TEMPERATURE DEPENDENCE OF JET SWELL AND MATERIAL FUNCTIONS
IN POLYMER MELT SYSTEMS
By
MALCOLM C. JOHNSON, JR.
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1977
To: Patsy for the encouragement to be myself,
Zeus,
Kilgore,
Our Family and Friends.
ACKNOWLEDGEMENT
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
facilities.
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.
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENT. . . . . . . ... . iii
LIST OF TABLES . . . . . . . . . vi
LIST OF FIGURES . . .. . . . . . vii
KEY TO SYMBOLS . . . . . . . . . .x .
ABSTRACT . . . . . . . . . . . xv
CHAPTERS:
I. INTRODUCTION AND BACKGROUND .. .... . .. 1
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
III. PREDICTION OF MATERIAL FUNCTIONS IN SIMPLE
SHEARING FLOW (SSF) . .. . . . 52
III.1 Constitutive Model . .. . .. 52
III.2 Constitutive Predictions for SSF . 56
III.3 Constitutive Parameter Fitting . . 58
iv
III.4 Temperature Superposition . . .
IV. RESULTS: COMPARISON OF EXPERIMENT TO THEORY.
IV.1 Material Functions . . . . .
IV.2 Jet Swell . . . . . . . .
V. CONCLUDING REMARKS . . . . ....
APPENDICES:
A. COMPUTER PROGRAM SOURCE LISTINGS . ..
A.1 Main Programs .
A.2 Subroutines .. .
B. COMPUTER PROGRAM OUTPUT
C. JET SWELL DATA. . . .
BIBLIOGRAPHY . . . . .
BIOGRAPHICAL SKETCH . . . .
Page
61
64
65
92
106
111
. . . 112
. 127
. 142
. . . 176
. . . 182
. . . 186
LIST OF TABLES
Table Page
II1 Capillary Dimensions. . . . . ... 35
IV1 PS#1 Constitutive and Material Parameters . 70
IV2 PS#2 Constitutive and Material Parameters . 70
LIST OF FIGURES
Figure Page
II1 ICR extrusion system schematic. . .. .29
II2 Cone/plate system schematic . . . ... .41
II3 ERD system schematic. . . .. . . . .48
IV1 Effect of temperature on ICR (solid points) 'and
RM5 (hollow points) viscosity versus shear rate
data for PS#1; () constitutive model predic
tions . . . . . . . . . . 71
IV2 Effect of temperature on RM5 PNSD versus shear
rate data for PS#1; () constitutive model
predictions . . . . . . . .. . 72
IV3 Effect of temperature on RVE elastic modulus
versus frequency data for PS#1; () constitu
tive model predictions .. . . . .. . 73
IV4 Effect of temperature on RVE viscous modulus
versus frequency data for PS#1; () constitu
tive model predictions. .. . . . . 74
IV5 Effect of temperature on RVE dynamic viscosity
versus frequency data for PS#1; () constitu
tive model predictions. . . . . .. 75
IV6 Effect of temperature on ICR (solid points) and
RM5 (hollow points) viscosity versus shear rate
data for PS#2; () constitutive model predic
tions . . . . . .. . . . . 76
IV7 Effect of temperature on RM5 PNSD versus shear
rate data for PS#2; () constitutive model
predictions . . . . .. . . .. . 77
IV8 Effect of temperature on RVE elastic modulus
versus frequency data for PS#2; () constitu
tive model predictions. . . . . . . 78
IV9 Effect of temperature on RVE viscosity modulus
versus frequency data for PS#2; () constitu
tive model predictions . . . .. . 79
vii
Page
IV10 Effect of temperature on RVE dynamic viscosity
versus frequency data for PS'2; () constitu
tive model predictions. . . . .. . 80
IV11 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
IV12 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
IV13 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
IV14 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
IV15 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
IV16 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
IV17 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
IV18 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
IV19 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
viii
Page
IV20 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
IV21 Effect of temperature on the constitutive
prediction of the recoverable shear versus
shear rate for PS1. . . ... . . . 93
IV22 Effect of temperature on the constitutive
prediction of the recoverable shear versus
shear rate for PS#2. .. ... . . . .. 94
IV23 Experimental JS magnitude versus the
temperatureshifted wall shear stress for
PS#1, measured at 473, 503 and 533 degrees K.,
shifted to 503 degrees K.. .. .. ... 97
IV24 Experimental JS magnitude versus the
temperatureshifted wall shear stress for
PS#2, measured at 473, 503 and 533 degrees K.,
shifted to 503 degrees K.. . . . . . 98
IV25 JS magnitude versus the recoverable shear at
the capillary wall as predicted by the various
elastic solid theories . . . . . .. 100
IV26 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
IV27 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
IV28 Effect of temperature on JS magnitude versus
shear rate data for PS#1; () constitutive
model predictions. . . . .. . . 104
