|  The fluid dynamics of a viscoelastic fluid dripping onto a substrate†
---|---
|  Konstantinos Zinelis∗a,b, Thomas Abadiec, Gareth H. McKinleyb, and Omar K.
Matara
|  Extensional flows of complex fluids play an important role in many
industrial applications, such as spraying and atomisation, as well as
microfluidic-based drop deposition. The Dripping-on-Substrate (DoS) technique
is a conceptually-simple, but dynamically-complex, probe of the extensional
rheology of low-viscosity, non-Newtonian fluids. It incorporates the
capillary-driven thinning of a liquid bridge, produced by a single drop as it
is slowly dispensed from a syringe pump onto a solid partially-wetting
substrate. By following the filament thinning and pinch-off process the
extensional viscosity and relaxation time of the sample can be determined.
Importantly, DoS allows experimentalists to measure the extensional properties
of lower viscosity solutions than is possible with commercially available
capillary break-up extensional rheometers. Understanding the fluid mechanics
behind the operation of DoS will enable users to optimise and extend the
performance of this protocol. To achieve this understanding, we employ a
computational rheology approach, using adaptively-refined time-dependent
axisymmetric numerical simulations with the open-source Eulerian code,
Basilisk. The volume-of-fluid technique is used to capture the moving
interface, and the log-conformation transformation enables a stable and
accurate solution of the viscoelastic constitutive equation. Here, we focus on
understanding the roles of surface tension, elasticity and finite chain
extensibility in controlling the Elasto-Capillary (EC) regime, as well as the
perturbative effects that gravity and substrate wettability play in setting
the evolution of the self-similar thinning and pinch-off dynamics. To
illustrate the interplay of these different forces, we construct a simple one-
dimensional model that captures the initial rate of thinning when the dynamics
are dominated by a balance between inertia and capillarity. This model also
captures the structure of the transition region to the nonlinear EC regime in
which the rapidly growing elastic stresses in the thread balance the capillary
pressure as the filament thins towards breakup. Finally, we propose a fitting
methodology based on the analytical solution for FENE-P fluids to improve the
accuracy in determining the effective relaxation time of an unknown fluid.

##

††footnotetext: a Department of Chemical Engineering, Imperial College London,
London SW7 2AZ, United Kingdom††footnotetext: b Department of Mechanical
Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.
††footnotetext: c School of Chemical Engineering, University of Birmingham,
Birmingham B15 2TT, United Kingdom

##  1 Introduction

The formation of liquid droplets results from flows featuring complex
topological changes of a deformable fluid via the creation and breakup of
filaments. Thread formation and droplet pinch-off are of central importance to
atomisation and sprays 1, 2, 3, inkjet printing 4, 5, 6, dripping 7, 8, 9,
agrochemicals 10, 11, coatings 12, and also features in physiological flows
such as sneezing and coughing 13. In a complex fluid, the extensional
kinematics in the filaments arise from the streamwise velocity gradients and
shear-free boundary conditions in the neck region as the breakup is
approached. The extensional viscosity, $\eta_{E}$, which resists the thinning
process is larger than its shear counterpart, $\eta_{s}$,
($\eta_{E}=3\eta_{s}$ for a Newtonian fluid 14), and the dynamics of the drop
formation process are determined by the combined effects of capillary,
inertial, and viscous forces. The addition of a high molecular weight flexible
polymer such as poly(ethylene oxide) (PEO) to a Newtonian solvent like water
or water-glycerol mixtures, has been observed experimentally to strongly
influence the neck evolution and thinning dynamics, delaying the final pinch-
off 7, 15, 16, 17. For dilute polymer solutions, the fluid viscoelasticity can
lead to a large increase of the extensional viscosity (in comparison to the
shear viscosity) by a factor of $10^{2}-10^{3}$. It is therefore important to
measure accurately parameters such as $\eta_{E}$ and the polymer relaxation
time $\tau$, which characterise the extensional rheology of the solution as
they influence the filament thinning process and the timescales controlling
droplet formation in complex fluids 18.

Figure 1: Simulation setup and interface evolution during two distinct stages:
(a) initially, gravity-driven axial drop elongation followed by, (b) spreading
over the solid substrate of prescribed wettability set by the equilibrium
contact angle $\theta_{E}$. The simulation domain is indicated in red.

Various bespoke extensional rheometers have been developed based on a variety
of configurations such as jetting, e.g. the Rayleigh Ohnesorge jetting
extensional rheometer (ROJER) device 19, 20, spinning, flow through
T-junctions and cross-slot configurations (e.g. OSCER), as well as boundary
separation techniques such as the Filament Stretching Extensional Rheometer
(FiSER) and the Capillary Breakup Extensional Rheometer (CaBER) 21, 22, 19,
23, 14. The CaBER rheometer design exploits the capillarity-driven thinning of
a stretched fluid bridge in the absence of imposed external forces and is
suitable for studying a range of mobile complex fluid samples (i.e. low and
moderate viscosity) over a range of Hencky strains. However, as the viscosity
of the test fluid is lowered towards that of water, inertial effects become
increasingly important and limit successful operation 24.

The recently developed Dripping-onto-Substrate (DoS) protocol 25, 26, 27,
which involves the slow dripping of a single drop through a narrow needle
(needle radius $R_{0}\sim 1$ mm) onto a partially-wetting substrate, as
illustrated in Figure 1(a), is suitable for extensional rheology measurements
of $O(\mu L)$ volumes of complex fluids, enabling tests on expensive/scarce
material samples, such as protein solutions 28 whilst mitigating undesirable
shear and inertial effects. The DoS rheometry technique also facilitates the
measurement of the extensional properties of lower-surface tension and lower-
viscosity fluids at higher attainable extension rates than commercially
available capillary breakup instrumentation (e.g. the CaBER device) 14.

Effective DoS measurements have been reported in studies of the extensional
rheology for various polymer solutions, suspensions, inks, viscoelastic
surfactant fluids, cosmetics and food materials, as well as protein solutions
and associative polysaccharide systems 26, 29, 30, 31, 32, 5, 33, 34, 35, 28,
14. DoS protocols have helped to elucidate the non-Newtonian response to
extensional deformations for fluids which do not exhibit any measurable
viscoelastic behaviour in conventional shear rheometer or extensional
rheometer experiments. By understanding the sequence of local balances among
the capillary, inertial, viscous, and elastic forces acting on the thinning
thread attached to the drop as it spreads laterally on the substrate 14, the
DoS process can be shown to be characterised by the sequential emergence of a
number of distinguished regimes and distinct neck shapes.

At early thinning times, and for low-viscosity Newtonian samples, an Inertio-
Capillary (IC) response described by power-law dynamics dominates the thinning
process leading to a conically-shaped neck, close to $R_{min}(t)$, as
indicated in Figure 1(b). In contrast, in fluids of higher viscosity, a
distinct Visco-Capillary (VC) regime is established in which the neck is
slender and almost cylindrical in shape and the radius $R_{min}(t)$ decreases
linearly in time. For viscoelastic fluids, an additional Elasto-Capillary (EC)
thinning regime is observed as thread pinch-off is approached. This is
characterised by an exponential decrease in $R_{min}(t)$ with time and the
formation of a thin and axially uniform cylindrical thread connecting the
deposited drop to the residual fluid that is pinned by the nozzle. In the
final stages of the breakup process, this EC regime gives way to a Terminal
Visco-Elasto-Capillary (TVEC) region in which the radius once again decreases
to zero linearly in time. This terminal regime is controlled by the finite
extensibility of the polymer chains that are dissolved in the low-viscosity
solvent.

In this work, we use time-dependent, free-surface numerical simulations of a
canonical constitutive model for polymer solutions (the FENE-P model 36) to
study filament deformation and the breakup dynamics that underpin DoS
rheometry. We extend previous experimental work examining the role of
substrate wetting and gravitational forces for wormlike micellar solutions 37
and inertio-capillary dynamics 28 to account for the effects of nonlinear
viscoelasticity and finite chain extensibility that are captured by the FENE-P
constitutive equation. The thinning fluid thread and the substrate wettability
are parameterised by the Ohnesorge, Deborah, and Bond numbers, as well as a
macroscopic contact angle, respectively. Our computational rheology
experiments help identify the existence of optimal parameter ranges in which
experimental inaccuracies for these small-scale, high-speed rheological
measurements are minimised.

The rest of this article is organised as follows. In Section 2, the problem
formulation and the numerical framework used for performing the simulations
are presented. We then investigate how the dynamics of DoS rheometry are
affected by substrate wettability and gravitational forces, and representative
results are discussed in Section 3. In Section 4, we use a simple one-
dimensional model to capture the main dynamical features identified by our
full numerical simulations and explore how the data obtained from systematic
parametric variations of the DoS rheometry protocol can improve the analysis
of extensional rheology measurements with unknown samples. Finally,
recommendations for better practices in DoS rheometry and concluding remarks
are presented in Sections 4 and 5, respectively.

