Group Title: Temperature dependence of jet swell and material functions in polymer melt systems /
Title: Temperature dependence of jet swell and material functions in polymer melt systems
Full Citation
Permanent Link:
 Material Information
Title: Temperature dependence of jet swell and material functions in polymer melt systems
Physical Description: xvii, 186 leaves : ill. ; 28 cm.
Language: English
Creator: Johnson, Malcolm Cloud, 1948-
Publication Date: 1977
Copyright Date: 1977
Subject: Rheology   ( lcsh )
Viscoelasticity   ( lcsh )
Polymers and polymerization   ( lcsh )
Chemical Engineering thesis Ph. D
Dissertations, Academic -- Chemical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 182-185.
Additional Physical Form: Also available on World Wide Web
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by Malcolm C. Johnson, Jr.
 Record Information
Bibliographic ID: UF00097490
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000063559
oclc - 04212642
notis - AAG8758


This item has the following downloads:

PDF ( 6 MBs ) ( PDF )

Full Text







To: Patsy for the encouragement to be myself,
Our Family and Friends.


The author wishes to express his gratitude and

appreciation to:

Dr. R. J. Gordon for suggesting the research topic

leading to this dissertation, and for his guidance and

encouragement throughout the author's graduate career.

Dr. R. W. Fahien, Dr. H. E. Schweyer, Dr. T. E.

Hogen Esch, and Prof. Robert D. Walker, Jr., for their

interest and suggestions.

Dr. F. Philips Pike for encouragement during the

author's career at the University of South Carolina.

Dr. Joseph Starita and Rheometrics of Union, New

Jersey, for allowing the author use of their experimental


Dr. A. Edward Everage, Jr.., of Monsanto, and Dr. Gary

Allen of Union Carbide for donating and characterizing the

samples used in this research.

Dr. C. Balakrishnan, Dr. C. S. Chiou, F. J. Consoli,

F. Y. Kafka, and Dan White, fellow graduate students, for

many enlightening discussions.

The Department of Chemical Engineering for financial

support and personal assistance from the faculty and staff.




ACKNOWLEDGEMENT. . . . . . . ... . iii

LIST OF TABLES . . . . . . . . . vi

LIST OF FIGURES . . .. . . . . . vii

KEY TO SYMBOLS . . . . . . . . . .x .

ABSTRACT . . . . . . . . . . . xv



I.1 Introduction . . .. . . . . 1

1.2 Summary of Previous Research . . . 4

I.2.A. Experimental . . . 4

I.2.B. Theoretical. . . . . .. 14

1.3 Summary of Dissertation . .. ... 26

II. EXPERIMENTAL. . . . . . . 27

II.1 Capillary Viscometer Measurements . 28

II.1.A Viscosity ... . . . . 28

II.l.B Jet Swell . ... . . . 38

11.2 Cone/Plate Measurements . .. . .40

11.3 Eccentric Rotating Disk Measurements.. 47

SHEARING FLOW (SSF) . .. . . . 52

III.1 Constitutive Model . .. . .. 52

III.2 Constitutive Predictions for SSF . 56

III.3 Constitutive Parameter Fitting . . 58


III.4 Temperature Superposition . . .


IV.1 Material Functions . . . . .

IV.2 Jet Swell . . . . . . . .




A.1 Main Programs .

A.2 Subroutines .. .












. . . 112

. 127

. 142

. . . 176

. . . 182

. . . 186


Table Page

II-1 Capillary Dimensions. . . . . ... 35

IV-1 PS#1 Constitutive and Material Parameters . 70

IV-2 PS#2 Constitutive and Material Parameters . 70


Figure Page

II-1 ICR extrusion system schematic. . .. .29

II-2 Cone/plate system schematic . . . ... .41

II-3 ERD system schematic. . . .. . . . .48

IV-1 Effect of temperature on ICR (solid points) 'and
RM5 (hollow points) viscosity versus shear rate
data for PS#1; (-) constitutive model predic-
tions . . . . . . . . . . 71

IV-2 Effect of temperature on RM5 PNSD versus shear
rate data for PS#1; (-) constitutive model
predictions . . . . . . . .. . 72

IV-3 Effect of temperature on RVE elastic modulus
versus frequency data for PS#1; (-) constitu-
tive model predictions .. . . . .. . 73

IV-4 Effect of temperature on RVE viscous modulus
versus frequency data for PS#1; (-) constitu-
tive model predictions. .. . . . . 74

IV-5 Effect of temperature on RVE dynamic viscosity
versus frequency data for PS#1; (-) constitu-
tive model predictions. . . . . .. 75

IV-6 Effect of temperature on ICR (solid points) and
RM5 (hollow points) viscosity versus shear rate
data for PS#2; (-) constitutive model predic-
tions . . . . . .. . . . . 76

IV-7 Effect of temperature on RM5 PNSD versus shear
rate data for PS#2; (-) constitutive model
predictions . . . . .. . . .. . 77

IV-8 Effect of temperature on RVE elastic modulus
versus frequency data for PS#2; (-) constitu-
tive model predictions. . . . . . . 78

IV-9 Effect of temperature on RVE viscosity modulus
versus frequency data for PS#2; (-) constitu-
tive model predictions . . . .. . 79



IV-10 Effect of temperature on RVE dynamic viscosity
versus frequency data for PS'2; (-) constitu-
tive model predictions. . . . .. . 80

IV-11 ICR (solid points) and RM5 (hollow points)
viscosity versus shear rate data for PS#1,
measured at 473, 503 and 533 degrees K.,
temperature shifted to 503 degrees K.; (-)
constitutive model predictions . .. . 82

IV-12 RM5 PNSD versus shear rate data for PS#1,
measured at 473, 503 and 533 degrees K.,
temperature shifted to 503 degrees K.; (--)
constitutive model predictions. . .. . 83

IV-13 RVE elastic modulus versus frequency data for
PSr1, measured at 473, 503 and 533 degrees K.,
temperature shifted to 503 degrees K.; (-)
constitutive model predictions .. . . . 84

IV-14 RVE viscous modulus versus frequency data for
PS#1, measured at 473, 503 and 533 degrees K.,
temperature shifted to 503 degrees K.; (-)
constitutive model predictions. . . . ... 85

IV-15 RVE dynamic viscosity modulus versus frequency
data for PSi1, measured at 473, 503 and 533
degrees K., temperature shifted to 503 degrees
K.; (-) constitutive model predictions . . 86

IV-16 ICR (solid points) and RM5 (hollow points)
viscosity versus shear rate data for PS#2,
measured at 473, 503 and 533 degrees K.,
temperature shifted to 503 degrees K.; (-)
constitutive model predictions . .. . . 87

IV-17 RM5 PNSD versus shear rate data for PS#2,
measured at 473, 503 and 533 degrees K.,
temperature shifted to 503 degrees K.; (-)
constitutive model predictions. . . . ... 88

IV-18 RVE elastic modulus versus frequency data for
PS#2, measured at 473, 503 and 533 degrees K.,
temperature shifted to 503 degrees K.; (-)
constitutive model predictions. . . . ... 89

IV-19 RVE viscous modulus versus frequency data for
PS#2, measured at 473, 503 and 533 degrees K.,
temperature shifted to 503 degrees K.; (-)
constitutive model predictions . . . . 90



IV-20 RVE dynamic viscosity versus frequency data
for PS#2, measured at 473, 503 and 533 degrees
K., temperature shifted to 503 degrees K.;
(-) constitutive model predictions. . ... 91

IV-21 Effect of temperature on the constitutive
prediction of the recoverable shear versus
shear rate for PS1. . . ... . . . 93

IV-22 Effect of temperature on the constitutive
prediction of the recoverable shear versus
shear rate for PS#2. .. ... . . . .. 94

IV-23 Experimental JS magnitude versus the
temperature-shifted wall shear stress for
PS#1, measured at 473, 503 and 533 degrees K.,
shifted to 503 degrees K.. .. .. ... 97

IV-24 Experimental JS magnitude versus the
temperature-shifted wall shear stress for
PS#2, measured at 473, 503 and 533 degrees K.,
shifted to 503 degrees K.. . . . . . 98

IV-25 JS magnitude versus the recoverable shear at
the capillary wall as predicted by the various
elastic solid theories . . . . . .. 100

IV-26 Experimental JS magnitude versus the
recoverable shear at the capillary wall
predicted from constitutive theory for PS#1;
(-) predictions of the elastic solid theory 102

IV-27 Experimental JS magnitude versus the
recoverable shear at the capillary wall
predicted from constitutive theory for PS#2;
(-) predictions of the elastic solid theory 103

IV-28 Effect of temperature on JS magnitude versus
shear rate data for PS#1; (-) constitutive
model predictions. . . . .. . . 104

IV-29 Effect of temperature on JS magnitude versus
shear rate data for PS#2; (-) constitutive
model predictions. .... .. . . . . 105


Ar = cross-sectional area of the capillary rheometer

reservoir (barrel).

a = centerline separation of eccentric rotating disk

rheometer plates.

aT = constitutive shift factor for shear rate and


B = jet swell magnitude.

Bm = jet swell magnitude measured at room temperature.

bT = constitutive shift factor for viscosity (shear

and dynamic) and the viscous modulus.

C = [C (2-E)]/2.
C = integration constant.

cT = constitutive shift factor for normal stress

functions and the elastic modulus.

D = capillary diameter.

D. = diameter of the recovered extrudate jet.
Dr = diameter of the capillary rheometer reservior


d = deformation tensor.

dn = differential molecular fraction.

F = measured capillary rheometer extrusion force.

Fb = force required to correct the extrusion force

for entrance (Bagley) losses.


Fbf = force required to correct the extrusion force

for barrel friction losses.

F = corrected capillary rheoineter extrusion force.

Fx = force measured in the eccentric rotating disk

rheometer, related to the viscous modulus.

F = force measured in the eccentric rotating disk

rheometer, related to the elastic modulus.

Fz = force measured in the cone/plate rheometer,

related to the primary normal stress difference.

G' = elastic (storage) modulus.

G" = viscous (loss) modulus.

h = gap setting in the eccentric rotating disk


I = unit tensor.

J = measured torque in the cone/plate rheometer,

related to the viscosity.

K = constant.

KE = constant.

k = Boltzmann's constant.

L = capillary length.

M = molecular weight.

M. = molecular weight of species i.

M = number averaged molecular weight.

Mw = weight averaged molecular weight.

N = number of molecular weight fractions.

Na = Avogadro's number.

N1 = primary normal stress difference.





















= reduced primary stress difference.

= radial distribution of the primary normal stress

difference evaluated at the capillary exit.

= primary normal stress difference at the wall of

the capillary exit.

= secondary normal stress difference.

= local slope of a log/log plot of shear stress

versus apparent shear rate.

= isotropic pressure.

= pressure drop across the capillary.

= plate radius in the cone/plate and eccentric

rotating disk rheometers.

= radial coordinate.

= final slope of a log/log plot of viscosity

versus shear rate, (l-a)/a.

= recoverable shear.

= absolute temperature.

= total stress tensor.

= radial distribution of total normal stress.

= radial distribution of total normal stress at

the capillary exit.

= time variable.

= velocity profile at the capillary exit.

= average velocity.

= velocity vector.

= capillary rheometer crosshead speed.


Vv = gradient of the velocity vector, v.

Vv = transpose of Vv.

x = reduced molecular weight, M/Mw.

x = Cartesian coordinates.

Z(.) = Riemann zeta function.

z = molecular weight variable or axial coordinate.

Greek Letters

a = constitutive parameter related to the power law


B = cone angle in the cone/plate rheoreter.