IV29 Effect of temperature on JS magnitude versus
shear rate data for PS#2; () constitutive
model predictions. .... .. . . . . 105
KEY TO SYMBOLS
Ar = crosssectional area of the capillary rheometer
reservoir (barrel).
a = centerline separation of eccentric rotating disk
rheometer plates.
aT = constitutive shift factor for shear rate and
frequency.
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 (2E)]/2.
A
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.
J
Dr = diameter of the capillary rheometer reservior
(barrel).
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.
x
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
rheometer.
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.
A
N1
N1(r,L)
N1(R,L)
N2
n
P
AP
R
r
S
SR
T
T
T11(r)
T11(r,L)
t
u(r,L)
V
v
VXH
= 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, (la)/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.
xii
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
slope.
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.
A
n = reduced shear viscosity.
6 = weight averaged molecular weight.
w
i = the i relaxation time of the j molecular
j
weight fraction.
o = angular coordinate.
TT = 3.14154...
p = material density.
pth
P = material density of the i species.
i
xiii
Pt
T
T12
(T12)wall
i
T.
~j
= 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
fraction.
= angular coordinate.
= angular velocity or frequency.
xiv
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
TEMPERATURE DEPENDENCE OF JET SWELL AND MATERIAL FUNCTIONS
IN POLYMER MELT SYSTEMS
By
MALCOLM C. JOHNSON, JR.
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 lengthto
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 ratetype 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
xvi
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
interest.
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.
xvii
CHAPTER I
INTRODUCTION AND BACKGROUND
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
4
and Merrington phenomenon, is considered historically
5
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 socalled 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 largescale 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, trialanderror 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
situations.
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 buildup to welldeveloped, 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 takeup device. This
variable takeup 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 lengthtodiameter (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
6
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
6
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
model.
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 steadystate 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 closetolerance 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
die.
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 solidlike
response, rotationaltype 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 wellmodeled, sophisticated
systems.
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 temperaturecontrolled 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 inphase
(elastic) or outofphase (viscous) contributions, depending
on the magnitude of the recorded torque and the phase lag.
The dynamic viscosity may be calculated from the viscous
modulus.
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 offset 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
10
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
steadystate mode as in the case of the rheometers discussed
above. Therefore, the careful experimentalist must insure
that measured variables represent the longtime 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 stressfree 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
Ii
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
industrialtype process where material is extruded directly
into air.
Collection techniques vary from a highly specialized
12
device capable of obtaining multiple samples rapidly2 to
simple, systematic cutting of the material by hand with
1315
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
15
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
16
total swelling is complete.6 From this viewpoint it is
necessary to obtain measurements away from the end of a
sample.
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 springadjustable micrometer.15'19'20 A
study directly comparing these two techniques found them to
21
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 stressfree
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
21
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.
