# Capillarity-driven thinning dynamics of entangled polymer solutions

Jianyi Du  [  Hiroko Ohtani  [  Alper Kiziltas  [  Kevin Ellwood  [  Gareth H.
McKinley  [<EMAIL_ADDRESS>
###### Abstract

We analyze the capillarity-driven thinning dynamics of entangled polymer
solutions described by the Doi-Edwards-Marrucci-Grizzuti (DEMG) model and the
Rolie-Poly (RP) model. Both models capture polymer reptation, finite rates of
chain retraction and finite extensibility of single polymer molecules, while
differing slightly in their final form regarding to the convective constraint
release. We calculate numerically the filament thinning profiles predicted by
the two models with realistic entanglement densities, assuming cylindrical
filament shapes and no fluid inertia. Both results reveal an early tube-
reorientation regime, followed by a brief intermediate elasto-capillary
regime, and finally a finite-extensibility regime close to the pinch-off
singularity. The results presented in this work reveal two critical features
in the transient extensional rheology of entangled polymer solutions that have
been reported from previous experimental studies, but are poorly described by
the widely-used FENE-P model. First, the relaxation time obtained from
capillary breakup extensional rheometry is notably smaller than that from
steady-shear rheometry. Their ratio can be expressed as a universal function
of the entanglement state and the polymer concentration, which agrees well
with the experimental data for a range of entangled polymer solutions. Second,
the filament thinning dynamics at sufficiently high polymer concentrations are
governed by the tube reorientation at intermediate strain-rates, and the
apparent extensional viscosity shows a noticeably rate-thinning response. We
finally evaluate the filament thinning dynamics of aqueous polyethylene oxide
solutions ($1\text{\,}\mathrm{M\text{Da}}\text{/}$) over dilute and entangled
regimes. As the concentration increases, the profiles deviate from the well-
studied exponential-thinning trends beyond the entangled threshold, becoming
increasingly power-law in character.

###### keywords:

American Chemical Society, LaTeX

Massachusetts Institute of Technology] Hatsopoulos Microfluids Laboratory,
Department of Mechanical Engineering, Massachusetts Institute of Technology,
Cambridge, MA 02139, United States Ford Motor Company]Ford Research and
Innovation Center, Dearborn, MI 48124, United States Ford Motor Company]Ford
Research and Innovation Center, Dearborn, MI 48124, United States Ford Motor
Company]Ford Research and Innovation Center, Dearborn, MI 48124, United States
Massacbhusetts Institute of Technology] Hatsopoulos Microfluids Laboratory,
Department of Mechanical Engineering, Massachusetts Institute of Technology,
Cambridge, MA 02139, United States

##  1 Introduction

It is well known that the extensional rheological behavior of entangled
polymer solutions or polymer melts exhibit distinct properties compared with
those of dilute polymer solutions 1. In these entangled systems, strong
intermolecular interactions result in more complex microstructural dynamics
that lead to distinct relaxation mechanisms over a wide range of time- and
lengthscales during material deformation. Consequently, most entangled polymer
systems exhibit strongly nonlinear rheological properties, such as rate-
thinning viscosities, stress overshoots following step-strains and onset of
transient elastic instabilities 1, 2. Numerous industrial processes depend on
understanding the non-linear extensional response of these materials,
including polymer fiber spinning, food processing, injection or plastic blow
molding and filament extrusion 3.

Over the past few decades, extensive efforts have been made to characterize
and model the rheological complexity of entangled polymer systems 1.
Significant progress has been made since the milestone work of reptation
theory, which provides a coarse-grained canonical framework to understand the
dynamical response of entangled polymer chains 4. Each reptating polymer chain
is envisioned to be fully or partially confined by surrounding polymer chains
in a mean-field imaginary tube. The entangled chains interact with each other
at their topological crossovers, or entanglements, and these geometrical
constraints inhibit the transverse motion of a single chain and only allow for
diffusive motion, or “reptation,” along the imaginary tube with a greatly
reduced diffusivity 5, 6. This low diffusivity in the tube results in a
significant slowdown in the dominant polymer relaxation dynamics. Based on
this concept, Doi and Edwards 7 proposed a full-dimensional constitutive model
for monodisperse entangled linear polymers derived from the Lodge rubberlike
liquid 8. In the original Doi-Edwards (DE) model, the tube segments are
reoriented in the flow direction based on the “independent alignment
approximation (IAA).” Under the IAA approximation, the stress originates from
the affine transformation of the tube segments instead of the tube stretch.
Because of the extensibility of tube segment is not considered, the primitive
chains between entanglements are simplified as rigid rods. The IAA only
applies to an intermediate strain-rate range of
$1/\lambda_{\mathrm{D}}\lesssim\dot{\gamma}\ll 1/\lambda_{\mathrm{R}}$ in the
stress relaxation process 1. Here, $\lambda_{\mathrm{R}}$ is the Rouse time
quantifying the time for retraction of a single polymer chain in the tube, and
$\lambda_{\mathrm{D}}$ is the disengagement time characterizing the timescale
over which a single polymer chain reptates out of its constraining tube. These
two timescales can be connected through the relationship 9, 10 as

$\displaystyle\lambda_{\mathrm{D}}$
$\displaystyle=p(Z)\lambda_{\mathrm{R}}=p(Z)\dfrac{\xi_{0}(Nb)^{2}}{6\pi^{2}kT},$
(1a) $\displaystyle p(Z)$
$\displaystyle=3Z(1-\dfrac{3.38}{\sqrt{Z}}+\dfrac{4.17}{Z}-\dfrac{1.55}{Z^{3/2}}),$
(1b)

where the Rouse time $\lambda_{\mathrm{R}}$ is explicitly defined as a
function of the monomeric friction coefficient $\xi_{0}$ and the statistical
contour length $Nb$ 11. $Z=M/M_{\mathrm{e}}$ is the number of entanglements
per polymer chain 2, and $M$ and $M_{\mathrm{e}}$ are the molecular weights of
a single polymer chain and one entangled tube segment, respectively. The form
of $p(Z)$ is a truncated Taylor expansion of a function of $Z^{-1/2}$
numerically calculated by Likhtman and McLeish 10 to incorporate the effects
of contour length fluctuation (CLF) in polymer chains of realistic lengths and
numbers of entanglements.

The DE model successfully captures a number of key rheological features of an
entangled polymer system, such as a molecular-weight independent storage
modulus, a non-trivial second-order normal stress difference in steady shear
flow and strain-softening behavior in a step-strain experiment at large
strains 12, 2. However, this model is well-known to underpredict the scaling
of the zero-shear viscosity with the molecular weight, giving $\eta_{0}\propto
M^{3}$, compared with $\eta_{0}\propto M^{3.4}$ from experiments. The DE model
also predicts excessive levels of shear- and extension-thinning in steady
shear and extensional flows respectively, leading to a strong susceptibility
to flow instabilities 2, 4. Subsequent studies have attributed these
deficiencies in the DE model to the absence of two non-reptative contributions
to the chain dynamics: (1) a finite rate of chain retraction on short
timescales or high strain rates (i.e., $t<\lambda_{\mathrm{R}}$ or
$\lambda_{\mathrm{R}}\dot{\gamma}>1$), and (2) convective constraint release
on longer timescales of a relaxation process (i.e., $t>\lambda_{\mathrm{D}}$
or $1/\lambda_{\mathrm{D}}<\dot{\gamma}<1/\lambda_{\mathrm{R}}$) 13. These
deficiencies have subsequently been accounted for in a number of more
sophisticated constitutive models based on the underlying micro-mechanical
framework of reptation, including the Doi-Edwards-Marrucci-Grizzuti model 13,
the GLaMM model 14 and the Rolie-Poly model 15.

Over the past twenty years, the capillarity-driven thinning technique has
provided an accurate and efficient protocol to probe the transient extensional
rheological properties of complex fluids. In this technique, a small initially
cylindrical fluid sample is placed between two coaxial discs and then
stretched by a fast axial strain. The resulting deformation - driven by the
Rayleigh-Plateau instability - produces a thin, elongated filament, which
subsequently undergoes a transient self-thinning process through the action of
capillarity. When inertial and gravitational effects are negligible, the
temporal evolution of the filament profile, resulting from the balance between
the capillary pressure and the viscoelastic response from the fluid, can be
used to extract a measure of the transient extensional rheological properties
of the material 16. A number of variations of the capillarity-driven thinning
technique are now available. Examples include the Capillary Breakup
Extensional Rheometer (CaBER) 16, the Rayleigh-Ohnesorge Jetting Extensional
Rheometer (ROJER) 17 and the Dripping-On-Substrate (DOS) method 18. These
techniques enable determination of the extensional rheological properties for
a number of complex material systems, including Newtonian fluids 19, dilute
polymer solutions 20, 21, 22, 23, yield-stress fluids and emulsions 24, 25,
26, particulate suspensions 27, 28, 29, magnetorheological fluids 30, as well
as more complex systems such as cellulose solutions 31, liquid food additives
32, 33 and cosmetic products such as nail varnishes 34.

However, an explicit calculation of the filament thinning dynamics for
entangled polymer systems has not been presented. A handful of studies 35, 36,
37, 38 have experimentally characterized the capillarity-driven thinning
dynamics of aqueous polyethylene oxide (PEO), polystyrene (PS), hydroxy-ethyl
cellulose (HEC), and wormlike micelle solutions across the dilute and
entangled regimes. As the concentration increases beyond the entanglement
threshold, the temporal evolution of the mid-filament radius measured
experimentally deviates notably from the well-studied exponential thinning
result described by an elasto-capillary balance of the Hookean dumbbell model
20, 39. Observations reveal two unique features in the transient extensional
rheological behavior of the entangled regime. First, the apparent extensional
relaxation time $\lambda_{\mathrm{e}}$ (fitted from the exponential-thinning
evolution of the filament radius) is evidently smaller than that independently
extracted from shear flow (which we denote here as $\lambda_{\mathrm{s}}$),
and the ratio of the two relaxation times
$\lambda_{\mathrm{e}}/\lambda_{\mathrm{s}}$ determined in shear and extension
is found to decrease at higher concentrations. Arnold et al. 35 have suggested
the use of a damping function of the K-BKZ form to reconcile the different
relaxation times in shear and extensional flows. However, additional fitting
parameters are introduced to the resulting non-linear constitutive models, and
it remains unclear if these parameters are robust to material variations and
concentration changes, without a clear physical interpretation of the damping
terms. Secondly, the temporal evolution of the filament thinning in the
entangled regime progressively shifts away from the exponential response
expected for a dilute solution 20 towards a power-law relationship between the
mid-plane radius and the time to singularity. This power-law relationship
concomitantly results in transient extensional-thinning behavior in the
extensional viscosity over a wide range of strain rates 16. A phenomenological
inelastic models have been applied by a recent study to explain this behavior
38. However, the microstructural origin of this rate-thinning trend has not
been considered, and a microscopically-orientated model that incorporates the
evolution in the polymer conformation inside the thinning fluid element is
lacking, inhibiting the extraction of accurate constitutive parameters from
the filament thinning response.

