Elasto-capillary thinning and breakup
of model elastic liquids
Shelley L. Anna
Division of Engineering and Applied Sciences, Harvard University, Cambridge,
Massachusetts 02138
Gareth H. McKinley
a)
Department of Mechanical Engineering, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139
(Received 14 June 2000; final revision received 9 October 2000)
Synopsis
We study the elasto-capillary self-thinning and ultimate breakup of three polystyrene-based ideal
elastic fluids by measuring the evolution in the filament diameter as slender viscoelastic threads
neck and eventually break. We examine the dependence of the transient diameter profile and the
time to breakup on the molecular weight, and compare the observations with simple theories for
breakup of slender viscoelastic filaments. The evolution of the transient diameter profile predicted
by a multimode FENE-P model quantitatively matches the data provided the initial stresses in the
filament are taken into account. Finally, we show how the transient uniaxial extensional viscosity of
a dilute polymer solution can be estimated from the evolution in the diameter of the necking
filament. The resulting ‘‘apparent extensional viscosity’’ profiles are compared with similar results
obtained from a filament stretching rheometer. Both transient profiles approach the same value for
the steady state extensional viscosity, which increases with molecular weight in agreement with the
Rouse–Zimm theory. The apparent discrepancy in the growth rate of the two transient curves can
be quantitatively explained by examining the effective stretch rate in each configuration. Filament
thinning studies and filament stretching experiments thus form complementary experiments that
lead to consistent measures of the transient extensional viscosity of a given test fluid. © 2001 The
Society of Rheology. 关DOI: 10.1122/1.1332389兴
I. INTRODUCTION
Small quantities of polymeric additives often have very pronounced effects on fluid
motion, and one particular type of flow that has been studied extensively in this regard is
the necking and breakup of polymeric liquid jets. Newtonian jets rapidly neck down
under capillary action and pinch off into evenly spaced droplets and interspersed satellite
droplets due to the well-known Rayleigh–Tomotika instability 关Rayleigh 共1878兲; Ray-
leigh 共1892兲; Tomotika 共1935兲兴, whereas even very dilute polymer solutions form stable
jets or ‘‘beads-on-a-string’’ structure that persist for a relatively long time 关Goldin et al.
共1969兲兴. Extensive reviews of the early literature in the subject are given by Bogy 共1979兲
and in the monograph by Yarin 共1993兲. A comprehensive review of more recent literature
on breakup phenomena in Newtonian fluid filaments and jets, including experimental
a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 1150148-6055/2001/45共1兲/115/24/$20.00
results, numerical simulations, stability analyses, and similarity solutions for the ap-
proach to breakup, is provided by Eggers 共1997兲. Early linear stability analyses of vis-
coelastic jets indicated that they were actually less stable than equivalent Newtonian jets
关Middleman 共1965兲; Goldin et al. 共1969兲; Goren and Gottlieb 共1982兲兴. However, Goldin
and co-workers noted that nonlinear phenomena become increasingly important as the
filament necks down, so that a linear analysis is no longer valid 关Goldin et al. 共1969兲兴.
Goren and Gottlieb also observed that allowing for a finite initial tension in the filament
increased the stability of the filament 关Goren and Gottlieb 共1982兲兴. By using a finite
element simulation to incorporate nonlinear effects, Bousfield and co-workers were able
to simulate the entire transient evolution of a viscoelastic jet as it formed the beads-on-
a-string configuration 关Bousfield et al. 共1986兲兴. More recently, Chang and co-workers
have presented an asymptotic analysis and numerical simulations of the elasto-capillary
necking dynamics of viscoelastic jets described by the Oldroyd-B model 关Chang et al.
共1999兲兴.
These stability analyses have shown that the extensional rheological response of a
fluid dictates whether or not it will form stable jets and filaments. Building on early work
by Chang and Lodge 共1972兲, Ide and White incorporated effects of surface tension and
nonlinear extensional rheology to predict the ‘‘spinnability’’ of viscoelastic fluids in
imposed uniaxial elongation 关Ide and White 共1976兲兴. Renardy 共1994, 1995兲 has presented
nonlinear analyses of the asymptotic approach to capillary-driven breakup of slender
Newtonian and viscoelastic jets using a Lagrangian formulation, and showed that vis-
coelastic constitutive models predicting unbounded stress growth during extension lead to
the formation of filaments that never break while other models with a finite steady state
uniaxial extensional viscosity will break in finite time. During the necking and breakup of
viscoelastic fluid filaments, elastic tensile stresses resist the pinching caused by capillary
action. As a result, tensile stresses in the viscoelastic fluid grow and polymer chains are
elongated in the local region near breakup. Since the local flow field in this region is
primarily extensional, analysis of necking phenomena provides a plausible way of mea-
suring the extensional response of viscoelastic fluids. Schu
¨
mmer and Tebel 共1983兲 de-
scribed a ‘‘free jet elongational rheometer,’’ in which the diameter of a periodically
forced fluid jet undergoing capillary thinning is monitored as a function of time. Another
filament breakup device, referred to as the ‘‘microfilament rheometer’’ was introduced in
1990 关Bazilevsky et al. 共1990兲兴, and is similar in operation to the filament stretching
device introduced at about the same time 关Matta and Tytus 共1990兲; Sridhar et al. 共1991兲;
Tirtaatmadja and Sridhar 共1993兲兴. The principal difference between the filament stretch-
ing rheometer and the filament breakup rheometer is that in filament stretching devices, a
cylindrical liquid bridge is formed between two rigid endplates, which are then actively
stretched apart with an exponentially increasing separation profile. In the filament
breakup device, these endplates are rapidly separated and then held at a fixed axial
separation, and the subsequent evolution of the midfilament diameter is monitored during
the process of necking and breakup.
