 6.3 Force meanirements kom the MOD405 1 g force transducer.

weights.

6.1.2 Noise reduction

Problems with noise first arose in Newtonian measurements because of the small

tensile forces. To put the noise level in perspective, 'blank' nuis were made, i-e., without

loading fluid samples between the disks. The force signal was recordeci, as shown in

Figure 6.3. Since there was no sample, the force reading should have been zero

throughout the process. Because the noise level is of order 104 N, and because the fll

range of the MOWOS transducer is 10'~ N (1 g force), the noise is well above the

acceptable level for accurate measurements of an extensionai property.

There are several possible sources of the noise. The accuracy itself is not the

reason since the resolution of the transducer is 15 pg force (1.47~ IO-' N), as specified in

the MOD405 manual. Another possible source is electncal interference during signal

transmission. The eIectrical signals for force, diameter, actuator position and speed

control etc. are transmitted through common cables, and so interference between them

could be a source of the noise. Noise could also result fiom mechanical vibration. This

possibility increases when one sees the noise signal kom the transducer: the magnitude of

the noise increases with speed. As shown in Figure 6.4, the scatter goes up to 5 x 1 o4 N. In

the rheometer, the DC motor is powered to provide speeds up to 3000 mm/sec, and so

vibration certainly o c a n . The force transducer was onginally fixed on the support track

of the actuator, which was co~ected to the motor. Vibration in the moving part of the

actuator was inevitably transmitted to the rest of the system, including the upper disk.

Figure 6.4 Signal from the force transducer (MOD405) at O = 5 5' during a 'biank'

mn.

Two ways to reduce the vibrations were considered. One was to d u c e the

vibration of the system as a whole, and the other was to separate the tmnsducer fiom the

rest of the system. In terms of system vibration, decreasing the vibration energy or

increasing the system mass were considered. Since exponentid acceleration and high

speeds were necessary, the total energy input by the motor could not be reduced. But

increasing the total mass of the rheometer was possible. However, the frst idea tried was

to separate the transducer fiom the vibration source, by mounting the transducer to a

nearby wall.

Trial experiments were nin after mounting the transducer ternporarily to a wall.

Output fkom the transducer showed significant noise reduction at al1 tested extensional

rates. Isolating the transducer was (total mass 105 g) much easier than making the whole

rheometer more rigid.

Hence the transducer was mounted on a two-dimensional translation stage

combined with an additional stage to provide movement in the third dimension. The two-

dimensional stage was fixed to the wall such that it was separate from al1 parts of the

rheometer.

Figure 6.5 is a force reading tiom the 1 g transducer MOD405 without a test

sampie after the separate mounting of the transducer. The plot shows that the noise is

reduced to a level of order of IO-' N, which is about +5% of the force measuring range.

This level was considered satisfactory.

Figure 6.5 Force transducer noise for MOD405 during test period after noise

reduction

6.2 Calibration of the diameter-measuring device and control of its position

6.2.1 Zum bach calibration

As descnbed previously, the filament diarneter was measured by the Zumbach

system, consisting of the measuring head (ODAC 16J) and the process unit (USYS 10).

They satisfy the rquuements of precision and fast response: the smallest meanirable

diameter is 0-OSmm, with a resolution of 0.001 mm and the device provides 240 sans or

80 readings per second.

The device can meanire both opaque and transparent cylinders and calibrations

were conducted for both cases. A calibration wire was placed in the centre of the

measuring field and then measured in a 'blank' run. The readings fiorn the Zumbach

were compared with the ideal diameter of the caiibration wires determined by a

micrometer (Figures 6.6 and 6.7). The measured d u e s represented by the symbols fa11

on the solid line representing the ideal diameter- The Zumbach measurements were found

to be accurate to within SYO for al1 diameters.

6.2.2 Calibration o f Zumbach positioning

As discussed previously, the rnid-point of the filament is the crucial location for

measurement. Locating this position is as important as measuring the diameter itself

The positioning systern for measuring the diameter, as discussed in Section 4.2,

was calibrated by comparing twice the value of the Zumbach position with the filament

length L(t). Good agreement was obtained, as shown in Figure 6.8.

Figure 6.6 Zumbach calibration for transparent objects

O 0.5 1 1 .S 2 2.5 3 3.5

Achirl diamater, mm

Figure 6.7 Zumbach calibration for opaque objects

6 3 Velocitycalibration

The rheometer is designed so that the desired extensional motion is achieved by

contro1ling the velocity, and therefore important to know how ciosely the actuator

velocity follows the commanded velocity. The control prograrn recorded the velocity of

the actuator during every run. The actual velocity as a fnction of time was then

compared with the commanded velocity, and an example is plotted in Figure 6.9. As

shown, good agreement was obtained.