F(z) = Gamma function of z.

y = shear rate.

y = reduced shear rate.

ya = apparent shear rate.

E = phenomenalistic constitutive parameter.

In = shear viscosity.

no = zero shear viscosity.

n' = dynamic viscosity.
n = reduced shear viscosity.

6 = weight averaged molecular weight.
i = the i relaxation time of the j molecular
weight fraction.

o = angular coordinate.

TT = 3.14154...

p = material density.

P = material density of the i species.





= material density at the temperature where JS

magnitude measurements were made.

= material density at the extrusion temperature.

= deviatoric stress tensor.

= shear stress.

= shear stress evaluated at the capillary wall.

= deviatoric stress tensor contribution of the

i time constant of the jth molecular weight


= angular coordinate.

= angular velocity or frequency.


Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy




June, 1977

Chairman: Dr. R. J. Gordon
Major Department: Chemical Engineering

An elastic effect commonly observed in the extrusion

of viscoelastic fluids is jet swell (JS), the expansion or

puffing up of material as it emerges from a capillary or die.

This phenomenon is generally attributed to the relaxation of

normal stresses developed in the fluid upstream of the die

exit. Advance knowledge of the magnitude of this effect is

important in the design of fabrication equipment used in the

polymer and plastic industry.

The prediction of JS magnitude was accomplished in

two steps. First, the recoverable shear was calculated at

the conditions of interest using a newly developed constitu-

tive theory. Secondly, this quantity was related to JS

magnitude using the previously developed, long length-to-

diameter (L/D) ratio elastic solid theories with appropriate

modifications. Predictions with respect to deformation rate

magnitude and temperature were possible with this technique.

The constitutive theory used in the calculation of the

recoverable shear was derived from a combination of continuum

and molecular approaches, thereby incorporating desirable

characteristics of both. This rate-type model contains four

adjustable parameters which were easily determined from

material functions measured in simple shearing flow (SSF).

Additionally, this model incorporates provisions for including

effects due to variations in material molecular weights.

Predictions of theological functions with respect to tempera-

ture arise naturally in the development.

Data in an SSF flow environment were obtained for

each of two similar polystyrene samples at 473, 503 and 533

degrees K. using three commercially available instruments.

Viscosity versus shear rate data were obtained using a capil-

lary viscometer. Viscosity and primary normal stress

difference (PNSD) versus shear rate data were measured with

a cone/plate device. Viscous and elastic moduli versus

frequency measurements were taken using an eccentric rotating

disk rheometer.

The JS magnitude data were obtained from extrudate

samples collected in conjunction with the capillary viscometer

measurements. A capillary series containing five L/D ratios

of 8.56 to 34.91 was employed to experimentally determine

the long capillary JS magnitude. Annealing of extrudate

samples to obtain maximum swelling produced a negligible

increase in JS magnitude, and the standard density correction

varied from 3 to 5 percent of the actual swell ratio. Data

indicated that JS magnitudes obtained from the longest

capillary ratio could be used for direct comparison with


predictions based on the elastic solid theories within the

limits of experimental accuracy.

Constitutive parameters were fit to each sample at

503 degrees K. using viscosity and PNSD data. Material

functions at additional temperatures were calculated from

the suggested temperature dependence included in the theory.

All measured material functions were in good agreement with

predictions, indicating that the constitutive model correctly

portrayed the relative forms of the theological functions of


The various elastic solid theories were found to

predict similar curves of JS magnitude versus recoverable

shear. The theory best representing the data was chosen

to make calculations of swell versus shear rate for each

sample at all temperatures. While predictions described

the qualitative relationships between JS magnitude, tempera-

ture,and shear rate to within 25 percent, discrepancies

increased with increasing temperature and deformation rate.

The most serious limitations of the current study

were the failure to predict JS magnitude as a function of

L/D ratio, and the limited shear rate range for which PNSD

measurements could be obtained.

The major contributions of this research are the

new capability of predicting JS magnitude as a function of

temperature, and the successful testing of a constitutive

theory capable of making improved predictions of material

functions in SSF.



I.1 Introduction

For a given flow situation the viscoelastic fluid

usually responds in a manner quite different from that

exhibited by a purely viscous Newtonian fluid. '2 A non-

Newtonian anomaly observed in extrusion is the pronounced

swelling of material exiting from an orifice, capillary,

or die. This effect is generally attributed to the relaxa-

tion of normal stresses present in the emerging free jet

that arise due to fluid deformations in the region upstream

of the die exit. Frequently referred to as "jet swell,"

"puff up," "memory," "post extrusion swell," "die swell,"

or "extrudate expansion," terminology which credits the

initial discovery to early investigators, namely Barus effect
and Merrington phenomenon, is considered historically
incorrect. The present work will refer to this elastic

manifestation as jet swell (heretofore JS) because this

designation appears to be an accurate and unambiguous descrip-

tion of the process in addition to its being widely used

in the current literature. The exact definition of JS

magnitude depends upon the particular geometry under

consideration, but it may be regarded as the unconstrained

and relaxed extrudate's dimension perpendicular to the

direction of flow (in the so-called gradient direction)

divided by this normal or gradient dimension inside the

die. In the case of capillary extrusion, for example,

this quantity is simply the ratio of the relaxed jet's

diameter to the diameter of the die.

Prediction of theological functions and elastic

behavior of molten plastics and polymers is important in

the development of large-scale industrial processes and

fabrication equipment. Prior knowledge of the amount of

swell that will occur in a particular extrusion situation

is highly desirable from viewpoints of product quality and

quantity. Additionally, trial-and-error system design may

be reduced or eliminated. Unfortunately, current JS estima-

tion techniques sometimes fail to accurately describe the

magnitude of this expansion for even the most simple flow

condition and geometry, and these theories generally cannot

be rigorously extended to more realistic and complicated


The general practice used in modeling the JS problem

is to consider a representative, idealized system with an

easily managed geometry for which meaningful experimental

data are relatively simple to obtain. This is accomplished

by modeling the Instron Capillary Rheometer (ICR) used in

the determination of shear viscosity. This isothermal

extrusion process may be viewed as occurring in three

parts: (1) a converging flow from a large reservoir into

a cylindrical die, (2) a build-up to well-developed, simple

shear flow (SSF) within the die, and (3) the molten jet

emerging from the die into the atmosphere, swelling as

stresses within it relax and hardening as it cools. It

is this final frozen jet diameter that is to be predicted

from system and material particulars. Although this

simplified approach would appear to be most applicable to

the extrusion of a single fiber, JS need not be rigidly

controlled in this process because the final filament

diameter may be more easily maintained by increasing or

decreasing the applied tension of the take-up device. This

variable take-up force is not easily adapted to the extru-

sion of larger objects such as solid rods where the current

predictions should prove to be most useful. It should be

noted here that only a minor modification of the above

simplified system geometry is required to estimate the

thickness of extruded tube walls and sheets.

Previous research indicates that the JS magnitude

of a material following SSF is an increasing function of

polymer molecular weight and of system throughput, usually

expressed in terms of the shear rate at the capillary wall.

It is found to be a decreasing function of temperature and

of material residence time in the capillary, usually expressed

as a length-to-diameter (L/D) ratio, up to an L/D of between

10 and 50 (depending on the material) where JS becomes

independent of the capillary length. Unfortunately, highly

sophisticated calculations are needed to completely charac-

terize the observed L/D effect. This would include the

modeling of the converging flow field and subsequent velo-

city profile buildup in the capillary entrance region. To

avoid these complexities, this study will modify the proposed

three region system mentioned above. This is accomplished

by utilizing capillaries of sufficiently large L/D ratios

such that JS magnitude is independent of capillary length.

Another problem is the tremendous amount of data and large

number of samples required to characterize any molecular

weight dependence. This will prohibit the current study

from drawing any conclusions related to this variable. In

view of these observations, the primary objectives of the

current research will be to improve and modify existing JS

theories in order to make better quantitative predictions of

shear rate and temperature dependence. Qualitative conclu-

sions with respect to the residence time effect will be

based solely on experimental observations.

1.2 Summary of Previous Research

Concentrated research into the mechanisms of extru-

date swelling became increasingly important with the discovery

and subsequent consumer demand for plastics and synthetic

fibers. This discussion of previous research separates

contributions into experimental and theoretical approaches,

although most studies contain both due to the importance of

obtaining confirmation of new predictive theories.

I.2.A Experimental

Previous experimental studies incorporating a

theological approach to the JS phenomenon will be discussed

in three parts. First, three testing devices commonly

used in the determination of material functions of polymer

melt systems will be briefly reviewed. Although the two

rotational instruments mentioned here are not used directly

in the measurement of JS magnitude, their importance in

obtaining the theological functions required for the predic-

tion of this effect will become obvious in Chapters III and

IV. This section is intended to be a short introduction to

these rheometers. Consult Chapter II of this thesis for a

more complex interpretation, including the straightforward

mathematical analysis required to obtain meaningful experi-

mental results.

Secondly,.experimental conditions and methods will

be compared to specific apparatus and techniques that have

been previously used in the experimental determination of

JS magnitude. Finally, qualitative findings and generally

observed trends in JS magnitude that can be related to the

current study will be analyzed.

The testing devices to be discussed here are (1) the

capillary rheometer, (2) the cone/plate rheometer, and (3)

the eccentric rotating disk (ERD) rheometer. Although these

instruments are designed to measure different theological

functions, each operates in a manner such that all measure-
ments are obtained in SSF environments. This equivalence

of flow fields permits direct comparisons of data among

the three instruments at the same deformation magnitude.

Stated differently, material particles undergoing SSF


experience identical deformation histories at a given shear

rate, regardless of which instrument is used for the measure-

ments. These comparisons will serve as an internal check of

both experimental consistency and validity of the constitutive


The capillary rheometer, or viscometer as it is

sometimes called, is currently the most common device used

for the measurement of the viscosity function of a material

undergoing SSF. The apparent shear rate, a measure of the

deformation intensity, may be calculated from a knowledge

of the mass flow rate and the dimensions of the system. The

wall shear stress and hence the viscosity function can be

related to the force required to maintain this flow. Early

workers modeled and characterized the required system well

enough that any number of commercially available instruments

are capable of obtaining accurate viscosity function data.

These devices fall into two general categories based upon

the mode of operation, namely constant .stress or constant

shear rate. In the case of the constant stress mode, a

given constant stress is applied to the sample and the

corresponding steady-state deformation rate is measured,

while in the constant shear rate case, a given deformation

rate is imposed and the resulting shear stress is calculated

from the applied load. The constant shear rate technique is

apparently the easier mode in which to obtain raw data

because no additional measurements are required to determine

the deformation rate. However, the data obtained in this

manner must be corrected for the nonparabolic profile that

is generally exhibited by a viscoelastic fluid undergoing

laminar tube flow.

Both types of capillary rheometers usually incor-

porate design concepts that permit maximum flexibility with

respect to deformation rate and temperature. These include

interchangeable capillaries that extend the range of both

flow resistances and attainable shear rates, and heating

blocks capable of close-tolerance temperature control.

