An interlaboratory comparison of measurements
from filament-stretching rheometers using common
test fluids
a)
Shelley L. Anna
Division of Engineering and Applied Sciences, Harvard University, Cambridge,
Massachusetts 02138
Gareth H. McKinley
b)
Department of Mechanical Engineering, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139
Duc A. Nguyen and Tam Sridhar
Department of Chemical Engineering, Monash University, Clayton, Victoria,
Australia
Susan J. Muller
Department of Chemical Engineering, University of California–Berkeley,
and Lawrence Berkeley National Laboratory, Berkeley, California 94720
Jin Huang and David F. James
Department of Mechanical and Industrial Engineering, University of Toronto,
Toronto M5S 3G8, Canada
(Received 14 June 2000; final revision received 9 October 2000)
Synopsis
Following development of a filament-stretching extensional rheometer at Monash University,
similar rheometers have been designed and built in other laboratories. To help validate the basic
technique, a collaborative program was undertaken to compare results from several instruments.
First, three test fluids prepared at the University of California at Berkeley were characterized in
steady and transient shear flows there and at the Massachusetts Institute of Technology 共M.I.T.兲, and
then tested in extensional rheometers at M.I.T., Monash and the University of Toronto. Each fluid
is a constant-viscosity solution of narrow-molecular-weight-distribution polystyrene dissolved in
oligomeric polystyrene. The solute molecular weights are 2.0, 6.5, and 20 million g/mol, and the
polymer concentration in each fluid is 0.05 wt. %. From linear viscoelastic measurements, the Zimm
relaxation times of the fluids are found to be 3.7, 31, and 150 s, respectively. The scaling of
relaxation times with molecular weight indicates better-than-theta solvent quality, a finding
a兲
This paper was presented by D. F. James at the Society of Rheology Annual Meeting in Madison, Wisconsin,
October 1999.
b兲
Author to whom correspondence should be addressed; electronic mail: gareth@mit.edu
© 2001 by The Society of Rheology, Inc.
J. Rheol. 45共1兲, January/February 2001 830148-6055/2001/45共1兲/83/32/$20.00
consistent with independent intrinsic viscometry measurements of equilibrium coil size. Each fluid
was tested in the three filament stretching rheometers at similar Deborah numbers. Despite
variations in instrument design and the general difficulty of the technique, transient Trouton ratios
measured in the three instruments are shown to agree quantitatively. © 2001 The Society of
Rheology. 关DOI: 10.1122/1.1332388兴
I. INTRODUCTION
The development of a filament stretching apparatus by Sridhar and coworkers has
enabled the transient extensional flow behavior of mobile fluids to be characterized,
particularly the behavior of dilute polymer solutions 关Sridhar et al. 共1991兲兴. Although the
importance of extensional rheology measurements has been recognized since the turn of
the century 关see 共Petrie 共1979兲 for a historical account兴, generating purely extensional
flows of mobile fluids has proven to be extremely difficult. Various devices have been
developed to measure the ‘‘extensional viscosity,’’
E
, of a mobile fluid, including
opposed jet devices, spin-line rheometers, and two- and four-roll mills. These techniques,
reviewed in detail elsewhere, all have serious drawbacks, among them an unknown
preshear history, ‘‘contamination’’ of the extensional flow field with regions of shearing,
and an inability to approach steady state conditions 关Gupta and Sridhar 共1988兲; James and
Walters 共1994兲兴. An international ‘‘round-robin’’ study in the Journal of Non-Newtonian
Fluid Mechanics in 1990 attempted to quantify the differences between many of these
devices by comparing measurements for the same fluid, denoted M1 关Sridhar 共1990兲兴.
This study showed that different devices can yield widely different results at a given
extensional rate. In fact, James and Walters compiled the data from this study and the
plot shows over three decades of variation in measurements of
E
as a function of
imposed strain rate
˙
关James and Walters 共1994兲兴. James and Walters note that the
‘‘values of
E
. . . are transient values and the disparity makes it clear how strongly
these values depend upon variables other than strain rate.’’ The round-robin study of the
M1 fluid demonstrated that a new device was needed which could overcome some of the
drawbacks of earlier devices and allow systematic study of other dependent variables.
