# Rapid Viscoelastic Spreading

Ambre Bouillant  Pim J. Dekker  Michiel A. Hack  Jacco H. Snoeijer  Physics of
Fluids Group, Mesa+ Institute, University of Twente, 7500 AE Enschede, The
Netherlands

###### Abstract

We investigate the rapid spreading dynamics of a viscoelastic drop on a solid.
Upon contact, surface tension drives a fast motion of the contact line along
the substrate. Here, we resolve this motion for viscoelastic liquids by
experiments on aqueous polymer solutions of varying concentration. It is found
that while the contact line motion is only mildly affected by the polymers,
the interface profile becomes much sharper owing to singular polymer stress.
This behaviour is reminiscent of that observed for viscoelastic drop
coalescence; in this light we revisit the analogy between spreading and
coalescence, which sheds new light also on the spreading of Newtonian liquids.

##  I Introduction

When a drop comes into contact with a surface, it starts to spread on the
solid. A liquid bridge connecting the drop to the substrate moves radially
outwards from the contact point, wetting a circular area of radius $r_{0}(t)$.
The fate of the drop after contact has been the subject of numerous studies,
focussing both on static and dynamical aspects. The statics deals with the
final shape of the drop and ultimate spreading extent, while the dynamics is
dedicated to how a contact line moves and how a drop reaches its final state
De Gennes (1985); Bonn et al. (2009); Snoeijer and Andreotti (2013). In
partial wetting conditions, the equilibrium contact angle $\theta$ sets the
final shape of the liquid drop. For the most common case of water drops, the
combination of limited spreading and low viscosity of the liquid means that
the spreading dynamics is resisted by inertia. A balance between the driving
capillary effect with the opposing inertia provides the relevant time scale,
namely, the inertio-capillary time $\tau=\sqrt{\rho R^{3}/\gamma}$, where
$\gamma$ denotes the liquid surface tension and $\rho$, the liquid density.
For a water drop with millimetric size $R$, $\tau\approx 3$ ms. When observed
with the naked eye, the drop thus reaches it final shape quasi-
instantaneously. In complete wetting conditions, by contrast, the liquid keeps
on spreading slowly making it easily observable. This late stage in the
spreading of Newtonian liquids was first studied by Tanner Tanner (1979), who
showed that the radius of the wetted area $r_{0}(t)$ grows as $t^{1/10}$. This
slow regime is dominated by strong viscous dissipation close to the moving
contact line Tanner (1979); Voinov (1976); Cox (1986). This law, however,
fails to capture the early stages of the spreading dynamics, where the
geometry of the liquid bridge is manifestly different from the slender wedge
assumption that underlies Tanner’s law.

The rapid spreading that occurs immediately after the drop is brought into
contact with spreading was addressed much more recently, starting with Biance
et al. Biance et al. (2004). They identified a regime at early time
($t<\tau$), where inertia balances capillarity. The size of the wetted area
$r_{0}(t)$ grows as

$\frac{r_{0}}{R}\sim\left(\frac{\gamma}{\rho
R^{3}}\right)^{1/4}\,t^{1/2}=\left(\frac{t}{\tau}\right)^{1/2},$ (1)

which involves the inertio-capillary time. This initial work focussed on
hydrophilic substrates that are completely wetting, and was followed by other
studies that varied the substrate properties using both experiments and
simulations Bird et al. (2008); Carlson et al. (2012a, b); Winkels et al.
(2012); Eddi et al. (2013a); Stapelbroek et al. (2014). While some works
indicated that properties of the substrate influence the dynamics Bird et al.
(2008); Carlson et al. (2012a, b), it was found that – at very early times –
the spreading dynamics is universal Winkels et al. (2012); Eddi et al. (2013a)
and described by the same scaling (1) as for hydrophilic surfaces.
Surprisingly, this universal spreading was found to be irrespective of the
substrate wettability and nature (whether it be completely or partially
wetting, or even soft), and despite the presence of chemical patterns or any
roughness (anisotropic, random or even controlled) Stapelbroek et al. (2014).