##  2 Formulation and Methodology

###  2.1 Flow configuration

Figure 1 shows the configuration of a typical DoS experiment. We use an
axisymmetric description of the filament shape $R(z,t)$ as a prolate pendant
drop of length $\ell_{0}$ is brought into contact with a partially-wetting
substrate. The fluid is an incompressible, viscoelastic solution containing
polymer chains of finite extensibility $L^{2}$ which issues from a nozzle of
length $\ell_{nozzle}$ and initial radius $R_{0}$. The fluid of density
$\rho_{l}$ and dynamic viscosity $\eta_{l}=\eta_{p}+\eta_{s}$ is surrounded by
a gas of density $\rho_{g}$ and dynamic viscosity $\eta_{g}$; here, $\eta_{p}$
and $\eta_{s}$ denote the polymer and solvent contributions to the total
viscosity, respectively. Additionally, we use $\beta=\eta_{s}/\eta_{l}$ to
express the relative contribution of the solvent viscosity $\eta_{s}$ to the
total dynamic viscosity of the polymer solution $\eta_{l}$. The droplet
issuing from the nozzle is brought into contact with a solid substrate located
at a distance $\ell_{0}$ below the nozzle exit resulting in a characteristic
aspect ratio of $\ell_{0}/2R_{0}$. The substrate wettability is parameterised
by a macroscopic equilibrium contact angle $\theta_{E}$.

The initial volume of the droplet $V_{drop}$ is ensured to be larger than the
corresponding volume of a fluid column of radius $R_{0}$ and length $\ell_{0}$
for stability considerations 15. As shown schematically in Figure 1(a), the
pendant drop develops a prolate amphora shape whose subsequent spreading
across the substrate is influenced by gravitational forces, wettability
effects, as well as its rheological response. The lateral spreading of the
droplet foot and conservation of volume leads to the development of a thin
polymeric thread of characteristic radius $R_{min}(t)$, which connects the two
hemispherical droplets that are attached to the nozzle and spreading across
the substrate, respectively, as illustrated in Figure 1(b). The thinning and
eventual break up of this thread is influenced by a delicate balance of
capillary, inertial, viscous, and elastic forces, as will be discussed below.
Understanding this evolving balance is what enables the DoS configuration to
be employed in extensional rheometry.

###  2.2 Governing equations and numerical method

In what follows we non-dimensionalise the governing equations by scaling with
the characteristic length $R_{0}$, an inertio-capillary velocity
$U_{R}=\sqrt{\gamma/\left(\rho_{l}R_{0}\right)}$, and the Rayleigh time
$t_{R}=R_{0}/U_{R}=\sqrt{\rho_{l}R_{0}^{3}/\gamma}$. All stresses and the
isotropic pressure thus can be made dimensionless using a pressure scale
$\rho_{l}U_{R}^{2}=\gamma/R_{0}$. The filament thinning dynamics are then
governed by the following dimensionless continuity and momentum conservation
equations:

$\bm{\tilde{\nabla}\cdot\mathbf{\tilde{u}}}=0,$ (1)
$\tilde{\rho}\left(\dfrac{\partial\mathbf{\tilde{u}}}{\partial\tilde{t}}+\mathbf{\tilde{u}}\cdot\tilde{\bm{\nabla}}\mathbf{\tilde{u}}\right)=-\tilde{\bm{\nabla}}\tilde{p}+Oh\left(\beta\tilde{\bm{\nabla}}\cdot\tilde{\bm{\sigma}}_{s}+\frac{\left(1-\beta\right)}{De}\tilde{\bm{\nabla}}\cdot\tilde{\bm{\sigma}}_{p}\right)+\tilde{\kappa}\tilde{\delta}\mathbf{n}+Bo\mathbf{\tilde{g}},$
(2)

where the tildes designate dimensionless variables; $\tilde{t}$,
$\tilde{\rho}$, $\mathbf{\tilde{u}}$, $\tilde{p}$, $\tilde{\bm{\sigma}}_{s}$,
$\tilde{\bm{\sigma}}_{p}$, $\tilde{\kappa}$, $\tilde{\delta}$, $\mathbf{n}$,
and $\mathbf{\tilde{g}}$ correspond to dimensionless time, density, velocity,
pressure, solvent and polymeric stress components, interfacial curvature, the
Dirac delta function, the outward-pointing unit normal vector to the
interface, and the gravitational acceleration, respectively. A one-fluid
Volume-of-Fluid (VOF) approach 38 is used, where the volume fraction $c$ of
the liquid phase is advected in every computational cell as
$\partial{c}/\partial\tilde{t}+\tilde{\mathbf{u}}\cdot\tilde{\bm{\nabla}}c=0$.
Additionally, the local density $\tilde{\rho}$ and viscosity $\tilde{\eta}$
are determined by:

$\displaystyle\tilde{\rho}$ $\displaystyle=$ $\displaystyle
c+(1-c)\dfrac{\rho_{g}}{\rho_{l}},$ (3) $\displaystyle\tilde{\eta}$
$\displaystyle=$ $\displaystyle c+(1-c)\dfrac{\eta_{g}}{\eta_{l}}.$ (4)

In Eq. (2), $De=\tau/(R_{0}/U_{R})=\tau/t_{R}$ represents an intrinsic Deborah
number, which captures the ratio of the relaxation time of the polymer $\tau$
to the Rayleigh flow time scale $t_{R}$; $Oh=\eta_{l}/\sqrt{\rho_{l}\gamma
R_{0}}$ is the Ohnesorge number that represents the relevant contribution of
capillary and viscous forces, and $Bo=\rho_{l}gR_{0}\ell_{0}/\gamma$ is a Bond
number that characterises the balance between gravity and capillarity in the
initial pendant drop as it touches the substrate.

The dimensional viscous stress tensor arising from the solvent is given by
$\bm{\sigma_{s}}=\eta_{s}(\bm{\nabla}\mathbf{u}+(\bm{\nabla}\mathbf{u})^{T})$,
and $\bm{\sigma_{p}}$ is the dimensional polymeric stress contribution to the
total stress given here by the FENE-P constitutive equation:

$\bm{\sigma_{p}}=\frac{\eta_{p}}{\tau}\left(\frac{\mathbf{A}}{1-\frac{\operatorname{tr}(\mathbf{A})}{L^{2}}}-\mathbf{I}\right),$
(5)

where $L^{2}$ is the polymer chain finite extensibility, and $\mathbf{A}$ is
the dimensionless chain conformation tensor whose transport is governed by the
following dimensionless equation:

$\frac{\partial\mathbf{A}}{\partial\tilde{t}}+\mathbf{\tilde{u}}\cdot\bm{\tilde{\nabla}}\mathbf{A}-\left(\bm{\tilde{\nabla}}\mathbf{\tilde{u}}\cdot\mathbf{A}+\mathbf{A}\cdot\bm{\tilde{\nabla}}\mathbf{\tilde{u}}^{T}\right)=-\frac{1}{De}\left(\frac{\mathbf{A}}{1-\frac{\operatorname{tr}(\mathbf{A})}{L^{2}}}-\mathbf{I}\right).$
(6)

We avoid numerical instabilities due to rapid growth of elastic stresses
during the formation of the elastic thread (High-Weissenberg Number Problem
39) via the log-conformation transformation 40. The local time-varying
Weissenberg number $Wi$ is the local strain rate normalised by the polymer
relaxation time 41:

$Wi=\tau\dot{\epsilon}_{min}=-2\tau\
\dfrac{d\left(\log(R_{min})\right)}{dt}=-2De\
\dfrac{d\left(\log(\tilde{R}_{min})\right)}{d\tilde{t}}.$ (7)

We use this dimensionless strain rate to assess the thinning rate of the
viscoelastic filament. The tildes, which designate dimensionless variables,
are suppressed henceforth.

Figure 2: Contour plots of (a) the volume fraction, (b) the axial velocity
component, and (c) the axial polymeric stress at different times which
correspond to the initialisation, inertio-capillary thinning regime, the onset
of the elasto-capillary regime, and the terminal thinning regime of the
viscoelastic filament (which is controlled by the polymer finite
extensibility). The formation of the characteristic thin viscoelastic thread
between two beads is highlighted at $t=14.8$. (d) Representation of the
evolution of the minimum filament $R_{min}$ in time, shifted by $t_{i}$ which
is the time when the influence of the initial condition ceases (here
$t_{i}=7.46$); The simulations capture the three main characteristic regimes
of filament thinning in DoS rheometry: i.e. Inertio-Capillary (IC), Elasto-
Capillary (EC), and Terminal Visco-Elasto-Capillary (TVEC) regimes. The
vertical dashed lines indicate the end of the IC regime and the subsequent
transition to the exponential thinning, the onset of the EC regime, and
finally the beginning of the TVEC thinning regime, respectively. (e) Temporal
evolution of the local dimensionless strain rate, $Wi$, which shows good
agreement between the numerical simulations and the theoretical prediction for
the initial inertio-capillary dynamics (blue dashed line), the constant $Wi$
plateau which corresponds to the expected thinning-rate ($\sim-1/(3De)$ 42,
43, 44, 41), during the exponential thinning, and the rapid terminal
divergence in $Wi$ when the polymer finite extensibility limit has been
reached 44, 15, 45. Here, $t_{max}-t_{i}=4.29$ (with $t_{max}=11.75$ and
$t_{i}=7.46$) corresponds to the time when the local $Wi$ attains its maximum
value; this, in turn, corresponds to the end of the IC regime and the
subsequent transition to the EC regime. The parameter values are listed in
Table 1. For the corresponding movie please see the Electronic Supporting
Information.