Figure 6.9 Velocity calibration

CHAPTER 7

TEST MATERIALS AND THEIR SHEAR CHARACTERIZATIONS

In this work, the test fluids were two standard Newtonian fluids and several

international non-Newtonian test fluids. The formulation and relevant physical properties

of these fluids, as well as their shear properties, are introduced in this chapter.

7.1 Newtonian fluids and their shear properties

A Newtonian fluid is an appropriate calibration fluid because its extensional

viscosity is known to be three times its shear viscosity, Le., its Trouton ratio is three,

independent of strain or strain rate. Measurements for two Newtonian fluids were carried

out to provide conf~dence in the extensional measurements for the later non-Newtonian

ffuids.

The two Newtonian fluids were Viscasil 12,500 and 30,000 silicone oils, two

fluids made for test purposes by the General Electric Company, U.S.A. The shear

viscosity of these fiuids, and the other physical properties used in this work, density and

57

surface tension coefficient, are listed in Table 7.1. The shear viscosity data were verified

using a Brooffield viscometer in our laboratory at roorn temperatures.

1 Fluid 1 Shear viscosity 1 Surface Tension Coefficient 1 Density

Table 7.1 Properties of the Newtonian test fluids, from General Electric Company

7.2 Non-Newtonian fluids properties

In this project, three Boger fluids were prepared as standard test fluids. They were

made by Prof Susan J. Muller in the Department of Chemical Engineering at the

University of California, Berkeley, and are referred to as SM fluids hereafler.

The solvent of the three SM fluids is oligomeric styrene (Piccolastic A5 fiom

Hercules). The relaxation time of the solvent is 2.7~ 104 sec and its shear viscosity is 34.0

Pa-s at 25 OC [Anna and McKinley, 19991. The solutes are polystyrene of various high

molecular weights, as listed in Table 7.2. The polymer concentration in each solution is

the same, 0.05 % by weight.

The MIT laboratory of Prof G. H. McKinley provideci complete data of the

physical and shear properties of the SM fluids. In Table 7.3, the zero-shear viscosity qo

and the relaxation time k of these fluids at 25 OC are presented. Both properties are

temperature dependent and the temperature effects can be accounted for using the shifi

factor a~ given by Eqn. S. 12. The shear viscosity data were verined using a Brooffield

viscorneter.

The Ml fluid was also a test fuid for the preliminary investigation because its

extensional viscosity has been investigated b y many researchers (the M 1 project). This

Fiuid

Ml Polyisobutylene Aldrich Chern. 0.244 %

(MW. 4-6x10~)

SM- 1

SM-2

Table 7.2 Composition and physical properties of the non-Newtonian test fluids, the

SM fluids and Ml wulier, 1997; Tirtaatmadja, 19931.

Solute

Polystyrene (M. W. 2 . 0 ~ 1 06)

Polystyrene (M. W. 6.9~ 1 06)

Table 7.3 Steady shear properties of the non-Newtonian test fluids at 25 OC [Anna

and McKinley, 1999; Binding et al, 1990; Tirtaatmadja, 19931

Concentration Surface tension

coefficient (Nh)

Fluid

SM- 1

SM-2

SM-3

Ml

Density

(ks/m))

0.05 wt %

0.05 wt %

MW (glmol)

2x 106

7 x 106

20x10~

4-6x 106

0.03

0.03

1 O00

1 000

rlo 0'a-s)

39

46

56

2.04

(sec)

3 -7

31.1

155

0.86

fluid was prepared in the Chernical Engineering Department at the Monash University in

1988. It is a 0.244 % solution of polyisobutylene in a sotvent consisting of 93%

pol ybut ylene and 7% kerosene (Shell) mguyen and Sridhar, 1 990). Other rheological

properties can be found in a special issue of foumai of Non-hrewtonian Fiztid Mechzics

[Vol. 3 5, 199 11. Its shear viscosity was measured in our laboratory using a Brookfield

viscometer, and the results are plotted in Figure 7.1.

100

Shear rate, 11s

10

u! Q

; 8 .- > L (P Q) r u3

Figure 7.1 Shear viscosity q of Ml from the Brookfield viscometer.