To address this limitation, in the present work, we calculate numerically the
capillarity-driven thinning dynamics of entangled polymer solutions governed
by two representative microstructural-based constitutive models derived from
reptation theory. We show that the resulting filament thinning profiles
exhibit a complex three-stage evolution dominated by tube reorientation,
polymer chain stretching and ultimately the finite extensibility of a single
polymer chain, respectively. The asymptotic solutions of the filament thinning
profiles in each stage are calculated analytically. The predictions for the
transient shear and extensional rheological response agree with the
experimental observations for a number of entangled polymer solutions reported
in previous studies without introducing additional fitting parameters. To
illustrate the general trends, we present a dimensional prediction of the
filament thinning profiles for a family of aqueous polyethylene oxide (PEO)
solutions with varying concentrations in a transient extensional flow below,
at, or above, the entangled threshold. The close similarity of the predicted
filament thinning profiles with published experimental data provides
additional physical insights into the process of the capillarity-driven
thinning for an expanded variety of complex polymeric fluids.

##  2 Numerical Model and Analysis

The geometrical configuration of the capillarity-driven thinning process is
illustrated in Fig. 1. A long thin cylindrical liquid filament that exceeds
the Rayleigh-Plateau stability limit for a given fluid volume and aspect ratio
40 is generated by a rapid separation from the two coaxial discs. The temporal
evolution of the filament radius $R(t)$ results from a balance between the
capillary pressure and the viscoelastic stress arising from the fluid
deformation.

Figure 1: Geometrical setup of the capillarity-driven thinning process for a
viscoelastic fluid. A thin cylindrical filament forms between two quasi-static
hemispherical reservoirs and undergoes a self-thinning process driven by the
capillary pressure and resisted by the viscoelasticity stress generated in the
deforming thread. The extensional rheological parameters of the fluid can be
extracted from measurements of the temporal evolution of the filament radius
$R(t)$.

In Fig. 1, the assumption of a slender cylindrical filament is valid when the
filament thinning process is dominated by an elasto-capillary balance, and the
stress components satisfy $\sigma_{zz}\gg\sigma_{rr}$ 20, 41. In the absence
of inertial and gravitational effects, an explicit solution of $R(t)$ can be
obtained by equating the material response to the filament thinning with the
external driving stresses as 19

$\eta_{\mathrm{e}}^{+}\dot{\epsilon}=N_{1}^{+}=\frac{F_{\mathrm{a}}(t)}{\pi
R^{2}}-\Gamma K.$ (2)

In this equation, the transient normal stress difference,
$N_{1}^{+}=\sigma_{zz}^{+}-\sigma_{rr}^{+}$, describes the overall resistance
to the filament thinning from the material response, which can be equivalently
represented by the transient extensional viscosity $\eta_{\mathrm{e}}^{+}$.
When the filament is close to cylindrical, the strain rate in a material
element of size $\mathrm{d}V=\pi R^{2}\mathrm{d}z$ at any $z$ along the
filament can be expressed 16 as

$\dot{\epsilon}=-\dfrac{2\dot{R}(t)}{R(t)}.$ (3)

The time-varying external axial force on the filament $F_{\mathrm{a}}(t)$ can
be expressed from dimensional analysis in the form $F_{\mathrm{a}}(t)\equiv
2\pi X\Gamma R(t)$, where $X$ is a dimensionless geometric correction factor
to account for the slenderness of filament shapes that are not perfectly
cylindrical (i.e., $X=1$ for cylindrical shapes). The capillary pressure is
derived from the surface tension $\Gamma$ and the mean curvature $K$ of the
liquid filament, which can be expressed as

$K=\dfrac{1}{R(1+R_{,z}^{2})^{1/2}}+\dfrac{R_{,zz}}{(1+R_{,z}^{2})^{3/2}}=\dfrac{1}{R},$
(4)

where $R_{,z}\equiv\partial R/\partial z=0$ and
$R_{,zz}\equiv\partial^{2}R/\partial z^{2}=0$ due to the assumption of
cylindrical filament shapes.

From Eq. 2, the evolution of the filament radius under an elasto-capillary
balance governed by the Hookean dumbbell model can be expressed 20, 21, 42 as

$R(t)=\left(\frac{GR_{0}^{4}}{2\Gamma}\right)^{1/3}\exp{\left(-\frac{t}{3\lambda}\right)},$
(5)

where $G$ and $\lambda$ are the modulus and relaxation time characterizing the
Hookean dumbbell constitutive equation, and $R_{0}$ is the filament radius at
the onset of filament thinning. The Hookean dumbbell model has been applied
extensively to describe the shear and transient extensional rheological
responses for a number of dilute polymer solutions 39, 43, 16, 22, 21. In
extensional flow at a constant strain rate, the Hookean dumbbell model shows
an unbounded stress growth, which results in an exponential thinning in the
radius $R(t)$ that is controlled by the relaxation time $\lambda$ as described
by Eq. 5. This timescale is also consistent with the relaxation timescale in
shear flow predicted by the same constitutive model. Finally, we note that
substituting Eq. 5 into the expression for $\dot{\epsilon}(t)$ given above
results in a constant strain rate $\dot{\epsilon}=2/(3\lambda)$, or in a
dimensionless form using the Weissenberg number as
$\mathrm{Wi}\equiv\lambda\dot{\epsilon}=2/3$.

###  2.1 Tube models

####  2.1.1 Doi-Edwards-Marrucci-Grizzuti (DEMG) model

Marrucci and Grizzuti 13 modified the original Doi-Edwards (DE) model by
introducing an additional evolution equation to describe the polymer chain
stretch. As a result, in conjunction with the tube reorientation and chain
reptation that are characterized by the disengagement time
$\lambda_{\mathrm{D}}$, stretching of the flexible polymer chain introduces a
second, independent relaxation process that becomes pivotal at higher strain
rates 44. This new relaxation process quantifies the stretching of the polymer
chain along its primitive path length within an orientated tube segment and is
characterized by the Rouse time of the polymer chain $\lambda_{\mathrm{R}}$.
Pearson et al. 45 proposed a closed differential form of the constitutive
equation for the DEMG model given by

$\displaystyle\bm{S_{(1)}}$ $\displaystyle=-2(\bm{\nabla
v}^{\mathrm{T}}:\bm{S})\bm{S}-\dfrac{1}{\Lambda^{2}\lambda_{\mathrm{D}}}(\bm{S}-\dfrac{1}{3}\bm{I}),$
(6a) $\displaystyle\dot{\Lambda}$ $\displaystyle=\Lambda(\bm{\nabla
v}^{\mathrm{T}}:\bm{S})-\dfrac{f(\Lambda)}{\lambda_{\mathrm{R}}}(\Lambda-1),$
(6b) $\displaystyle\bm{\sigma}$
$\displaystyle=3G_{\mathrm{N}}f(\Lambda)\Lambda^{2}\bm{S}.$ (6c)

In Eq. 6, $\bm{S}=\left<\bm{uu}\right>$ is the ensemble average of the tube
orientation tensor, where $\bm{u}$ is the end-to-end vector for each tube
segment. The scalar $\Lambda$ describes the overall strain imposed on the
polymer chains within the tubes. The subscript “$(1)$” indicates the first-
order upper-convected derivative 8, and $\bm{\nabla v}$ is the velocity
gradient tensor. The finite extensibility factor for a single polymer chain
$f(\Lambda)$ is given by the inverse Langevin function which can be
approximated in a simple explicit form using Cohen’s Padé approximation 46 as

$f(\Lambda)=\frac{1-1/\Lambda_{\mathrm{m}}^{2}}{3-1/\Lambda_{\mathrm{m}}^{2}}\cdot\frac{3-\Lambda^{2}/\Lambda_{\mathrm{m}}^{2}}{1-\Lambda^{2}/\Lambda_{\mathrm{m}}^{2}},$
(7)

where $\Lambda_{\mathrm{m}}$ describes the maximum polymer extensibility which
scales with the molecular weight as $\Lambda_{\mathrm{m}}\sim M^{1/2}$. From
Eq. 6(a), it can be seen that the evolution of the orientation tensor $\bm{S}$
is a result of the interplay between the imposed convective flow and the tube
relaxation characterized by the disengagement time $\lambda_{\mathrm{D}}$. By
definition, the tube orientation tensor is constrained by
$\operatorname{tr\hskip 1.0pt}\bm{S}\equiv 1$ and $\bm{S}:\bm{S}\leq 1$. When
$\bm{S}=\bm{I}/3$, all tube segments are, on average, randomly oriented 13.

