# Probing dissipation in spreading drops with granular suspensions

Alice Pelosse  Matière et Systèmes Complexes, CNRS UMR 7057, Université Paris
Cité, 10 Rue A. Domon et L. Duquet, 75013 Paris, France  Élisabeth Guazzelli
Matière et Systèmes Complexes, CNRS UMR 7057, Université Paris Cité, 10 Rue A.
Domon et L. Duquet, 75013 Paris, France  Matthieu Roché  Matière et Systèmes
Complexes, CNRS UMR 7057, Université Paris Cité, 10 Rue A. Domon et L. Duquet,
75013 Paris, France<EMAIL_ADDRESS>
(today)

###### Abstract

In this article, we study the spreading of droplets of density-matched
granular suspensions on the surface of a solid. Bidispersity of the particle
size distribution enriches the conclusions drawn from monodisperse experiments
by highlighting key elements of the wetting dynamics. In all cases, the
relation between the dynamic contact angle and the velocity of the contact
line follows a similar relation as that of a simple fluid, despite the
complexity introduced by the presence of particles. We extract from this
relation an apparent wetting viscosity of the suspensions that differs from
that measured in the bulk. Dimensional analysis supported by experimental
measurements yields estimate of the size of the region inside the droplet
where the value of the dynamic contact angle depends on a balance of viscous
dissipation and capillary stresses. Depending on how particle size compares
with this viscous cut-off length seems crucial in determining the value of the
apparent wetting viscosity. With bimodal blends, the particle size ratio can
be used to show the effects of the local structure and volume fraction at the
contact line, both impacting the value of the corresponding wetting viscosity.

##  I Introduction

A liquid wetting the surface of a solid is a common observation in daily
experiences. A classical example is that of raindrops hitting a glass window
and falling along its surface, leaving trails of water. The physics underlying
this supposedly simple situation is in fact quite rich as the properties of
all the involved media matter, including those of the surrounding atmosphere.
Moreover, length scales from the molecular up to the millimetre range must be
considered, leading to a high level of theoretical complexity. A drop of
simple fluid spreading onto a solid substrate is subjected to three competing
physical mechanisms, namely gravity, capillarity, and viscosity [see e.g. 2].
The shape and dynamics of the droplet depend on the length scale at which they
are analysed with respect to the capillary length, $\ell_{c}=(\gamma/\rho
g)^{1/2}$, with $\gamma$ the surface tension of the solid, $\rho$ its density,
and $g$ the acceleration of gravity. If we consider droplets with a radius
$R>>\ell_{c}$, the balance between gravity and capillarity controls the shape
and dynamics at length scales comparable to $R$. At microscopic scales near
the triple-phase contact line, the balance involves instead viscosity and
capillarity. As the height diminishes at the approach of the contact line,
viscous stresses diverge when using the classical no-slip boundary condition
on the substrate. A variety of models have been proposed to regularise this
non-integrable singularity. A common way to circumvent this divergence issue
has been to replace the no-slip boundary condition by a Navier slip condition,
i.e. to introduce a microscopic cut-off length scale, $\lambda$, in the
continuum modelling [21, 11]. For the intermediate scales, gravity and
viscosity as well as capillarity participate into the dynamics of the drop.

The above picture becomes more involved when the spreading fluid is complex,
such as a polymer solution [25], a polymeric melt [35], or a viscoplastic
material [38]. Such fluids can have intrinsic timescales (e.g. relaxation,
agitation) or characteristic length scales (e.g. particle size, persistence
length) that need to be accounted for in the spreading dynamics. The present
work focuses on the spreading of dense granular suspensions of large spherical
particles that do not experience Brownian motion. These particulate systems
are characterised by an additional length scale, the particle size, that must
be addressed in the multiscale description of an advancing contact line. The
interplay of the vanishing height of the flow and the finite size of the
particles should control the distance at which the particles can approach the
contact line. This view has been confirmed in our previous study of the motion
of the triple-phase contact line surrounding a droplet of monomodal granular
suspension [44]. Interestingly, the relation between the dynamic contact
angle, i.e. the angle between the liquid-gas interface and the solid-liquid
interface measured in the droplet, and the speed of the contact line happens
to be similar to that found for a simple liquid, known as the Cox-Voinov law
[41, 10]. In dimensionless form, this speed was reported in term of the
capillary number, $Ca=\eta U/\gamma$, measuring the relative effect of viscous
to surface tension forces, where $\eta$ is the viscosity of the liquid and $U$
the velocity of the contact line. We found that the viscosity involved in this
capillary number differed from the widely-studied bulk viscosity of
suspensions as it depended on particle diameter, $d$, in addition to particle
volume fraction, $\phi$. This observation resulted from the aforementioned
ability of the particles to approach the contact line close enough to affect
dissipation. In particular, we showed that the apparent viscosity reduced to
the viscosity of the suspending fluid when the particle size became larger
than approximately 100 $\mathrm{\SIUnitSymbolMicro m}$. This study suggests
that a granular suspension may be an interesting system for the study of
wetting as the discrete nature of this complex fluid can be used to probe the
size of the domain in which the Cox-Voinov relation is a valid description of
the relation between the dynamic contact angle and the velocity of the contact
line.