###  2.3 Numerical set up

The simulations are performed with the open-source code Basilisk 46, 47, 48,
49, 50. The interface is reconstructed with a piecewise linear interface
calculation (PLIC) technique 38, 47, and the height-function method is used to
calculate the geometrical properties of the interface; this ensures accurate
modelling of the capillarity-driven thinning of the polymeric solution 38, 51.
The simulation domain is a square of dimensions $1.67\pi R_{0}\times 1.67\pi
R_{0}$, which results in an aspect ratio $\approx 3$. The left boundary is the
axial symmetry axis, while no-penetration Dirichlet conditions are considered
for all the velocity components at the upper and lower boundaries, where a
nozzle and solid substrate are considered, respectively. In addition, we
follow the implementation by López-Herrera _et al._ 47 for the boundary
conditions for all the polymeric stress tensor components.

The contact line at the upper plane is pinned at $r=R_{0}$ to account for the
nozzle exit, following the approach applied by Sakakeeny and Ling 52 so that
drop oscillations can be effectively handled. The wetting of the solid
substrate at the lower plane is modelled with a macroscopic contact angle
boundary condition in combination with the height-function method 53. Although
a no-slip boundary condition is imposed on the substrate, the velocity field
for the interface advection is located at the centre of the cell faces and
therefore results in an implicit slip condition at the contact line with a
slip length which scales with the minimum grid size $\Delta
r_{minimum}/2\leavevmode\nobreak\ $ 53, 54. Therefore, it is critical to
specify small enough grid cell sizes to ensure that the macroscopic dynamics
remain unaffected by the microscopic contact line dynamics (further details of
the mesh convergence study are provided in the Electronic Supporting
Information). Additionally, for the initial conditions of the DoS process at
$t=0$, which is set when the polymeric droplet first touches the solid
substrate, we consider that $r=R_{0}$ corresponds to the nozzle radius in the
upper panel, and the polymeric chains are considered to be initially at rest
which corresponds to $A_{zz}=A_{rr}=1$ everywhere within the liquid phase. The
initial shape of an elongated polymeric droplet between the nozzle exit and
the solid substrate is given by an elliptical equation:

$\left(\dfrac{z-2.45}{2.83}\right)^{2}+\left(\dfrac{r}{2}\right)^{2}-1=0.$ (8)

The grid cells in the simulation domain are refined according to a quadtree
adaptive mesh refinement (AMR) scheme available in Basilisk 38 based on the
location of the interface, as well as on the regions where the gradients of
the axial elastic stress are large. This allows for sufficient resolution of
the extremely thin polymeric thread, which is formed between the two primary
hemispherical beads (as also observed in experiments 25). The adaptive mesh
refinement enables accurate simulation of the thinning dynamics as breakup of
the filament ($R_{min}(t)\rightarrow 0$) is approached. Starting from a base
grid resolution of $512\times 512$ square cells for the whole domain, the
adaptive scheme refines the cells down to a minimum square cell of size
$\Delta r_{minimum}=0.0026$.

Figure 3: (a) Contour plots of the axial elastic stress component for
different Deborah numbers, $De=0$, 0.2, 0.5 and 1, at dimensionless times that
characterise different minimum radii of the filament, $R_{min}=0.55$, 0.27,
0.15 and 0.05. Also shown are enlarged profiles of the viscoelastic stress
distribution in the thread at $R_{min}=0.05$ for $De=0.2$ and $1$,
highlighting the effect of elasticity and the axially-uniform distribution of
the polymeric stress in the thread for $De=1$. (b) Temporal profiles of the
dimensionless minimum filament radius and the local dimensionless strain rate
$Wi=\tau\dot{\epsilon}$ for the same range of Deborah numbers, highlighting
the emergence of a distinct Elasto-Capillary (EC) thinning regime at
$De\gtrsim 0.5$ in which a constant dimensionless strain-rate $Wi=2/3$ is
expected. The rest of the parameters remain unchanged from Table 1. For the
corresponding movie please see the Electronic Supporting Information.

##  3 Results and Discussion

In this section, we provide a discussion of our results obtained from the
numerical simulations of Dripping-Onto-Substrate (DoS) rheometry.
Specifically, the effects of the parameters $De$, $L^{2}$, $Bo$, and
$\theta_{E}$ on the thinning dynamics are analysed, and connections with the
effectiveness of DoS rheometers are also established.

###  3.1 Numerical results

Figure 4: (a) Contour plots of the axial stress distribution in the viscoelastic thread at $R_{min}=0.55$, 0.27, 0.15, and 0.05, for four representative polymer finite extensibility values $L^{2}=400$, 1600, 2500, and 10000 at fixed Deborah number ($De=1$). (b) Profiles of the viscoelastic filament at time $t^{*}$ when its minimum radius reaches a value of $5\%$ of the nozzle radius. (c) Temporal evolution of the minimum filament radius $R_{min}(t)$, and (d) evolution in the local strain rate within the neck for $L^{2}=400$, 900, 1600, 2500, and 10000. The rest of the parameters remain unchanged from Table 1. Table 1: Simulation parameters of DoS rheometry. Gravitational effects are initially neglected ($Bo=0$). The density and viscosity ratios $\rho_{g}/\rho_{l}$ and $\eta_{g}$/$\eta_{l}$ are the same as those used by Zinelis _et al._ 50. $De$ | $Oh$ | $\beta$ | $L^{2}$ | $Bo$ | $\theta_{E}$ | $\rho_{g}/\rho_{l}$ |  $\eta_{g}$/$\eta_{l}$
---|---|---|---|---|---|---|---
1 | 0.2 | 0.85 | 10000 | 0 | $30^{\circ}$ | 0.01 | 0.01

Figure 2 shows the simulation results for a weakly viscoelastic fluid
characterised by $De=1$ and large finite polymer chain extensibility
$L^{2}=10000$, with additional material properties as defined in Table 1,
where the density and viscosity ratios are the same as in Zinelis _et al._ 50.
Figure 2(a)-(c) presents the evolution of the interfacial shape along with
contour plots of the volume-fraction, axial velocity component, and axial
polymeric stress, respectively, coloured by the corresponding magnitude of
each quantity. These figures demonstrate the evolution of a DoS experiment
from the time when the droplet first touches the solid substrate ($t=0$) until
the establishment of a fully-developed viscoelastic thread ($t=14.8$) when
$R_{min}$ reaches the minimum resolvable value in our simulation
($R_{min}\approx 0.05$). The snapshots are taken at times which correspond to
the initialisation of the numerical simulation ($t=0$), to the subsequent
establishment of the inertio-capillary (IC) ($t=9$) and elasto-capillary (EC)
($t=12$) regimes, and finally to the ultimate thinning regime when the finite
extensibility limit of the polymeric chains is reached ($t=14.8$).

The contours of axial velocity presented in Figure 2(b) show the initial slow
drainage of the filament which eventually leads to the final rapid breakup of
the viscoelastic thread and the subsequent formation of two hemispherical
beads (of different volumes) attached to the nozzle and the lower substrate.
The contour plots of the axial polymeric component in Figure 2(c) reveal the
importance of elastic stresses during the formation of the polymeric thread.
In particular, at early times, the axial polymeric stresses are weak, and
inertial and capillary forces drive the filament thinning process. However, at
intermediate times ($t\geq 12$), the axial polymeric stress increases rapidly
as the filament radius continues to thin. This large elastic stress acts to
stabilise the thinning thread and retard its eventual breakup.

The IC, EC, and TVEC regimes are all indicated in Figure 2(d), which uses a
semi-log representation of the evolution in the minimum filament radius
$R_{min}(t)$; here, the dimensionless time scale on the abscissa has been
shifted by an initial offset, $t_{i}$ which denotes the time at which the
residual effects of the initial spreading process become negligible, i.e.,
when $R_{min}\leq 1$ and the initial prolate drop has established a concave
necked configuration. For these specific parameter conditions, we determine
$t_{i}=7.46$. In Figure 2(d), the development of the exponential elasto-
capillary thinning process is also evident, as expected 25, until the finite
extensibility of the polymer chain (set by the magnitude of the parameter
$L^{2}$) has been reached.

To examine the accuracy with which the simulation describes the thinning of
the polymeric filament, we also monitor the temporal evolution of the local
Weissenberg number $Wi$ in Figure 2(e) which evolves in a non-monotonic way
41, 9. Inspection of this profile confirms the fact that the simulations
capture accurately the IC regime during the early times of the thinning
process along with the subsequent exponential decrease of the filament radius
in the EC regime. This is characterised by the expected dimensionless thinning
rate of $-2\dot{R}_{min}/R_{min}=2/(3De)$ which corresponds to a constant
local Weissenberg number of $Wi=2/3$. The final steep increase of $Wi$ at the
very end of the simulation indicates that the polymeric chains become fully
extended and the filament approaches the “pinch-off" interfacial singularity.
Accurate resolution of this TVEC regime requires very high levels of mesh
refinement 50, but is not essential for successfully identifying and utilising
the exponential EC regime. We thus cease our simulations when the minimum
thread radius falls below $R_{min}\leq 0.05$ and the strain rate starts to
rapidly increase again. We proceed now with studying the competing roles of
elasticity, finite extensibility, gravity, and substrate wettability, in
designing robust and efficient DoS rheological measurements.