O Temp = 25 O C A Temp = 20 OC

a

ff

CHAPTER 8

EXTENSIONAL RESULTS

In this chapter, values of extensional viscosity obtained from the filament

stretching rheometer for both Newtonian and non-Newtonian fluids will be presented in

terms of the Trouton ratio. Constant extensional rates were rnaintained for the duration of

the expenments for al1 four non-Newtonian fluids- The reproducibility of the extension&

data, as weil as the errors involved, wiil be discussed. Finally, the extensional

rneasurernents for S M 4 and SM-2 will be comparai with data fiom two other

laboratories.

8.1 Newtonian calibrations

Stress in Newtonian fluids, unlike non-Newtonian fluids, does not depend on

deformation history. Their Trouton tatio is three, independent of strain or strain rate.

Thus it is not necessary to obtain a constant extensional rate for these fluids. For the two

Newtonian fluids tested, Viscasil 12,500 and 30,000, experimental data were obtained at

nominal extensional rates of 5 s" and 10 S? The diameter and force traces for these two

iiquids are s h o w in Figures 8.1-8.4. In each case, the diameter starts at 3 mm, the size of

Figure 8.1 Data for the Newtonian fluid, Viscasil 12,500, at a nominal extensional

rate of d =S s-': (a) diameter profile (b) force history

Figure 8.2 Data for Viscasil 12,500 at a nominal extensional rate of t 4 0 5':

(a) diameter profile (b) force history

Figure 8.3 Data for Viscasil 30,000 at a nominal extensional rate of f. =S S-':

(a) diameter profile (b) force history

lime. s

Figure 8.4 Data for Viscasil 30,000 at a nominal extensional rate of 1 =10 s-':

(a) diameter profile (b) force history

the disks. Even without correction, the diameter plots are close to the ideal shape: a single

slope on a serni-log scale. In the force plots obtained, the force rises quickiy at the start of

stretching and then decays exponentially, as seen fiom the straight lines on the semi-log

force versus tirne plots. The Newtonian measurements were possible after the transducer

noise was reduced.

The figures show more scatter at the end of the force plots, which is related to the

force transducer. As described in Section 6.1, the noise Ievel for the MOD405 transducer

is of the order of 10" N. Near the end of the stretching process, the filament is very thin

and the measurable force decreases to 0(10'~) N, quivalent to the noise level. Therefore,

the noise added a random signal of the same amplitude to the measured force.

Figure 8.5 presents the transient Trouton ratio for both Newtonian fluids at the

two nominal extensional rates. It is encouraging that the Trouton ratio values are not only

constant, but also fall very close to the predicted ratio of 3. The error band is I 0.5. The

accurate extensional measurements for Newtonian fluids provide a solid basis to continue

with non-Newtonian measurements. The scatter is greatest at the end of the Trouton ratio

plot, corresponding to the larger scatter in the force plot.

Tests were conducted With the Newtonian fluids to determine the effect of initial

aspect ratio. As discussed in Chapter 4, usually I\o a 1 is necessary to prevent sagging and

breakup of the initial filament. However, in this rheometer & < 1 could not be achieved

due to the loading procedure. Initial distances of Lo = 1-5 mm and 2.0 mm, equivdent to

aspect ratios of 1 and 413, were used in testing Viscasil 30,000 at two nominal

extensional rates. As shown in Figure 8.6, the Trouton ratio values are identical at both

Figure 8.5 Transient Trouton ratio for Newtonian fluids

rates. Moreover, the Trouton ratios with Lo = 1.5 mm show more scatter than those with

La = 2 mm close to the end of testkg. This is reasonable because the sample size is

smaller with =1.5 mm, resulting in a smaller tensile force. Based on these results, the

initial aspect ratio of 4/3 was used in subsequent experiments.

8.2 Extensional results for Non-Newtonian fluids

There were four non-Newtonian test fluids investigated in this work Their

extensional data - diameter, force and Trouton ratio - are presented in this section. Since

temperature control was not avadable at this stage, experiments were undertaken at the

laboratory temperatures, and the parameters such as shear viscosity, Deborah number and

the relaxation time, were correcteci to the reference temperature, 25 OC, using the shift

factors given in Section 5 -4.1.