Another feature specific to the spreading of a granular suspension is that
there is a self organisation of the advancing front rows of particles near the
rim of the drop [44]. To give a full picture, the close vicinity of the
contact line is a region devoid of particles. Behind this particle-depleted
region, a few layers of crystallised beads are observed. As the height
increases further, the particles switch from a crystal-like to a disordered
structure. This ordering is particularly seen for dense suspensions, e.g. for
packing fractions, $\phi$, of 40% or above. Confinement by the free interface
seems to be responsible of this observed organisation. However, the role of
this ordered particle phase in energy dissipation and its effects on the
dynamics of spreading are still open problems that require scrutiny. A simple
model that matches the shape of the droplet in the depleted region to that of
a particle-rich region with no constraint on the ordering of particles fails
at describing the relation between contact angle and velocity.

Particles near a receding contact lines have been experimentally studied when
a plate is withdrawn from a bath of non-Brownian suspension. In this
configuration named dip-coating, the key result is that the plate can be
coated by particles when the entrained film is thicker than the particle
diameter. More precisely, increasing the withdrawal speed leads to different
regimes of coating going from a film devoid of particles, to an heterogeneous
monolayer of particles, and finally a thick film of suspension entrained by
the plate [16, 27]. When the suspension contains two sizes of particles, the
small particles are first entrained on the plate to form an heterogeneous
monolayer and the large particles begin to be entrained only with a further
increase of withdrawal speed [23]. In the heterogeneous regime, confined
particles gather in clusters without clear ordering in contrast to the highly-
structured phase observed near the advancing contact line during the spreading
of suspension drops on a solid substrate [44].

In this work, we examine the advancing contact line of a granular suspension,
with the aim of clarifying the origin of the difference between bulk viscosity
and its counterpart extracted from wetting experiments. In § II, we discuss
the range of interface heights for which we should expect the viscous-
capillary balance at play in the Cox-Voinov relation to be valid. The
materials and methods used in the experiments are described in § III. In § IV,
we test experimentally the predictions of § II with simple fluids. Then, we
resort to the discrete nature of the particles in a suspension to probe
dissipation in the vicinity of the contact line. These experiments are another
test of the outcomes of § II. Results obtained for monodisperse and bidisperse
suspensions consisting of different particle combinations are presented in §
V. While bidisperse suspensions seem to be a more complicated system, they
offer the possibility to use two different sizes for scrutinising dissipation.
We discuss our findings and provide concluding remarks in § VI.

##  II Elements of the wetting theory

###  II.1 General framework

Figure 1: Sketch of a fluid droplet spreading on a rigid substrate.

The classical situation of a fluid droplet of radius $R$ and central height
$h_{0}$ that spreads onto the surface of a rigid substrate is depicted in
figure 1. The fluid has a dynamic viscosity $\eta$, a density $\rho$, and a
surface tension $\gamma$. In most of the situations, inertia is negligible,
i.e. the Reynolds number is small, ${Re}=\rho Uh_{0}/\eta\ll 1$, where $U$ is
the characteristic spreading velocity. Since $h_{0}\ll R$, the lubrication
approximation can be applied. As the flow is axisymmetric, we use cylindrical
coordinates with the $z$-axis being the axis of symmetry. Under these
assumptions, the height of the air-liquid interface, $h(r)$, is a solution of
the free-surface, thin-film equation,

$\frac{3\eta}{\gamma}\partial_{t}h+\frac{1}{r}\partial_{r}\left[h^{2}(h+3\lambda)r\partial_{r}\left(\partial_{rr}h+\frac{1}{r}\partial_{r}h-\frac{\rho
g}{\gamma}h\right)\right]=0,$ (1)

where $g$ is the gravitational acceleration [19, 20, 33]. A particular feature
of the above equation (1) is that a characteristic length scale, $\lambda$,
has been introduced to relax the no-slip boundary condition at the line of
contact between the droplet, the substrate, and the surrounding gas [21, 11].
In the context of the present work, we grant $\lambda$ its usual role of a
slip length and we take it to be of the order of the size of a few nanometers,
comparable to that of a molecule of the liquid.

The resolution of (1) requires appropriate boundary conditions to be specified
as

$h(r\rightarrow R)=0,\;\partial_{r}h(r\rightarrow
R)=\theta_{m},\;\mbox{and}\;\int_{r=0}^{R}h(r)2\pi rdr={V}_{0},$ (2)

where $V_{0}$ is the volume of the droplet and $\theta_{m}$ is the microscopic
contact angle.