Curiously enough, the spreading law (1) is identical to the growth of the
bridge of during the coalescence of two drops, in the inviscid regime Eggers
et al. (1999); Duchemin et al. (2003); Thoroddsen et al. (2005); Aarts et al.
(2005); Paulsen et al. (2011); Sprittles and Shikhmurzaev (2014). The analogy
between rapid spreading and coalescence has proven very robust, both in the
low-viscosity limit Biance et al. (2004); Courbin et al. (2009); Winkels et
al. (2012) and in the viscous limit Eddi et al. (2013a). Indeed, when a drop
is brought into contact with a solid wall or with another drop, in both cases
the dynamics evolves through a singular liquid bridge that is vanishingly
small at the moment of contact. The success of the spreading-coalescence
analogy has been attributed to the fact that the coalescence geometry exhibits
a mirror-plane that mimics the role of the substrate in spreading. One should
bear in mind, however, that it is not obvious that the flow along the plane of
symmetry is similar in both cases, as for the spreading over a substrate one
expects the no-slip condition to impede the motion of the contact line Carlson
et al. (2012a, b); Johansson and Hess (2018). Still, the spreading-coalescence
analogy offers a rationalisation as to why rapid drop spreading of Newtonian
liquids is insensitive to substrate properties.

In the present study we investigate the rapid spreading of viscoelastic
liquids, focusing on polymer suspensions. Viscoelastic liquids are hybrid in
nature: they can flow like ordinary liquids, yet respond elastically when
excited by rapid deformations Tanner (2000). This is illustrated by saliva and
most biological fluids that contain long and flexible polymers that are
viscoelastic in nature. When a drop of saliva sandwiched between two fingers
is drawn away, it can stretch by several times its radius without breaking, an
elastic response that leads to a highly elongated threads Anna and McKinley
(2001). However, when loaded over a long timescale, such a drop flows as a
liquid and spreads out on the substrate. This viscoelastic behaviour
originates from the polymer chains that are suspended in the Newtonian
solvent. At rest, the polymer chains tend to form coils, for which the free
energy is minimal. When stretched by the flow, however, the polymers elongate
and exert an additional stress onto the solvent. Once the stretching is turned
off, polymers relax back to the coil-like structure, which comes with a
charactersistic relaxation time $\lambda$. Viscoelastic liquids are ubiquitous
in biofluids and in technologies such as coating, printing, aerosol generation
and polymer processing, and their flow raises many challenges. Of particular
interest is how viscoelastic liquids behave near singularities Renardy (2000),
such as flows around sharp edges Hinch (1993); Renardy (1993); Evans and
Sibley (2008), bubble cusps Astarita and Apuzzo (1965); Liu et al. (1995);
Zenit and Feng (2018) or during the breakup of drops Bazilevskii et al.
(1990); Entov and Hinch (1997); Anna and McKinley (2001); Amarouchene et al.
(2001); Clasen et al. (2006); Eggers et al. (2020); Deblais et al. (2020).
Such flows involve regions of extreme polymer stretching – as is also expected
to be the case for rapid viscoelastic spreading.

Figure 1: (a) Side-view schematic of the experimental setup: a drop of PEO is
slowly brought into contact with a smooth, transparent and clean glass plate.
(b) Upon contact, a water drop rapidly spreads over the solid. The white
dashed line indicates the position of the substrate; the lower half of the
image is a reflection. The contact line motion is described by the rapid
growth of the contact radius $r_{0}$. (c) Spreading of a 2.0 wt% PEO drop. The
wetting dynamics is weakly slowed down and the interface profiles are much
sharper, as compared to the case of water. The scale bar, common to both
series of snapshots, indicates 500 $\mu$m. (d) Series of snapshots taken from
below of a drop spreading on a (transparent) substrate. Prior to contact, the
thin air gap separating the drop from the solid produces an interference
pattern. After contact, the liquid quickly wets the solid: the dark area
expands radially by typically 200 $\mu$m in 50 $\mu$s, resulting in spreading
velocities of about 5 m/s. The scale bar indicate 200 $\mu$m.