To calculate the capillarity-driven thinning dynamics of a thin fluid filament
described by the DEMG model, we substitute Eq. 6 into Eq. 2. For mathematical
simplicity, we non-dimensionalize the expression according to $\hat{R}\equiv
R/R_{0}$ and $\hat{t}\equiv t/\lambda_{\mathrm{R}}$, where $R_{0}=R(0)$ is the
filament radius at $t=0$. The dimensionless stress balance equation can be
expressed as

$\dfrac{1}{\hat{R}}=3\mathrm{Ec}_{0}f(\Lambda)\Lambda^{2}\Delta S,$ (8)

where $\Delta S\equiv S_{zz}-S_{zz}$ is the magnitude of the tube
reorientation calculated from the difference of the $zz$\- and $rr$-components
of the reorientation tensor, and $\mathrm{Ec}_{0}\equiv
G_{\mathrm{N}}R_{0}/\Gamma$ is the intrinsic elasto-capillary number of the
fluid, where $G_{\mathrm{N}}$ is the viscoelastic plateau modulus of the
entangled fluid system. Because of the axisymmetric shape of the filament and
the irrotational nature of the extensional flow, the $rr$\- and
$\theta\theta$-components of the reorientation tensor are identical in
magnitude. Therefore, the trace of the reorientation tensor satisfies
$\operatorname{tr\hskip 1.0pt}\bm{S}=S_{zz}+2S_{rr}=1$, and the temporal
evolution of $\Delta S$ can be rewritten as

$\Delta S_{,\hat{t}}=\mathrm{Wi}(\Delta S+1)-2\mathrm{Wi}\Delta
S^{2}-\dfrac{1}{\Lambda^{2}p(Z)}\Delta S.$ (9)

Here, the Weissenberg number quantifying the strength of the flow is given by
$\mathrm{Wi}\equiv\lambda_{\mathrm{R}}\dot{\epsilon}=-2\hat{R}_{,\hat{t}}/\hat{R}$
according to Eq. 3. The ratio of timescales in Eq. 9 as
$p(Z)=\lambda_{\mathrm{D}}/\lambda_{\mathrm{R}}$ can be expressed as a
function of the number of entanglements per polymer chain $Z$ as calculated
from Eq. 1.

####  2.1.2 Rolie-Poly (RP) model

In addition to the finite rates of polymer chain stretching and retraction
within the tube that are incorporated in the DEMG model, convective constraint
release (CCR) has been recognized more recently as another critical non-
reptative mechanism that alters the non-linear rheology of entangled polymer
systems in both steady and transient flows 47, 48, 14, 11. CCR is manifested
at $\dot{\epsilon}>1/\lambda_{\mathrm{D}}$, when polymer disengagement from
the physically entangled network becomes faster than reptation due to the
surrounding polymer molecules being convected away from the chain of interest
by the strong flow. To obtain a more comprehensive understanding of how CCR
potentially affects the dynamics of capillarity-driven thinning, we adopt the
more sophisticated Rolie-Poly (RP) model proposed by Likhtman and Graham 15,
and compare the resulting temporal evolution of the filament thinning profiles
with those obtained from the DEMG model. A specific form of the constitutive
equation similar to Eq. 6 is taken that separates the tube reorientation and
the polymer chain stretch 11 according to

$\displaystyle\bm{S_{(1)}}$ $\displaystyle=-2(\bm{\nabla
v}^{\mathrm{T}}:\bm{S})\bm{S}-\dfrac{1}{\Lambda^{2}}\left[\left(\dfrac{1}{\lambda_{\mathrm{D}}}+2\beta
f(\Lambda)\dfrac{1-1/\Lambda}{\lambda_{\mathrm{R}}}\Lambda^{\delta}\right)(\bm{S}-\dfrac{1}{3}\bm{I})\right],$
(10a) $\displaystyle\dot{\Lambda}$ $\displaystyle=\Lambda(\bm{\nabla
v}^{\mathrm{T}}:\bm{S})-\dfrac{f(\Lambda)}{\lambda_{\mathrm{R}}}(\Lambda-1)-\left(\dfrac{1}{\lambda_{\mathrm{D}}}+2\beta
f(\Lambda)\dfrac{1-1/\Lambda}{\lambda_{\mathrm{R}}}\Lambda^{\delta}\right)\dfrac{\Lambda^{2}-1}{2\Lambda},$
(10b) $\displaystyle\bm{\sigma}$
$\displaystyle=3G_{\mathrm{N}}f(\Lambda)\Lambda^{2}\bm{S},$ (10c)

where the notations of $\bm{S}$, $\Lambda$ and $\bm{\nabla v}$ are identical
to those in Eq. 6, and the Padé approximation for the inverse Langevin
function $f=f(\Lambda)$ is evaluated from Eq. 7. The dimensionless
coefficients $\beta$ and $\delta$ are two model parameters that quantify the
magnitude of the CCR effect and the contribution of the polymer chain stretch
to the CCR effect, respectively. Previous studies have suggested that
$\beta=1$ and $\delta=-0.5$ produce the best-fitting results with the full
kinetic model over a wide range of shear rates 47, 15, and we thus adopt these
values in our calculation and asymptotic analysis.

Following the numerical procedure for the DEMG model, the differential
equation for the temporal evolution of the axial averaged tube orientation
$\Delta S\equiv S_{zz}-S_{rr}$ can be rewritten in a similar form as Eq. 9
with the addition of the CCR term as

$\Delta S_{,\hat{t}}=\mathrm{Wi}(\Delta S+1)-2\mathrm{Wi}\Delta
S^{2}-\dfrac{1}{\Lambda^{2}}\left[\dfrac{1}{p(Z)}+2\beta
f(\Lambda)(1-1/\Lambda)\Lambda^{\delta}\right]\Delta S,$ (11)

where the time-varying Weissenberg number is identically defined as in Eq. 9,
and the stress balance given by Eq. 8 remains unmodified.

###  2.2 Capillarity-driven thinning dynamics

Numerical calculations of the capillarity-driven thinning dynamics for the two
selected constitutive models have been performed for a range of representative
material parameters. The initial conditions for the reorientation tensor and
the chain stretch following the initial axial step strain that form that
liquid bridge are shared by both models. The reasons for these choices will be
explained in detail in the rest of this manuscript, and they are specified
according to the magnitude of the elasto-capillary number ($\mathrm{Ec}_{0}$)
as

$\displaystyle\mathrm{Ec}_{0}\geq\dfrac{1}{3}:$ $\displaystyle
S_{rr}(0)=\dfrac{1}{3}\left(1-\dfrac{1}{3\mathrm{Ec}_{0}}\right),$
$\displaystyle
S_{zz}(0)=\dfrac{1}{3}\left(1+\dfrac{2}{3\mathrm{Ec}_{0}}\right),$
$\displaystyle\Lambda(0)=1,$ $\displaystyle\mathrm{Ec}_{0}<\dfrac{1}{3}:$
$\displaystyle S_{rr}(0)=0,$ $\displaystyle S_{zz}(0)=1,$ $\displaystyle
f[\Lambda(0)]\Lambda(0)^{2}=\dfrac{1}{3\mathrm{Ec}_{0}}.$

In Fig. 2, the temporal evolution of the filament radius, the tube
reorientation and the chain stretch for both models (dashed line: DEMG model;
solid line: RP model) are presented. Here the elasto-capillary number
$\mathrm{Ec}_{0}$ varies from $\mathrm{Ec}_{0}=$0.12$$, and the number of
entanglements is fixed at $Z=10$. Here we initially set
$\Lambda_{\mathrm{m}}\rightarrow\infty$ because the finite extensibility of
the polymer chain only affects the filament thinning behavior very close to
the filament breakup and is thus not depicted in this figure. The overall
trends observed in the filament radius described by the two models are
similar, except that the RP model predicts a noticeably faster filament
thinning due to the additional CCR effect. By inspecting the temporal
evolution of the filament thinning profiles at $\mathrm{Ec}_{0}=2$ (dark
purple lines), we find two distinct filament thinning regimes. At an early
time, the filament radius decays slowly. In this stage, the magnitude of tube
reorientation $\Delta S\equiv S_{zz}-S_{rr}$ increases progressively to unity,
indicating full alignment in the extensional direction; the polymer molecules
however remain initially unstretched. As the magnitude of the tube
reorientation saturates to unity, the filament radius starts to decay at a
much faster rate in an exponential manner. The filament kinematics in this
fast-thinning stage can be attributed to the rapid polymer chain stretch. When
$\mathrm{Ec}_{0}=0.1<1/3$ (orange lines), the normal stress generated by the
tube orientation does not suffice to balance the driving capillary pressure
(i.e., Eq. 8 cannot be satisfied as written), and a nontrivial polymer stretch
at $t=0$ is necessary to produce a larger normal stress difference to balance
the driving capillary pressure, as specified from Eq. 12. Consequently, only
the second exponential-thinning regime is manifested.

Figure 2: Capillarity-driven thinning dynamics described by the two selected
models (dashed line: DEMG model; solid line: Rolie-Poly model) with varying
elasto-capillary numbers $\mathrm{Ec}_{0}=1/10$, $1/3$, $1$ and $2$ at a fixed
number of entanglements per polymer chain, $Z=10$
($\lambda_{\mathrm{D}}/\lambda_{\mathrm{R}}=8.97$). Here infinite
extensibility of the polymer chain ($\Lambda_{\mathrm{m}}\rightarrow\infty$)
is chosen for numerical simplicity. (a) The temporal evolution of the
dimensionless filament radius $\hat{R}\equiv R/R_{0}$ against $\hat{t}\equiv
t/\lambda_{\mathrm{R}}$ on a logarithmic scale predicted by both selected
models. When $\mathrm{Ec}_{0}>1/3$, the filament thinning profiles exhibit two
distinct stages. All the curves show an exponentially-thinning trend when the
filament radius is sufficiently small. (b) The temporal evolution of the
dimensionless radius ($\hat{R}\equiv R/R_{0}$) predicted by both selected
models on a linear scale. The thinning profile expected for a Newtonian fluid,
using the apparent viscosity from Eq. 13 is plotted for $\mathrm{Ec}_{0}=1$
and $\mathrm{Ec}_{0}=2$. (c) The temporal evolution of the tube orientation
$\Delta S\equiv S_{zz}-S_{rr}$ predicted by both selected models. (d) The
temporal evolution of polymer chain stretch $\Lambda$ showing an exponential
growth at long times, which is delayed when $\mathrm{Ec}_{0}>1/3$.