14
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 longerthannecessary period of time,
although these types of effects are easily separated from
desired behavior due to the large difference in time scales
15
of the two processes. While the semiempirical elastic
theories used to predict JS magnitude assume that this
stressfree 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
farreaching effect on JS magnitude, but it has been found
20
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
23
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
13,15,20
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 rotationaltype 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
welldeveloped 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 realworld 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 flowinduced stress tensor T, by
T pi + T (I1)
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) (12)
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) (13)
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 (12) are calculated to be
T11 T12 0
T = 112 T22 0. (14)
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 welldefined 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 sheartype flow, is defined as the ratio of
the shear stress, T12' to the shear rate, y
T12
n (15)
Y
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 (16)
S= T22 T33 (7)
where the N.'s are the PNSD and SNSD respectively.
1
When a Newtonian fluid undergoes SSF, the viscosity
function is a constant and both normal stress differences
vanish.
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 nonNewtonian viscosity is constant at very
small deformations in the socalled 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 onefifth to onehalf the magnitude for polymer
2527
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,
D.
B = (18)
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} (19)
(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). (I10)
Combining equations (19) and (110) lends to
B 8 fD/2 N1(r,L) 1/2
B (VD) r [u(r,L) ]dr (I11)
(VD) P
Inasmuch as the velocity profile at the tube exit has been
assumed to be welldeveloped and laminar, the integral
in equation (I11) 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 (I11) predicts an "expansion" of
B = 0.866 (112)
2
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
29,30
expansion of about 10 percent.2930
Equation (I11) 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
15
result in errors of up to seven orders of magnitude. The
most obvious reason for this .inadequacy is the assumption
of welldeveloped flow up to the capillary exit, and the
assumption that the SNSD vanishes.
A more recent study utilizing mass, momentum and
31
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
28
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 solidlike responses are the socalled
elastic solid (ES) theories. Semiempirical 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
33
or the theory of rubber elasticity,33 it is possible to relate
SR to the PNSD, N1, and shear stress, T12
RN
S (113)
R T
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
N1
SR 1 (I14)
12
Thus equation (114) 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
die.
The earliest ES theory, developed by Spencer and
37
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 (115)
R B
Previous studies have discussed critically the assumptions
leading to equation (115) and have concluded that these
mechanisms would result in telescopictype deformations of
the unconstrained free jet rather than the desired radial
swelling.
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. (116)
B
38
Bagley and Duffy used a oneconstant stored
energy function of a rubberlike solid to obtain
4 1 1)/2
SR = (B 2 (117)
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 (I18)
R 2
15 19
Graessley, et al., and Vlachopoulos, et al.,
extended this work using rubber elasticity theory together
with a semiempirical 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. (119)
14
Mendelson, et al.,1 assumed slightly different
constants for the stored energy function of rubberlike
solids, and obtained
SR = (B21n B)1/2. (120)
18
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 (122)
(SR)wall 1
White and Roman utilized an integraltype 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
42
McKelvey2 to relate JS magnitude to SR using the recovery
of an elastic fluid after a sheartype deformation
2 1 3/2 1 1/2
B = { SR(1 + )2 2 (123)
R R
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
47
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 Newtonianjet expansion
following a viscometric tube flow has been developed
48,49
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.
26
12,19
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 Summaryof 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
explored.
Chapter III introduces the four parameter ratetype
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 temperatureshifted 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.
CHAPTER II
EXPERIMENTAL
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 threeregion extrusion system is presented in Figure
II1. 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 xy 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
(15) 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
I
~_~ ~~ __ I

t
O
an ICR. This is easily accomplished by simplifying the
axial (z) and radial (r) components of the equations of
52
motion written in cylindrical coordinates for the SSF
region shown in Figure IIl. The usual assumptions are
made:
(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 welldeveloped flow
is attained, and this flow is maintained up to
the capillary exit.