Additionally, standard designs insure that most of the flow

resistance is offered by the capillary, although techniques

have been developed to correct extrusion forces for losses

that occur due to pressure drop in the reservoir and presum-

ably for losses resulting from the converging flow into the


Although relatively rapid characterizations of the

material's viscous response are possible with capillary

viscometers, these devices are not capable of measuring

elastic behavior directly. To measure this solid-like

response, rotational-type instruments must be used. Modern

instrumentation, together with the evolution of the earliest

of these devices which related elastic effects to recoil,

has led to the development of well-modeled, sophisticated


In the cone/plate system, the sample is placed in

the gap between a fixed, flat plate and a truncated cone

having a very small included angle, typically five degrees

or less. The temperature-controlled system operates in

two different modes to obtain data in either an SSF environment

or oscillatory shear flow when a sinusoidal deformation is

used. For SSF the viscous response is calculated from the

torque required to rotate the cone at the desired angular

velocity. The shear rate is calculated from this angular

velocity and the cone angle. The primary normal stress

difference (PNSD), a direct measure of elasticity, may be

related to the thrust or force generated in the sample

that tends to push the cone away from the plate. The small

angle of the cone insures that the shear rate will be

constant throughout the gap region, and the cone diameter

may be varied with the elasticity of the sample to increase

or decrease the measured thrust as required by the force

transducer sensitivity. In the dynamic mode, the plate is

not fixed but free to rotate, and small amplitude oscilla-

tions at the cone are transmitted through the sample to the

lower plate where they are measured as being in-phase

(elastic) or out-of-phase (viscous) contributions, depending

on the magnitude of the recorded torque and the phase lag.

The dynamic viscosity may be calculated from the viscous


The ERD rheometer is similar to the cone/plate

device in appearance, except that the cone is replaced by

a plate identical to the lower one. The plates' centerlines

are off-set a small distance and both are rotated simulta-

neously in the same direction. The viscous and elastic

moduli, and hence dynamic and complex viscosities, may be

calculated from a knowledge of the forces required to main-

tain the separation of the plates as they rotate.

Although rotational instruments are necessary for

the measurement of the elastic portion of material response

which is not obtainable with capillary rheometers, they

suffer from at least two serious drawbacks. First, they

require considerable expertise on the part of the instrument

operator before consistent and reliable data may be obtained.

Secondly, they are unable to operate at the large deformation

rates encountered in commercial processes.

Any rheometer used in the testing of viscoelastic

materials is subject to a transient response effect that is
characteristic of this class of materials. These transient

responses, while an indication of assorted elastic and viscous

effects, are undesirable when measurements are needed in a

steady-state mode as in the case of the rheometers discussed

above. Therefore, the careful experimentalist must insure

that measured variables represent the long-time response

of the material, particularly at the lower deformation rates

where this effect is most noticeable.

Essentially all of the studies concerned with the

experimental determination of JS magnitude utilize the

capillary rheometer extrusion system. Hence, the remainder

of the discussion pertaining to previous investigations will

be confined to this particular type of instrument. Inasmuch

as this device generally operates in a vertical position,

it is important to employ a technique that minimizes errors

in JS magnitude that may arise due to necking or sagging of

the extruded material under its own weight. It is also

important that the collection and/or measurement techniques

employed be reproducible in nature and representative of

the modeled process. The JS magnitude may be referred back

to the extrusion temperature by means of a density correction

if measured at a different temperature. Additionally, the

sample may be annealed to a stress-free state if this

particular condition is to be met for direct comparison with

theory. Finally, the researcher must decide whether or

not these conditions mentioned above must be met in order

to obtain useful predictions of relevant commercial processes.

One technique used to minimize errors due to the

material sagging under its own weight that occur when

extruding into air is to force the sample directly into a
fluid having the same density as the extrudate. The

material is usually annealed in this fluid at the extrusion

temperature during the process and measured simultaneously

by a photographic scheme. While there are inherent advantages

in this particular method, it is quite obviously a different

physical situation from that usually encountered in an

industrial-type process where material is extruded directly

into air.

Collection techniques vary from a highly specialized
device capable of obtaining multiple samples rapidly2 to

simple, systematic cutting of the material by hand with
scissors. 15 Regardless of the degree of sophistication

employed, it is important to develop methods that minimize

distortion of the material. This has been accomplished

previously by cutting the emerging material as close to

the exit as possible, extruding approximately one inch of

new material, then cutting the sample as close to the exit
as possible. This method tends to eliminate unwanted

material sagging that occurs in the collection of longer

samples, while making it possible to measure the sample's

diameter away from an end distorted by the cutting. It

has been demonstrated that material emerging from a capillary

required a length equivalent to several die diameters before
total swelling is complete.6 From this viewpoint it is

necessary to obtain measurements away from the end of a


Once the sample has been collected, measurement

may be accomplished by either of two commonly used methods.

A photographic technique in which the samples are measured

by comparison to a reference standard generally affords

good accuracy but requires relatively longer times to accom-

plish at greater expense.12'17'18 A faster, more convenient

method better suited to handling the large number of samples

required in a study of this nature is to measure the samples

by hand with a spring-adjustable micrometer.15'19'20 A

study directly comparing these two techniques found them to
agree to within an average of 2 percent,21 certainly within

the normal variation of the samples themselves.

Results obtained by previous investigators concerning

the annealing of collected extrudates to the stress-free

state are both puzzling and contradictory. The general

process employed heats the sample to a point approaching

the glass transition temperature of the material and allows

it to remain in that state until it attains maximum swell.

Two possible annealing environments have been utilized in

previous studies, namely gas, generally air or nitrogen,

and liquid, usually a silicon oil with its density chosen
to minimize sagging effects. Nakajima and Shida21 used a

nitrogen environment to obtain significant swelling of

high density polyethylene (HDPE) samples. Graessley,

et al.,15 used an air environment which produced no additional

swelling of polystyrene (PS) samples, but small amounts

of swell were obtained for the same samples when using a

silicon oil bath although they were too small to be reported.
Mendelson, et al., annealed HDPE samples in silicon oil

and noticed additional swelling of up to 100 percent. Others

report similar findings.11'12,20 Based on these limited

data it would appear that polyethylenes exhibit measurable

annealing effects while PS swells very little, regardless

of the medium used. It is important to point out that

spurious surface tension effects may appear if annealing is

continued for a longer-than-necessary period of time,

although these types of effects are easily separated from

desired behavior due to the large difference in time scales
of the two processes. While the semi-empirical elastic

theories used to predict JS magnitude assume that this

stress-free condition is met, it is doubtful that a material

undergoing large extrusion rates in an industrial process

meets this particular criterion. Hence, the applicability

of these methods is questionable.

There is general agreement among current experimen-

talists that JS magnitude is an increasing function of

deformation rate regardless of the material tested or system

geometry utilized in the determination. It is an accepted

fact that the variables which tend to increase or decrease

the viscosity of the material have an analogous effect on

JS magnitude. Factors pertinent to this study include

variations in molecular weight and temperature. However,

some materials, such as polyvinylchloride, exhibit strong

chain/chain interactions and often do not obey these typical

findings. Generalizations to such systems should be made

with caution.

The effects of varying the geometry of the extrusion

system, namely the capillary entrance angle and L/D ratio,

should be mentioned briefly here. In capillary rheometry

it is common to see the die entry angle vary from a flat or

180 degree configuration to as small as 30 degrees. At

first glance this entrance geometry might appear to have a

far-reaching effect on JS magnitude, but it has been found
that it is of little if any consequence. Apparently this

difference in geometries results only in a slightly modified

entrance correction and a subsequent increase in the maximum

throughput attained before the appearance of melt fracture22

which has been defined previously as the gross distortion
of the emerging extrudate.23 Therefore the entrance geometry

will have no effect on JS magnitude and its consequence

on the actual extrusion force may be accounted for with the

entrance correction.

The aforementioned L/D effect is characterized by

a noticeable decrease of JS magnitude up to an experimentally

determined value where swelling becomes a constant regardless

of capillary length. For PS and HDPE the L/D ratio required

for this condition has been reported to be from 10 to
25,131520 while for other materials, such as polyesters,

the JS magnitude may be a function of die length well past

L/D ratios of 100.24 Obviously this L/D effect is an unknown

function of the material, the temperature, and the throughput,

and until theories capable of predicting this effect are

developed, experimental determination of the L/D behavior

will be a necessity. Studies which base experimental

conclusions on one large L/D ratio to obtain JS magnitude

measurements are almost as suspect as those doing so with

one relatively short L/D ratio capillary. To assure greatest

accuracy, a complete series of capillaries of various lengths

with identical diameters are needed to make the required

extrapolation to long L/D ratio JS magnitude.

I.2.B Theoretical

Although JS was observed more than 100 years ago,5

a theory capable of accurate quantitative predictions of JS

magnitude for any of the variety of flow conditions and

geometries has not as yet been advanced. This dilemma is

due in part to the unknown but apparently complex deformation

histories which do not lend themselves to simplistic modeling

and are not easily measured. More specifically, however,

there is a general lack of understanding of the basic process

itself which implies that an exact solution to the problem

would not be forthcoming even if detailed knowledge of the

flow fields were available.1

The apparent nonexistence of early theoretical

work in this area may be attributed to the insignificant

amount of polymer fabrication taking place prior to World

War II. With the onset of increased importance of the

polymer and plastics industry following the war came an

increase in research. The elasticity concept developed

in the rubber industry was extended to melts. A measure

of this property was the material's time constant or relaxa-

tion time obtained from early rotational-type viscometers.

Attempts made to correlate this parameter to observed

elastic behavior, specifically JS, were apparently useful

only for qualitative predictions with respect to shear rate

and temperature. Some limited success was attained in

relating the swelling of polymer melts to blow molding of

bottles. These earliest attempts resulted only in empirical

correlations and assorted qualitative predictions.1

In the last 15 years there have been numerous attempts

to predict JS based on assorted theological approaches to

the problem. It is these most recent advancements that will

be discussed in detail here. These theories assume isothermal

and incompressible flow conditions. Additionally they assume

well-developed SSF at the capillary exit, negligible surface

tension and gravitational effects, and negligible viscous

drag at the extrudate/air interface. Although these assump-

tions prevent the application of results to every commercial

process of interest, the real-world system is too complex

to model at this time. Hopefully these required assumptions

will be justified by the results.

It is convenient at this point to review some basic

concepts and definitions that will be helpful in the discus-

sion of previous theoretical approaches.

By definition, the total stress tensor, T, is related

to the deviatoric or flow-induced stress tensor T, by

T -pi + T (I-1)

where I is the unit tensor and p is the isotropic pressure.

The generalized velocity field components in SSF may

be written as

(vl v2, v3) = (yx2, 0, 0) (1-2)

where the x.'s describe the system coordinates, the v.'s are
1 1
the velocity components in the flow, gradient, and neutral

directions respectively, and ? denotes the velocity gradient

or shear rate.

The deformation tensor, d, a measure of the velocity

gradients for a given flow situation, arises naturally from

continuum mechanics concepts. The components of d describe

the rate of separation of material particles in the deforming

fluid. It is this separation which results in the development

of internal stresses. By definition

d .(Vv + Vyt) (1-3)

where the del operator denotes the gradient of the vector

quantity y, and the superscript t represents the matrix

transpose operation.