As a part of this 1990 study, a ‘‘falling plate’’ device was introduced by Matta and
Tytus 关Matta and Tytus 共1990兲兴 which would subsequently evolve into the filament
stretching apparatus. By simply placing a small amount of fluid between two circular
plates and allowing the bottom one to fall under gravity, Matta and Tytus were able to
generate a nearly pure extensional flow and to calculate the tension in the fluid filament
during stretching. Motivated by this work, Tirtaatmadja and Sridhar designed a device to
control the separation of the two endplates such that the fluid is subjected to nearly ideal
uniaxial extension while the force on one plate is measured. In this device, the evolution
of fluid stress from equilibrium to steady state can be followed 关Sridhar et al. 共1991兲;
Tirtaatmadja and Sridhar 共1993兲兴. Several groups have subsequently built similar filament
stretching devices and reported results for various test fluids 关Kro
¨
ger et al. 共1992兲; Ooi
and Sridhar 共1993兲; Berg et al. 共1994兲; Ooi and Sridhar 共1994兲; Solomon and Muller
共1996b兲; Spiegelberg et al. 共1996兲; Spiegelberg and McKinley 共1996兲; van Nieuwkoop
and Muller von Czernicki 共1996兲; Jain et al. 共1997兲; Verhoef et al. 共1999兲兴. The flow
kinematics, including the effect of the nonideal flow conditions near the rigid endplates,
have been explored by researchers, and still others have used data from filament stretch-
ing devices to make quantitative comparisons with constitutive models 关Shipman et al.
共1991兲; Tirtaatmadja and Sridhar 共1995兲; Gaudet et al. 共1996兲; Kolte et al. 共1997兲;
Sizaire and Legat 共1997兲; Szabo 共1997兲; Remmelgas et al. 共1998兲; Yao and McKinley
共1998兲; Olagunju 共1999兲; Yao et al. 共2000兲兴. In addition, the device has been used to
84 ANNA
ET AL.
examine stress relaxation and birefringence in uniaxial extension 关Orr and Sridhar
共1996兲; Spiegelberg and McKinley 共1996兲; Doyle et al. 共1998兲; Sridhar et al. 共2000兲兴.
The goal of the current study is to investigate the reliability of filament stretching
rheometers and to determine their utility as a quantitative method for characterizing the
transient extensional rheology of viscoelastic liquids. In the following sections, we com-
pare the transient stress growth in well-characterized test fluids measured in three differ-
ent filament stretching devices that have been built independently in different laborato-
ries. We highlight important practical considerations and procedures needed to achieve
good agreement in results from these different devices.
In Sec. II, we describe the composition and properties of the three test fluids used in
this interlaboratory project. Measurements of intrinsic viscosity, steady and dynamic
shear rheology, and transient startup and cessation of shear flow are presented. Further-
more, it is found that the fluid response is consistent with a Zimm spectrum of relaxation
times, and we demonstrate remarkably good agreement between scaling exponents ob-
tained from intrinsic viscosity measurements and small-amplitude oscillatory shear data.
With this complete characterization in shear flows as a foundation, we proceed with
measurements of extensional viscosity and compare data from the three different instru-
ments for the three test fluids. In Sec. III, we describe the three filament stretching
devices, and compare the fluid response in different ways to show that nearly ideal
kinematics are achieved. Finally, in Sec. IV we present measurements of the transient
extensional rheology of the three test fluids for a range of imposed strain rates. We hope
that this study will provide a guide for experimenters who wish to use filament stretching
rheometry as a standard characterization technique.
II. CHARACTERIZATION OF COMMON TEST FLUIDS
A. Fluid composition
The three fluids used in this study, denoted SM-1, SM-2, and SM-3, are dilute polymer
solutions with nearly constant shear viscosities, which enable the effects of elasticity to
be isolated from those of shear thinning 关Boger 共1977/78兲; Boger and Nguyen 共1978兲兴.
Each solution has the same concentration, c ⫽ 0.05 wt. %, of high molecular weight
polystyrene 共Pressure Chemical, P.D.I.