When plotted on a linear scale as shown in Fig. 2(b), the temporal evolution
of the filament radius exhibits a linear decrease for $\hat{R}\gtrsim 0.4$
when $\mathrm{Ec}_{0}$ is sufficiently large. Notably, the filament kinematics
in the early thinning regime are similar to those predicted from a Newtonian
fluid 19 with a thinning rate given by
$\dot{R}\sim-\Gamma/\eta_{\mathrm{app}}$. In practice, a measure of the shear
viscosity at low strain rates can be obtained from this linear trend assuming
the filament to be of cylindrical shape. This apparent shear viscosity can be
analytically calculated by imposing $\Lambda=1$ in the constitutive equations
(Eq. 6 and Eq. 10) as

$\mathrm{Tr}_{\mathrm{app}}\equiv\dfrac{\eta_{\mathrm{app}}}{G_{\mathrm{N}}\lambda_{\mathrm{D}}}=1+\dfrac{1}{6\mathrm{Ec}_{0}}-\dfrac{2}{9\mathrm{Ec}_{0}^{2}}.$
(13)

Eq. 13 is further plotted against $\mathrm{Ec}_{0}$ in Fig. 3. From this
figure, we notice a threshold of
$\mathrm{Ec}_{0}>\mathrm{Ec}_{0}^{*}=8/(3+\sqrt{297})\approx 0.395$ when a
valid apparent viscosity (positive) is obtained. This condition is
substantially identical to the piecewise initial conditions in Eq. 12, where
the tube reorientation dominates the filament thinning dynamics in the early
thinning regime for $\mathrm{Ec}_{0}>1/3$. As $\mathrm{Ec}_{0}$ increases,
this apparent shear viscosity $\eta_{\mathrm{app}}$ approaches the zero-shear
viscosity given by $\eta_{0}=G_{\mathrm{N}}\lambda_{\mathrm{D}}$.

Figure 3: Apparent shear viscosity $\eta_{\mathrm{app}}(\mathrm{Ec}_{0})$
obtained from the filament thinning profiles in the early thinning regime. A
meaningful viscosity reading (which must be positive) is obtained for
$\mathrm{Ec}_{0}>\mathrm{Ec}_{0}^{*}$ (indicated by the thin dashed line). As
$\mathrm{Ec}_{0}$ increases, the apparent shear viscosity measured by the
filament thinning process approaches the zero-shear viscosity
$\eta_{0}=G_{\mathrm{N}}\lambda_{\mathrm{D}}$ (thin solid line). Inset:
Schematic of the evolution of filament radius for
$\mathrm{Ec}_{0}<\mathrm{Ec}_{0}^{*}$ (I) and
$\mathrm{Ec}_{0}\geq\mathrm{Ec}_{0}^{*}$ (II). The dashed lines show reference
lines where $R(t)$ decreases linearly with $t$.

The exponential-thinning regime observed from the filament thinning profiles
of both models at intermediate time, as shown in Fig. 2, is reminiscent of the
filament thinning dynamics under the elasto-capillary balance predicted by the
Hookean dumbbell model for dilute polymer solutions. Nevertheless, it remains
unclear if the polymer relaxation process in the exponential-thinning trend
presented in this figure is consistent with that expected for the dilute
polymer systems. To obtain an accurate measure of the extensional rheological
properties from the capillarity-driven thinning dynamics, we revisit the
filament thinning profiles of both selected models with varying finite
extensibilities $\Lambda_{\mathrm{m}}$, a fixed elasto-capillary number
$\mathrm{Ec}_{0}=1$ and a fixed number of entanglements per polymer chain,
$Z=10$. As shown in Fig. 4, a consistent exponential-thinning trend can be
observed at intermediate time for the both models with an identical thinning
rate. From previous discussions, this exponential-thinning trend in the
filament radius is dominated by the exponentially increasing polymer chain
stretch $\Lambda$ induced by a strong extensional flow. As $\Lambda$
approaches the specified finite maximum chain stretch $\Lambda_{\mathrm{m}}$,
the temporal evolution of the filament radius substantially deviates from the
exponential thinning trend and the thread breaks in finite time.

Figure 4: Capillarity-driven thinning dynamics described by the two selected
models (dashed line: DEMG model; solid line: Rolie-Poly model) with varying
finite extensibility factors $\Lambda=$1050100$$ and
$\Lambda_{\mathrm{m}}\rightarrow\infty$ at a fixed elasto-capillary number,
$\mathrm{Ec}_{0}=1$ and a fixed number of entanglements per polymer chain,
$Z=10$. (a) The temporal evolution of the dimensionless filament radius
$\hat{R}\equiv R/R_{0}$ on a logarithmic scale at intermediate times. A
consistent exponential thinning trend is manifested for both models at
intermediate times; at longer times, the filament radius deviates
substantially from the exponential-thinning trend and rapidly approaches
singularity. An apparent extensional relaxation time
$\hat{\lambda}_{\mathrm{e}}$ (nondimensionalized by $\lambda_{\mathrm{R}}$)
can be obtained from fitting the exponentially-decaying region. A practical
optical resolution limit is indicated at $\hat{R}=0.001$. (b) The temporal
evolution of the polymer chain stretch $\Lambda$ on a logarithmic scale at
intermediate times. Initially the stretch increases exponentially; close to
the point of filament breakup, the polymer chain stretch $\Lambda$ approaches
the specified maximum stretch of $\Lambda_{\mathrm{m}}$ (thin horizontal
dashed lines).

While the capillarity-driven thinning dynamics predicted by these tube models
for entangled polymer solutions are distinct from those predicted by the
Hookean dumbbell models for dilute polymer solutions, it is common in practice
to extract an extensional relaxation time from the exponential-thinning
regime. To mitigate the ambiguity in the interpretation of the measurements
from the application of different models, we define an apparent extensional
relaxation time $\hat{\lambda}_{\mathrm{e}}$ (nondimensionalized by
$\lambda_{\mathrm{R}}$) that is obtained from fitting the data using the
Hookean dumbbell model, such that in the exponential-thinning regime,
$\hat{R}(\hat{t})\sim\exp[-\hat{t}/(3\hat{\lambda}_{\mathrm{e}})]$. For each
selected models, an analytical solution of this exponential-thinning process
can be obtained by solving Eq. 6, Eq. 8 and Eq. 10 in the limit of
$\mathrm{Ec}_{0}\rightarrow 0$ (negligible tube orientation) and
$\Lambda_{\mathrm{m}}\rightarrow\infty$ (infinite extensibility for polymer
chains). The apparent extensional relaxation times characterizing the
exponential thinning process in each model are thus found to be

$\displaystyle\hat{\lambda}_{\mathrm{e,DEMG}}$
$\displaystyle\equiv\dfrac{\lambda_{\mathrm{e,DEMG}}}{\lambda_{\mathrm{R}}}=\dfrac{1}{2},$
(14a) $\displaystyle\hat{\lambda}_{\mathrm{e,RP}}$
$\displaystyle\equiv\dfrac{\lambda_{\mathrm{e,RP}}}{\lambda_{\mathrm{R}}}=\dfrac{1}{2+1/p(Z)},$
(14b)

where $p(Z)=\lambda_{\mathrm{D}}/\lambda_{\mathrm{R}}$ is the ratio of the
disengagement time and the Rouse time expressed in Eq. 1. These expressions
are plotted in Fig. 5. It is evident that the apparent extensional relaxation
times $\lambda_{\mathrm{e}}$ measured from the filament thinning profiles are
primarily determined by the magnitude of the Rouse time $\lambda_{\mathrm{R}}$
in the entangled fluid. As $Z\rightarrow\infty$, both expressions for
$\lambda_{\mathrm{e}}$ approach an identical limit of
$\lambda_{\mathrm{R}}/2$. The apparent extensional relaxation time from the RP
model includes a coupling term that depends on the disengagement time
$\lambda_{\mathrm{D}}$ (through $p(Z)$). This term arises from CCR but only
becomes comparable in magnitude with the Rouse time contribution when $Z$ is
relatively small. In this limit, polymer chain relaxation is further promoted
by the convected displacement of surrounding polymer molecules in the
transient extensional flow.

Figure 5: The dimensionless apparent extensional relaxation time
$\hat{\lambda}_{\mathrm{e}}=\lambda_{\mathrm{e}}/\lambda_{\mathrm{R}}$
extracted from the exponential-thinning regime of the filament thinning
dynamics for both selected models. For the DEMG model (dashed line),
exponentially fitting always gives
$\lambda_{\mathrm{e}}=\lambda_{\mathrm{R}}/2$. For the Rolie-Poly model (solid
line), a monotonic trend is observed, approaching the DEMG model as
$Z\rightarrow\infty$. The result for $Z<2$ is shown as a dashed line due to
the dubious accuracy of the Taylor series expansion (Eq. 1) in this limit 10.
The blue marker shows the result for the Rolie-Poly model at $Z=10$ used in
Fig. 2, 4 and 6.

Finally, we numerically calculate the filament thinning profiles close to the
final breakup event, where the polymer chain stretch $\Lambda$ approaches the
maximum extensibility $\Lambda_{\mathrm{m}}$ and the temporal evolution of the
filament radius deviates from the exponential-thinning trend. For clarity in
the subsequent analysis, we transform the time axis by plotting the results
against $\hat{\tau}\equiv\hat{t}_{\mathrm{C}}-\hat{t}$, where
$\hat{t}_{\mathrm{C}}$ is the time of filament breakup (nondimensionalized by
$\lambda_{\mathrm{R}}$) which can be numerically obtained from extrapolation
of the filament radius to $\hat{R}=0$. As shown in Fig. 6, the temporal
evolution of $\hat{R}(\hat{t})$ with the same parameter values as in Fig. 4 is
plotted for the two selected models. As the filament approaches singularity
($\hat{t}\rightarrow\hat{t}_{\mathrm{C}}$, or $\hat{\tau}\rightarrow 0$),
finite polymer extensibility progressively dominates the filament thinning
dynamics. Consequently, the filament radius undergoes a linear decay, and the
asymptotic predictions of the two models at identical values of finite
extensibility $\Lambda_{\mathrm{m}}$ overlap with each other.