The nonzero components of the total stress tensor

for the SSF in equation (1-2) are calculated to be

T11 T12 0
T = 112 T22 0. (1-4)

O 0 T3 3
As previously demonstrated, any velocity field

giving rise to these same nonzero components of the stress

tensor is an example of an SSF. Further, it may be shown

that quantities measured in any of these equivalent SSF

situations may be compared directly at equal gradients.10

For this particular flow field, the deformation history of

a given material in a well-defined system is a unique func-

tion of the deformation gradient only. From a rheologist's

point of view, estimation of the stress tensor is a necessity

in order to make predictions of a material's response to a

given flow situation. To accomplish this task, it is

advantageous to introduce three new quantities, namely the

viscosity function, and the primary and secondary normal

stress difference (SNSD) functions. The viscosity or

momentum transport coefficient, a measure of a material's

resistance to a shear-type flow, is defined as the ratio of

the shear stress, T12' to the shear rate, y
n (1-5)
It is beneficial to deal with normal stress functions that

are independent of the isotropic pressure because of the

difficulties that arise when attempting to relate this

quantity to any measurable variable. Hence, by definition

N T11 T22 (1-6)

S= T22 T33 (-7)

where the N.'s are the PNSD and SNSD respectively.
When a Newtonian fluid undergoes SSF, the viscosity

function is a constant and both normal stress differences


In the case of a viscoelastic material undergoing

SSF, the viscosity function is dependent upon the deformation

magnitude, and the normal stress differences are generally

nonzero. The non-Newtonian viscosity is constant at very

small deformations in the so-called zero shear region, and

decreases as the rate of shear increases until the power

law or high shear region is reached. Such material behavior

is termed shear thinning. The PNSD is usually of the same

order of magnitude as the shear stress at moderate shear

rates, while the SNSD is found to be opposite in sign and

approximately one-fifth to one-half the magnitude for polymer
melts and solutions alike. The basis of the theories

developed for the estimation of JS magnitude center around

the prediction of the PNSD in SSF.

One of the first predictive attempts, advanced by

Metzner, et al.,28 was moderately successful for the predic-

tion of stress levels in laminar jets of various polymer

solutions. Although the original intent of the study was

the development of a technique capable of high shear rate

estimation of the PNSD from JS magnitude data, the reverse

calculation is also possible. First, JS magnitude, B, is

defined as the ratio of the recovered jet diameter, D., to

the diameter of the die, D,
B = (1-8)

Next, a momentum balance is written between the die exit,

where flow is considered to be fully developed SSF, and the

recovered extrudate

B 8 rD/2 T11(r,L) -1/2
B = ( r [u(r,L) ]dr} (1-9)
(VD) o P

Here u(r,L) and T l(r,L) denote the radially dependent velocity

and total stress, respectively, at the die exit, and p and

V represent the fluid density and average velocity. If the

SNSD at the wall is assumed to vanish at the tube exit, and

all stresses are referred to zero centerline pressure at the

tube exit, then

T11(r,L) = NI(r,L). (I-10)

Combining equations (1-9) and (1-10) lends to

B 8 fD/2 N1(r,L) -1/2
B (VD) r [u(r,L) ]dr (I-11)
(VD) P

Inasmuch as the velocity profile at the tube exit has been

assumed to be well-developed and laminar, the integral

in equation (I-11) is easily evaluated. The velocity profile

and the PNSD can be calculated from material functions

predicted from an appropriate constitutive equation whose

parameters have been fit with theological data. Hence the

prediction of JS magnitude is possible.

In the case of a Newtonian fluid undergoing laminar

tube flow, equation (I-11) predicts an "expansion" of

B = 0.866 (1-12)

implying a slight contraction.8 Although this approximate

value has been observed at large Reynolds numbers, experi-

mental studies at low flow rates indicate a contradictory
expansion of about 10 percent.2930

Equation (I-11) may be rearranged in order to directly

calculate N1(R,L).28 Experimental results on polymer solu-

tions indicate that stress differences obtained in this

manner compare favorably with PNSD's obtained by extrapolating

cone/plate rheometer data to high shear rates.1528 Unfor-

tunately, similar predictions of PNSD's for polymer melts
result in errors of up to seven orders of magnitude. The

most obvious reason for this .inadequacy is the assumption

of well-developed flow up to the capillary exit, and the

assumption that the SNSD vanishes.

A more recent study utilizing mass, momentum and
energy balances has been advanced by Bird, et al. These

balances are also written between the die exit region and

the fully recovered jet, and the momentum and energy balances

include the viscous dissipation terms not considered by
Metzner, et al. With this approach it has been possible

to acquire additional insight into the mechanisms of JS,

including the strong dependence of JS magnitude on the PNSD,

and the much weaker effect of the SNSD. Unfortunately, this

theoretical technique requires assumptions concerning the

velocity profile inside the die, its rearrangement near the

tube exit, and the subsequent free jet expansion mechanics

needed to describe the shape of the extruded material. As

in the case of the pure momentum balance approach, predic-

tions at large flow rates compare well with experiment, while

poor agreement is attained with analysis of the slow flows

important in polymer melt processing.

A general class of theories which consider the

material to exhibit solid-like responses are the so-called

elastic solid (ES) theories. Semi-empirical in nature, they

relate JS magnitude to a new quantity, the recoverable shear

(SR), which itself is related to a ratio of the elastic

to viscous stresses. Hence, if this stress ratio is known

for a given flow situation, JS may be estimated from any of

the number of relationships described later in this section.

The quantity SR may be thought of as the stored elastic

energy present in a constrained material undergoing a shear-

type deformation. The ES theories assume that this energy

is fully recovered when the material becomes unconstrained

and relaxes outside the tube exit.

Using either Hooke's Law for a purely elastic solid32
or the theory of rubber elasticity,33 it is possible to relate

SR to the PNSD, N1, and shear stress, T12
S (1-13)
R '112
An alternate approach more applicable to polymer

melt systems has been derived for viscoelastic fluids by

using either shear compliance concepts19'34 or modified,

single time constant constitutive theory3536
SR 1 (I-14)
Thus equation (1-14) may be used in conjunction with any of

the variety of ES theories to predict JS magnitude at the

relevant system conditions if the PNSD and shear stress

functions are known. Many theories of this type have been

developed. They generally utilize an elastic energy or

force balance to relate the strained material inside the

die to the unconstrained and relaxed material outside the


The earliest ES theory, developed by Spencer and
Dillon,37 related the JS magnitude, B, to the recoverable

shear, SR, using Hooke's Law in shear as applied to an

elastic solid
S=2 1
S B (1-15)
Previous studies have discussed critically the assumptions

leading to equation (1-15) and have concluded that these

mechanisms would result in telescopic-type deformations of

the unconstrained free jet rather than the desired radial


The theory advanced by Nakajima and Shida21 calculated

the strain required to stretch the fully recovered extrudate

lengthwise until its diameter necked down to that of the die.

This strain was in turn equated to the recoverable shear
S 2 1
R 4. (1-16)

Bagley and Duffy used a one-constant stored

energy function of a rubber-like solid to obtain
4 1 1)/2
SR = (B 2 (1-17)

Additionally they used an elastic energy balance

to obtain an expression equivalent to Graessley, et al.,1
4 2 1/2
S (B4 + 2 3)2 (I-18)
R 2
15 19
Graessley, et al., and Vlachopoulos, et al.,

extended this work using rubber elasticity theory together

with a semi-empirical equation of state to predict the shear

compliance, hence recoverable shear, at the die wall. There-

fore, it was possible to relate the recoverable shear at the

wall to the average value of the recoverable shear that

appears in the ES theories

(SR)wall = r SR. (1-19)
Mendelson, et al.,1 assumed slightly different

constants for the stored energy function of rubber-like

solids, and obtained

SR = (B21n B)1/2. (1-20)
Mori and Funatsu8 assumed a form of purely elastic

deformation in shear to obtain

S = [B(1 + dln B 2 1] (-21)
R dln 12

although their derivation apparently contains either typo-

graphical or algebraic errors.

A group of theories developed along somewhat different

lines utilized direct application of assorted constitutive
theories to predict JS magnitude directly. Tanner35 related
theories to predict JS magnitude directly. Tanner related

swelling of a BKBZ fluid3 undergoing SSF in a cylindrical

die to the recoverable shear

(S [2(B6 1)/2 (1-22)
(SR)wall 1
White and Roman utilized an integral-type constitu-

tive equation with appropriate memory function to relate JS

magnitude to assorted empirical parameters fit directly from

experimental data specific to one system.

Cogswell39 utilized concepts developed earlier by

Rigbi40 and apparently by McIntosh41 as discussed by
McKelvey2 to relate JS magnitude to SR using the recovery

of an elastic fluid after a shear-type deformation
2 1 3/2 1 1/2
B = { SR(1 + )2 2 (1-23)
13 43 44
Studies by Bagley, et al., Chapoy and Nakajima

are empirical attempts to predict JS magnitude as a function

of system variables including the L/D effect. These theories

developed models containing adjustable parameters which were

fit to the experimental data. Unfortunately, parameters fit

to a given extrusion system cannot be rigorously applied to

other materials and flow situations for predictive purposes.

A recently developed technique allows for prediction

of elastic behavior solely from a knowledge of viscosity
45 46
function data.4546 Using a memory integral expansion as
suggested by earlier researchers,47 several interesting

relationships involving theological functions of interest

may be derived, including a result allowing the estimation

of the PNSD from a knowledge of the viscosity function.

Hence, the prediction of the recoverable shear and therefore

JS magnitude from small amounts of easily obtained, funda-

mental data is possible. Apparently this technique has

been tested for several systems with reasonable success.

A numerical technique used to solve the equations

of motion for an unconstrained Newtonian-jet expansion

following a viscometric tube flow has been developed
recently.4 9 This method utilizes a finite element approach

to solve the subsequent coupled differential equations. The

solution to this system apparently verifies experimental

findings with respect to the low flow JS magnitude of 1.13.30

Additionally, entrance pressure drop measurements that have

been obtained by an alternate theoretical approach5 are

predicted quite well with the numerical solution. Unfor-

tunately, extension of this particular solution method to

the viscoelastic fluid is considerably more complex than

the Newtonian case.

It is generally accepted experimentally that low

Reynolds number flow of a Newtonian fluid emerging from die

where a SSF took place results in a JS magnitude of 1.10
29 30
to 1.13.29,30 Recent data indicate that this same expansion

is observed in the slow flows of viscoelastic materi-

als.11'12151819 Some researchers suggest the importance

of adjusting experimental measurements by small amounts so

that their experimental measurements will reflect the

contributions due to elastic response only. These corrections

have been accomplished by either dividing the experimentally

derived JS magnitude by 1.115 or by subtracting 0.135 from

it. Most researchers ignore this small correction.


Two recent papers219 review many of the current

ES theories discussed above and have compared the various

predictions with available experimental data. The major

conclusions that may be drawn from all of these comparisons

is that no theory is able to make consistently accurate

predictions of JS magnitude.

1.3 Summary-of Dissertation

In Chapter II the experimental apparatus and tech-

niques utilized to determine the required material functions

and the methods employed to afford accurate and precise

measurement of JS magnitude are discussed. Viscosity data

in SSF is obtained with an Instron Capillary Rheometer, and

extrudate samples collected in the process are measured

with a micrometer to determine JS magnitude at the given

flow conditions. Additional viscosity measurements and

PNSD data are acquired with a Rheometrics Mechanical

Spectrometer operated in SSF only. Finally, a Rheometrics

Viscoelastic Tester is used to estimate the elastic and

viscous moduli in the ERD mode. Data on each of two

different commercial grade PS's are taken at temperatures

of 473,503 and 533 degrees K. over as wide a range of

deformation rates as possible. The effect of annealing

extrudate samples to obtain maximum swelling is briefly


Chapter III introduces the four parameter rate-type

constitutive model used to predict material functions. The

advantages of this particular model are discussed, and the

equations' solutions in SSF are presented and compared with

desired functional dependence. The graphical techniques

used to fit required constitutive parameters are advanced,

and temperature superposition predictions are derived.