⬍
1.20) dissolved in oligomeric styrene 共Picco-
lastic A5 Resin, Hercules兲. The mass-average molecular weights of the polystyrene are
2.0, 6.5, and 20⫻ 10
6
g/mol, respectively, and the density of each fluid is 1.02 g/cm
3
.
The manufacturer-reported values of the polydispersity indices for the three solutes are
1.03, 1.05, and 1.20, respectively. The three fluids were prepared at the University of
California at Berkeley. In each case, the high molecular weight polymer was dissolved
directly in the oligomeric resin, and the fluid was gently rolled once a day for 2 months
to ensure a homogeneous mixture.
The first step of the characterization was measurement of quiescent solution proper-
ties, including polymer molecular weight, coil size, and solvent quality, which were
determined through a combination of intrinsic viscometry, gel permeation chromatogra-
phy 共GPC兲, and static light scattering at UC Berkeley. Manufacturer-reported molecular
weights of the starting polymers were confirmed by GPC in tetrahydrofuran and by static
light scattering in dioctyl phthalate at 22 °C, the theta point for polystyrene. Static light
scattering also yielded the radii of gyration of the coils under theta conditions R
g,
;
details of these measurement techniques are reported elsewhere 关Solomon and Muller
共1996a兲; Lee et al. 共1997兲兴. The measured radii of gyration under theta conditions were in
excellent agreement with the values calculated from the molecular weight and the char-
acteristic ratio C
⬁
as
85FILAMENT-STRETCHING RHEOMETERS
R
g,
2
⫽ C
⬁
nl
2
, 共1兲
where C
⬁
⫽ 9.85 关Brandrup et al. 共1999兲兴, n is the number of carbon–carbon bonds in
the backbone, and l ⫽ 1.54 Å is the length of a carbon–carbon bond 关Flory 共1953兲兴.
Direct measurements of coil sizes and solvent quality in these solutions via static light
scattering could not be made because of insufficient contrast in the scattering between the
oligomeric styrene solvent and the high molecular weight polystyrene solute. Instead,
these measurements were made through intrinsic viscometry as described below.
For each molecular weight, a series of dilutions were prepared from a 0.05 wt. %
master solution and viscosities were measured in a capillary viscometer or in a Rheomet-
rics RMS-800 mechanical spectrometer using cone-and-plate fixtures. The limiting vis-
cosity number 共or intrinsic viscosity兲
关
兴 is defined through an expansion of the solution
viscosity
in concentration c as
⫽
s
共
1⫹
关
兴
c⫹k
H
关
兴
2
c
2
⫹ ...
兲
, 共2兲
and was determined by a dual Huggins–Kramer extrapolation of the viscosity data to
infinite dilution. In the above equation,
s
is the 共oligomeric styrene兲 solvent viscosity
and k
H
is the Huggins coefficient. The resulting values of
关
兴 are given in Table I, along
with other properties of the three solutions. The radius of gyration R
g
for each fluid was
then determined from the Flory–Fox equation:
关
兴
⫽ ⌽
0
R
g
3
/M
w
, 共3兲
where ⌽
0
is a universal constant, equal to 3.67⫻ 10
24
mole
⫺ 1
.
The size of a polymer coil in solution may be related to the solvent quality through
either a coil expansion parameter
␣
defined as
␣
⬅
R
g
R
g,
共4兲
or through an excluded volume exponent
where
R
g
⬀ M
. 共5兲
For a theta solvent,
␣
is unity by definition and
⫽ 1/2. For good solvents,
␣
is greater
than unity and 0.5
⬍
⬍
0.6.
Alternatively, since R
g,
⬀ M
1/2
, one may define a coil expansion exponent
␥
from
Eq. 共4兲 above, i.e.,
␣
⬀ M
␥
where
␥
⫽
⫺ 1/2 . The value of the exponent
␥
typically
varies from zero for a theta solvent to 0.1 for a good solvent. As shown in Table I, the
TABLE I. Equilibrium properties and molecular weight scaling exponents for SM Boger fluids. The concen-
tration c of all three solutions is fixed at c ⫽ 0.000 51 g/mL. The excluded volume exponent is determined to
be
⫽ 0.52⫾ 0.015, and the coil expansion exponent,
␥
⫽ 0.02⫾ 0.015.