The linear decay of the filament radius close to the filament breakup has also
been observed for the FENE-P model 42. This asymptotic linear variation is
consistent with the capillarity-driven thinning dynamics of a highly
anisotropic Newtonian fluid 19. Consequently, we can extract the terminal
extensional viscosity $\eta_{\mathrm{e,\infty}}$ predicted by the models in
this linearly-thinning regime by rewriting Eq. 2 and Eq. 8 in form of Taylor
expansion regarding to $1/\Lambda$ (where $\Lambda<\Lambda_{\mathrm{m}}$).
Consequently, an analytical expression for the terminal extensional viscosity
can be obtained as

$\eta_{\mathrm{e,\infty}}=\dfrac{3G_{\mathrm{N}}\lambda_{\mathrm{R}}\Lambda_{\mathrm{m}}^{2}}{1-1/\Lambda_{\mathrm{m}}}.$
(15)

The asymptotic solutions for the filament radius using
$\eta_{\mathrm{e,\infty}}$ with varying finite extensibilities
$\Lambda_{\mathrm{m}}$ are shown in Fig. 6 as thin dashed lines, and they are
consistent with the full numerical calculations of the filament thinning
dynamics close to breakup.

Figure 6: Evolution of the filament radius close to breakup with varying
finite extensibility factors $\Lambda_{\mathrm{m}}=$1050100$$ at a fixed
elasto-capillary number $\mathrm{Ec}_{0}=1$ and fixed number of entanglements
per polymer chain $Z=10$. The filament radius is plotted against
$\hat{\tau}\equiv\hat{t}_{\mathrm{C}}-\hat{t}$, where $\hat{t}_{\mathrm{C}}$
is critical time of breakup (nondimensionalized by $\lambda_{\mathrm{R}}$).
Linear decrease of the filament radius is observed close the breakup,
consistent with the filament thinning response of a Newtonian fluid. As a
result, a terminal extensional viscosity $\eta_{\mathrm{e,\infty}}$ can be
obtained analytically close to singularity. Asymptotic solutions of the
filament radius are plotted as thin dashed lines. A practical optical limit of
$\hat{R}=0.001$ is also indicated.

##  3 Discussion

###  3.1 Ratio of the apparent extensional and shear relaxation times

In practice, the shear relaxation time $\lambda_{\mathrm{s}}$ is routinely
obtained from the small amplitude oscillatory shear (SAOS) as
$1/\omega_{\mathrm{c}}$, where $\omega_{\mathrm{c}}$ is the characteristic
angular frequency at which the storage and loss moduli cross. For model
dilute, monodisperse polymer solutions that can be well described by the
Hookean or FENE dumbbell model, there is a single characteristic relaxation
time that characterizes the polymer chain dynamics regardless of the flow type
(shear or extension). However, a number of previous studies of the
capillarity-driven thinning dynamics observed in different entangled polymer
solutions have revealed distinctly different measures of the relaxation time
in shear and extensional flow 35, 36, 37, 38. A simple dumbbell model cannot
readily reconcile these differences in the reported relaxation times from a
physical perspective.

In this section, we establish a theoretical framework using the tube models
that allows us to clearly interpret the difference in shear and extensional
relaxation times observed in entangled polymer solutions. We have shown in
Section 2.2 that the temporal evolution of the filament radius resulted from
capillarity-driven thinning features an exponential-thinning trend at
intermediate time. In this stage, chain extension and the Rouse time
$\lambda_{\mathrm{R}}$ dominate the filament thinning process and controls the
magnitude of the apparent extensional relaxation time. In contrary, the shear
relaxation time can be analytically calculated by imposing a small-amplitude
oscillatory shear strain to the DEMG and the RP models. If only the leading-
order terms are retained, both tube models reduce to the Maxwell model with a
longest relaxation time $\lambda_{\mathrm{D}}$ and modulus
$G=3G_{\mathrm{N}}$. This result is not surprising, because at small strains,
the rheology is dominated by the processes of tube deformation and
reorientation, which is consistent with the relaxation mode described by the
disengagement time $\lambda_{\mathrm{D}}$. To summarize, we compare the
expected apparent shear ($\lambda_{\mathrm{s}}$) and extensional
($\lambda_{\mathrm{e}}$) relaxation times of the Hookean dumbbell model and
the two selected tube models in Table 1. Without introducing additional
fitting parameters, we can quantify the ratio of the two relaxation times
using the constitutive parameters of the tube models.

Table 1: Dynamic moduli $G^{\prime}$ and $G^{\prime\prime}$, the crossover angular frequency $\omega_{\mathrm{c}}$ where $G^{\prime}=G^{\prime\prime}$, and the apparent shear and extensional relaxation times $\lambda_{\mathrm{s}}=1/\omega_{\mathrm{c}}$ and $\lambda_{\mathrm{e}}$ for the Hookean dumbbell model, the DEMG model, and the Rolie-Poly model. The apparent extensional relaxation time in the limit of $Z\rightarrow\infty$ is also shown. Model |  $G^{\prime}(\omega)$ |  $G^{\prime\prime}(\omega)$ |  $\omega_{\mathrm{c}}$ |  $\lambda_{\mathrm{s}}$ |  $\lambda_{\mathrm{e}}$ |  $\lambda_{\mathrm{e}}(Z\rightarrow\infty)$
---|---|---|---|---|---|---
Hookean dumbbell |  $\dfrac{G\lambda^{2}\omega^{2}}{1+(\lambda\omega)^{2}}$ |  $\dfrac{G\lambda\omega}{1+(\lambda\omega)^{2}}$ |  $\dfrac{1}{\lambda}$ |  $\lambda$ |  $\lambda$ |  $\lambda$
DEMG |  $\dfrac{G_{\mathrm{N}}\lambda_{\mathrm{D}}^{2}\omega^{2}}{1+(\lambda_{\mathrm{D}}\omega)^{2}}$ |  $\dfrac{G_{\mathrm{N}}\lambda_{\mathrm{D}}\omega}{1+(\lambda_{\mathrm{D}}\omega)^{2}}$ |  $\dfrac{1}{\lambda_{\mathrm{D}}}$ |  $\lambda_{\mathrm{D}}$ |  $\dfrac{1}{2}\lambda_{\mathrm{R}}$ |  $\dfrac{1}{2}\lambda_{\mathrm{R}}$
Rolie-Poly |  $\dfrac{G_{\mathrm{N}}\lambda_{\mathrm{D}}^{2}\omega^{2}}{1+(\lambda_{\mathrm{D}}\omega)^{2}}$ |  $\dfrac{G_{\mathrm{N}}\lambda_{\mathrm{D}}\omega}{1+(\lambda_{\mathrm{D}}\omega)^{2}}$ |  $\dfrac{1}{\lambda_{\mathrm{D}}}$ |  $\lambda_{\mathrm{D}}$ |  $\dfrac{\lambda_{\mathrm{R}}}{2+1/p(Z)}$ |  $\dfrac{1}{2}\lambda_{\mathrm{R}}$

For entangled polymer solutions, the molecular weight between entanglements
$M_{\mathrm{e}}(c)$ is, of course, larger than reported for polymer melts due
to dilution of the polymer chains. To incorporate the dependence on polymer
concentration in our expression for the relaxation time ratio, we refer to
Pearson et al. 49 and Ravindranath et al. 50 and calculate the effective
molecular weight between entanglements in the entangled solutions as

$M_{\mathrm{e}}(c)=M_{\mathrm{e,0}}c^{\frac{1}{1-3\nu}},$ (16)

where $c$ is the mass fraction of the polymers in solution, and
$M_{\mathrm{e,0}}$ is the molecular weight between entanglements in the
polymer melts ($c=1$). The excluded volume parameter $\nu=0.5$ for
$\theta$-solvents, and $\nu=0.6$ for good solvents. From Eq. 1, the number of
entanglements per chain in the entangled polymer solutions can be there
expressed as

$Z_{\mathrm{sol}}(c)=\dfrac{M}{M_{\mathrm{e,0}}}c^{\frac{1}{3\nu-1}}\equiv
Z_{0}c^{\frac{1}{3\nu-1}}.$ (17)

From Eq. 17, we can thus evaluate the evolution in relaxation time ratio
$\lambda_{\mathrm{e}}/\lambda_{\mathrm{s}}$ as a function of the number of
entanglements per chain $Z_{\mathrm{sol}}(c)$ as

$\dfrac{\lambda_{\mathrm{e}}}{\lambda_{\mathrm{s}}}=\dfrac{\hat{\lambda}_{\mathrm{e}}[Z_{\mathrm{sol}}(c)]}{p[Z_{\mathrm{sol}}(c)]}=\begin{cases}\dfrac{1}{6Z_{\mathrm{sol}}\bigg{(}1-\dfrac{3.38}{\sqrt{Z_{\mathrm{sol}}}}+\dfrac{4.17}{Z_{\mathrm{sol}}}-\dfrac{1.55}{Z_{\mathrm{sol}}^{3/2}}\bigg{)}},&\text{(DEMG)}\\\
\dfrac{1}{6Z_{\mathrm{sol}}\bigg{(}1-\dfrac{3.38}{\sqrt{Z_{\mathrm{sol}}}}+\dfrac{4.17}{Z_{\mathrm{sol}}}-\dfrac{1.55}{Z_{\mathrm{sol}}^{3/2}}\bigg{)}+1},&\text{(Rolie-
Poly)}\\\ \end{cases}$ (18)

where the dimensionless apparent extensional relaxation time
$\hat{\lambda}_{\mathrm{e}}$ is expressed explicitly in Eq. 14b, and the ratio
of the disengagement time and the Rouse time $p(Z)$ is expressed in Eq. 1.

To substantiate the analytical prediction of Eq. 18, we revisit the
experimental data from two previous studies 35, 36 with two different polymer
solutions at varying molecular weights and concentrations. To ensure the
validity of the tube models, only measurements reported in the entangled
regime ($c>c_{\mathrm{e}}$) are considered. The material properties and the
experimental measurements of the selected systems are shown in Table 2. The
molecular weight between entanglements $M_{\mathrm{e}}$ is evaluated at a
temperature of $20\text{\,}\mathrm{\SIUnitSymbolCelsius}\text{/}$, and the
number of entanglements per chain for the corresponding polymer melts $Z_{0}$
is calculated based on the assumption of monodispersity.