The velocity compensation for this fluid was perforrned using the method

describeci in Chapter 4 and singie-dope diameter profiles were obtained. Figwe 8.7 a

shows the diameter profile for SM- L after correction, at De = 19.5. The extensional rate

& =3 .O S-' was determineci from the best fit of a straight line through the data The ideal

diameter is the solid line and the symbols represent the rneasured diameter values f?om

the Zumbach. The graph confirrns that the actuai diameter is close to ideai. Analysis of

the data showed that the diameter

Deborah numbers. The minimum

lower limit of the Zumbach-

was within B % of the ideai diameter for SM-1 at ail

diameter rneasured was 0.10 mm, which is above the

O Lw1.5 mm, at 10 11s Lo=2 mm, at 10 1/s

A Lo=1.5 mm, at 5 1/s x Lw2 mm. at 5 Ifs

0.1 1 0.5 0-7 0.9 1.1

Time, s

Figure 8.6 Comparison of Newtonian data with two initial aspect ratios, for two

nominal extensional rates.

Figure 8.7 Data for SM-1 at De = 19.5 ( 6 =3 8): (a) diameter (b) force

Since the fluid's shear viscosity is Iow, 39 Pas ai 25 OC, the MOD405 force

transducer was used for the force measuremeats. Extensional data for SM-1 were

obtained at De numbers of 10.7, 19.5 and 46.7. The tensile force trace corresponding to

the diameter in Figure 8.7.a is shown in Figure 8.7.b. The minimum force measured d e r

the start-up was 5 x 1 0 ~ N, above the noise level of the MOD405 transducer used. As

stretching commenced the force increased rapidly to a maximum and then decreased

when further stretched. This initial rapid increase is sirnilar to what is found for

Newtonian fluids. The peak is the viscous response of the fluid to a sudden change fiom

rest. The force reached a minimum before approaching a second maximum at high

strains. During the initial extensional motion, the polymer molecules in the solution are

aiigned dong the flow direction and stretched from their equilibriurn conformatioas.

After alignment, the tensile force decreases with the filament diameter because the stress

in the filament remains unchangecl. At large mains, large hydrodyoamic forces cause the

molecular chahs to uncoil and the resistance increases rapidly. This is reflected in an

increase in the tende force while the diameter continues to decrease. The second peak in

the force plot has a larger magnitude than the fint one. At the higher extensional rates for

SM-1, the force decreases exponentially after reaching the maximum, as illustrated in

Figure 8.7.b. This exponential decrease implies that a steady state has been achieved in

the extensional stresses, Le., the moldes are w longer extending and have fixed

conformations.

At a smaller extensional rate, & = 1.32 sml, a plateau instead of a peak was found

in the force diagram for al1 runs while the diarneter decreased exponentially, as shown in

Figure 8.8.

Results of transient Trouton ratio versus Hencky strain for SM4 at three Deborah

numbers-10.7, 19.5 and 46.7-are shown h Figure 8.9. Generally, the Trouton ratio rises

rapidly to a value close to 3 within 0.1 strain unit and remains roughly constant until the

strain goes above 2. Experimentaiiy, the Trouton ratio values start at 2.3 to 2.9 for De =

19.5 and 46-7, whereas the value is 3-4 for De = 10.7. verall, these values Ml in the

reasonable range of 2.3 to 3.4. The values frther indicate of the accuracy of the results.

As the fluid is stretched m e r , the extensional stress in the filament Increases

dramaticaily. In this region, the significant stress growth is a result of macromolecules

being extendeci from their equilibrium confguration~~ As shown in the figure, the

transient Trouton approaches steady-state values at large strains. For SM-1, the steady-

state vdue was found to be around 900 for De = 19.5 and 46.7; for De = 10.7, the steady

state value was fond to be 1500, somewhat higher than the other two. These values are,

however, lower than the steady-state values obtaned for other fluids. Tirtaatmadja and

Sridhar [1993] reported steady-suite values of 2-5x10~ for Ml fluid and two other high

molecular weight P B solutions, and van Nieuwkoop et al [1996] obtained values as high

as 6x 103 for some similar dilute PIB solutions. A Hencky strain of 6.4, i-e., L& = e6-q

was achieved for this fluid,

Several runs were made at each Deborah number. The reproducibility is

illustrated in Figure 8.10, which is for De = 10.7.

1 O

o measured diameter - O=Doexp(-1 -32U2)

Figure 8.8 Data for SM- 1 at De = 10.7 (0 4-32 s-'): (a) diameter (b) force history

Figure 8.9 Cornparison of Transient Trouton ratio for SM-1 at al1 De numbers

8.2.2 Fluid SM-2

Extensionai data at four Deborah numbers were obtained for SM-2. This fluid is

more viscous than SM-1, with a shear viscosity of 46 Pas at 25 OC.