Extensional rheometry utilizing these two devices has progressed in parallel since the
early 1990s. Several groups have developed filament stretching rheometers based on the
original design of Sridhar and co-workers, and a significant amount of work has been
performed to quantify the extensional rheology of dilute polymer solutions as well as to
define appropriate experimental technique and interpretation of the resulting measure-
ments 关Berg et al. 共1994兲; Solomon and Muller 共1996兲; Spiegelberg et al. 共1996兲;
Spiegelberg and McKinley 共1996兲; van Nieuwkoop and Muller von Czernicki 共1996兲;
Kolte et al. 共1997兲; Szabo 共1997兲; Anna et al. 共1999兲; Orr and Sridhar 共1999兲; Verhoef
et al. 共1999兲兴. Most recently, an interlaboratory comparison of filament stretching data
116 ANNA AND MCKINLEY
for three common test fluids was presented, in which the authors demonstrated that
agreement between different designs of filament stretching rheometers is excellent 关Anna
et al. 共2000兲兴.
Using the filament breakup device, Bazilevsky and co-workers 共1990,1997兲 have pre-
sented measurements for various polyacrylamide and polyethylene oxide-based polymer
solutions, and compared the results with simple theories for the approach to breakup of
Newtonian and viscoelastic filaments. Liang and Mackley 共1994兲 presented similar self-
thinning measurements for polyisobutylene solutions, and used the evolution in the di-
ameter profile to obtain a characteristic relaxation time for these fluids. They found that
the relaxation time
R
obtained from filament thinning measurements was nearly a factor
of 3 larger than the mean relaxation time
¯
obtained from shear rheology measurements.
More recently, Stelter and co-workers 共2000兲 presented thinning measurements for poly-
acrylamide solutions, and used the transient diameter profiles to compute a relaxation
time and an apparent elongational viscosity for these semidilute solutions. Entov and
Hinch 共1997兲 used a multimode FENE model to predict the breakup profile of very
slender viscoelastic fluid filaments. The authors analyzed the asymptotic behavior in three
different regimes of the elasto-capillary flow and predicted the time dependence of the
midfilament diameter in each regime. In a recent experimental and numerical study, Kolte
and Szabo 共1999兲 found that quantitative agreement with experimental data could not be
achieved using this theory unless axial variations in the filament profile and radial varia-
tions in the tensile stress difference were also incorporated. Finally, McKinley and
Tripathi 共2000兲 also used a filament breakup device to compare measurements with
viscous Newtonian fluids to the appropriate similarity solution for the approach to
breakup and showed that, for Newtonian filaments, quantitative agreement with the one-
dimensional theory of Papageorgiou 共1995兲 can be achieved by correctly accounting for
the self-similar configuration of the necking filament.
Having demonstrated that measurements of capillary breakup in Newtonian fluids can
be successfully described by a one-dimensional theory, we now seek to investigate if the
appropriate one-dimensional analysis 关Entov and Hinch 共1997兲兴 can be used to extract
quantitatively accurate values of material parameters for viscoelastic liquids. The analysis
of Entov and Hinch for capillary thinning and breakup of elastic filaments is based on a
multimode formulation of the FENE-P dumbbell theory, which accurately describes the
rheology of infinitely dilute monodisperse homopolymer solutions. Unfortunately, almost
all experimental analyses of elasto-capillary thinning to date have used semidilute or
concentrated solutions of polydisperse macromolecules. The shear rheology of such ma-
terials is typically poorly described by idealized models such as the FENE constitutive
equation unless unrealistically low values of the relevant model parameters are chosen
关Entov and Hinch 共1997兲兴. It is thus not possible to ascertain whether material properties
extracted from measurements of elasto-capillary thinning are consistent with the values
of molecular parameters on which the original model and analysis were based. The goal
of the present study is thus to investigate the elasto-capillary drainage and breakup of
dilute monodisperse polymer solutions which have already been well characterized in
viscometric flows using conventional shear rheometry and also in transient uniaxial ex-
tension using filament stretching rheometry.