In Chapter IV experimental results are compared

with material function predictions, and JS magnitude values

calculated from some of the pertinent large L/D ratio ES

theories are tested against the measurements described in

Chapter II. Constitutive parameters are fit to both

materials at 503 degrees K. using a combination of viscosity

and PNSD data. These parameters are used to predict (1)

all measured material functions and recoverable shear at

that temperature as an internal test of the model, and

(2) the material functions and recoverable shear at the

remaining two temperatures using temperature-shifted para-

meters. A theoretical value of JS magnitude may then be

calculated from the recoverable shear. Additionally, all

material functions should be temperature superimposable as

described in Chapter III.

Chapter V critically discusses shortcomings of the

current study, the existing theories and experimental

techniques, and suggests future studies to elaborate on

the current treatment.


The purpose of experimentation in the current research

is twofold. First, any new approach to be used for JS magni-

tude prediction should be tested rigorously by direct

comparison with experimentally measured values. Secondly,

the theories developed in this research necessitate experi-

mental determination of material function parameters used

in conjunction with the predictions.

The three specific commercial testing devices used

for the purpose of theological characterizations are (1) the

Instron Capillary Rheometer (ICR), a capillary viscometer,

(2) the Rheometrics Mechanical Spectrometer (RM5), a

cone/plate device, and (3) the Rheometrics- Viscoelastic

Tester (RVE), and ERD rheometer. Additionally, the ICR is

used to obtain JS magnitude samples simultaneously with

viscosity data.

This chapter outlines the theoretical analysis and

operating techniques of these three devices as related to

the current research. The methods required for extrudate

sample collection, measurement, and annealing are developed

and described in detail.

II.1 Capillary Viscometer Measurements

The ICR is a constant shear rate viscometer which

is operated in an SSF mode with a capability of extruding

polymer melts at elevated temperatures (roughly 320 to 620

degrees K.). Deformation magnitudes of up to three decades

are attainable for a given capillary diameter. A schematic

of the three-region extrusion system is presented in Figure

II-1. The sample is loaded into the heated reservoir

(barrel) and forced out through an interchangeable capillary

by a plunger driven with the moving crosshead. The force,

F, required to extrude the material at a chosen crosshead

speed, VXH, is measured by a load cell and displayed against

time on an x-y chart recorder. Chapter II.1.A describes

the available methods of correcting the extrusion force for

losses in the reservoir and die entry region. This corrected

value is then directly related to the shear stress at the

capillary wall by a momentum balance. The standard method

used to relate the apparent shear rate to the true wall

shear rate is also presented. In Chapter II.l.B, techniques

developed to process extrudate samples in order to obtain

meaningful JS magnitude data are discussed in detail.

II.1.A Viscosity

The viscosity function may be calculated from equation

(1-5) if the shear stress and true shear rate at the wall

are known. This section will discuss the origins of previ-

ously summarized relationships 242,51 which are used to

calculate these two quantities given raw data obtained with



~_~ ~~ __ I



an ICR. This is easily accomplished by simplifying the

axial (z) and radial (r) components of the equations of
motion written in cylindrical coordinates for the SSF

region shown in Figure II-l. The usual assumptions are


(1) The flow is steady, laminar, and isothermal.

The axial component of velocity is a function

of r only.

The radial and tangential velocities are zero.

The fluid is incompressible.

(2) The Bagley correction applies to entrance losses.

The tube is long enough that well-developed flow

is attained, and this flow is maintained up to

the capillary exit.

(3) Gravity and surface tension effects are negligible

(4) The no-slip boundary condition applies at the

tube wall.

It is convenient to denote the flow, gradient, and

neutral coordinate directions of this particular system,

namely z, r, and 6, respectively, as

(z,r,6) = (1,2,3). (II-1)

Simplifying the z component of the equations of
motion,52 and further assuming that the order of differentia-

tion with respect to z and r may be interchanged, yields

@ C. (11-2)
Hence, the pressure gradient in the axial direction
is a constant (C) and independent of r.

Applying the above simplifications to the r component

of the equations of motion52 results in the following shear

stress distribution:
Cr (11-3)
12 2
Determination of the constant, C, and subsequent

evaluation of equation (11-3) at the wall conditions yields

(AP)D (II-4)
12 wall 4 L

where D and L are capillary dimensions and AP is the pressure

drop across the capillary. It is convenient to drop the wall

subscript from the shear stress as long as it is remembered

that this quantity has been measured at that particular


If the reference or zero pressure is equated to that

at the capillary exit, the pressure drop term, AP, becomes

the pressure at the capillary entrance. The pressure is

more conveniently expressed as a force term which is easily

accomplished because the cross-sectional areas of the

capillary and reservoir are known. In order to relate this

force to the experimentally determined extrusion force, F,

it is necessary to compensate for the loss due to friction

in the reservoir and for the loss resulting from the entrance

flow into the capillary. These corrections, when subtracted

from the measured extrusion force, yield the desired quantity,

namely the force required to extrude the material through

the capillary in an SSF.

This study measured the so-called barrel friction

forces directly by extruding samples at each flow rate with

no capillary attached.5 This force was subtracted from

extrusion forces measured with a capillary in place. This

correction is generally quite small.

In order to correct for entrance losses, a Bagley-
type method was employed. Simply stated, at each extrusion

rate a series of constant diameter capillaries of various

lengths was utilized to measure the extrusion force, F.

After correction for the above-mentioned barrel friction

losses, a plot of this corrected force versus L/D ratio was

extrapolated to zero L/D. The force at this point is the

Bagley correction which is subtracted from the extrusion

force measured with each capillary at the given crosshead

speed. In mathematical terms, the corrected extrusion

force, Fc, equals the ICR extrusion force, F, minus the

contributions due to barrel friction, Fbf, and entrance or

Bagley losses, Fb:

F = F F Fb. (11-5)

Equation (11-4) becomes
T c (II-6)
12 4A L
where A is the reservoir cross-sectional area.
The calculation of the viscosity function from shear

stress data required that the shear rate at the capillary

wall be evaluated. Assuming as a first approximation that

a Newtonian velocity profile exists in the capillary, it

is possible to calculate the so-called apparent shear rate,

ya. Differentiating the parabolic velocity distribution

with respect to r, and evaluating the resulting expression

at the wall conditions leads to

a (11-7)

where V is the average velocity in the capillary. From the

equation of continuity, it is possible to express ya in terms

of measurable ICR system parameters, namely the crosshead

speed, VXH, and the reservoir diameter, D as

8VH D2
Ya r (11-8)
Unfortunately, the velocity profile of a material with a

shear rate dependent viscosity deviates somewhat from the

Newtonian case. However, the true wall shear rate, y, may

be related to the apparent value in equation (11-8) as

originally suggested by Rabinowitsch7

S= y 3n+ (11-9)
a 4n

where n is the local slope of a double logarithmic plot of

measured wall shear stress versus apparent wall shear rate.

Although the necessary barrel friction corrections

are conveniently made by hand, the Bagley and Rabinowitsch

corrections are more easily accomplished with a digital

computer. Hence, the data in this current study were

processed and corrected with the FORTRAN language computer

program entitled BAGLEY/RM CORRECTION. A listing of the

program appears in Appendix A. Required inputs, namely

system and capillary dimensions, and extrusion forces,

corrected for barrel friction together with the corresponding

crosshead speeds, are explained in considerable detail

with comment statements. The important program outputs are

viscosity, wall shear stress, and true wall shear rate.

Tabulated program outputs appear in Appendix B.

The ICR used in this study was actually an Instron

Universal Testing Instrument, Floor Model Type TT, with an

Instron RMCR anvil attachment added to hold the capillary

extrusion system. This consists of a reservoir which is

0.375 inches in diameter, surrounded by a heating block

containing insulation with imbedded resistance heaters.

The heaters are connected to an Instron Temperature

Programmer which maintains the selected temperature setpoint

with a feedback three-action controller. Although designed

to control the setpoint to within + 0.5 degree K. and the

axial temperature gradient to + 1.0 degree K., the actual

setpoint and longitudinal tolerances achieved were + 1.0

degree K. and + 2.0 degrees K. respectively. This variation

is considered negligible in this study.

The capillary series required for the determination

of viscosity and Bagley corrections was machined from a

stainless steel rod and bored out so that 0.125 inch OD

seamless tubing having an 0.027 inch ID could be inserted

and welded into it. This particular method of manufacture

is preferred over drilling the desired hole diameter directly

into the capillary blank because of unwanted grooves that

occur on the interior surface as a result of the drilling.

The capillary blanks were fabricated to prescribed Instron'

tolerances with entry and exit planes faced to a flat or

180 degree configuration. The interior dimensions are

described in Table II-1.



Capillary Length-L L
Number inch D

1 0.2312 8.56
2 0.3245 12.02
3 0.4837 17.91
4 0.6997 25.91
5 0.9422 34.90

Capillary Diameter, D, is 0.0270 inch.

The crosshead speed could be varied from two to

0.0002 inches per minute by selecting the desired value on

the front control panel. The crosshead is driven by an

asynchronous motor which maintains the constant rate regard-

less of the force encountered. The plunger is machined to

close tolerances and has provisions for Teflon "o" rings

to insure a snug fit in the barrel to prevent sample leakage.

The plunger is driven by an extension attached to a load

cell which is mounted to the moving crosshead. Several load

cells are available depending on the range of forces to be

measured. For the current study, a load cell range of zero

to 1000 pounds force proved sufficient for all runs. A

load cell is essentially a balancing resistor bridge, and

its output is recorded on a moving chart. The recorder

speed may be varied in order to facilitate decisions regarding

attainment of a steady force trace, and hence steady state

capillary flow during extrusion.

A scenario of an ICR run is outlined below. The

reader requiring more detailed knowledge is referred to

the Instron- manual.53

(1) The desired extrusion temperature is chosen

and this setpoint is selected on the Temperature

Programmer. The block is allowed sufficient

warm up time before adjustment of the various

heaters around the reservoir is made. The

three-action controller settings are adjusted

to return the system quickly to the setpoint

when an upset is created in the system such as

occurs when a sample is loaded into the barrel.

(2) The load cell is balanced and calibrated in

tension using the appropriate attachments and

weights as manufactured by Instron.

(3) The sample, usually in pellet form, is poured

into the top of the barrel and tightly packed

with a tamping rod to remove all pockets of air.

(4) The run begins with the first crosshead speed

and is continued until a constant force trace

is obtained. As soon as this steady flow condi-

tion is attained, the force is recorded, the

next crosshead speed is selected, and the

process is repeated. The general order of

crosshead speeds is chosen such that one speed

is repeated several times during the course of a

run to test for thermal degradation of the sample.

After all crosshead speeds are completed, the

remaining material is forced out of the reser-

voir. The barrel is then cleaned by pushing a

small cloth patch through it with the brass rod

supplied with the machine. Generally, all

desired speeds could be completed with a single

load of sample in the barrel.

Some specific problems encountered in a run resulting

in unreliable data are air in the sample and thermal degrada-

tion of the sample in the barrel. Trapped air occurs as a

direct result of poor sample packing and appears as loud

popping sounds in the emerging extrudate. Detection of air

resulted in the immediate discarding of the run. Thermal

degradation of a heated sample in capillary rheometry would

appear as a marked decrease in the required extrusion force

with time for a given crosshead speed. By utilizing a check-

point speed, this reduction in extrusion force is easily


It should be noted here that the selection of

appropriate temperatures and capillary dimensions plays an

important role in a study of this nature. Temperatures need

to be high enough for the material to be well above its

glass transition temperature so that it is well behaved,

but not so high that thermal degradation occurs. Capillary

dimensions need to be chosen in conjunction with the tempera-

ture and the appropriate viscosity of the material. Diameters

small enough to yield the high shear rates observed in

commercial processes are necessary, but it must be remembered

that the resistance to flow also increases as the diameter

decreases. This larger flow resistance increases the

possibility of melt fracture, an undesirable flow condition.