Figure 5: (a) Contour plots of the axial polymeric stress component at
$R_{min}=0.55$, 0.27, 0.15 and 0.05 for different $Bo=0$, 0.1, 0.4 and 0.8,
highlighting the influence of the gravitational forces in the initial and
terminal thinning dynamics. (b) Filament shape at $R_{min}=0.05$ for the same
set of Bond numbers. Temporal profiles of (c) the dimensionless filament
radius $R_{min}$ and (d) the local dimensionless strain rate $Wi$,
highlighting the self-similar exponential thinning during the Elasto-Capillary
(EC) regime for a range of different Bond numbers ($0\leq Bo\leq 1$). The rest
of the parameters are the same as given in Table 1. For the corresponding
movie please see the Electronic Supporting Information.

In Figure 3 we explore the level of elasticity in the polymeric solution above
which elastic effects are detectable in DoS rheometers. Specifically, in
Figure 3(a), we show contour plots and interface profiles for $De=0$, 0.2,
0.5, and 1, respectively. It can be observed that for $De=0$ and 0.2, an
approximately conical neck is formed, whilst, for $De\geq 0.5$, the thread
becomes thin and of uniform thickness as also highlighted at the right of
panel (a). In Figure 3(b), at the left we show the exponential thinning of the
filament radius $R_{min}(t)$ for $De=0,0.2,0.5$ and 1. A steeper slope of the
filament thinning is observed with an increasing Deborah number, as expected
according to the Oldroyd-B predictions ($\sim-1/(3De)$ 42, 43, 44, 41). For
this reason, according to Eq. (7) the dimensionless strain rate $Wi(t)$
written as a function of dimensionless parameters ($Wi(t)=-2De\
d\left(\log(R_{min}(t))\right)/dt=2De/(3De)=2/3$), is independent of the
Deborah number during the EC regime. Hence, we expect a plateau in the
dimensionless strain-rate at $Wi\approx 2/3$ when an EC regime is established.
At the right of panel (b) this is observed only for $De=0.5$ and $De=1$.
However, at $De=0.2$ the transition to an exponential thinning can barely be
detected. Thus, there appears to be a critical value of the Deborah number in
the range $0.2<De\leq 0.5$ beyond which an EC regime is established. Careful
inspection of Figure 3 (b) also reveals that the presence of viscoelasticity,
characterised by any finite $De$, leads to a slightly accelerated initial
thinning in the IC regime in comparison to the Newtonian case (solid black
line) for which $De=0$.

Figure 6: Variation of the scaled characteristic times $\tilde{t}_{i}$,
$\tilde{t}_{max}$ and $\tilde{t}^{*}$ with Bond number. Here, we consider the
zero gravity case ($Bo=0$) as a reference to quantify the effect of
gravitational forces (finite Bond numbers) on the onset of the inertio-
capillary thinning ($\tilde{t}_{i}$), the subsequent transition to the Elasto-
Capillary (EC) regime ($\tilde{t}_{max}$) until the final filament breakup
($\tilde{t}^{*}$)

.

Figure 7: (a) Contour plots of the axial stress component at $R_{min}=0.55$,
0.27, 0.15 and 0.05 for four contact angles $\theta_{E}=30^{\circ}$,
$45^{\circ}$, $60^{\circ}$ and $80^{\circ}$, covering different levels of
substrate wettability. (b) The filament interface at times $t^{*}$ when
$R_{min}=0.05$ for the same range of contact angles. (c) Evolution in the
minimum filament radius $R_{min}(t-t_{i})$ and (d) evolution in the local
dimensionless strain rate $Wi(t)$ for substrate wettabilities in the range of
$30^{\circ}\leq\theta_{E}\leq 80^{\circ}$. The rest of the parameters remain
unchanged from Table 1. For the corresponding movie please see the Electronic
Supporting Information.

Having identified the threshold $De$ above which effective measurements of the
elastic properties of polymeric solutions are possible, we now examine the
role of the chain finite extensibility parameter $L^{2}$. Figure 4 shows that
higher values of $L^{2}$ permit the development of larger axial polymeric
stresses during the EC regime. However, there are no apparent differences in
the interface evolution and the thread thickness, which is confirmed by
inspection of Figure 4(b). The temporal evolution of the filament radius
$R_{min}$ in Figure 4(c) shows that for small $L^{2}$ values ($L^{2}=400$ and
900), a much steeper (but still approximated exponential) decrease of the
filament radius is observed. The profile of the corresponding local
dimensionless strain-rate $Wi$ in Figure 4(d) confirms the faster rate-of-
thinning of the filament radius for $L^{2}=400$ and $L^{2}=900$, which
correspond to higher $Wi$ values ($Wi\geq 1$). For these relatively small
values of the polymer finite extensibility, the stretching limit of the
polymeric chains in the thinning thread is approached rapidly soon after the
onset of the EC regime, which consequently results in a very short exponential
thinning period. In contrast, the duration of the exponential filament
thinning regime increases with $L^{2}$ and at sufficiently high
extensibilities (here we regain the Oldroyd-B limit $Wi=2/3$ for
$L^{2}=10000$). These results demonstrate that a sufficiently large chain
extensibility is needed for accurate rheological measurements in DoS
rheometers.

Figure 8: Variation of the scaled characteristic times $\tilde{t}_{i}$,
$\tilde{t}_{max}$ and $\tilde{t}^{*}$ with the static contact angle
$\theta_{E}$. Here, we consider the case of the fastest substrate spreading
($\theta_{E}=30^{\circ}$) as a reference to quantify the effect of substrate
wettability on the initial onset of the inertio-capillary thinning
($\tilde{t}_{i}$), the subsequent transition to the Elasto-Capillary (EC)
regime ($\tilde{t}_{max}$) and the approach to the final filament breakup
($\tilde{t}^{*}$).

We now study the effect of gravity on the interfacial dynamics during dripping
onto a substrate. Figure 5(a) presents contour plots of the axial polymeric
stress component for $Bo=0$, 0.1, 0.4, and 0.8, respectively, with a constant
fluid contact angle $\theta_{E}=30^{\circ}$. Inspection of this figure reveals
that the development of the elastic stress is only weakly dependent on gravity
over the range of $Bo$ values that we examine. Furthermore, a quantitative
comparison of the filament profiles as shown in Figure 5(b) demonstrates that
increasing $Bo$ leads to slightly longer filaments and an increase in the drop
footprint across the solid substrate. In addition, Figures 5(c) and (d) show
that even though an increase in the gravitational body force leads to faster
thinning in the IC regime as well as a more rapid transition to the onset of
the characteristic exponential thinning regime, the dynamics in the EC regime
are not influenced by variations in the Bond number. These results demonstrate
that DoS rheological measurements of viscoelastic fluid properties, such as
the relaxation time, remain unaffected by gravitational perturbations in the
range $0\leq Bo\leq 1$.

To better understand and quantify the influence of gravity on the thinning
dynamics, it is worth examining the dependence of important features of the
filament thinning profile, such as the characteristic times $t_{i}$ (which
indicates when initial geometric effects become negligible), $t_{max}$ (which
corresponds to the local maximum filament strain rate and the subsequent onset
of the EC regime), and $t^{*}$ (which represents the thinning time, when the
thread can be considered to be fully established and is about to undergo
breakup, here when $R_{min}=0.05$). The dependence of $t_{i}$, $t_{max}$, and
$t^{*}$ on $Bo$ is presented in Table 2.

In addition, in Figure 6 we show the variation of these characteristic times
with Bond number. Specifically, we plot the scaled quantities
$\tilde{t}_{i}=t_{i}(Bo)/t_{i}(Bo=0)$,
$\tilde{t}_{max}=t_{max}(Bo)/t_{max}(Bo=0)$,
$\tilde{t}^{*}=t^{*}(Bo)/t^{*}(Bo=0)$ where we use the zero Bond number case
as a reference to determine the effect that gravitational forces exert on the
initial phase ($t_{i}$) in DoS rheometry, the transition from the inertio-
capillary to elasto-capillary dynamics ($t_{max}$) and finally to the time of
the “pinch-off" singularity ($t^{*}$). Figure 6 confirms the initial
observation that gravity accelerates the filament thinning, where the onset of
the power-law and exponential filament thinning regimes at larger Bond numbers
($Bo\approx 1)$ are up to $40\%$ and $20\%$ faster, respectively.