Table 2: Material properties, concentrations and experimentally measured ratios of the apparent shear and extensional relaxation times $\lambda_{\mathrm{e}}/\lambda_{\mathrm{s}}$ for the selected material systems from the previous studies. Materials |  $M_{\mathrm{e}}$ (g/mol) |  $M$ (g/mol) |  $c$ (wt%) |  $\lambda_{\mathrm{e}}/\lambda_{\mathrm{s}}$
---|---|---|---|---
PEO/Water 35 |  $2200$ 51 |  $1\times 10^{6}$ |  $2.5$ |  $0.168\pm 0.042$
$3.0$ |  $0.197\pm 0.049$
$2\times 10^{6}$ |  $1.5$ |  $0.125\pm 0.031$
$2.0$ |  $0.104\pm 0.026$
PEO/Water 36 |  $2200$ |  $1\times 10^{6}$ |  $2.0$ |  $0.34$
$2.5$ |  $0.24$
$3.0$ |  $0.24$
$3.5$ |  $0.15$
$2\times 10^{6}$ |  $1.5$ |  $0.12$
$2.0$ |  $0.10$
$4\times 10^{6}$ |  $1.0$ |  $0.10$
$1.5$ |  $0.05$
$2.0$ |  $0.02$
PS/DEP 36 |  $16600$ 52 |  $3\times 10^{6}$ |  $3.0$ |  $0.71$
$4.0$ |  $0.50$
$5.0$ |  $0.24$
PS/DEP 53 |  $16600$ |  $13.2\times 10^{6}$ |  $1.41$ |  $0.36$
$1.77$ |  $0.31$

We compare the experimental data reported in Table 2 with the analytical
prediction of Eq. 18 for the two tube models (dashed line: DEMG model; solid
line: RP model) in Fig. 7. Both models predict an overall descending trend for
the ratio of the extensioanl and shear relaxation times with increasing number
of entanglements per chain. Both predictions are applicable for entangled
polymer solutions within the range of $2<Z_{\mathrm{sol}}(c)<100$ due to the
validity of Eq. 1 10. As $Z_{\mathrm{sol}}(c)$ increases, the two predictions
coincide and the ratio of relaxation times asymptotically approaches a scaling
relation of $\lambda_{\mathrm{e}}/\lambda_{\mathrm{s}}\sim
Z_{\mathrm{sol}}^{-1}$. From Fig. 7, it is clear that the experimental data of
varying polymers, molecular weights and concentrations are broadly
characterized by a single decreasing trend, which agrees well with the
predictions from the two selected tube models without any additional fitting
parameters. This overall consistency between the experimental data and the
analytical prediction lines provides a physical insight into the origin of the
substantial differences in the shear and extensional relaxation times reported
in earlier studies: The tube networks constitute the primary topology of the
entangled polymer systems and effectively shield the contribution to the shear
rheology from a single polymer chain at small shear strains. However, the
integrity of these tube structures becomes increasingly susceptible to a
strong extensional flow as the tubes are forced to align. During this process,
the free energy of each polymer chain decreases, and the material relaxation
becomes progressively independent of the tube structures that are manifested
at a larger length scale.

Figure 7: The ratio of apparent extensional and shear relaxation times
$\lambda_{\mathrm{e}}/\lambda_{\mathrm{s}}$ against the number of
entanglements in entangled polymer solutions
$Z_{\mathrm{sol}}(c)=Z_{0}^{1/(3\nu-1)}$. The dashed and solid lines
correspond to the prediction lines from the DEMG and the Rolie-Poly models,
respectively, within the range of $2<Z_{\mathrm{sol}}(c)<100$. The
experimental data for varying polymers, molecular weights and concentrations
of the selected material systems broadly collapse onto a monotonically
descending master curve, and are consistent with the predictions from both the
selected tube models.

###  3.2 Extensional-thinning for highly entangled polymer systems

Another important feature extracted from measurements of capillarity-driven
thinning dynamics of the entangled polymer solutions is the rate-thinning
extensional viscosity 38, and such extensional thinning can be increasingly
pronounced as the polymer concentration further increases 1. This trend can be
effectively captured by the selected tube models. In Fig. 8, we show the
transient extensional viscosity that would be computed from the temporal
evolution of the filament radius with the same parameter variation as in Fig.
4. The Trouton ratio is defined as the (transient) extensional viscosity
scaled by the zero-shear viscosity $\eta_{0}$. For both single-mode tube
models studied in this work, this zero-shear viscosity can be analytically
calculated as $\eta_{0}=G_{\mathrm{N}}\lambda_{\mathrm{D}}$. The strain rates
on the abscissa are nondimensionalized by the disengagement time
$\lambda_{\mathrm{D}}$ to accommodate the DE model in which polymer chain
stretch is absent. When
$\mathrm{Wi}_{\mathrm{D}}\equiv\lambda_{\mathrm{D}}\dot{\epsilon}\rightarrow
0$, the Trouton ratio approaches a constant of $\mathrm{Tr}_{0}=3$.

Figure 8: The apparent transient Trouton ratio
$\mathrm{Tr}_{\mathrm{app}}=\eta_{\mathrm{e,app}}/\eta_{0}=\eta_{\mathrm{e,app}}/(G_{\mathrm{N}}\lambda_{\mathrm{D}})$
against the Weissenberg number
$\mathrm{Wi}_{\mathrm{D}}=\dot{\epsilon}\lambda_{\mathrm{D}}$ extracted from
the capillarity-driven thinning dynamics of the Rolie-Poly model (thick solid
lines) with varying finite extensibilities $\Lambda_{\mathrm{m}}=$1050100$$ at
a fixed elasto-capillary number $\mathrm{Ec}_{0}=1$ and a fixed number of
entanglements $Z=10$. The transient extensional viscosities in both models
show rate-thinning behavior at small strain rates
($\mathrm{Wi}_{\mathrm{D}}\lesssim 10$) due to tube reorientation. As
$\mathrm{Wi}_{\mathrm{D}}$ increases, the polymer chain stretch becomes
dominant in the extensional rheology, such that $\mathrm{Tr}_{\mathrm{app}}$
rapidly increases until reaching the terminal extensional viscosity set by the
finite extensibility according to Eq. 15 (horizontal dashed lines). Two
reference lines are drawn from the steady extension predicted by the RP model
(thin black solid line) and the original Doi-Edwards model (thin black dotted
line). An infinite extensibility is imposed to the polymer chain stretch for
the RP model. The prediction from steady extension of the RP model with chain
stretch is broadly consistent with those extracted from the capillarity-driven
thinning in the rate-thinning region, except for a different critical
$\mathrm{Wi}_{\mathrm{D}}^{*}$ when the extensional viscosity diverges. In
contrary, the prediction from the DE model (which does not describe additional
chain stretch) illustrates the extensional rate-thinning trend expected from
tube reorientation.

The transient Trouton ratio extracted from the capillarity-driven thinning
dynamics predicted by the two tube models exhibit a complex and non-monotonic
rate-dependent behavior. As we have illustrated in the previous sections, when
$1\ll\mathrm{Wi}_{\mathrm{D}}\ll p(Z)$, chain reptation driven by Brownian
motion overcomes the stretching induced by extensional flows. As a result, the
tubes are forced to align progressively towards the extensional direction.
This tube-alignment mechanism generally leads to a decreasing extensional
viscosity with increasing strain rate due to increased chain mobility in a
relatively straightened tube network. In contrary, in a sufficiently fast
extensional flow where $\mathrm{Wi}_{\mathrm{D}}\gg p(Z)$, the finite rate of
polymer chain retraction dominates the polymer conformation. Beyond this
point, the tube has been completely reoriented and hence does not effectively
contribute to the extensional rheology of the systems. Instead, the resistance
arising from the polymer chain stretch acting as entropic springs adds up to
the overall viscoelastic response and thus increases the extensional
viscosity. Eventually, as the chains reach full extension, the extensional
viscosity reaches a terminal value given by Eq. 15.

The transient apparent extensional viscosity $\eta_{\mathrm{e,app}}$ extracted
from the time-dependent capillarity-thinning dynamics can be further compared
with the response predicted from a steady homogeneous extensional flow. As
shown in Fig. 8, the steady extensional viscosity from the RP model (black
solid line) is broadly consistent with the transient predictions (thick
colored lines). The rate of extensional-thinning predicted by the DEMG model
(not displayed in the figure) is subtly distinct due to the additional CCR
effect. A major discrepancy between the steady and transient responses is the
critical Weissenberg number $\mathrm{Wi}_{\mathrm{D}}^{*}$ where the
extensional viscosity diverges. From analytical calculations, the steady
extension predicts that $\mathrm{Wi}_{\mathrm{D}}^{*}=p(Z)+1/2$, while for the
transient viscosity, a delayed response is identified as
$\mathrm{Wi}_{\mathrm{D}}^{*}=4p(Z)/3$. This increased front factor can be
attributed to a non-vanishing rate of chain stretch $\Lambda$ during the
temporal evolution of the filament radius.

In practice, if the rate-thinning behavior is prominent in the capillarity-
driven thinning dynamics for a tested fluid, the elasto-capillary number is
reasonably large. As a result, the exponential-thinning regime of the filament
radius derived from the polymer chain stretch is unlikely to be observed
within the measurable range. To parsimoniously describe the rate-thinning
kinematics, we numerically calculate the original Doi-Edwards (DE) model with
no polymer chain stretch. A differential form of the constitutive equation is
taken from Eq. 6 by setting $\Lambda=1$. In this scenario, the magnitude of
reorientation $\Delta S$ has an analytical solution, and a modified version of
the DE model with the addition of the solvent viscosity can be obtained to fit
the experimental data without approaching singularity at high strain rates.