Figure 8.1 1 a shows a representative corrected diameter profile for this solution,

at De = 3 1.8. The extensional rate 6 = 1.65 S-' was determineci by a best-fit straight line to

the diameter data. Verification of this extensional rate was carried out by plotting hD/D

vs. time. The maximum error was &4%.

The corresponding force history is presented Figure 8.1 1 b. Similar to SM-1, the

force shoots up quickly, deches exponentidly, and then increases again. However, no

second peak was observed for SM-2 at any Deborah number. It was observed that the

filament detached fkom one of the disks at a high strain but before a second peak would

have been reached. Detachment of the filament terminated the stretching, in contrast to

termination caused by either the motor speed limit or the length of travei, as happened

with SM- 1.

Figure 8.12 presents Trouton ratio renilts for SM-2 at De numbers of 20.9, 3 1.8,

and 58.5, which were chosen to be dose to values obtained in the other two Iaboratories

for cornparison purposes. The start-up value is 2.8 for De = 31-8 and 58.5, and Tr

increases slightly fkom this value with Hencky strain. A sharp rise occurs at a strain of

about 1.8. For De = 20.9, the Trouton ratio starts at a lower value of 2.2 but increases

faster than the other two for the first stage, Le., there is no fkst plateau. Because of the

detachment, the maximum strain is 4.1, in contrast to the maximum strain of 6.4 achieved

with SM- 1. However, the maximum Trouton ratio for SM-2 is 2200, somewhat larger

Figure 8.10 Transient Trouton ratio measurement at De = 1 0.7 ( 5 = 1 -32 5') for SM- 1.

Figure 8.1 1 Data for SM-2 at De = 3 1.8 ( =2.8 S-'): (a) diameter @) force history

than the value of 2000 for SM-1.

The reproducibility is given by a representative example, at De = 58.5, as shown

in Figure 8.13. For the four runs, the values of the start-up plateau agree well, and there is

some scatter in the following rapid growth section. The data agree more closely again at

high strains.

8.2.3 Fluid SM-3

SM-3 is the most viscous of the three SM fluids, with a shear viscosity of 56 Pa-s

at 25 OC. Measurements were made at a single De number, 1 10.

Diameter and force plots for this fluid are shown in Figure 8.14. The extensional

rate obtained fkom the best-fit tine is 1.6 s-'. The maximum error between the measured

diameter and the ideal diameter in this case was *7%.

Since the maximum force with this fluid exceeds the measuring range of the

MOD405 force transducer, the MOD404 transducer was used. As happened with SM-2,

the filament detached fiom one of the disks before the force reached a second maximum.

The extensional stress growth of SM-3 is similar to the other two. As Figure 8.15 shows,

the Trouton ratio value rises fiom 2.6 in the linear viscoelastic region and remains

approximately at this value until the strain is around 1.3. It then increases rapidly to

values in excess of IO'. in accordance with the force measurements, no steady suite is

seen in the Trouton ratio plot for this fluid.

The four nuis at De = 110 (t =1.6 sa') in Figure 8.15 indicate the reproducibility

of data for SM-3.

O diameter

- =Doexp(-l.6VZ)

Figure 8.14 Data for SM-3 at De = 1 10 (1 = 1 -6 s*'): (a) diameter (b) force history

O Run 1 o Run 2 A Run 3 o Run 4

Figure 8.15 Transient Trouton ratio for SM-3 at De = 1 10 (t = 28.6 OC, 4.6 S-')

8.2.4 Fluid M l

Fluid Ml was the standard fluid in the Ml project for the extensional viscosity

study about a decade ago. Its pnmary rheological properties were found by many

researchers, and Tirtaatmadja and Sridhar [ 19931 made TE measwements. in this work,

extensional measurements were made for Ml at two extensional rates. Its shear viscosity

is 2.04 Pas at 25 OC, which is an order of magnitude smailer than the viscosities of the

SM fluids presented earlier.

Corrected diameter plots of Ml for two extensionai rates are shown in Figures

8.16 and 8.17. The extensional rates of 5.6 S-' and 10.0 S-' are obtained fiorn the best-fit

lines of the diameter plots. High extensional rates are necessary with this fluid because of

its low viscosity. Plots of ADD versus time show that the actual diarneter data are within

B % of the ideal diameter, for both extensional rates.

Tensile force measurements were made using the MOI3405 (1 g) force transducer

because Ml is less viscous. The force profile is generally similar to those of the SM