The best trade-off of temperature and capillary dimension

considerations generally results in (1) a temperature low

enough so that thermal degradation is negligible, (2) the

onset of melt fracture occurs only at the lowest of tempera-

tures at the highest of extrusion rates and (3) the low end

of the range of commercial shear rates of interest is


II.l.B Jet Swell

The experimental determination of JS magnitude may

be separated into four categories: (1) collection, (2)

annealing, (3) measurement and (4) density correction.

Samples are gathered concurrently with ICR viscosity data.

The primary objective in sample collection is to

avoid errors due to the extrudate's sagging under its own

weight, thus yielding inconsistent JS magnitude data. The

current techniques have evolved naturally from previous
experimental studies.

Upon reaching a steady flow condition, the extrudate

is cut as close to the capillary exit as possible and

discarded. Approximately one inch of new material is

extruded and then cut near the capillary. (A sharp pair of

ordinary scissors is sufficient for this purpose.) Too

long a sample results in undesirable elongation, while too

short a sample may result in a measurement taken on a surface

that has been distorted by the cutting.

Annealing in the current study consisted of taping

the samples to a support and placing the entire apparatus

in an air-environment oven for 15 to 20 minutes at tempera-

tures of approximately 400 to 450 degrees K. The samples

were measured both before and after the treatment by the

technique described in the following paragraph. The samples

were placed in the oven both horizontally and vertically to

characterize the possible sagging effect previously described
by other investigators.141521

The reported sample diameter is measured approxi-

mately 0.125 to 0.500 inch from the leading end of the sample.

Measurements to tolerances of 0.0002 inch are made with a

spring-adjustable micrometer so as to apply the same force

to each sample. Two measurements are taken for each sample,

namely the largest and smallest diameters at a given cross

section. Current published data suggest that the JS

magnitudes of samples may vary approximately 5 to 20

percent.1115'8-21 Generally, two to five samples at

each flow condition are gathered, measured, and averaged

to calculate the reported JS magnitude value.

Finally, the averaged measurement is corrected back

to the extrusion temperatures, since it was measured at

room conditions. As suggested by previous researchers, it

is possible to calculate the corrected magnitude, B, from

density measurements assuming that the extrudate expands

equally in all directionsl3,14
Pm 1/3
B = B (-) (II-10)

where B denotes the measured JS magnitude, and p and p

are the material densities at the extrusion temperature,

and the temperature at which measurements were made,


Density measurements are easily made in the ICR

barrel at the desired temperature. First, it is necessary

to plug the bottom of the reservoir with a capillary blank.

Then, a known sample mass may be placed into the barrel and

the crosshead moved downward until the plunger contacts the

sample. The volume and hence the density of the sample may

then be calculated.

11.2 Cone/Plate Measurements

The RM5 cone/plate rheometer is useful for rheolog-

ical characterizations of polymer melts when either viscosity

measurements at low deformations or elastic material func-

tions are required. A schematic of the instrument is presented

in Figure 11-2. The system is contained in an environmental

chamber capable of controlling the temperature over the

range of approximately 150 to 570 degrees K. Although five

decades of shear rates are attainable with this device, it

is generally impractical to operate at the large deformation

levels normally encountered in industrial processes of

interest because of difficulties in maintaining the sample

in the gap between the cone and the plate. While it is


possible to operate the instrument in either a dynamic

oscillatoryy) or a steady SSF, the current study requires

the latter characterization only.

The development of the necessary working equations

presented here is also discussed elsewhere.2,52 It is

convenient to define the flow, gradient, and neutral

coordinate directions for the required spherical coordinates,

0, 0, and r, in a manner analogous to equation (II-1)

(p, 0, r) = (1, 2, 3). (11-11)

The velocity field may be represented as

(l' v2' v ) = 2 ,' 0, 0] (11-12)

where w is the angular velocity and is the cone angle.

The coordinate 0 is measured from the cone axis downward,

perpendicular to the plane of the plate.

Calculation of the shear rate, y, for a small cone

angle, , yields

Y =


This result indicates that the shear rate, and thus the

stress tensor, are independent of position in the gap region.

In order to relate the shear stress to the measured

torque, J, it is necessary to integrate T12 over the cone

surface, again for small cone angles, to obtain
T12 =-3 (II-14)

where R is the plate radius. The viscosity function, n,

is then calculated using equation (1-5).

The procedure required to obtain the PNSD is some-

what more complicated. From the r component of the equations

of motion with the gravitational and inertial terms

neglected, it is possible to obtain the radial distribution

for the total normal stress in the flow direction, namely

T11(r), as

Tll(r) = T 1(R) + (N + 2N2)ln(r/R) (11-15)

where T11(R) represents the total normal stress in the flow

direction evaluated at the air/sample interface where r = R.

By integrating T11(r) over the cone surface, again for small

cone angles, this normal stress distribution may be related

to the generated thrust, F
F = T 11(r) 27rdr. (11-16)
z 11

Combining equations (11-15) and (11-16), performing the

necessary operations, and referring all pressures to the

radial component of total normal stress evaluated at r = R,

it is possible to show
N -z. (11-17)
1 2T

Here Fz represents the experimentally measured force applied

by the RM5 to maintain the spacing between the cone and the


The RM5 used in connection with this research was the

demonstration model belonging to Rheometrics, Inc., located

in Union, New Jersey. The plate diameter was 0.984 inch, and

the cone angle was 0.1 radian. These particular dimensions

were recommended by the Rheometrics- staff as being suitable

system parameters for the current polymer melt characteriza-

tions. As mentioned previously, the plate surface area

should be chosen such that the forces generated by the sample

are of the appropriate magnitude to be easily measured by the

force transducers. Likewise, the cone angle should be small

enough to justify the assumptions leading to equations (11-13)

through (11-17).

A typical cone used with the RM5 is actually truncated

to avoid the possible damage that could occur with the metal-

to-metal contact of the cone point with the plate. Hence,

an important part of machine operation consists of correctly

setting the gap or distance required between this truncated

cone and the fixed plate. This dimension is chosen such that

an identical, untruncated cone would make slight contact

with the plate if substituted for the truncated cone. A

gap setting of 0.01 inch was necessary for the particular

cone used in this research.

A scenario of an RM5 cone/plate rheometer run in the

steady SSF mode is outlined below. For additional detail,

the reader is referred to the appropriate RheometricsG

(1) The force transducers, temperature controllers,

and rheometer alignment were maintained and

calibrated by Rheometrics personnel.

(2) The desired run temperature is selected and the

cone/plate apparatus is placed inside the

environmental chamber. When thermal equilibrium

is reached, the gap required for the cone in

use is adjusted.

(3) In order to facilitate sample loading, a thin

strip of metal shim material is placed around

the plate by utilizing the built-in clamp

attachment on the instrument's lower spindle.

An excess amount of sample, usually in pellet

form, is placed inside the cylindrical space

formed by the shim and the plate. The entire

apparatus is returned to the chamber and heated

to the run temperature. The chamber is then

reopened, the shim material removed, and the

apparatus is again placed into the oven. When

the run temperature is reached, the correct gap

is set by squeezing the excess material from

between the cone and plate. The chamber is

opened only long enough to trim away this surplus

material (which is discarded) before being

reclosed. This procedure results in the build

up of a fairly large normal force, Fz, in the

system which would interfere with PNSD measure-

ments. Hence, it is necessary to wait until

this force decays to near zero at which time

the run may begin.

(4) Starting with the lowest angular velocity

desired, each rotation speed is allowed to

continue until a steady torque and normal

force are recorded on the two channel strip

chart. Upon reaching this steady flow condition,

the cone rotation is reversed and again contin-

ued until the steady state conditions are

reached. The measured torque reverses signs

with the direction of rotation, thus permitting

a more accurate calculation of this quantity

since the zero baseline of the transducer need

not be estimated. The measured normal force

is not a function of the direction of rotation,

and therefore cannot be calculated with the

same accuracy as torque.

(5) Subsequent rotation speeds are chosen and step

four above is repeated. It would be desirable

to repeat a given angular velocity several times

during the course of a run as a check for thermal

degradation. Unfortunately, this was not

possible because a noticeable amount of material

was lost from the gap region before all desired

angular velocities had been tested. When this

sample loss was observed, the run was halted

and the remaining material in the instrument

was removed and discarded.

Obviously, material loss severely limits the range

and accuracy of measurements taken with the RM5 device. This

loss, coupled with the additional factors of possible thermal

degradation of the sample and the slow relaxation of developed

normal forces created in preparing the sample, restricts the

usefulness of this particular rheometer. Apparently,

this material loss factor will result in greater errors in

the reported viscosity than in the measured PNSD. This

occurs because of the absence of contributions at large

values of radial position which account for a large fraction

of the total measured torque. Finally, the errors associated

with an inexperienced operator are more pronounced with this

instrument than either the ICR or the RVE because of the

critical nature of the tasks involved, namely setting the

gap, trimming the excess sample, and ascertaining the steady

state forces.

11.3 Eccentric Rotating Disk Measurements

The automated RVE rheometer, a descendent of the

earlier Maxwell Orthogonal Rheometer, is useful when charac-

terization in an oscillatory (dynamic) mode is required.

A schematic of the instrument is presented in Figure 11-3.

The RVE device is essentially identical to the RM5 cone/plate

rheometer with these exceptions: (1) two identical plates

are used rather than the cone/plate combination, and (2) the

particular instrument used in this study directly calculated

and outputted the desired material functions, namely the

viscous and elastic moduli, from the measured run data.

Additionally, it is possible to obtain data over the entire

range of angular velocities automatically rather than

choosing each speed separately.

The analysis required to relate the measured system

parameters to the desired result is quite tedious and time-

consuming. Therefore, only the techniques commonly utilized

Top View


Side View

upper plate


lower plate

Figure 11-3. ERD system schematic.

to derive these expressions will be presented here. A more

complex description of the device and the methods required
in the derivation are found elsewhere. 56

The general modeling practice suggests that the

system may be represented with cylindrical coordinates,

denoting the flow, gradient, and neutral directions as


(6, r, z) = (1, 2, 3). (11-18)

The velocity field is simply expressed as

(vl' v2' v3) = (rw, 0, 0) (11-19)

where each sample layer parallel to the rheometer plates

rotates about an imaginary axis connecting the plates'


For convenience, the velocity components are trans-

formed into Cartesian coordinates. Next, it is necessary

to assume an appropriate constitutive model in order to

calculate the stress tensor for this particular flow field.

Calculation of these quantities for the current flow situa-

tion indicates a direct correspondence between the elastic

(G') and viscous (G") moduli as calculated for any oscilla-

tory flow field, leading to the working equations
F h
G' = --- (II-20)
fTR a
F h
G" 2 (11-21)
7TR a
where F and F are measured according to Figure 11-3, R
x y
denotes the plate radius, a the centerline separation, and

h the gap setting. The dynamic viscosity, n', may be

calculated from the definition
n' = (11-22)

The RVE rheometer used in connection with this

research was the demonstration model belonging to Rheometrics-,

Inc. The plate diameter was 0.984 inch and the gap setting

was 0.1 inch. The strain could be varied manually from 2

to 20 percent, although 10 percent generally gave the best

results. The environmental chamber and method of sample

preparation are identical to the RM5 except that the gap

was set automatically. Although the angular velocities

could be present for fully automated processing, this study

utilized part automated and part manual operation so that

a given speed could be rerun in order to check for thermal

degradation of the sample. The rheometer's central pro-

cessing unit output a printed tape containing the internally

calculated values from equations (11-20) and (11-21). A

detailed description of operational procedures is found
in the rheometer manual.