Table 2: Dependence of the characteristic $t_{i}$, $t_{max}$ and $t^{*}$ timescales on the Bond number. These times correspond to the onset of the Inertio-Capillary (IC) regime, the onset of the elasto-capillary dynamics, and the time when the minimum filament radius decreases to $5\%$ of its initial value ($R_{min}(t^{*})\rightarrow 0.05$). $Bo$ | $t_{i}$ | $t_{max}$ | $t^{*}$
---|---|---|---
0 | 7.46 | 11.75 | 14.11
0.1 | 7.16 | 11.27 | 13.78
0.2 | 6.68 | 10.59 | 13.16
0.3 | 6.34 | 10.10 | 12.75
0.4 | 6.01 | 9.68 | 12.38
0.6 | 5.51 | 8.99 | 11.84
0.8 | 4.87 | 8.22 | 11.24
1 | 4.51 | 7.74 | 10.76

The solid surface wettability, characterised by the macroscopic contact angle
$\theta_{E}$, influences the lateral spreading of the drop that is deposited
on the substrate which, in turn, can play a role in driving the filament
thinning dynamics. We study partially wetting solid substrates with
wettabilities characterised by $\theta_{E}=30^{\circ},\leavevmode\nobreak\
45^{\circ},\leavevmode\nobreak\ 60^{\circ},\leavevmode\nobreak\ {\rm
and}\leavevmode\nobreak\ 80^{\circ}$, which correspond to hydrophilic and
partially hydrophobic substrates at the lower and higher ends of the contact
angle range, respectively. Here, we have also set $Bo=0$ to isolate the
effects of wettability from those associated with gravity. The contour plots
in Figure 7(a) show that increasing substrate wettability (by decreasing
$\theta_{E}$) leads to an enhancement in the degree of spreading along the
substrate, resulting in larger drop footprints and also longer connecting
filaments. In contrast, more hydrophobic substrates lead to bead-like drops
with smaller footprints and shorter connecting filaments, as can also be seen
in Figure 7(b). However, in contrast to the gravitational effects discussed
above, the increase in $\theta_{E}$ is seen to retard significantly the onset
of the EC regime for $\theta_{E}\geq 60^{\circ}$, as shown in Figure 7(c) and
(d), although once established, the exponential EC thinning regime retains the
theoretically expected rate-of-thinning of $-1/(3De)$ for all $\theta_{E}$
values studied.

Table 3: Dependence of the characteristic timescales $t_{i}$, $t_{max}$ and $t^{*}$ on the substrate wettability (represented by $\theta_{E}$). $\theta_{E}[^{\circ}]$ | $t_{i}$ | $t_{max}$ | $t^{*}$
---|---|---|---
30 | 7.46 | 11.75 | 14.11
45 | 7.81 | 12.44 | 14.58
50 | 7.87 | 12.58 | 14.72
60 | 8.05 | 13.27 | 15.18
70 | 8.37 | 14.74 | 16.26
75 | 8.63 | 16.18 | 17.40
80 | 11.04 | 20.99 | 21.78

Similarly to our analysis of the gravitational effects, here we also examine
the impact of decreasing substrate wettability on the characteristic
timescales $t_{i}$, $t_{max}$ and $t^{*}$. The results for these three
timescales as a function of the contact angle $\theta_{E}$ are provided in
Table 3. In addition, Figure 8 shows the evolution of the scaled
characteristic times
$\tilde{t}_{i}=t_{i}(\theta_{E})/t_{i}(\theta_{E}=30^{\circ})$,
$\tilde{t}_{max}=t_{max}(\theta_{E})/t_{max}(\theta_{E}=30^{\circ})$,
$\tilde{t}^{*}=t^{*}(\theta_{E})/t^{*}(\theta_{E}=30^{\circ})$, selecting the
lowest contact angle ($\theta_{E}=30^{\circ}$) as the reference case for
quantifying the role of substrate wetting. Figure 8 confirms the observation
of Figure 7: The use of more hydrophobic substrates causes significant
retardation of the thinning dynamics, including the duration of the initial
geometric rearrangement, the transition to the exponential EC region, and the
final breakup of the viscoelastic thread. The trends presented in Figure 8
show that increasing the contact angle $\theta_{E}$ results in an opposite
trend from the increase in Bond number. There is a significant delay in the
transition to the EC regime, of up to $23\%$ for more hydrophobic substrates.

These results demonstrate that the operating range in DoS rheometry can be
increased by tuning the substrate wettability. Specifically, increasing the
substrate wettability leads to a more rapid initial necking during the IC
regime. The capillarity-driven drainage of fluid away from the thinning neck
leads to more rapid growth of the elastic stresses in the thinning thread and
an earlier onset of the EC regime (at slightly larger length scales). This is
the configuration we seek to establish in DoS rheometry because it corresponds
to a constant imposed strain rate and facilitates determination of the
(unknown) relaxation time of a single fluid.

##  4 Extensional Rheometry with DoS

In this section, we explore conditions for optimising the operational range of
DoS rheometry. We also present a simplified one-dimensional model of the
thinning dynamics whose predictions are compared to those from the full
numerical simulations. Finally, by comparing our results to the predictions of
the FENE-P model in a capillary thinning flow we provide a methodology for
improving the analysis of transient extensional rheometry data obtained from
DoS experiments near the limit of finite time breakup.

###  4.1 One-dimensional thinning model

We begin with the development of a one-dimensional model for describing the
thinning dynamics in a DoS flow configuration. This is inspired by the one-
dimensional models developed by Tirtaatmadja _et al._ 41 and Wagner _et al._
55, but we seek to extend these descriptions to incorporate the perturbative
roles of gravitational body forces and substrate wettability. We first
consider the force balance for a slender viscoelastic thread expressed in
dimensionless form 41:

$\dfrac{1}{R_{min}(t)}=C_{1}\left(-\dot{R}_{min}(t)\right)^{2}+Oh\left[6\beta\dfrac{\dot{R}_{min}(t)}{R_{min}(t)}+\left(1-\beta\right)\dfrac{1}{De}\Delta\sigma_{p}(t)\right],$
(9)

where the over-dot designates a total derivative with respect to time. In Eq.
(9), the term on the left-hand side corresponds to the capillary pressure
contribution driving the thinning. On the right-hand-side, the first term
describes inertial acceleration in which $C_{1}$ is an adjustable pre-factor
which captures the effects of gravitational body forces and substrate
wettability on the initial inertio-capillary regime (as reported in Section
3). The second term on the right-hand-side of Eq. (9) expresses the
contribution of viscous forces, where $\beta=\eta_{s}/\eta_{l}$ is the
relative contribution of the Newtonian viscous solvent, and the last term,
given by

$\Delta\sigma_{p}(t)=\left[A_{zz}-A_{rr}\right]/\left({1-{tr\left(\mathbf{A}\right)}/{L^{2}}}\right),$
(10)

corresponds to the dimensionless polymeric normal stress difference that
develops in the thinning filament.

If the contribution of the viscous stresses is negligible ($Oh\ll 1$) compared
to the capillary and elasticity contributions, then Eq. (9) can be re-written:

$\dfrac{1}{R_{min}(t)}\sim
C_{1}\left(-\dot{R}_{min}(t)\right)^{2}+\left(1-\beta\right)\dfrac{Oh}{De}\dfrac{\left(A_{zz}-A_{rr}\right)}{\left[1-\frac{tr\left(\mathbf{A}\right)}{L^{2}}\right]}.$
(11)

As the filament necks down, we have seen from our simulations (cf. Figure
3(a)) that the axial stress component eventually becomes dominant, so that
$A_{zz}\gg A_{rr}$. Furthermore, in the limit of large polymer chain
extensibility, before the effects of finite extensibility become important, we
can take $1/\left(1-tr\left(\mathbf{A}\right)/L^{2}\right)\approx 1$. Thus,
provided $1\leq A_{zz}\ll L^{2}$, Eq. (6) combined with Eq. (7) reduces to a
single evolution equation for $A_{zz}(t)$ and $R_{min}(t)$:

$\dot{A}_{zz}\approx-\left(4\frac{\dot{R}_{min}(t)}{R_{min}(t)}+\frac{1}{De}\right)A_{zz}.$
(12)

Separation of variables and integration of this equation yields an expression
for $A_{zz}$ in terms of $R_{min}(t)$ and $De$ 44:

$A_{zz}(t)=\dfrac{A_{zz}^{0}}{R_{min}^{4}(t)}\exp{\left(-\frac{t}{De}\right)},$
(13)

where $A_{zz}^{0}$ is an integration constant that can be set to unity
corresponding to an initially unstretched polymer chain. Substituting Eq. (13)
into Eq. (11), leads to the following nonlinear evolution equation for
$R_{min}(t)$:

$\dfrac{1}{R_{min}(t)}\approx
C_{1}\left(-\dot{R}_{min}(t)\right)^{2}+\left(1-\beta\right)\dfrac{Oh}{De}\dfrac{1}{R_{min}^{4}(t)}\exp{\left(-\frac{t}{De}\right)}.$
(14)

Equation (14) incorporates the competing effects of fluid inertia,
capillarity, and nonlinear fluid elasticity on the dynamics of filament
thinning. At very short times, before elastic stresses grow to enter the
dominant balance, balancing just the capillary and inertial terms in the IC
regime, we have $1/R_{min}(t)\approx C_{1}(-\dot{R}_{min}(t))^{2}$.
Integration of this equation from $R_{min}=1$ at $t=0$ results in the
expression:

$R_{min}(t)=\alpha(t_{max}-t)^{2/3}.$ (15)

The radius therefore initially decreases as a power law in time. Here,
$\alpha$ is a dimensionless pre-factor which determines the rate-of-thinning
during the inertio-capillary balance 56, 57, 27, 35 and is related to the pre-
factor $C_{1}$ in the original force balance by the expression
$C_{1}=9/(4\alpha^{3})$. In the absence of fluid elasticity $t_{max}$ is the
time to breakup; however substituting Eq. (15) into Eq.(13 it is clear that
the polymer stretch grows extremely rapidly during this inertio-capillary
thinning phase until elastic stresses become large enough to enter the
dominant balance in Eq. (11). At this time (denoted $t_{max}$), the thinning
rate drops dramatically, inertial effects become progressively less important
and the transition to the EC regime begins. In the EC regime, the dominant
balance between capillarity and elasticity in Eq.(14) gives:

$\dfrac{1}{R_{min}(t)}\approx(1-\beta)\left(\dfrac{Oh}{De}\leavevmode\nobreak\
\dfrac{1}{R_{min}^{4}(t)}\right)\exp(-t/De),$ (16)

Rearranging we obtain the following expression for the evolution of the
filament radius in the EC regime:

$R_{min}(t)\approx\left[(1-\beta)\dfrac{Oh}{De}\right]^{1/3}\exp(-t/3De),$
(17)

which is consistent with the characteristic exponential rate-of-thinning in
the viscoelastic filament confirmed notably by Entov and Hinch 44, Amarouchene
_et al._ 58, Deblais _et al._ 17. At very long times, as the filament thins to
very small scales, $A_{zz}(t)$ approaches the finite extensibility limit
$L^{2}$ and the thread enters the TVEC regime 35. In this regime, the radius
once again decreases linearly in time and the strain rate grows rapidly. The
dynamics of this regime have been considered by Wagner _et al._ 55 but are
very challenging to resolve numerically and are beyond the scope of the
simulations in the present paper.