Here we reveal how the addition of polymers influences the early stages of
drop spreading on a solid substrate. Given the spreading-coalescence analogy
discussed above, the coalescence of viscoelastic drops naturally serves as a
benchmark for rapid viscoelastic spreading Varma et al. (2020); Dekker et al.
(2022). Surprisingly, the bridge temporal evolution during coalescence was
found to be only mildly affected by the addition of polymer Dekker et al.
(2022): the bridge growth is nearly identical to that for Newtonian fluids of
low viscosity Eggers et al. (1999); Duchemin et al. (2003); Thoroddsen et al.
(2005); Aarts et al. (2005); Ristenpart et al. (2006); Paulsen et al. (2011);
Eddi et al. (2013b), and was observed to follow (1) for spherical drops.
However, the spatial features of the coalescing bridges are modified: profiles
are sharper and the bridge curvature is enhanced. In the present study we
experimentally probe the rapid spreading of polymeric drops. Specifically, we
resolve the spatio-temporal dynamics of the bridge growth, and inquire whether
the spreading-coalescence analogy is robust in the context of viscoelastic
liquids.

##  II Experimental procedure

Spreading experiments are performed on partially wetting glass substrates by
studying drops of polymer solutions of low viscosity and varying concentration
$c$. To that end, we use solutions of polyethylene oxide (PEO,
$M_{W}=4.0\times 10^{6}$ g/mol, Sigma-Aldrich) in distilled water. The
solution concentration is varied from 0.01 to 2.0 wt%, across $c^{*}\approx$
0.1–1 wt%, the threshold over which polymer coils start to overlap
(Supplementary Information SI ). Each solution is mixed with a magnetic
stirrer for at least 24 hours and degassed to get rid of air bubbles.

Calibrations of the polymeric solutions used in this study are identical to
those in Dekker et al. (2022), repeated here for completeness in the
Supplementary Information SI . The shear viscosities ($\eta$) of the solutions
are measured using a rheometer (MCR 502 with CP50-1, Anton Paar), the surface
tension ($\gamma$) and the relaxation times ($\lambda$) are deduced from
pendant drop experiments, by fitting the drop shape and looking at the
thinning of the connecting thread with the dispensing needle Anna and McKinley
(2001), respectively. As reported in the SI, the polymer relaxation time
$\lambda$ ranges from 1 to 50 ms in the explored range of concentrations
$c=0.01-2.0$ wt%, and crosses the inertio-capillary time $\tau=\sqrt{\rho
R^{3}/\gamma}\,\approx 3$ ms. Of particular importance is the values taken at
the early stages of spreading of the ratio $\lambda/t$, a “local instantaneous
Deborah number”, previously defined in Dekker et al. (2022) which reflects the
deformation rate. This number takes values as high as 102 up to 104 at the
smallest timescale resolved in our experiments when the polymer concentration
is increased from 0.01 to 2.0 wt%. A strong polymer stretching and elastic
effects at the earliest stages of spreading are thus anticipated.