Although the flow in a filament stretching rheometer and in a capillary breakup rhe-
ometer are slightly different, both devices generate a uniaxial extensional flow. In the
filament stretching device, the characteristic time scale of the flow is well defined,
t
flow
⫽ (
˙
0
)
⫺ 1
, and thus the flow kinematics and fluid response are relatively straight-
forward to interpret. However, an ideal filament stretching experiment is practically dif-
117ELASTO-CAPILLARY THINNING AND BREAKUP
ficult to implement, and several limitations arise from mechanical constraints as well as
gravitational sagging effects and elastic instabilities 关Anna et al. 共2000兲兴. By contrast, a
filament breakup experiment is relatively simple to perform; however, since the fluid
column is allowed to spatially rearrange and select its own time scales, the dynamics of
the fluid response can be quite complicated and the kinematics are typically time depen-
dent. Analysis and interpretation of filament thinning data is therefore not as straightfor-
ward as that of filament stretching data. However, the response of a given fluid to these
two different flows depends only on its microstructure as encoded in the constitutive
equation, and thus we expect a correct analysis of each experiment to yield consistent
rheological information for a given fluid.
In this study, we examine the filament stretching and breakup responses of three
different polystyrene-based Boger fluids, in which the polymer concentration is fixed at
0.05 wt % and the polymer molecular weights are 2.0, 6.5, and 20 million g/mol. These
three Boger fluids are the same well-characterized test fluids used in the interlaboratory
comparison of filament stretching devices mentioned above 关Anna et al. 共2000兲兴.Inthe
following section, we present measurements of the evolution in the midfilament diameter
profiles D
mid
(t) observed during elasto-capillary thinning experiments for the three dif-
ferent Boger fluids. We compare the characteristic relaxation time obtained from this data
to the relaxation times obtained from conventional viscometric experiments. In Sec. III,
we compare these experimental observations to theoretical predictions using a multimode
FENE model, and in Sec. IV, we show how the force balance on the filament can be used
to reexpress midfilament diameter profiles in the form of a transient extensional viscosity.
Finally, we compare the molecular weight dependence of the steady-state extensional
viscosity obtained from the asymptotic behavior of filament breakup experiments to the
behavior predicted from kinetic theory for bead-spring chains.
II. FILAMENT THINNING AND BREAKUP EXPERIMENTS
A. Experiment setup
Although several devices have been developed to specifically make measurements of
visco-elasto-capillary thinning 关Bazilevsky et al. 共1990兲; Kolte and Szabo 共1999兲;
McKinley and Tripathi 共2000兲; Stelter et al. 共2000兲兴, the filament breakup experiments
performed in the present study were actually carried out in a filament stretching apparatus
that has been described in detail elsewhere 关Anna et al. 共1999, 2000兲兴. A schematic
diagram of the experimental configuration is shown in Fig. 1. A nearly cylindrical sample
FIG. 1. Schematic diagram of a capillary thinning experiment.
118 ANNA AND MCKINLEY
of fluid initially fills the gap between two rigid, circular endplates with diameter D
0
and
initial separation L
0
. The endplates are then moved apart to a final separation L
f
with an
exponentially increasing separation profile, L
p
(t) ⫽ L
0
exp(⫹E
˙
t) , where E
˙
is the stretch
rate. During the short period of stretching, both the midfilament diameter D
mid
and the
tensile force at the endplate F
p
are monitored but once the endplates have stopped
moving, the force rapidly decays to zero, and the midfilament diameter alone is moni-
tored until the fluid column breaks. The filament stretching apparatus was used so that the
rate of separation of the endplates could be precisely controlled, and so that the initial
stresses in the filament could be monitored. However, we have also performed analogous
filament breakup experiments on the same test fluids in a microfilament rheometer de-
signed by Entov and co-workers 关Bazilevsky et al. 共1990兲兴 in which the rate of axial
separation is not precisely controlled, and the transient midfilament diameter profiles
following cessation of stretching agreed quantitatively between the two devices. The
initial axial separation of the plates may thus be viewed as the elongational equivalent of
a ‘‘step–strain’’ experiment, provided that the separation rate E
˙
is much faster than the
viscoelastic relaxation time in the fluid (E
˙
Ⰷ 1/
1
) and the timescale for capillary drain-
age (E
˙
Ⰷ 2
/
0
D
0
).
The diameter sensor used in these experiments is a laser micrometer which computes
the size of an object in its path based on the intensity of light entering the sensing element
共Omron Z4LA兲. This sensor is calibrated periodically using transparent optical fibers that
mimic the opacity of the fluid filaments. The calibrated resolution of the sensor is ap-
proximately 20
m, and a representative calibration plot for the sensor is shown in Fig.
2. The Hencky strain (t) experienced by the fluid element at the axial midplane at time
t can be defined using the midfilament diameter, in the same way as in a filament
stretching experiment, by
⫽ 2ln
共
D
0
/D
mid
共
t
兲兲
. 共1兲
FIG. 2. Calibration of diameter sensor using optical fibers of known size. The measured voltage 共䊉兲 increases
linearly as diameter increases, as shown by the solid fitted line.
119ELASTO-CAPILLARY THINNING AND BREAKUP