Fig. 9 shows the dimensionless shear (blue) and extensional (black)
viscosities of the RP model (solid line), the DEMG model(dashed line) and the
Doi-Edwards model (dotted line) at $\mathrm{Ec}_{0}=1$, $Z=10$ and
$\Lambda_{\mathrm{m}}\rightarrow\infty$ (if applicable). A range of
$0.1<\mathrm{Wi}_{\mathrm{D}}<10$ is observed to be consistent with the
extensional-thinning region from the capillarity-driven thinning dynamics. All
three prediction lines of the shear viscosity show a consistent shear-thinning
trend, which is resulted from the tube reorientation. The extensional
viscosity, on the contrary, shows a diverse trend between the two tube models
with the effect of the polymer chain stretch, and the DE model. Nevertheless,
all three curves exhibit a similar scaling relation in most of the
extensional-thinning regime. The asymptotic solutions of the shear and
extensional viscosities as $\mathrm{Wi}_{\mathrm{D}}\rightarrow\infty$ from
the DE model are representative of the rate-dependent material response from
the tube reorientation. They are analytically calculated up to the second
dominant order as

$\displaystyle\dfrac{\eta}{G\lambda_{\mathrm{D}}}$
$\displaystyle=\dfrac{3}{12^{1/3}\mathrm{Wi}_{\mathrm{D}}^{4/3}}\dfrac{1}{1-\dfrac{4}{12^{2/3}\mathrm{Wi}_{\mathrm{D}}^{2/3}}}\sim\mathrm{Wi}_{\mathrm{D}}^{-4/3},$
(19a) $\displaystyle\mathrm{Tr}_{\mathrm{D}}$
$\displaystyle=\dfrac{9}{3\mathrm{Wi}_{\mathrm{D}}+1}\sim\mathrm{Wi}_{\mathrm{D}}^{-1}.$
(19b)

Therefore, the shear and extensional viscosities asymptotically approach a
scaling law of $\mathrm{Wi}_{\mathrm{D}}^{-4/3}$ and
$\mathrm{Wi}_{\mathrm{D}}^{-1}$, respectively, as $\mathrm{Wi}_{\mathrm{D}}$
increases. The larger-than-unity power in the scaling from the shear flow is
well-known to produce flow instability 11. Nevertheless, the linearly inverse
extensional-thinning trend predicted by the DE model shows a pronounced
strain-rate dependence and provides a feasible explanation to the extensional
rheology of a wider range of entangled polymer solutions.

Figure 9: Dimensionless shear and extensional viscosities, $\eta/\eta_{0}$
(blue lines) and $\mathrm{Tr}_{\mathrm{app}}$ (black lines) against the
Weissenberg number $\mathrm{Wi}_{\mathrm{D}}$ for the Rolie-Poly model (solid
line), the DEMG model (dashed line) and the Doi-Edwards model (dotted line) at
$\mathrm{Ec}_{0}=1$, $Z=10$ and $\Lambda_{\mathrm{m}}\rightarrow\infty$ if
applicable. The shear viscosities among the three tube models shows a
consistent shear-thinning trend until close to $\mathrm{Wi}_{\mathrm{D}}=10$.
In extensional flow, a broadly consistent extensional-thinning trend is
observed at low $\mathrm{Wi}_{\mathrm{D}}$, followed by distinctly different
trend for $\mathrm{Wi}_{\mathrm{D}}\gg 1$. The two tube models with the
additional polymer chain stretch predict a rapidly increasing extensional
viscosity, while the DE model predicts a steadily decreasing trend. The
asymptotic solutions of both the shear and extensional viscosities for the DE
model at high $\mathrm{Wi}_{\mathrm{D}}$ are plotted as thin lines, which
exhibit a power-law scaling of $\mathrm{Wi}_{\mathrm{D}}^{-4/3}$ and
$\mathrm{Wi}_{\mathrm{D}}^{-1}$, respectively.

###  3.3 Demonstration: Polyethylene oxide

We choose a specific polymer solution, polyethylene oxide (PEO) at a molecular
weight of 1 MDa to demonstrate the evolution of the capillarity-driven
thinning dynamics over a wide range of the concentrations covering both the
dilute and the entangled regimes. Table 3.3 shows the properties and
configurations of the aqueous PEO solutions.

Table 3: Properties and configurations of the aqueous PEO solutions applied to
the numerical calculation of the capillarity-thinning dynamics at varying
concentrations.

Properties |  Values
---|---
Molecular weight $M$ ($\mathrm{g}\text{/}\mathrm{mol}$) |  $10^{6}$
Monomer size $M_{0}$ ($\mathrm{g}\text{/}\mathrm{mol}$) |  $44$
Degree of polymerization $N$ |  22727tnote:dop
Molecular weight between entanglements in melts $M_{\mathrm{e}}$ ($\mathrm{g}\text{/}\mathrm{mol}$) |  2200 51
Characteristic ratio $C_{\infty}$ |  5.6 6
Monomer length $l$ ($\mathrm{nm}\text{/}$) |  0.35 6
Kuhn step $N_{K}$ |  4058tnote:ks
Kuhn length $b_{K}$ ($\mathrm{nm}\text{/}$) |  2tnote:kl
End-to-end distance $\left<R_{0}^{2}\right>^{1/2}$ ($\mathrm{nm}\text{/}$) |  125tnote:ete
Statistical length $b$ ($\mathrm{nm}\text{/}$) |  0.83tnote:sl
Temperature for solutions $T$ ($\mathrm{\SIUnitSymbolCelsius}\text{/}$) |  25
Surface tension $\Gamma$ ($\mathrm{mN}\text{/}\mathrm{m}$) |  62.2 54
Initial filament radius $R_{0}$ ($\mathrm{mm}\text{/}$) |  1
Zimm time $\lambda_{\mathrm{Z}}$ ($\mathrm{ms}\text{/}$) |  0.51 22
Friction coefficient of monomer $\zeta_{0}$ |  $1.4\times 10^{-11}$
Rouse time $\lambda_{\mathrm{R}}$ ($\mathrm{ms}\text{/}$) |  20.2tnote:rt
Overlap concentration $c^{*}$ ($\mathrm{\text{wt\%}}\text{/}$) |  0.161 22
Entanglement concentration $c_{\mathrm{e}}$ ($\mathrm{\text{wt\%}}\text{/}$) |  1.7 36

  1. a

$N=M/M_{0}$.

  2. b

$N_{K}=N/C_{\infty}$ and $N_{K}=\Lambda_{\mathrm{m}}^{2}$ in dilute polymer
solutions 6.

  3. c

$b_{K}=C_{\infty}l$ 6.

  4. d

$\left<R_{0}^{2}\right>^{1/2}=\sqrt{N_{K}}b_{K}$ 6.

  5. e

$b=b_{K}\sqrt{N}$ 11.

  6. f

$\lambda_{\mathrm{R}}=\zeta_{0}N^{2}b^{2}/(6\pi^{2}kT)$ 11.

The modulus in the FENE-P for the dilute polymer solutions can be calculated
from the kinetic theories as $G=nKT=ckT/\left<R_{0}^{2}\right>^{3/2}$ 42. The
plateau modulus of the entangled polymer solutions can be calculated from the
Graessley-Fetters definition 11 as

$G_{\mathrm{N}}(c)=\dfrac{4\rho_{\mathrm{sol}}cRT}{5M_{\mathrm{e}}(c)},$ (20)

where $\rho_{\mathrm{sol}}$ is the solution density, and the molecular weight
between entanglements for solutions $M_{\mathrm{e}}(c)$ is expressed in Eq.
16. The constitutive equation for semi-dilute polymer solutions is taken from
Prabhakar et al. 55 by adding a concentration- and strain-dependent correction
factor to the relaxation time in the FENE-P model as

$\lambda(c/c^{*},\Lambda)=\nu(c/c^{*},\Lambda)\lambda_{\mathrm{Z}},$ (21)

where $c^{*}$ is the overlap concentration, and $\lambda_{\mathrm{Z}}$ is the
Zimm time that describes the relaxation time in the regime of infinite
dilution. The polymer chain stretch $\Lambda$ is expressed 56, 39, 42, 55 as

$\Lambda=\dfrac{\sqrt{\operatorname{tr\hskip
1.0pt}\left<\bm{Q}\bm{Q}\right>}}{\left<R_{0}^{2}\right>^{1/2}},$ (22)

where $\bm{Q}$ is the end-to-end vector of a single polymer molecule, and the
angle brackets correspond to the ensemble average.

To illustrate the filament thinning profiles, six concentrations are selected
in the dilute and semi-dilute regimes
($c=$0.0016\text{\,}\mathrm{\text{wt\%}}\text{/}0.081\text{\,}\mathrm{\text{wt\%}}\text{/}0.24\text{\,}\mathrm{\text{wt\%}}\text{/}$$;
predicted by the FENE-P model with a corrected relaxation time; marked in
blue) as well as the entangled regime
($c=$2.0\text{\,}\mathrm{\text{wt\%}}\text{/}2.5\text{\,}\mathrm{\text{wt\%}}\text{/}3.0\text{\,}\mathrm{\text{wt\%}}\text{/}$$;
predicted by the RP model; marked in yellow and red). Based on the properties
listed in Table 3.3, we further calculate the concentration-specific
parameters that are used for the numerical calculation as shown in Table 4. It
is noted that both the (plateau) moduli and the number of entanglements
increase with the concentration, while the maximum polymer chain stretch
decreases as the polymers are increasingly entangled, thus shortening the
length of one tube segment.