As illustrated in figure 1(a), the drop is slowly grown using a dispensing
needle connected to a syringe pump (PHD 2000, Harvard Apparatus), pumping at a
rate $<0.1\;\mu$L/min. The needle is placed about $2\rm\;mm$ above the
horizontal substrate, in order to produce drops with radius $R\approx
0.9\rm\;mm$. Experiments are performed on clean transparent glass slides
(supplied from Menzel-Gläser), on which we measure static advancing contact
angles of 52${}^{\circ}\pm 4^{\circ}$. It is well-known that the exact nature
of the substrate, specifically its wettability, has no influence on the early
dynamics of a spreading droplet Winkels et al. (2012). Upon contact with the
substrate, the liquid spreads. The deformations induced by the needle are
local and far from the contact zone, without disturbing the liquid bridge. The
very slow increase in the drop volume enables approach velocities on the order
of $1\;\mu$m/s, that is about three orders of magnitude smaller than typical
bridge velocities The spreading dynamics is recorded both from the side and
from the bottom using synchronized high speed cameras (Nova S12, Photron). The
side-view camera is equipped with a zoom lens (Navitar lens), allowing for
frame rates ranging from 100 to 200kfps and a spatial resolution of typically
2 $\mu$m/pixel. Typical snapshots of the early stage of the spreading of a
drop with radius $R=0.9\pm 0.1$ mm are shown in Fig. 1(b-c) for water and for
a PEO solution of 2wt%, respectively. Time increases from left to right with
intervals of 125 $\mu$s between each image. Note that the upper half of the
image represents the drop, while the lower half (below the white dotted line)
is the reflection in the substrate.

Side views are complemented with synchronized bottom views. The bottom-view
camera, mounted on an inverted microscope with a zoom lens (with 10$\times$
Olympus lens), shoots at 250kfps, which enables a better spatial and temporal
resolutions than for the side-view. We display in Fig. 1(d) series of
snapshots showing the liquid progression. Prior to contact, the thin air gap
separating the drop from the solid produces an interference pattern, where
fringes scroll as the drop approaches the substrate. After contact, the liquid
quickly wets the solid: the dark area expands radially by approximately 200
$\mu$m in 50 $\mu$s, resulting in spreading velocities of about 5 m/s.

We focus on the spatio-temporal evolution of the bridge profile, $r(x,t)$,
defined in Fig. 1 as the distance parallel to the substrate from the drop
vertical axis of symmetry (located in $r=0$) and the liquid-air interface. We
also denote as $r_{0}(t)$ the radius of the wetted area, which corresponds to
$r_{0}(t)=r(0,t)$. The profiles are extracted using a sub-pixel interface
tracking code. Finding the precise moment of first contact between the drop
and the substrate is crucial to determine the growth exponent $\alpha$. Here
$t=0$ is determined by the bottom view, whose resolution is typically 5
$\mu$s, and further refined by extrapolating a power-law $r_{0}\propto
t^{\alpha}$, following the protocol as described in Dekker et al. (2022).

##  III Rapid spreading dynamics

###  III.1 Temporal dynamics

The effect of polymers on the contact radius temporal growth is inconspicuous.
Figure 2 presents the temporal evolution of the contact radius $r_{0}(t)$ for
spreading drops of varying polymer concentrations, both on linear (a) and
logarithmic (b) scales. In view of the known behavior for water drops, the
horizontal axis is scaled by the inertio-capillary time $\tau$ and the
vertical axis by the drop radius $R$. Interestingly, the data exhibit a nearly
perfect collapse for all polymer concentrations, especially at early times.
Only at later times some differences become apparent. To further quantify this
behavior, we fitted the growth of the liquid bridge using a power-law of the
type $r_{0}/R=K(t/\tau)^{\alpha}$. For water, the spreading exponent is known
to be $\alpha=1/2$, while the numerical prefactor reported in literature is
$K=1.2$ Biance et al. (2004); Winkels et al. (2012). The fitted values of
$\alpha$ and $K$ in our experiments on polymer solutions are presented in Fig.
2(c,d), for all tested concentrations. We find that the spreading exponent is
consistent with $\alpha=1/2$, for all polymer concentrations. Likewise, the
prefactor $K$ is in close agreement with the value measured for water.