(3) Gravity and surface tension effects are negligible
(4) The noslip 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). (II1)
Simplifying the z component of the equations of
52
motion,52 and further assuming that the order of differentia
tion with respect to z and r may be interchanged, yields
@ C. (112)
az
Hence, the pressure gradient in the axial direction
A
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:
A
Cr (113)
12 2
A
Determination of the constant, C, and subsequent
evaluation of equation (113) at the wall conditions yields
(AP)D (II4)
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
location.
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 crosssectional 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 socalled 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
8
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 abovementioned 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. (115)
Equation (114) becomes
F D
T c (II6)
12 4A L
r
where A is the reservoir crosssectional area.
r
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 socalled apparent shear rate,
ya. Differentiating the parabolic velocity distribution
with respect to r, and evaluating the resulting expression
at the wall conditions leads to
8V
a (117)
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 (118)
D
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 (118) as
originally suggested by Rabinowitsch7
S= y 3n+ (119)
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 threeaction 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 II1.
TABLE II1
CAPILLARY DIMENSIONS
Capillary LengthL 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
threeaction 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
recognized.
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 tradeoff 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
attained.
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
1315
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 airenvironment 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
14,15,21
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
springadjustable 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.1115'821 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 () (II10)
mpt
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,
respectively.
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 112. 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
SdW
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 (II1)
(p, 0, r) = (1, 2, 3). (1111)
The velocity field may be represented as
(l' v2' v ) = 2 ,' 0, 0] (1112)
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 =
(1113)
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
3J
T12 =3 (II14)
where R is the plate radius. The viscosity function, n,
is then calculated using equation (15).
The procedure required to obtain the PNSD is some
what more complicated. From the r component of the equations
52
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) (1115)
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
z
R
F = T 11(r) 27rdr. (1116)
z 11
Combining equations (1115) and (1116), 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
2F
N z. (1117)
1 2T
Here Fz represents the experimentally measured force applied
by the RM5 to maintain the spacing between the cone and the
plate.
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 (1113)
through (1117).
A typical cone used with the RM5 is actually truncated
to avoid the possible damage that could occur with the metal
tometal 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
54
manual.
(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 builtin 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 113.
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
a
Side View
upper plate
sample
lower plate
Figure 113. ERD system schematic.
to derive these expressions will be presented here. A more
complex description of the device and the methods required
55,56
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
before
(6, r, z) = (1, 2, 3). (1118)
The velocity field is simply expressed as
(vl' v2' v3) = (rw, 0, 0) (1119)
where each sample layer parallel to the rheometer plates
rotates about an imaginary axis connecting the plates'
centers.
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' =  (II20)
2
fTR a
F h
G" 2 (1121)
7TR a
where F and F are measured according to Figure 113, 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
G"
n' = (1122)
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 (1120) and (1121). A
detailed description of operational procedures is found
57
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
measurements.
CHAPTER III
PREDICTION OF MATERIAL FUNCTIONS IN
SIMPLE SHEARING FLOW (SSF)
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
5864
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 shearthinning 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 (beadspring) model for dilute
solutions of linear macromolecules. The continuum modifica
tion suggested by Gordon and Schowalter559 introduced an
alternate, semiempirical expression for the rate of change
of the molecular endtoend vector. This led to more
desirable material function forms for dilute, monodisperse
polymer solutions undergoing SSF, including a shearthinning
60,61
viscosity, a positive PNSD, and a negative SNSD.60
This model was extended to a multibead/spring or
RouseZimm approach by inclusion of multiple relaxation
times. Considerable improvement in dynamic viscosity predic
62
tions was attained with this modification.
Further model revision incorporated a SchultzZimm
molecular weight distribution to account for effects
61
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
i
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 weightaveraged 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 2a
1 + D 2 n p I j=l ... ,N (III1)
jj j Dt [Z(a)1] pM 1+j ~ '
N.