M
w
共g/mol兲
关
兴
共mL/g兲
R
g
共nm兲 L
c
1
*
共g/mL兲
c
2
*
共g/mL兲 c/c
2
*
SM-1
2.0⫻ 10
6
120 41 88 0.0083 0.0011 0.44
SM-2
6.5⫻ 10
6
250 78 164 0.0040 0.000 58 0.87
SM-3
2.0⫻ 10
7
430 133 277 0.0023 0.000 34 1.50
86 ANNA
ET AL.
values of
and
␥
for our polystyrene–共oligomeric styrene兲 system were found to be 0.52
⫾ 0.015 and 0.02⫾ 0.015, respectively, indicating that the styrene oligomer is slightly
better than a theta solvent.
The above measurements also enable assessment of the diluteness of the three test
fluids. Since the polystyrene concentrations are the same in each case, the diluteness
decreases with increasing molecular weight. Several measures of diluteness have been
proposed in the literature, and a very thorough discussion of these is given by Graessley
and by Harrison and co-workers 关Graessley 共1980兲; Harrison et al. 共1998兲兴. The most
common method of assessing diluteness depends on the magnitude of the intrinsic vis-
cosity. A critical concentration c
1
*
can be defined as
c
1
*
⫽ 1/
关
兴
, 共6兲
although Graessley notes that a proportionality factor of 0.77 is more rigorous. The
values of c
1
*
given in Table I are well above the concentration of 0.000 51 g/mL of our
solutions, showing that all three fluids lie well within the dilute regime according to this
definition. A second, more conservative measure of diluteness is based on the concentra-
tion at which the coils at equilibrium begin to physically overlap. This critical concen-
tration c
2
*
is given by
c
2
*
⫽
M
w
4
3
R
g
3
N
A
, 共7兲
where N
A
is Avogadro’s number 关Graessley 共1980兲兴. By this measure, the SM-3 fluid lies
in the semidilute regime, since c ⬇ 1.5c
2
*
. A plot of the fluid viscosity versus concen-
tration also shows significant curvature at concentrations above 0.034 wt %, indicating
that SM-3 is above a nominal value of c
*
, as defined in terms of the relative magnitude
of the quadratic term in Eq. 共2兲. However, in this study, SM-3 is considered to be a dilute
solution, because we found that the behavior of the fluid in viscometric experiments was
consistent with that of a dilute solution.
B. Steady and dynamic shear rheology
The rheology of the three fluids in both steady and dynamic shear flows was charac-
terized at the Massachusetts Institute of Technology 共M.I.T.兲 using a TA Instruments
AR1000N cone-and-plate rheometer. First normal stress differences were also measured
using a Rheometric Scientific RMS-800 cone-and-plate rheometer to obtain values of the
first normal stress coefficient ⌿
1
(
␥
˙
) . Controlled shear stress measurements were carried
out at three temperatures: 15, 25, and 35 °C, and time–temperature superposition was
used to obtain master curves for the steady and dynamic material functions over 6 de-
cades in shear rate. Throughout this text, the official Society of Rheology nomenclature is
used for the material functions studied 关Dealy 共1995兲兴.
The temperature dependence of the viscometric properties was determined by continu-
ously increasing the temperature from 15 to 35 °C over the period of1hataconstant rate
of (⌬T/⌬t) ⫽ 0.056 K/s while keeping the applied shear stress fixed at 10.0 Pa. No
hysteresis was observed in subsequent tests with a decreasing temperature ramp. This
confirms that evaporation of the viscous oligomeric styrene solvent is negligible over this
temperature range and the temperature ramp is slow enough that the process is quasi-
static. The measured viscosity curve was then normalized by the viscosity at the reference
temperature T
0
⫽ 25 °C to obtain the shift factor a
T
as a function of temperature T. The
87FILAMENT-STRETCHING RHEOMETERS