Figure 2: Temporal spreading dynamics. (a) Contact radius $r_{0}$ (scaled by
the drop radius $R$), as a function of time $t$ (scaled by the inertio-
capillary time $\tau$) for water and PEO solutions of varying concentrations.
(b) Same data on a logarithmic scale. (c) Fitted exponent $\alpha$ extracted
from $r_{0}/R=K(t/\tau)^{\alpha}$ as a function of $c$, the PEO concentration;
the dashed line indicates the exponent expected for Newtonian liquids,
$\alpha=1/2$. (d) Prefactor $K$ as a function of a function of $c$ ; the
dashed line indicates the value expected for Newtonian liquids, $K=1.2$ Biance
et al. (2004); Winkels et al. (2012). The error bars on $\alpha$ and $K$ arise
from the averaging different experiments with the same liquid.

Remarkably, we find that the early stages of spreading of PEO solutions is
nearly identical to that of pure water. Even though there is a very rapid
motion, which certainly induces large polymer stretching, the temporal
dynamics is hardly affected by the presence of polymer. At most, we observed a
weak trend for the prefactor $K$, indicating a minor slowing down. We note
that the highest concentration $c=2\mathrm{wt}\%$ departs slightly from the
water exponent and prefactor, which can possibly be attributed to the fact
that the concentration exceeds the overlapping concentration $c^{*}$
(estimated around $1\mathrm{wt}\%$ SI ); this mixture is clearly not in the
dilute regime, which might be the cause why $\alpha$ differs slightly from the
value observed at lower concentrations.

An important comment to make is that the spreading of PEO solutions is very
different from the spreading observed for Newtonian liquids of matching shear
viscosity. For example, for $c=2\mathrm{wt}\%$ the shear viscosity is
$\eta\approx 10$ Pa.s (see SI), and the corresponding Newtonian drop would be
deeply in the viscous regime. The scaling in the viscous regime is linear with
a logarithmic correction, as $r_{0}\sim t\ln t$, which results into apparent
spreading exponents close to 0.8 Eddi et al. (2013b). However, the data in
Fig. 2 shows no trend towards an increase of the exponent, from the inertial
1/2 toward the viscous scaling. We thus conclude that in spite of the strongly
enhanced shear viscosity of the polymer solutions, the behaviour for $\alpha$
and $K$ remains “inertial” in nature.

###  III.2 Spatial structure

Though the _temporal_ dynamics of the spreading is hardly affected, the
presence of polymer strongly modifies the _spatial_ structure of the bridge.
This can be seen from the profiles $r(x,t)$ plotted in Fig. 3, taken for
different times after contact. The profiles appear nearly symmetric in the
vertical direction, owing to the reflection in the substrate that is included
in the plot. From these images it is evident that viscoelastic bridges are
much sharper: while the water profiles are rather shallow (panel a), the
interface exhibits a more pronounced curvature due to the addition of polymers
(panels b,c). Remarkably, this effect is highly localised at the center of the
bridge; the effect of polymer fades away at farther distance from the
substrate.

Figure 3: Evolution of the bridge profiles (including the reflection in the
substrate and the profile prior to contact in light blue) for spreading drops
of (a) water, (b) PEO with 1wt%, and (c) PEO with 2wt%, shown at different
times (with $t/\lambda=0.0007\to 0.02$ for the 1 wt% PEO solution and
$t/\lambda=0.0003\to 0.01$ for the 2 wt%). Viscoelastic solutions exhibit a
much sharper bridge than water.

We now further quantify the sharpness of the bridge. First, we remark that the
apparent contact angle of the interface with the substrate is very close to
90∘. This can be concluded from plotting $r(x,t)$ as a function of $x^{2}$.
This plot gives a purely linear trend without any offset or asymmetry (data
available in the SI (SI, )), which confirms that, on the scale of our
experimental resolution, the apparent angle cannot be distinguished from 90∘.
Subsequently, we can quantify the sharpness of the bridge by inspecting the
bridge curvature $\kappa$ close to the substrate. The curvature is defined as
$\kappa=r_{xx}(t)$, which can be extracted at each instant from the slope of
the curve of $r(x,t)$ versus $x^{2}$, in the vicinity of $x=0$ (cf. SI (SI,
)). The curvature is of particular interest, since it gives direct information
of the stress inside the liquid, according to Laplace’s law of capillarity De
Gennes et al. (2004). Hence, the enhanced curvature is a direct measure for
the increase in stress due to polymer stretching.