T = I + I T1 (III2)
i j=1
DT aT
+ v. (Vv Ed)T T(Vv Ed) (1113)
2 j
M M.( 2
l = j 1[ I (III5)
wt J
Z(a) = (III6)
M=l M1
55
where the constitutive parameter e is subject to the condition
0 5 e < 1. Here, pi and M. represent the density and molecule
.t1
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 (13), (III1), and (III3), 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 RouseZimm relationship for the differential molecular
fraction, dn
+2 z+ (M (z+2)9 11
dn = +2 l] ( ) dM (1117)
w w
where z, a measure of the width of the molecular weight
distribution, is defined by
M
z + 2 w (
z + 1 M
n
Here, M and M represent the weight and number averaged
w n
molecular weights, respectively.
This set of relationships will be designated as the
GordonSchowalterEverage (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 (1119)
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 (III10)
no [Z(O)ljr(z+2) 0 j=2
1
2a+(2C6 yx )
M (III11)
w
C = [C (2E)]1/2. (III12)
Here F(z) denotes the Gamma function.
Additionally, the SNSD, N2, is found to be
N N
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
.a
.( 2 dx
J2a (2a 2 2
N
exp[(z+2)xl Y
j=2
1
j 2a+(2 a0 wx2 )
w
(III13)
(III14)
(III15)
The viscous or loss modulus, G", may be calculated
from a knowledge of n' and the frequency, w
G" = n'w. (III16)
The GSE model predicts the following desirable
material function characteristics in SSF:
(1) The viscosity is shearthinning 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 onehalf the
magnitude of the PNSD, depending on the value
of E.
Additionally, the model predicts the following
experimentally observed similarities in material function
shapes:
(1) The viscosity/shear rate curve and the dynamic
viscosity/frequency curve are superimposable
according to
n(9) = n'(Cw). (11117)
(2) The PNSD/shear rate curve and the elastic modulus/
frequency curve are superimposable according to
2
2 N () = G'(Cw) (III18)
where C is defined by equation (III12).
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
64
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
parameters:
(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
(III9) for large deformations to obtain
1a
S (III19)
(3) It is convenient to introduce the dimensionless
A A
variables n and N1 in order to make the right
hand sides of equations (III9) and (III10)
unique functions of a dimensionless shear rate,
A
Y
A
n (11120)
0
Sw N1
N1 = N (III21)
Y = C w y. (III22)
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
w
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
64
zero shear viscosity, n
oKEMw
S= (III23)
w N pkT(1c)2a(Z(a)1)
a.
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 (III23) may be written as
T1o
6 = K (III24)
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
temperature.
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 (III24) to equations (III9), (III10),
(III14), and (III15). 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,
respectively,
(
(
(N
(
(G
(n
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
(PT)ref
c^ = 
cT
(pT)
new
(III25)
(III26)
(III27)
(III28)
(III29)
(III30)
(III31)
(11132)
(III33)
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
63
parameters and equations (1119) to (III16). Additionally,
all material function data were shifted to master curves at
503 degrees K. by using the superposition theory and shift
factors developed above.
CHAPTER IV
RESULTS: COMPARISON OF EXPERIMENT TO THEORY
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
65
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
65
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 Arrheniustype 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
66
percent with density data published elsewhere. The shift
factor cT, calculated from equation (III33), 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 IV1 and IV2.
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 IV1
through IV10. 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
temperature.
(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
70
Table IV1
PS_1 Constitutive and Material Parameters
T
degrees K.
473
503
533
p
gm/cc.
0.95
0.94
0.93
no
poise
7.20 x 104
1.50 x 104
4.00 x 103
6
w
sec.
0,158*
0.0313
0.00796*
*Temperature Shifted Parameter
Temperature Independent Parameters
a = 5.2, e = 0.52
M = 2.40 x 105 M =7.40 x 104
w n
W
3.243, z = 0.554
n
Table IV2
PS#2 Constitutive and Material Parameters
T
degrees K.
473
503
533
p
gm/cc.