This simple one-dimensional model captures the overall dynamics of the
filament thinning process but is not complete enough to describe the
additional driving forces provided by the perturbative effects of gravity or
substrate wettability. However, the contribution of these effects to the
inertio-capillary thinning process (observed in Figures 5 and 7) can be
captured through the changes in the pre-factor $\alpha$ defined in Eq. (15)

Figure 9: Temporal variation of $R^{1.5}_{min}$ for $0\leq Bo\leq 1$ and the
corresponding linear fits (blue solid lines) allow determination of the pre-
factor $\alpha$, according to the power-law associated with the inertio-
capillary dynamics given by Eq. (15). The initial onset of the IC thinning
regime can be identified to be at $R^{1.5}_{min}\approx 0.8$ (or equivalently
$R_{min}\approx 0.85$) (dashed line).

To illustrate this, following the graphical approach suggested by Day _et al._
56 we use a re-arranged form of Eq. (15),
$R_{min}^{3/2}(t)=-\alpha^{3/2}t+\alpha^{3/2}t_{max}$, and we plot in Figure 9
the temporal evolution of $R_{min}^{3/2}$ on a semi-log scale; this allows us
to rapidly determine the value of the constant $\alpha$ for each simulation
and how it varies with the Bond number. It is clear that for all $Bo$
examined, there is a distinct range $0.1\leq R_{min}^{3/2}\leq 0.8$, where
$R_{min}^{3/2}$ decreases linearly in time, consistent with the IC regime. We
indicate the onset of the IC regime at a specified time, named $t_{IC}$ which
corresponds to $R_{min}\approx 0.85$ (or equivalently $R_{min}^{3/2}\approx
0.8$), which is a value that can be easily monitored in the experiments.
Fitting of the linear portions of the data in Figure 9 accounting for the re-
arranged form of Eq. 15, yields $0.355\leq\alpha\leq 0.425$ for $0\leq Bo\leq
1$.

We follow the same procedure for quantifying the effect of the substrate
wettability on the inertio-capillary thinning, which yields
$0.355\leq\alpha\leq 0.212$ for $30^{\circ}\leq\theta_{E}\leq 80^{\circ}$. We
provide the values of the pre-factor $\alpha$ as a function of the Bond number
and the contact angle $\theta_{E}$ in Tables 4 and 5, respectively.

In addition, we also plot in Figure 10(a) and (b) the variation of this
numerical pre-factor $\alpha$ as a function of the Bond number and the cosine
of the contact angle ($cos\left(\theta_{E}\right)$), which provides a measure
of the force driving the lateral spreading of the fluid across the solid
substrate. The pre-factor $\alpha$ in Eq. (15) follows a power-law evolution
with $Bo$, which can be approximated by
$\alpha(Bo)\approx\alpha_{0}\left(1+0.223Bo^{2/3}\right)$, where $\alpha_{0}$
corresponds to the value of $\alpha$ for $Bo=0$. The inertio-capillary scaling
of the initial thinning regime is thus only weakly modified by gravity (for
$Bo\leq 1$). Inspection of Figure 10(b) shows that as the wettability of the
substrate is reduced the rate of spreading is also reduced. The dependence of
$\alpha$ on the contact angle can be closely approximated by
$\alpha(\theta_{E})\approx 0.377\cos(\theta_{E})^{1/3}$.

Table 4: Variation of the pre-factor $\alpha$ for the Inertio-Capillary (IC) regime as a function of the Bond number. $Bo$ | 0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.6 | 0.8 | 1
---|---|---|---|---|---|---|---|---
$\alpha$ | 0.355 | 0.369 | 0.381 | 0.392 | 0.401 | 0.412 | 0.424 | 0.425
Table 5: Dependence of the pre-factor $\alpha$ for the Inertio-Capillary (IC) regime on the substrate wettability characterised by the contact angle $\theta_{E}$. $\theta_{E}$ | $30^{\circ}$ | $45^{\circ}$ | $50^{\circ}$ | $60^{\circ}$ | $70^{\circ}$ | $75^{\circ}$ | $80^{\circ}$
---|---|---|---|---|---|---|---
$\alpha$ | 0.355 | 0.338 | 0.329 | 0.304 | 0.266 | 0.241 | 0.212
Figure 10: Variation of the pre-factor $\alpha$ with (a) the Bond number and
(b) the substrate wettability characterised by $\cos(\theta_{E})$, is
characterised by a power law fit of index 2/3 and 1/3, respectively. Figure
11: One-dimensional model predictions (solid lines) and numerical simulation
results (symbols) for the temporal evolution of (a) the viscoelastic filament
radius $R_{min}$ and (b) the dimensionless strain rate $Wi$ in the neck. Here
$Bo$ varies as in Figure 5 and the rest of the parameters remain unchanged.
The exponential filament thinning rate in the Elasto-Capillary (EC) regime is
also shown in (a) and the corresponding constant $Wi=2/3$ plateau (dashed
lines) is shown (b).

Having determined the variation of $\alpha$ with the Bond number and the
macroscopic contact angle, we compare the predictions of the one-dimensional
model in Eq. (14) with the results of the numerical simulations, for $0\leq
Bo\leq 1$ at a fixed contact angle $\theta_{E}=30^{\circ}$. We show in Figure
11(a) and (b) the results of the comparison between the full numerical
simulations and the solution of Eq. (14) (using the Matlab ODE23t package),
for both the temporal evolution of the filament radius $R_{min}(t)$ and the
corresponding evolution of the dimensionless strain rate $Wi(t)$,
respectively. Here, the time $t$ has been shifted by an initial offset
$t_{IC}$ which corresponds to the time when $R_{min}=0.85$ after the initial
reconfiguration process (during which the prolate drop first touches the
substrate and rapidly spreads laterally to form a necked droplet). Figure 9
clearly shows that below a radius $R_{min}\approx 0.85$ (or
$R_{min}^{3/2}\approx 0.8)$ the initial phase of filament thinning is well
described by the inertio-capillary scaling. Inspection of Figure 11(a) and (b)
reveals that the simplified one-dimensional model accurately captures both the
IC and the EC thinning regimes. However, the transition between the two
regimes is less abrupt in the full simulations as the shape of the filament
rearranges and transient two-dimensional effects cannot be neglected. Similar
observations have been made regarding the effect of the substrate wettability
59.

###  4.2 Improvement of thinning predictions with the FENE-P analytical
solution

Table 6: Determination of the three fitting parameters in Eq. (19): Deborah number (De), elasto-capillary number (Ec), and polymeric finite extensibility ($L^{2}$); Here, $(De_{input},Ec_{input},L^{2}_{input})$ are the known (ground truth) values that are used as inputs to the numerical simulations; $De^{[Oldroyd-B]}_{fit}$ and $De^{[AM]}_{fit}$ are the values of the dimensionless relaxation time obtained when we use the asymptotic Oldroyd-B result: $R_{min}(t)\sim\exp(-t/(3De)$ (labelled “Oldroyd-B Fitting") and the “Anna-McKinley" model provided in Eq. (18) (labelled “Anna-McKinley Fitting") to determine the dimensionless relaxation time of the dilute polymer solution from the data in the EC regime (here, we consider the data generated by the numerical simulations). $(De^{[FENE]}_{fit},Ec^{[FENE]}_{fit},L^{2}{}^{[FENE]}_{fit})$ are the parameters obtained when Eq. (19) is fitted to the available data for the filament radius (labelled “FENE-P Fitting”). The errors $\epsilon^{[Oldroyd-B]}$, $\epsilon^{[AM]}$ and $\epsilon^{[FENE]}$ are the discrepancies in the prediction of the dimensionless polymeric relaxation time $De_{fit}$ when either the Oldroyd-B result or Eq. (18) or Eq. (19), respectively are used. Input | Oldroyd-B Fitting | Anna-McKinley Fitting | FENE-P Fitting |  Error for $De_{fit}$
---|---|---|---|---
$De_{input}$ | $Ec_{input}$ | $L^{2}_{input}$ | $De^{[Oldroyd-B]}_{fit}$ | $De^{[AM]}_{fit}$ | $De^{[FENE]}_{fit}$ | $Ec^{[FENE]}_{fit}$ | $L^{2}{}^{[FENE]}_{fit}$ | $\epsilon^{[Oldroyd-B]}$ | $\epsilon^{[AM]}$ | $\epsilon^{[FENE]}$
1 | 0.03 | 400 | 0.60 | $0.56\pm 0.006$ | $0.70\pm 0.006$ | 0.003 | $10^{4}$ | $-40\%$ | $-44\%$ | $-30\%$
1 | 0.03 | 900 | 0.66 | $0.80\pm 0.001$ | $0.85\pm 0.010$ | 0.430 | $10^{2}$ | $-34\%$ | $-20\%$ | $-15\%$
1 | 0.03 | 1600 | 0.94 | $0.90\pm 0.001$ | $0.91\pm 0.006$ | 0.680 | $10^{2}$ | $-6\%$ | $-10\%$ | $-9\%$
1 | 0.03 | 2500 | 1.00 | $0.96\pm 0.007$ | $0.98\pm 0.020$ | 0.740 | $10^{2}$ | 0% | $-4\%$ | $-2\%$