The resulting $\kappa(t)$ is plotted in Fig. 4, for different concentrations.
In Fig. 4(a), the curvature $\kappa$ is made dimensionless with the drop
radius $R$ 111Normalisation with the dominant curvature, $R/r_{0}^{2}$, is
perhaps more natural, but it markedly increases the noise in the data in the
limit where $r_{0}\rightarrow 0$., while time is scaled with the inertio-
capillary time $\tau$. For all concentrations, the bridge curvature is found
to decay algebraically, with the smallest curvatures observed for water. The
exponent does not appear to take on a distinct universal value, but changes
with the nature of the liquid. We see a transition of the exponent from around
$-1$ for water to lower apparent exponents, towards approximately $-1/2$ for
the higher PEO concentrations. Note that though rapid spreading of water drops
is extensively studied, the bridge curvature is usually not considered; hence
we are not aware that the behavior $\kappa\sim 1/t$ for water has been
previously reported. In an attempt to account for the viscoelastic nature of
the solutions, we also report the data by normalising time after contact with
the polymer relaxation time $\lambda$, shown in Fig 4(b). This scaling gives a
much better grouping of the data, which suggests that the dynamics of the
curvature is indeed governed by polymer stretching, and its subsequent
relaxation. The effective exponent is found to be below $-1$, though we cannot
identify a universal exponent.

Figure 4: Bridge curvature $\kappa$ (normalised by the drop radius $R$)
plotted as a function of time $t$. (a) Time is normalised by the inertio-
capillary time $\tau$. (b) Same data, but time is normalised by the polymer
relaxation time $\lambda$. The suggested exponents are indicated as guide for
eyes.

##  IV The spreading-coalescence analogy revisited

We have thus found that the effect of viscoelasticity on rapid drop spreading
is rather intricate. On the one hand, polymers hardly affect the motion of the
contact line. Specifically, the spreading exponent remains identical to that
for the initial spreading of water drops. On the other hand, the shape of the
liquid-vapour interface in the bridge is strongly affected, as it exhibits
much larger curvatures. This dual spatio-temporal influence of polymers is
very strongly reminiscent to that observed recently in viscoelastic
coalescence Dekker et al. (2022). In what follows we will explore the analogy
between rapid spreading and coalescence in the context of viscoelastic drops.

Figure 5: Direct evidence for the spreading-coalescence analogy for the
temporal evolution of the bridge. The three panels directly compare the
dynamics of the bridge radius $r_{0}(t)$ for coalescence (triangles, taken
from Dekker et al. (2022)) and spreading (circles). The data for spreading and
coalescence are indistinguishable for (a) water, (b) PEO with 1wt% and (c) PEO
with 2wt%.

###  IV.1 Spatio-temporal dynamics of the liquid bridge

We first focus on the time evolution of the bridge radius $r_{0}(t)$, reported
in Fig. 5, for both spreading and coalescence. The data for coalescence are
taken from the same experiments as those presented in Dekker et al. (2022).
The results are again presented in dimensionless form, as $r_{0}/R$ versus
$t/\tau$, for drops of water (panel a), PEO with 1wt% (panel b), and PEO with
2wt% (panel c). Clearly, in all cases, the data for coalescence (triangles)
fall perfectly on top of the data for spreading (circles). Hence, coalescence
and spreading exhibit indistinguishible temporal dynamics, both in the
Newtonian and viscoelastic cases. This implies that the presence of the
substrate, in particular of the moving contact line, does not seem to have any
influence on the wetting dynamics $r_{0}(t)$ during spreading; irrespectively
of the presence of polymers inside the liquid.