Table 4: Selected concentrations of the aqueous PEO solutions to illustrate the filament thinning dynamics with concentration-specific properties. The colored lines next to the concentrations are consistent with Fig. 10. Concentrations ($\mathrm{\text{wt\%}}\text{/}$) |  $c/c^{*}$ |  $c/c_{\mathrm{e}}$ |  $G$ or $G_{\mathrm{N}}$ (Pa) |  $Z_{\mathrm{sol}}$ |  $\Lambda_{\mathrm{m}}^{2}$
---|---|---|---|---|---
0.0016 ( ) |  $9.9\times 10^{-3}$ |  $9.4\times 10^{-4}$ |  0.02 |  N/A |  4058
0.081 ( ) |  0.50 |  0.048 |  1.06 |  N/A |  4058
0.24 ( ) |  1.49 |  0.14 |  3.17 |  N/A |  4058
2.0 ( ) |  12.42 |  1.17 |  135.59 |  1.5 |  1187
2.5 ( ) |  15.53 |  1.47 |  224.02 |  2.3 |  898
3.0 ( ) |  18.63 |  1.76 |  337.63 |  3.4 |  715

In Fig. 10, the time axis is dimensional to provide an estimate of the breakup
time. As the concentration increases from below $c^{*}$ to above
$c_{\mathrm{e}}$, the filament thinning is progressively retarded by several
orders of magnitude. In Fig. 10(b) and (c), we show the two exponential-
thinning asymptotes using the Zimm time (black dashed lines,
$\lambda_{\mathrm{e}}=\lambda_{\mathrm{Z}}$) and the Rouse time (gray dashed
line, $\lambda_{\mathrm{e}}=\lambda_{\mathrm{R}}/2$ from Table 1),
respectively. These two asymptotes set two lower bounds for the temporal
evolution of the filament radius for dilute and entangled polymer solutions,
and can be practically used to identify the most appropriate constitutive
model to extract accurate extensional rheological properties. The kinematics
of the filament radius evolve from an exponential-thinning or slightly-
deviated exponential-thinning trend under the elasto-capillary balance to a
smoother thinning trend at concentrations above $c_{\mathrm{e}}$. In this
stage, the initial slow thinning becomes more prominent as the highly
entangled tube structures increasingly impede the disengagement of the polymer
chains in extensional flow. In practice, the measurement of the filament
radius is limited by the resolution of the image capturing devices. A
measurable range that covers three orders of magnitude down to $R/R_{0}<0.001$
is less accessible. As a result, the slow-thinning trend that occurs at the
beginning of the filament thinning is more evident in the experimental data.
To show this, we mark the time $t_{\mathrm{V\mbox{-}E}}$ when the transient
strain rate of the filament reaches the critical strain rate under an elasto-
capillary balance at $\mathrm{Wi}=2/(3\hat{\lambda}_{\mathrm{e}})$ in the
entangled regime (circles). Beyond this point, the filament radius decays in
an exponential trend until the polymer chain stretch approaches the maximum
stretch, as illustrated in Fig. 4. As shown in Fig. 10(d), where the filament
radius is plotted in a linear scale, the transition to the exponential
thinning occurs close to the filament pinch-off. The majority of the filament
thinning profile agrees well with the prediction from a Newtonian fluid, of
which the shear viscosity is identical to the apparent zero-shear viscosity
from Eq. 13.

Figure 10: Numerical calculations of the temporal evolution of the filament
radius for aqueous polyethylene oxide (PEO) solutions over the dilute and
entangled concentrations. The time axis is dimensional, and the filament
radius is nondimensionalized by the initial radius specified in Table 3.3. (a)
The temporal evolution of the filament radius in all concentration regimes
with both axes plotted in logarithmic scales. The filament thinning is
progressively retarded as the concentration increases, when the initial slow-
thinning becomes more evident. (b) The temporal evolution of the filament
radius in the dilute and semi-dilute regimes described by the FENE-P model
with a corrected relaxation time. Only the ordinate is plotted in a
logarithmic scale. The asymptote using the Zimm time
($\lambda_{\mathrm{e}}=\lambda_{\mathrm{Z}}$) is plotted in black dashed line
to show the filament thinning in the limit of infinite dilution. (c) The
temporal evolution of the filament radius in the entangled regime described by
the Rolie-Poly model. Only the ordinate is plotted in a logarithmic scale. The
asymptote using the Rouse time ($\lambda_{\mathrm{e}}=\lambda_{\mathrm{R}}/2$)
is plotted in gray dashed line to show the filament thinning in the limit of
no tube reorientation. (d) The temporal evolution of the filament radius in
the entangled regime described by the Rolie-Poly model plotted in a linear
scale. The dashed dotted lines show the predictions from a Newtonian fluid
with the apparent zero-shear viscosity from Eq. 13. The markers in (a), (c)
and (d) for the entangled solutions indicate the transition of the filament
thinning to an exponential-thinning trend under an elasto-capillary balance.

From the numerical calculation of the filament thinning profiles in the
entangled regime, we can quantify how the filament thinning dynamics evolve
with the concentration by inspecting three timescales: (1) the transition time
$t_{\mathrm{V\mbox{-}E}}$ when the exponential-thinning trend is manifested at
$\mathrm{Wi}=2/(3\hat{\lambda}_{\mathrm{e}})$, following visco-capillary
thinning behavior; (2) the filament breakup time $t_{\mathrm{C}}$ obtained by
extrapolating the prediction line from the RP model to $R=0$; (3) the filament
breakup time $t_{\mathrm{C,V}}$ predicted by the Newtonian fluid with the
apparent zero-shear viscosity in Eq. 13. As shown in Fig. 11, we observe that
the filament breakup time $t_{\mathrm{C}}$ increases in a broadly exponential
trend with the concentration above $c_{\mathrm{e}}$. At
$c/c_{\mathrm{e}}\approx 1.7$, the filament breakup time $t_{\mathrm{C}}$ is
approximately identical to $t_{\mathrm{C,V}}$, where the rate-thinning effect
induced by the tube reorientation cancels off with the rate-thickening effect
from the polymer chain stretch. Beyond this point, the elasto-capillary
transition time $t_{\mathrm{V\mbox{-}E}}$ and the Newtonian breakup time
$t_{\mathrm{C,V}}$ defined previously become increasingly close to
$t_{\mathrm{C}}$, while
$t_{\mathrm{C,V}}>t_{\mathrm{C}}>t_{\mathrm{V\mbox{-}E}}$. The evolution of
the three timescales illustrates a delayed transition to the exponential
thinning regime under an elasto-capillary balance as the concentration
increases. At sufficiently large concentrations, the tube reorientation that
induces a rate-thinning behavior is increasingly critical in determining the
final breakup time than the polymer chain stretch where the extensional
viscosity increases with the strain rate. Therefore, we can identify the key
features in the complex capillarity-driven thinning dynamics of the entangled
polymer solutions more accurately.

Figure 11: The dimensional filament breakup time $t_{\mathrm{C}}$ (black line)
by extrapolating the prediction line from the RP model to $R=0$. Another two
timescales $t_{\mathrm{V\mbox{-}E}}$ (blue) and $t_{\mathrm{C,V}}$ (red) are
plotted by taking the ratio to $t_{\mathrm{C}}$. The three selected
concentrations in the entangled regime illustrated in Fig. 10 are marked.

##  4 Conclusion

We numerically calculate the capillarity-driven thinning dynamics of the
entangled polymer systems by the application of two selected tube models: the
Doi-Edwards-Marrucci-Grizzuti (DEMG) model and the Rolie-Poly (RP) model. Both
models incorporate the interaction of polymer chains through reptative motion,
as well as the finite rates of retraction of a single polymer molecule. The RP
model adds another non-reptative mechanism of the convective constraint
release (CCR) effect. The two models are substituted in the stress balance
equation in the filament thinning configuration assuming cylindrical filament
shapes and no fluid inertia. By inspecting the kinematics of the resulting
filament thinning profiles, we find three distinct regimes of the dynamics
that are dominated by different constitutive parameters.

At the beginning of the filament thinning, the tube networks are forced to
reorient under a moderate extensional rate, and the polymer chains within the
tube remain unstretched. This mechanism results in slowed filament thinning
with a rate-thinning trend in the extensional viscosity. An analytical
solution for the rate of the shear- and extensional-thinning triggered by the
reptation can be obtained from the Doi-Edwards model. As the filament thinning
progresses, the stretch of the polymer chain stretch becomes increasingly
dominant in the extensional rheology. As a result, the temporal evolution of
the filament radius asymptotically approaches an exponential-thinning decay,
similar to the filament thinning trend predicted by the FENE-P model under the
elasto-capillary balance for dilute polymer solutions. An apparent extensional
relaxation time can be extracted from the exponential-thinning decay and is
found to approach half the Rouse time when the number of entanglements per
chain grows sufficiently large. This extensional relaxation time is notably
smaller than the shear relaxation time, i.e., the disengagement time. This
finding contradicts with the predictions by the constitutive models that
characterize dilute polymer solutions such as the FENE-P model, in which the
measured relaxation times are independent of the flow types. Such discrepancy
in the relaxation times reveals a different mechanism in the entangled polymer
solutions that predominates in the polymer relaxation in a strong extensional
flow. We further obtain an analytical expression for the ratio of the shear-
and extensional relaxation times that is free of additional fitting
parameters. This analytical result is substantiated by a number of different
entangled polymer solutions from the previous studies. For both the selected
tube models, as the extensional strain accumulates and approaches the finite
extensibility of a single polymer chain close to the filament breakup, the
temporal evolution of the filament radius deviates from the exponential trend.
A constant terminal extensional viscosity can be obtained that incorporates
the contribution from the polymer chain alignment, and its magnitude is much
higher than the zero-rate viscosity.

Finally, the capillarity-driven thinning dynamics of the aqueous polyethylene
oxide solutions with a molecular weight of
$1\text{\,}\mathrm{M}\mathrm{D}\mathrm{a}$ are numerically calculated based on
the real physical properties. A wide range of the concentrations over the
dilute, semi-dilute and entangled regimes described by multiple constitutive
models are compared in the same plot. As the concentration increases, the
filament breakup is progressively retarded, and the filament thinning
kinematics deviate from the well-studied exponential-thinning trend under the
elasto-capillary balance predicted by the FENE-P model. The rate-thinning
behavior that is prominent in a highly elastic entangled polymer system
becomes increasingly evident within the measurable range. We expect that the
new capillarity-driven thinning dynamics predicted by the tube models
presented in this work will lead to more accurate extensional rheological
characterizations of the entangled polymer solutions. We hope that new
insights can be added to better understand and optimize a number of industrial
applications that involve the processing of these complex polymer systems.

J.D. and G.H.M. would like to thank Ford Motor Company for the financial
support on this project.

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