We now revisit the role of the polymer finite extensibility $L^{2}$ on the
dynamics in the EC thinning regime that is shown in Figure 4. Our simulations
revealed that low values of the polymer extensibility ($L^{2}<1600$) resulted
in more rapid thinning rates in the EC regime and very short-lived periods of
exponential thinning. Similar observations are obtained in experimental
measurements by Gaillard _et al._ 60. Directly fitting a simple exponential to
such data leads to significant bias in DoS rheometry measurements of polymer
solutions, such as an underestimation of the polymer relaxation time. The
resulting error can be as large as approximately $40\%$ for $L^{2}=400$ if the
analytical expression for exponential filament thinning predicted by the
Oldroyd-B model, i.e., $R_{min}\sim\exp(-t/(3De)$ 44, 61, 9, is used to
extract the thinning rate or the apparent relaxation time. To address this
bias, Lauser _et al._ 28 and Jimenez _et al._ 34 use a four-parameter
empirical model originally proposed by Anna and McKinley 62 to more
systematically regress experimental measurements of filament thinning close to
breakup. In this model, the time-evolving neck radius is written in the form
of a four-parameter fit:

$R_{min}(t-t_{1})=A\exp{(-B(t-t_{1}))}-C(t-t_{1})+D,$ (18)

where the dimensionless time $t$ is shifted by the dimensionless value $t_{1}$
which corresponds to the time at which the EC regime begins. The parameter $B$
is the thinning rate (expected to be $B=1/3De$) and the transition to a linear
dependence in the final TVEC region close to the breakup is captured by the
final terms ($C,D$)). Lauser _et al._ 28 and Jimenez _et al._ 34 show that a
close agreement can be obtained between this functional form and experimental
data. Here, we explore how well this formulation can also characterise the
dynamics predicted by our numerical simulations (where the “ground truth" is
already known), as well as an alternative form motivated by the (known)
analytic form of the one-dimensional filament thinning process for the FENE-P
model.

Wagner _et al._ 55 derived an implicit analytical expression describing the
filament thinning and breakup of FENE-P fluids. Their expression accounts for
the thinning dynamics in the EC and the terminal (TVEC) regimes where finite
extensibility effects become important. In our notation this analytic solution
can be written in the form:

$t-t_{1}=-De\frac{\left(L^{2}+2\right)}{\left(L^{2}+3\right)^{2}}\Biggl{\\{}\frac{1}{1+EcR_{1}\left(L^{2}+3\right)}-\frac{1}{1+\xi
EcR_{1}\left(L^{2}+3\right)}\\\ +3\ln\left(\frac{1+\xi
EcR_{1}\left(L^{2}+3\right)}{1+EcR_{1}\left(L^{2}+3\right)}\right)+4EcR_{1}\frac{\left(L^{2}+3\right)}{\left(L^{2}+2\right)}\left(\xi-1\right)\Biggr{\\}}.$
(19)

Here the time-evolving dimensionless filament radius $R_{min}(t)$ is re-scaled
with the dimensionless value $R_{1}$ which represents the minimum filament
radius at the onset of the EC regime. Thus at time $t=t_{1}$ the re-scaled
radius is $R_{min}/R_{1}=1$. This scaling leads to a new dimensionless
filament radius $\xi=R_{min}/R_{1}$ and the dimensionless time ($t-t_{1}$)
characterises the entire evolution of the EC thinning regime. In addition,
$Ec=GR_{0}/\gamma$ is the elasto-capillary number which provides the
appropriate dimensionless measure of the fluid elasticity (i.e. the elastic
modulus $G$ of the polymer solution) scaled with the capillary pressure
$\gamma/R_{0}$. Equation (19) includes three independent fitting parameters:
the relaxation time $\tau$ or (equivalently the Deborah number $De$ in
dimensionless form), the elasto-capillary number $Ec$ and the finite
extensibility $L^{2}$ of the dissolved polymer. In a DoS rheometry experiment
with an unknown sample, we ideally seek to simultaneously determine all three
fitting parameters. Because we know in advance the ground truth values of
$De$, $Ec$ and $L^{2}$ (which act here as an input to our numerical
simulations), we can explore the suitability of using Eq. (19) to accurately
retrieve these three material parameters from measurements of the filament
midpoint radius $R_{min}(t)$.

Figure 12: Estimated $De_{fit}$ values from Table 6 for the range
$L^{2}_{input}=\\{400,900,1600,2500\\}$ using the “Oldroyd-B” (blue circles),
the “Anna-McKinley” (red squares) and the “FENE-P” (green diamonds) fitting
approaches. The dashed black line shows the ground-truth value of
$De_{input}=1$.

We provide in Table 6 the results of the fitting process for each
$(De_{input},Ec_{input},L^{2}_{input})$ to explore how accurately we can
determine the (dimensionless) relaxation time by fitting the neck evolution
during the thinning process. We consider first the well-known asymptotic
Oldroyd-B result: $R_{min}\sim\exp{(-t/3De)}$ 44, 61, 9, labelled “Oldroyd-B
Fitting" here. Subsequently, we fit the same data using the semi-empirical
“Anna-McKinley" model described by Eq. (18). These results are labelled by the
superscript $[AM]$. Finally, we also employ the FENE-P analytical solution
provided in Eq. (19) to fit the different computational datasets. This is
labelled “FENE-P Fitting”. Table 6 highlights the high bias error obtained for
low polymer extensibilities ($L^{2}_{input}=400$ and 900) when the
conventional “Oldroyd-B Fitting" is performed. On the other hand, it is
evident that for $L^{2}_{input}=400$ the use of Eq. (19) reduces considerably
the estimated error for $De_{fit}$ by approximately $10\%$. However, the use
of Eq. 18) results in a slightly worse estimated error in the value determined
for $De_{fit}$ compared to the “Oldroyd-B Fitting". For a moderate value of
extensibility ($L^{2}_{input}=900$) the “Anna-McKinley Fitting" and “FENE-P
Fitting" both result in reductions in the error incurred in determining the
fluid relaxation time by $14\%$ and $19\%$, respectively. Surprisingly,
however, these two methods of analysing Dripping-Onto-Substrate thinning data
are both found to lead to marginally worse predictions of $De_{fit}$ for
$L^{2}_{input}=1600$ and 2500. We graphically summarise in Fig. 12 the
estimated values of $De_{fit}$ using the “Oldroyd-B", “Anna-McKinley" and
“FENE-P" methods of analysis. It is evident that using the “FENE-P" solution
provides more accurate results than the “Anna-McKinley" expression in all
cases. In particular, it is seen that for low extensibility fluids
($L^{2}_{input}\leq 900$) using Eq. (19) provides the most accurate
determination of the characteristic fluid relaxation time.

We now focus in more detail on the “FENE-P Fitting" approach. Careful
inspection of Table 6 shows that using this fitting method leads to values of
$Ec^{[FENE]}_{fit}$ and $L^{2}{}^{[FENE]}_{fit}$ with pronounced deviations
from the corresponding input values. We investigate this surprising result
further by first plotting in Figure 13 the evolution of the dimensionless
filament radius $\xi(t-t_{1})=R_{min}(t-t_{1})/R_{1}$ with time $t-t_{1}$ to
focus on the thinning during the elasto-capillary regime. Specifically, we
plot (i) the results of the numerical simulations (symbols) for the four
different values $L^{2}_{input}=\\{400,900,1600,2500\\}$ at fixed
$De_{input}=1$ and $Ec_{input}=0.03$, (ii) the predictions of Eq. (19) for
${De_{input},Ec_{input},L^{2}_{input}}$ (solid black lines), and (iii) the
results of Eq. (19) for
$(De^{[FENE]}_{fit},Ec^{[FENE]}_{fit},L^{2}{}^{[FENE]}_{fit})$ (dashed red
lines). Inspection of Figure 13 highlights the significant deviation from the
simple exponential Oldroyd-B asymptotic result
($\xi\approx\exp(-(t-t_{1})/3De)$) due to the effects of polymer finite
extensibility. This deviation is most pronounced for $L^{2}_{input}=400$, for
which the exponential region is very short and restricted to times
$t-t_{1}\lesssim 1$. As a consequence of this truncated exponential thinning,
robust estimation of the characteristic relaxation time for a dilute polymer
solution with molecules of limited extensibility is very challenging.
Moreover, we also observe that both the solid black and red dashed lines
overlap substantially and both accurately capture the results of the numerical
simulations (symbols), with significant deviations only being observed for the
lowest extensibility case ($L^{2}_{input}=400$).