Figure 6: Evolution of the bridge for coalescing drops of water, PEO with 1wt%
and PEO with 2wt%. Bridge profiles (rotated by 90∘) are shown at different
times matching those in figure 3, with $t/\lambda=0.0007\to 0.02$ for 1 wt%
PEO and $t/\lambda=0.0003\to 0.01$ for 2 wt% PEO. Viscoelastic solutions
exhibit a much sharper bridge than water.

On the face of it, the spatial structure of the liquid bridge observed during
spreading is also analogous to that observed during coalescence of two drops.
As shown in Fig. 6, the bridge profiles during coalescence are sharper when
increasing the PEO concentration, in a way that is similar to drop spreading
(for comparison see Fig. 3). The two processes thus have in common that the
stress inside the bridge is strongly enhanced due to the stretching of the
polymer chains.

However, the central bridge curvature $\kappa(t)$ turns out to be essentially
different. To make this apparent, we plot the temporal evolution of the
curvature in Fig. 7, using the same rescalings as in Fig. 4. Let us first
focus on the case of pure water. For coalescence, we find a scaling behavior
that is close to $\kappa\sim 1/t^{3/2}$ [Fig. 7(a)]. However, for spreading of
water drops we found $\kappa\sim 1/t$ [Fig. 4(a)]. The curvature data of
coalescence clearly does not overlap witho that of spreading, which points to
a breakdown of the spreading-coalescence analogy. A similar breakdown of the
analogy is observed for polymer solutions. For this, one can directly compare
Fig. 7(b) and Fig. 4(b), where time is normalised by the polymer relaxation
time. For the case of coalescence, it appears that at early times there is a
universal scaling $\kappa\sim 1/t$, which again differs substantially from
$\kappa(t)$ observed during spreading. Hence, while there is a perfect
spreading-coalescence analogy for $r_{0}(t)$, this analogy does not hold to
the local curvatures. This implies that the local stresses at the interface
are different for spreading as compared to coalescence.

Figure 7: Coalescence bridge curvature $\kappa$ (normalised by the drop radius
$R$) plotted as a function of time $t$. (a) Time is normalised by the inertio-
capillary time $\tau$. (b) Same data, but time is normalised by the polymer
relaxation time $\lambda$.

###  IV.2 Is the bridge profile self-similar?

To conclude the discussion on the spreading-coalescence analogy, we finally
question whether the dynamics of the liquid bridge exhibits self-similarity.
For coalescence of Newtonian drops, it is known that the bridge profiles can
be collapsed upon appropriate rescaling of spatial coordinates with time.
Specifically, the scaling of the radial coordinate $r/r_{0}(t)$ and the axial
coordinate $xR/r_{0}^{2}(t)$ gives a collapse of profiles during the
coalescence of water drops Eddi et al. (2013b). In Fig. 8 we therefore
represent the spatio-temporal evolution of the bridge in viscoelastic
spreading and coalescence according to this Newtonian scaling.

We first focus on the case of pure water (left column of Fig. 8). Profiles for
water coalescence nicely collapse (top-left panel), testifying that the
dynamics is indeed self-similar. Specifically, the bridge profile follows a
dynamics given by $r(x,t)=r_{0}\mathcal{H}(xR/r_{0}^{2})$, where $\mathcal{H}$
is a universal function. This data collapse for water coalescence also
explains the observed scaling law for curvature. Namely, one evaluates
$\kappa=r_{xx}\sim R^{2}/r_{0}^{3}$, so that, using $r_{0}\sim t^{1/2}$, we
readily find $\kappa\sim 1/t^{3/2}$. Such a self-similarity is however not
observed for the spreading of water drops (bottom-left panel). We recall that
in this case we measured $\kappa\sim 1/t$, which explains why the attempted
rescaling in Fig. 8 naturally fails. This lack of self-similarity confirms
that the spreading-coalescence analogy breaks down, even for water drops, when
considering the spatial structure of the bridge.