In no case was the sample lost from the gap as with

the RM5. Additionally, an inexperienced operator could

obtain useful data readily without gross errors. The only

problem associated with human error was the trimming of the

sample, but apparently this factor did not influence results

as drastically as in the case of the RM5 cone/plate viscosity




Advance knowledge of the stress tensor and material

functions for a given flow condition is of extreme importance

to the rheologist. Predictions of this nature are usually

accomplished with a constitutive model which relates the

material's state of stress to its deformation history. A

good model should accurately predict observed behavior for

a variety of flow conditions and utilize a minimum number of

parameters that are easily determined. Finally, a model

derived from basic principles should not require large

amounts of experimental measurements to totally characterize

a material theologically. Hopefully, predictions based on

theory may be shifted or generalized to the bulk of applicable

flow fields and conditions.

III.1 Constitutive Model

A constitutive model that satisfies the requirements

of a suitable predictive equation has been developed recently
from a combination of continuum and molecular approaches. 64

Although relatively untested, it appears to accurately describe

commonly observed phenomena for materials in SSF conditions.

These include a shear-thinning viscosity, nonzero normal

stress differences, and transient responses to applied

deformations. Additionally, effects of molecular weight

distributions and temperature dependence arise naturally

from fundamental considerations used in the development.

The evolution of the current theory, applicable to

polydisperse polymer melt systems, began with the development

of the elastic dumbbell (bead-spring) model for dilute

solutions of linear macromolecules. The continuum modifica-

tion suggested by Gordon and Schowalter559 introduced an

alternate, semi-empirical expression for the rate of change

of the molecular end-to-end vector. This led to more

desirable material function forms for dilute, monodisperse

polymer solutions undergoing SSF, including a shear-thinning
viscosity, a positive PNSD, and a negative SNSD.60

This model was extended to a multi-bead/spring or

Rouse-Zimm approach by inclusion of multiple relaxation

times. Considerable improvement in dynamic viscosity predic-
tions was attained with this modification.

Further model revision incorporated a Schultz-Zimm

molecular weight distribution to account for effects
associated with variations in molecular weights. This

introduced a new but independently measurable parameter

requiring knowledge of both weight and number averaged

molecular weights of the macromolecules. This increase in

model complexity was accompanied by a comparable increase

in the accuracy of predictions.

Lastly, this theory was extended to apply to concen-

trated solutions and melt systems using the concept of a

uniform, effective friction coefficient.6364 Additionally,

it was assumed that the solvent viscosity appearing in the

dilute solution theory could be taken as zero and that the

polymer concentration could be replaced by the melt density.

These simplifications, coupled with the previous results, led
to the following set of equations for T the contribution

to thedeviatoric stress tensor from the ith relaxation time

of the jth molecular weight fraction. The four adjustable

constitutive parameters are: no, the zero shear viscosity;

e the weight-averaged relaxation time; a, the parameter

related to the power law exponent; and E, the phenomenalistic

parameter introduced by the continuum modification. These

parameters completely specify the solution to the following

system of equations:

S1 2 2-a
1 + D 2 n p I j=l ... ,N (III-1)
jj j Dt [Z(a)-1] pM 1+j ~ '

T = I + I T1 (III-2)
i j=1

+ v.- (Vv Ed)-T T-(Vv Ed) (111-3)

2 j
M M.( 2
l = j 1[ I (III-5)
wt J

Z(a) = (III-6)
M=l M1


where the constitutive parameter e is subject to the condition

0 5 e < 1. Here, pi and M. represent the density and molecule
weight of the i fraction, respectively.

The solutions to this system of equations is accom-

plished in three steps. First, the flow field model is

substituted into equations (1-3), (III-1), and (III-3), and

the resulting simultaneous equations (either differential or

algebraic) are solved for the individual contributions to the

stress tensor components. Next, the contributions of the

individual relaxation times are summed over all values to

obtain the pertinent components of the stress tensor for each

molecular weight fraction. Finally, these equations are

integrated over all molecular weight species using the well-

known Rouse-Zimm relationship for the differential molecular

fraction, dn

+2 z+ (M (z+2)9 11
dn -= +2 l] ( ) dM (111-7)
w w
where z, a measure of the width of the molecular weight

distribution, is defined by
z + 2 w (
z + 1 M
Here, M and M represent the weight and number averaged
w n
molecular weights, respectively.

This set of relationships will be designated as the

Gordon-Schowalter-Everage (GSE) constitutive model after the

researchers responsible for its development. Although the

solution to this system of equations appears quite formidable,

Chapter III.2 will demonstrate that SSF predictions of

material functions with this theory may be made in a straight-

forward manner.

III.2 Constitutive Predictions for SSF

By substituting a particular velocity field into the

GSE constitutive model, it is possible to calculate the

stress tensor using the techniques outlined in the preceding

section. The two flow fields of interest in this study are

SSF's in (1) the capillary and cone/plate rheometers, and

(2) the ERD rheometer. From the stress tensor it is possible

to calculate the particular theological functions of interest

which are (1) the viscosity, n, and PNSD, N1, for the steady

SSF, and (2) the dynamic viscosity, n', and the elastic or

storage modulus, G', in the oscillatory SSF.

For the steady SSF case, the results are

z+3 N +2
n- (z+2)z+3 z+2 N
S= Z(a)-1]f(z+3) x exp[-(z+2)x] I (111-9)
o 0 j=2

J )2dx
j2 +(2 Cewyx )

N1 21+(z+2)z+ 2 2N
1 wY z+4 N
f x exp[-(z+2)x] Y (III-10)
no [Z(O)-ljr(z+2) 0 j=2

2a+(2C6 yx )

M (III-11)

C = [C (2-E)]1/2. (III-12)

Here F(z) denotes the Gamma function.

Additionally, the SNSD, N2, is found to be

2 1 N

For the oscillatory SSF case, the results are

T' (z+2)z+3 r z+2 N
n = [Z(a)-l](z+3) jx exp[-(z+2)x] 2
o 0 j=2

G (z+2) *-e^ 2
G 2a (z+2)z+2) 2
G w z+4
So0 [Z(a)-llr(z+2)O x
o 0

.( 2 dx
J2a (2a 2 2

exp[-(z+2)xl Y
j 2a+(2 a0 wx2 )




The viscous or loss modulus, G", may be calculated

from a knowledge of n' and the frequency, w

G" = n'w. (III-16)

The GSE model predicts the following desirable

material function characteristics in SSF:

(1) The viscosity is shear-thinning with zero shear

and power law regions.

(2) The PNSD is positive with a limiting slope at

large deformations.

(3) The SNSD is negative and less than one-half the

magnitude of the PNSD, depending on the value

of E.

Additionally, the model predicts the following

experimentally observed similarities in material function


(1) The viscosity/shear rate curve and the dynamic

viscosity/frequency curve are superimposable

according to

n(9) = n'(Cw). (111-17)

(2) The PNSD/shear rate curve and the elastic modulus/

frequency curve are superimposable according to

2 N () = G'(Cw) (III-18)

where C is defined by equation (III-12).

The equations presented in this section are used to

calculate the SSF material functions required in this study.

A listing of the computer program utilized for this purpose,

POLY GSE/SSF, appears in Appendix A.l. The approximations

used to solve the infinite summations are outlined in comment

statements throughout the program. A more detailed description
of these simplifications appears elsewhere. The infinite

integrals are calculated by using a standard 32 point Gauss-

Laguerre quadrature numerical scheme.

III.3 Constitutive Parameter Fitting

It is highly desirable that the constitutive model

accurately portray the material functions of interest for a

variety of flow fields and conditions. It is equally impor-

tant that the adjustable parameters used in the constitutive

model be easily calculated from a minimum amount of theological

data. This latter goal is accomplished with the GSE model

by utilizing fitting techniques which combine both numerical

and graphical schemes, thus permitting relatively fast

estimation of the four unknown parameters. These methods

require the experimental determination of any two independent

theological functions, preferably over the range of deforma-

tion rates where predictions are desired. The current study

has chosen the viscosity and PNSD functions for this task,

primarily because the prediction of JS magnitude centers on

knowledge of these particular quantities. The weight and

number averaged molecular weights must be independently

measured or estimated in advance to calculate the molecular

weight variable z which is used in conjunction with the

polydisperse constitutive theory.

The following scenario outlines the general methods

employed in this study to estimate the four GSE model


(1) If appropriate data are available, the zero

shear viscosity, no, should be estimated from

either fully logarithmic plots of n vs. y,

n' vs. w, or a combination of the two.

(2) The parameter a may be related to the high

shear rate slope, S, of a fully logarithmic

plot of n vs. y. This is accomplished by

utilizing an asymptotic expansion of equation

(III-9) for large deformations to obtain
S (III-19)

(3) It is convenient to introduce the dimensionless
variables n and N1 in order to make the right-

hand sides of equations (III-9) and (III-10)

unique functions of a dimensionless shear rate,

n (111-20)

Sw N1
N1 = N- (III-21)

Y = C w y. (III-22)

It is possible to generate fully logarithmic

plots of these generalized functions with the

previously mentioned computer program POLY

GSE/SSF as listed in Appendix A.1 using the

values of a and z specified above.

(4) The generalized viscosity plot is superimposed

on a plot of the experimentally determined

viscosity data. The vertical shift is used to

calculate the zero shear viscosity, no, unless

this variable was determined in one above, in

which case the vertical shift is fixed by this

value. The horizontal shift specifies the

quantity CO w

(5) The generalized PNSD plot is superimposed on a

plot of the experimentally determined PNSD data.

The horizontal shift is fixed by the value of

Ce obtained in four above. The vertical
shift specifies the quantity C2 w/n o thereby

allowing the calculation 6w and C (hence E) from

the values of n and Ce found above.
O w

These techniques are used to calculate the GSE model

parameters for each of two PS samples at 503 degrees K.

III.4 Temperature Superposition

Prediction of JS magnitude temperature dependence

with the various ES theories requires that the temperature

behavior of the viscosity (or shear stress) and PNSD be well

characterized. A major advantage of the GSE model over

similar constitutive theories is that mechanisms which specify

these particular relationships arise naturally from funda-

mental considerations in the model development.

An intermediate result in the original model deriva-

tion related the weight averaged relaxation time, 9 to the
zero shear viscosity, n

S=- (III-23)
w N pkT(1-c)2a(Z(a)-1)
Here Na is Avogadro's number, k is Boltzmann's constant, T is

the absolute temperature, and KE is an unknown constant. If

the constitutive parameters a and e are assumed to be

temperature independent, equation (III-23) may be written as
6 = K (III-24)
w pT

where K is a constant. Therefore, a knowledge of the GSE

model parameters at a single temperature coupled with

measurements of the zero shear viscosity and material density

at any other temperature of interest permits the estimation

of the relaxation time and material functions at the new


An equivalent method of temperature prediction is

obtained by superimposing the desired material functions to

master curves calculated from the shift factors suggested

by comparing equation (III-24) to equations (III-9), (III-10),

(III-14), and (III-15). The following equations specify the

relationships for the shifting of data between a reference

state where the material functions are known, to a new state

where knowledge of these functions is desired. Using the

subscripts ref and new to denote reference and new states,







The shift factors, aT,

Y)ref = aT()new

)ref = bT(n)new
J)ref = T (Nl)new
1~ref = aT()new

)ref = T(G')new

)ref = bT()new.

bT, and CT, are specified by

(ro /pT)
o new
aT =
(r /pT)ref
(,o ref

b = oref
T (o)new

c^ = ----










This study measured the material functions and

densities of two PS samples at temperatures of 473, 503, and

533 degrees K. The constitutive parameters were fit at 503

degrees K. and the time constants and material functions at

473 and 533 degrees K. were calculated using the shifted


parameters and equations (111-9) to (III-16). Additionally,

all material function data were shifted to master curves at

503 degrees K. by using the superposition theory and shift

factors developed above.