Figure 13: Simulation data denoted by diamond symbols for different values of
polymer finite extensibility $L^{2}_{input}=\\{400,900,1600,2500\\}$ and fixed
$De_{input}=1$, $Ec_{input}=0.03$. The solution of Eq. (19) for the same
values $(De_{input},Ec_{input})$ and range
$L^{2}_{input}=\\{400,900,1600,2500\\}$ as in the numerical simulations are
shown by solid black lines. Dashed red lines correspond to the predictions of
Eq. (19) where the results of the “Improved fitting" columns in Table 6 for
$(De^{[FENE]}_{fit},Ec^{[FENE]}_{fit},L^{2}{}^{[FENE]}_{fit})$ are considered.
The prediction of the exponential Oldroyd-B decay during the Elasto-Capillary
(EC) regime 44, 61, 9 is also indicated by the dashed black line.

8I

Figure 14: Contour plot of the logarithm of the Mean Squared Error
($\log_{10}MSE$) when we fit Eq. (19) with fixed value $De^{[FENE]}_{fit}=1$
and for a range $Ec_{test}$ and $L^{2}_{test}$ values to the data generated by
numerical simulations of the FENE-P model for $L^{2}_{input}=2500$. The
colorbar corresponds to the magnitude of $\log_{10}MSE$ for each
$(Ec_{test},L^{2}_{test})$ combination. The white circle and triangle symbols
indicate respectively the known input values
$(Ec_{input}=0.03,L^{2}_{input}=2500)$, and the optimal fitted values as
predicted by Eq. (19) $(Ec^{[FENE]}_{fit}=0.58,L^{2}{}^{[FENE]}_{fit}=115.7)$
for fixed $De^{[FENE]}_{fit}=1$.

To understand in more detail the deviations observed in Table 6 for the fitted
values of $(Ec^{[FENE]}_{fit},L^{2}{}^{[FENE]}_{fit})$ given the known (input)
values of $(Ec_{input},L^{2}_{input})$, we also compute the solution of Eq.
(19) with fixed $De_{fit}{}^{[FENE]}=1$ and for various elasto-capillary
number and finite extensibility combinations, represented by $Ec_{test}$ and
$L^{2}_{test}$, respectively. To capture the range of values studied by Wagner
_et al._ 55 as well as the input values ${Ec_{input},L^{2}_{input}}$ utilised
in this work, we explore a representative range of $10^{-2}\leq Ec_{test}\leq
1$ and $10^{2}\leq L^{2}_{test}\leq 10^{4}$. In Figure 14 we show (in log
scale) a contour plot of the log Mean Squared Error ($\log_{10}MSE$) of the
output of Eq. (19) when it is fitted to test data over this range as a
function of $Ec$ and $L^{2}$. We compute the error as
$\log_{10}MSE=\sum_{i=1}^{n}(\xi(t-t_{1})_{i}-\hat{\xi}(t-t_{1})_{i})^{2}/n$;
here, $\xi(t-t_{1})_{i}$ is the actual output of the FENE-P analytical
solution for the i-th instance when ${Ec_{input}=0.03,L^{2}_{input}=2500}$ are
considered; $\hat{\xi}(t-t_{1})_{i})$ is the predicted value of Eq. (19) for
each test combination $Ec_{test},L^{2}_{test}$ at time $(t-t_{1})_{i}$, and
$n$ is the number of available data points. We also indicate in Figure 14 the
known input combination $(Ec_{input}=0.03,L^{2}_{input}=2500)$ with a white
circle symbol, and the “best’" fit (minimum $MSE$) values
$(Ec^{[FENE]}_{fit}=0.58,L^{2}{}^{[FENE]}_{fit}=115.7)$ with white triangle
symbol. This best fit value is obtained from fitting Eq. (19) with known
$De^{[FENE]}{}_{fit}=1$ to the available simulation data obtained with
$L^{2}_{input}=2500$. Figure 14 shows the existence of a broad trough in
$Ec,L^{2}$ space, where the fitting algorithm determines a local minimum in
the mean square error. The strong (anti)-correlation between the values
$\\{Ec_{test},L^{2}_{test}\\}$ that minimise the MSE arises from the
functional form of Eq. (19). Close inspection shows that the terms $Ec$ and
$L^{2}$ appear repeatedly as products. Regression can identify a locally-
optimal value of $\\{Ec\times L^{2}\\}$ that minimises the error in fitting
$R_{min}(t)$, but uniquely determining optimal individual values of $Ec$ and
$L^{2}$ is more challenging. However, imprecise determination of the best-fit
values of $Ec$ and $L^{2}$ does not corrupt the (enhanced) robustness in the
determination of the fluid relaxation time $\tau$ (or equivalently the
dimensionless value of $De$) from experimental observations of the filament
neck $R_{min}(t)$. Therefore, the improved values of $De_{fit}$ obtained from
regressing Eq. (19) to experimental test data obtained in DoS (as compared to
fitting the asymptotic Oldroyd-B result or using the Anna-McKinley form (Eq.
(18)) provide a promising direction for ensuring more robust extensional
rheological measurements of dilute polymeric solutions with Dripping-onto-
Substrate rheometry.

##  5 Conclusions

We have studied the thinning and breakup of a liquid droplet that is dripping
onto a partially wetting substrate through axisymmetric numerical simulations,
using a FENE-P constitutive relation which accounts for finite polymer chain
extensibility. We have used the open-source code Basilisk, which is based on a
Volume-of-Fluid interface-capturing methodology, and utilises adaptive mesh
refinement for accurate and efficient free surface flow solutions. The
simulation begins as a prolate drop of fluid is brought into contact with the
solid substrate at its base, establishing a macroscopic contact angle as the
boundary condition that parameterises the substrate wettability. The numerical
simulations account for capillary, gravitational, inertial, and viscous forces
as well as substrate wettability effects and capture successfully the two main
regimes that drive the filament thinning in Dripping-onto-Substrate (DoS)
rheometry: (1) the dominance of inertial and capillary forces at early times,
which result in the radius of the neck decreasing as a $2/3$ power-law in time
and, (2) at later times, the establishment of a characteristic elasto-
capillary balance which leads to a period of exponential filament thinning, in
which nonlinear fluid elasticity stabilises the thinning process, resulting in
thin long-lived viscoelastic ligaments.

Successful DoS experiments require that the fluid under study exhibits
sufficiently large viscoelasticity (i.e. $De=\tau/\sqrt{\rho
R^{3}_{0}/\gamma}\geq 0.5$) and possess a large enough extensibility of the
polymeric chains (i.e. $L^{2}\geq 1600$) for a clear elasto-capillary balance
to be established. Moreover, for the first time, the role of gravity
(parameterised by a dimensionless Bond number, $Bo=\rho
gR_{0}\ell_{0}/\gamma$), and the wettability of the substrate (parameterised
by the macroscopic contact angle $\theta_{E}$) have both been investigated
systematically. Increasing the Bond number and the substrate wettability both
are seen to accelerate the initial rate of inertio-capillary necking and the
transition to the elasto-capillary balance, without affecting the subsequent
evolution of the exponential decrease in filament radius during the elasto-
capillary thinning. Our computations show that robust measurements of the
polymer relaxation time can be determined in DoS rheometry over a range of
conditions but that operational limits of DoS measurements can be optimised by
tuning the relative magnitudes of the Bond number and the substrate
wettability. In addition, we have also developed a simple one-dimensional
thinning model which is shown to be capable of capturing the initial Inertio-
Capillary (IC) and the subsequent characteristic Elasto-Capillary (EC)
thinning regimes. With this model, we can describe the weak but systematic
dependence of the inertio-capillary balance on both gravitational body force
and the macroscopic contact angle through the variation in the pre-factor
$\alpha$ of the inertio-capillary balance that results in Eq. (15). We also
report simple power-law correlations of this pre-factor with variations of
$Bo$ and wettability (as represented by the factor $cos(\theta_{E})$).
Finally, for fluids with limited extensibilities that do not exhibit very
clear exponential thinning regimes, we have proposed a model-fitting framework
which is based on the analytical 1D solution for the thinning and breakup of
FENE-P filaments. This improves determination of the true relaxation time of
an unknown polymer solution through DoS Rheometry (which can be challenging
when polymeric solution samples with limited finite extensibilities are
involved).

By considering the critical ranges of key dimensionless parameters $(De$,
$L^{2}$, $Bo$, $\theta_{E})$ that govern the performance of DoS rheometry
using time-resolved free surface numerical simulations, the design of optimal
DoS experiments is now possible. Computational Rheometry thus provides a
useful digital design tool for enhancing our ability to successfully measure
the transient extensional response of low-viscosity complex fluids.

## Conflicts of interest

There are no conflicts of interest to report.

## Acknowledgements

This work is supported by the Engineering and Physical Sciences Research
Council, United Kingdom, through the EPSRC PREMIERE (EP/T000414/1) Programme
Grant. Support from Johnson Matthey for Konstantinos Zinelis is also
gratefully acknowledged.

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