The central and right panels in Fig. 8 correspond to bridge dynamics of
viscoelastic drops. For both coalescence (upper plots) and spreading (lower
plots), we find that the rescaling gives a grouping of the profiles, yet, with
a collapse that is less convincing than for water coalescence (top-left). In
particular the central region of the bridge does not collapse, which is in
line with the differences in scaling of the bridge curvature $\kappa(t)$. A
comparable breakdown with respect to the Newtonian self-similarity was found
in Dekker et al. (2022) for the coalescence of _sessile viscoelastic drops_ ,
which are resting on a solid surface before merging. For these sessile drops,
the polymer stress was proposed to scale as
$\sigma\sim\bar{\eta}_{\infty}\dot{r}_{0}/r_{0}$, where $\bar{\eta}_{\infty}$
represents the extensional viscosity at high rates. For any power-law dynamics
$r_{0}\sim t^{\alpha}$, this implies an elastic stress $\sigma\sim 1/t$. As
discussed in Dekker et al. (2022) for viscoelastic coalescence, the strong
polymer stress in the bridge center gives rise to a new type of self-
similarity, with scaling laws that arise from balancing polymer stress with
surface tension. While for spherical viscoelastic drops the stress singularity
during coalescence also appears to be $\sigma\sim 1/t$ (since $\kappa\sim
1/t$), we have not been able to identify a scaling that collapses the
spherical coalescence (nor spreading).

Figure 8: (a-c) Bridge profiles obtained for coalescing drops of water, PEO
with 1wt% and PEO with 2wt%, shown at different times, coinciding with Fig. 6.
Profiles are rescaled accordingly to the Newtonian self-similar scaling, that
is by normalising the horizontal and vertical bridge extend $r$ by $r/r_{0}-1$
and $x$ by $xR/r_{0}^{2}$. (d-f) Bridge profiles obtained for spreading drops
of water, PEO with 1wt% and PEO with 2wt%, shown at different times,
coinciding with Fig. 3, obtained by the same rescaling as in the top row.

##  V Discussion

We have investigated the rapid spreading dynamics of a viscoelastic drop on a
solid substrate, focusing on the spatiotemporal dynamics of the liquid bridge
that forms immediately after contact. It was found that the time evolution of
the contact radius was almost completely insensitive to the presence of
polymer, and closely follows the purely inertial spreading law (1). This,
however, does not mean that the polymers do not affect the spreading: the
interface profile of the liquid bridge is much sharper than the Newtonian
case, owing to singular polymer stress. These observations are of interest for
applications such as printing and spray deposition, where polymeric additives
can be used to control the impact process.

Our experimental findings for rapid viscoelastic spreading are strongly
reminiscent to those observed for viscoelastic drop coalescence. Indeed, just
like for Newtonian drops, it is found that the bridge radius $r_{0}(t)$ is
identical when comparing rapid spreading to coalescence. Importantly, however,
we have found an important breakdown of the spreading-coalesence analogy. This
is most directly appreciated from the bridge profiles in Fig. 8, by comparing
the top row (coalescence) to the bottom row (spreading). From these rescaled
profiles it can be seen the the bridges for spreading are systematically
sharper than for coalescence – pointing to a larger stress inside the bridge.
In the case of pure water, we anticipate that this can be attributed to a
viscous boundary layer associated to the substrate’s no-slip boundary
condition. In the case of polymer solutions, this effect could also be due to
a viscous boundary layer or due to the enhanced stretching of the polymers by
shear. While we have provided an interpretation for the observed curvatures
during coalescence, the experimental findings for $\kappa(t)$ during spreading
offer an interesting question for future studies. It also remains to be
explained why, in all these cases, the temporal growth of the bridge is
remarkably insensitive to the singular stress inside the central region of the
bridge.

###### Acknowledgements.

The authors thank Charu Datt and Walter Tewes for fruitful discussions. We
acknowledge support from NWO Vici (No. 680-47-63), and from an Industrial
Partnership Programme of NWO, co-financed by Océ-Technologies B.V., University
of Twente, and Eindhoven University of Technology.

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