It is important to compare theoretical predictions

with experimentally determined functions whenever possible.

It is also necessary to estimate approximate error magnitudes

associated with measurement and prediction of desired

variables. This discussion of results is conveniently

separated into two sections. First, the accuracy of material

functions as related to experimental measurement and data

correction and reduction is presented, and the parameter

fitting required to ultimately predict the temperature and

shear rate dependence of the recoverable shear is reviewed.

Comparisons of experimentally measured material functions

with those predicted from fitted constitutive parameters

together with the predicted temperature superposition will

be used as guidelines by which the accuracy of the current

constitutive theory may be evaluated. The importance of

the corrections employed with the ICR data and the effects

of thermal degradation of the sample in all rheometers

used in the determination of material functions is considered.

The perceived range of applicability of recoverable shear

predictions is estimated.

Secondly, error estimation of experimentally obtained

values of JS magnitude will be discussed from viewpoints of


sample variation, measurement, annealing, and density correc-

tions, and the determination of the long L/D ratio material

swelling will be reviewed. These long capillary JS magnitude

values are compared with predictions based on the various

ES theories utilizing recoverable shear as calculated from

fitted GSE parameters at the conditions of interest.

IV.1 Material Functions

The discussion in this section focuses on results

related to material function measurements and their'subsequent

prediction. First, errors arising from sources related to

experimental techniques used in the determination of material

functions with the three rheometers are examined. Secondly,

constitutive predictions of theological functions of interest

will be compared with experimentally determined values and

the temperature superposition predictions tested. Finally,

the derived recoverable shear function to be used in conjunc-

tion with the various ES theories to predict JS magnitude

will be presented.

Possible error sources in the capillary rheometer

experiments pertinent to this discussion are measurement of

the extrusion force, and determination of the barrel friction

and Bagley corrections. Careful calibration techniques

insured that errors in the measured extrusion force due to

load cell inaccuracies were no more than 1 percent for a full

scale reading. The force trace for a given run normally

varied less than 5 percent. The barrel friction correction

was found to be an increasing function of crosshead speed,

and averaged 2 percent of the total extrusion force. The

Bagley correction, employed to account for the loss that

occurs when the sample flows into the capillary from the

reservoir, was calculated and applied to the data in the

computer program BAGLEY/RM CORRECTION which is listed in

Appendix A. As expected, the Bagley plots as described in

Chapter II.1 were linear, implying that spurious effects due
to sample compressibility are negligible.65 The correction,

which varied from 3 to 20 percent of the total extrusion

force, was found to be an increasing function of crosshead

speed. Estimation of this term at the lower speeds was

hindered because of data scatter combined with the small

extrusion forces to be corrected. In no case, however,

should this correction introduce errors in the reported

extrusion force of more than 5 percent.

The average variation of calculated wall shear

stress at a given crosshead speed was 10 percent over the

range of capillary L/D's contained in the series. The

maximum viscosity variance that can be attributed to

temperature fluctuations in the equipment was estimated

at 10 percent using the experimentally determined zero

shear viscosities with an Arrhenius-type relationship.

Additionally, no thermal degradation was discernable as

determined from data points repeated over the course of a

run which generally lasted two hours. In view of these facts,

it is reasonable to assume that the ICR viscosity data

presented here are accurate to within 20 percent, a typical

tolerance for measurements of this type.

The force transducers used with the RM5 were accurate

to within 5 percent. The major problems encountered with

this rheometer were the long times required to attain steady

state at a given speed, the difficulty in estimating the

PNSD and viscosity measurements from the variable force

trace, and problems in maintaining the sample in the gap

region at the higher deformation rates. These problems

lead to significant doubts as to the accuracy of viscosity

and normal stress data from the viewpoint of machine operation

No thermal degradation would be expected in the RM5 because

the length of a run (about one hour). is less than that

required for the ICR where no such effects were observed.

The RVE device also utilized force transducers

accurate to within 5 percent. Data sampling and moduli

calculations were automatically accomplished by the rheometer'

computer, and there was no difficulty in maintaining the

sample in the gap. Repetition of selected data points

indicated that no thermal degradation occurred during the 20

to 30 minute runs. Reproducibility of a repeated run indicate

an average variation of about 20 percent.

Measurement of the material density at the extrusion

temperature is required in the current study for temperature

superposition theory calculations and JS magnitude density

corrections. The accuracy of these measurements obtained

with the ICR as described in Chapter II.1.B was apparently

very good. Measurements made on PS samples were reproducible

to within 1 percent. Density measurements obtained on a

polyethylene sample with this technique agreed to within 1
percent with density data published elsewhere. The shift

factor cT, calculated from equation (III-33), varied from

approximately 0.95 to 1.05 over the temperature range used

in this study. Hence, for these PS samples over this small

temperature interval, it could have been neglected. However,

a larger temperature interval or a material with a density

more sensitive to temperature would necessitate its inclusion.

Consequently, the shift factors aT and b are dominated by

the zero shear viscosities, and the JS magnitude density

correction will be small in the current systems.

The four constitutive parameters, no, ew, a, and e,

were fit to each of the PS samples at 503 degress K. by

applying the generalized plot method described in Chapter

III.3 to the ICR viscosity and RM5 PNSD function data.

Although the shape of the entire ICR viscosity data versus

shear rate curve was closely approximated by the calculated

generalized viscosity function, the RM5 PNSD data coincided

with the generalized PNSD plot over the intermediate range

of shear rates only. The zero shear viscosities at all

three experimental temperatures were estimated using a

combination of ICR viscosity and RVE dynamic viscosity data

extrapolated to the low deformation rate range. The tempera-

ture dependent relaxation time, e fit at 503 degrees K.,

was calculated at 473 and 533 degrees K. according to the

temperature shift theory advanced in Chapter 111.4. The

parameters a and E are assumed to be temperature independent.

These fitted parameters appear in Tables IV-1 and IV-2.

The molecular weight characterizations used in conjunction

with the polydispersed GSE model were supplied with the

samples courtesy of Union Carbide.

The experimentally determined material functions for

both PS samples together with GSE constitutive predictions

at the three temperatures are presented in Figures IV-1

through IV-10. The following general trends are observed

in both samples:

(1) Viscosity predictions show excellent agreement

with ICR viscosity data.

(2) Constitutive predictions of the viscosity func-

tion and the ICR viscosity data are generally

larger than the RM5 viscosity data. These

discrepancies increase with both shear rate and


(3) RM5 ENSD predictions are in agreement with

experimental data only in the intermediate

range of shear rates. Disagreement at the

lower rates is apparently attributable to force

transducer sensitivity. The observed data

trends at the higher rates are an indication

that material was lost from the gap region

during the experiment. These facts, coupled

with the apparently poor RM5 viscosity


Table IV-1

PS_1 Constitutive and Material Parameters

degrees K.









7.20 x 104

1.50 x 104

4.00 x 103





*Temperature Shifted Parameter

Temperature Independent Parameters

a = 5.2, e = 0.52

M = 2.40 x 105 M =7.40 x 104
w n
3.243, z = -0.554

Table IV-2

PS#2 Constitutive and Material Parameters

degrees K.








n o

1.00 x 105

1.80 x 104

4.80 x 103

*Temperature Shifted Parameter

Temperature Indenpendent Parameters

a = 3.5, e = 0.815
5 104
M =2.75x 10, M = 9.00 x 10
w n
= 3.056, z = -0.514





/ 0

0/ -.


0 cq

cq o

// 0

0-C) -H H

.0 .
/ L so-

0 < <0 r- U

I = 0
0 1 1 -O O
S- r-

S4 O 0 0

0 0 0 0
/~ t. D

o 3o

WO 7 O
0 0 3
C) O -1
0 (

(asTod) a

o C

C) Sri




H 0
S1 0

0^ H


C() '(

4u 0 P:-4

wo/us? T0

0rl 0


C 0 0 >0

0 0
H0 4 S::l

00 H
Do n c/


0 i0
.\ z


\ 4 s

0 0
0 0 0
T- I l -

(3 LUO/up) ,q

( uo/euAp) ,,O

0 0
0 0

0 co

CD co

U i


> 0




0 3
0 0


> 0

> 0.
4~ .
E ) *C

0 0 H

0 ^


- 0

0 0 0
- il r



4 00
l- H o>


-0 >
/ W


0 0 0

d(D d
Q / <

0 4-)
Cc -P )
0 a(

0 E;

I 0 I

O 0
/= Ln Ln -S ^S.


(asTOd) UL

0 0
1-1 1-1

(esTod) L



o 0



2 o
C. O


9 0 --

Q'3 0

C rO


0 -P



i-l r








ct 0

I 4-
C i




S 0
c jnr 0

So/ou)p) T x


( w/zp oN
~ ~ r -O"^H -

Ln -^i
0 0 0
i- O-

- 0



0 0

r~ '-

U) 0

r-- *-

0 -P

E "-

-P .0 0






'H .
0 --
C -
0 3*'-
o 7"

(9 Uo/aup) ,9

0 0 0
Ho H

( wo/oaup) ,,o


0 P,


) >
> *r


M 0




M >
*i l -
> -
CM -^
> *^,
ff +-


-- rO1




"H -


> 0



S0 -

Q -P

Q) w C)
LH0 V0

a4c 0
s < -

acP M

d- o,
'H (D 0
M > S

(GsTod) ,LU

0 0 0
Tr- l r-1

measurements, indicate some degree of experi-

mental inaccuracy with this rheometer.

(4) RVE moduli and dynamic viscosity predictions

agree quite well with experimental values

except at the very large deformation rates.

The temperature-superimposed material functions

calculated in the FORTRAN program TEMP SHIFT are presented

in Figures IV-11 through IV-20. The experimentally determined

data have been shifted to the predicted master curve at 503

degrees K. using the shift factors presented in Chapter III.4.

Reasonably good agreement is again obtained with the same

general trends present as noted in one through four above.

These results, when used as an internal test of the

GSE constitutive model, demonstrate that this particular

theory is capable of predicting SSF functions quite accurately

First, the model accurately predicts the dynamic functions

obtained with the RVE device using parameters fit to a

combination of ICR and RM5 data. Secondly, the predicted

temperature superpositions represent the experimental values

quite well. These findings indicate that the current consti-

tutive predictions of the relative forms of material function

magnitudes and temperature dependence in SSF are reasonably

successful. This implies that the GSE model could be used

to completely characterize a sample's SSF material functions

with respect to both temperature and shear rate from a

relatively small amount of theological data.

3 O












Q) ,-f
K 'A
0 *,

I 0 'H
> 0 -

( n d.

*il L *^

O 0

r-) 4-c 4
0: 0

0 e <
i- 0n

H -d

Od cM4-
'- CO' i

0 ( Ma

ev EC
^ 0) *

a -P E <
U if k ^
r1 c1 -P O

(@sTod) U q

-u~- ----I-~------- I -1 c

( 9ww/aup) TN ,


- r--


e- -


U- -P 0

1 +-o
O0 -4


O -

'*i3 <
r P ?-
ic ?-
rt ( -








University of Florida Home Page
© 2004 - 2010 University of Florida George A. Smathers Libraries.
All rights reserved.

Acceptable Use, Copyright, and Disclaimer Statement
Last updated October 10, 2010 - - mvs