0.95
0.93
0.92
n o
poise
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
M
= 3.056, z = 0.514
n
sec.
0.280*
0.0485
0.0126*
O
7/1
/ 0
0/ .
o"
0 cq
///
cq o
4'
// 0
0C) H H
.0 .
/ L so
0 < <0 r U
I = 0
S0
0 1 1 O O
S r
0
S4 O 0 0
0 0 0 0
SI
/~ t. D
o 3o
WO 7 O
CD C)
0 0 3
C) O 1
0 (
(asTod) a
o
o C
SMO
C) Sri
4)
P
V)
4'
H 0
S1 0
0^ H
0
S0
CH eCO
C() '(
4u 0 P:4
F ATC
wo/us? T0

0rl 0
73
C 0 0 >0
0 0
H0 4 S::l
Co
00 H
Do n c/
OcO
0 i0
.\ z
0
\ 4 s
\rl
0 0
0 0 0
T I l 
(3 LUO/up) ,q
G
( uo/euAp) ,,O
0 0
0 0
0 co
CD co
O O
LO LO
U i
Cl
0
> 0
Co
>0
Ec1
O
oH
0 3
0 0
O
> 0
CH
> 0.
4~ .
E ) *C
0 0 H
0 ^
CHOW
I
0
SI
0
 0
CD
I I
0 0 0
 il r
>0
cE
4 00
l H o>
0r
0 >
/ W
30
0 0 0
0
d(D d
Q / <
0 4)
Cc P )
0 a(
C
0 E;
I 0 I
O 0
/= Ln Ln S ^S.
/O I CT
(asTOd) UL
0 0
11 11
(esTod) L
Cl
*r
0
o 0
>
U)
0
2 o
C. O
%O
Od
0
9 0 
Q'3 0
*HQ1
M+>
C rO
3)
ri
0 P
CO
H
il r
0
O
H
^^
o
I
0
c11
O
T\
0
O O
H
ct 0
a
I 4
C i
cn
C4
V
S 0
c jnr 0
So/ou)p) T x
I I
O O
( w/zp oN
~ ~ r O"^H 
I I
Ln ^i
0 0 0
i O
C,1
 0
ri
0
3
0 0
r
r
O
r~ '
U) 0
>E
r *
0 P
0"
E "
4)
00
c,0
P .0 0
M0
0
OH
dO
EN
aOd
CO
bi)
'H .
0 
C 
PO
0 3*'
o 7"
(9 Uo/aup) ,9
I I I
LO 1 CO
0 0 0
Ho H
( wo/oaup) ,,o
U
0
0 P,
0
0
) >
> *r
>
.)
M 0
E
0
00
o
r
Ec
0
M >
*i l 
> 
CM ^
> *^,
ff +
Ma
CM1
C0
r1
 rO1
O
0
0
rH
4
"H 
0P
G O
> 0
0)
H
SCH
S0 
Q)C)
Q P
Q) w C)
LH0 V0
a4c 0
s < 
acP M
(UM
d o,
'H (D 0
M > S
(GsTod) ,LU
LO C
0 0 0
Tr l r1
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 temperaturesuperimposed material functions
calculated in the FORTRAN program TEMP SHIFT are presented
in Figures IV11 through IV20. 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
O
I O
00
<0
O
0
O1
0
<0
O
o
C,')
0
c"1
0
0
H
0(
0
Hl
0
ri
Q) ,f
K 'A
0
0 *,
S4)
I 0 'H
> 0 
( n d.
OM C)
C,
*il L *^
OE0
O 0
OcM
r) 4c 4
0: 0
E C
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 ,
rI
r4
 r
C
bc
e 
0
Q)O
WO
(U0
0
U P 0
1 +o
C\)
O0 4
o
'
*r
O 
LCO
'*i3 <
r P ?
QriC
ic ?
rt ( 
M3P
.)
C1
Edi
I
0
r
O
1<
i
~I