Rather than providing detailed derivations of the solutions of (1) with these
boundary conditions (2) [see e.g. 20, 10, 36, 32], we give some insight into
the different length scales involved in the problem described by (1) with (2).
Besides the slip length, $\lambda$, we also identify the capillary length,
$\ell_{c}=(\gamma/\rho g)^{1/2}$, for which gravitational forces balance
capillary forces, and the radius of a sphere having the same volume ${V}_{0}$
as that of the droplet, $R_{0}=(3{V}_{0}/4\pi)^{1/3}$. In experiments, it
usually happens that $\ell_{c}\gg\lambda$ and $R_{0}\gg\lambda$ by several
orders of magnitude. There is thus a region in the close vicinity of the
contact line in which the force balance only involves viscous and capillary
forces, with slip being dominant and gravity negligible. This viscous-
capillary region will be introduced in § II.2. On the opposite range, the
large scales are governed by the balance between gravity and capillarity. This
capillary-gravity region will be presented in § II.3. In between, the three
contributions of the competing viscous, capillary, and gravity forces must be
retained as developed in § II.4.

###  II.2 Viscous-capillary region

This region corresponds to the very close vicinity of the contact line, i.e.
at length scales much smaller than $\ell_{c}$. As mentioned earlier in § II.1,
the introduction of a Navier slip is an important ingredient as viscous
dissipation would diverge as $h\rightarrow 0$ otherwise. Resolution of the
reduced equation (1) where the gravity term has been dropped, with the
additional assumption that spreading is quasi-steady, leads to a relation
between the velocity, $U$, of the contact line and the dynamic contact angle,
$\theta_{app}(x)$, measured at a distance $x=R-r$ from the moving rim of the
droplet,

$\theta_{app}^{3}(x)=\theta_{m}^{3}+9{Ca}\log{\left(\frac{x}{\lambda}\right)},$
(3)

where $Ca=\eta U/\gamma$ is the capillary number. In the case of viscous
liquids, the microscopic contact angle, $\theta_{m}$, is close to the static
value [2]. For (nearly) perfectly wetting liquids, it is found that
$\theta_{m}\ll 1$ therefore negligible in (3). Equation (3) is often referred
to as the Cox-Voinov law [41, 10, 37, 2]. It indicates that $\theta_{app}(x)$
is an increasing function of the distance $x$ to the contact line at a given
capillary number, ${Ca}$. The interface in this viscous-capillary region must
then have a positive curvature when measured in the frame defined in figure 1.

###  II.3 Capillary-gravity region

The opposite limit corresponds to the region where $r\rightarrow 0$, i.e.
$x\sim R\sim R_{0}$. Moreover, we are interested in the limit where
$R_{0}>>\ell_{c}$. At these scales, viscous forces are insignificant and the
time-derivative term in (1) can be dropped. Moreover, in this macroscopic
region, $\lambda$ can be neglected in front of $h$ which is of the order of
$h_{0}$. Hence, we are left with a capillary-gravity balance, and (1) reduces
to

$\partial_{r}\left[h^{3}r\partial_{r}\left(\partial_{rr}h+\frac{1}{r}\partial_{r}h-\frac{\rho
g}{\gamma}h\right)\right]=0.$ (4)

This equation can be integrated once. Using the boundary condition $h(r=R)=0$,
the integration constant is seen to be zero. At that stage of the calculation,
it is convenient to normalise the $r$\- and $h$-scales as $r=R_{0}\tilde{r}$
and $h=h_{0}\tilde{h}$ where $R_{0}$ and $h_{0}$ are the radius of the
spherical drop and the characteristic interface height in this region,
respectively. Performing another integration leads to

$\partial_{\tilde{r}\tilde{r}}\tilde{h}+\frac{1}{\tilde{r}}\partial_{\tilde{r}}\tilde{h}-{Bo}\tilde{h}=C,$
(5)

where $C$ is a constant and $Bo=\rho
gR_{0}^{2}/\gamma=\left(R_{0}/\ell_{c}\right)^{2}$ the Bond number of the
droplet. The exact solution of (5) with the boundary conditions (2) is

$\tilde{h}(\tilde{r})\propto\frac{I_{0}\left(Bo^{1/2}\frac{R}{R_{0}}\right)-I_{0}\left(Bo^{1/2}\tilde{r}\right)}{I_{2}\left(Bo^{1/2}\frac{R}{R_{0}}\right)},$
(6)

where $I_{n}(x)$ is the $n$-th modified Bessel function of the first kind
[20]. The solution (6) reads with dimensional coordinates

$h(r)=\frac{{V}_{0}}{\pi
R^{2}}\frac{I_{0}\left(\frac{R}{\ell_{c}}\right)-I_{0}\left(\frac{r}{\ell_{c}}\right)}{I_{2}\left(\frac{R}{\ell_{c}}\right)}.$
(7)

Droplets are spherical caps when $Bo\ll 1$ whereas their top surface flattens
and they look like puddles in the limit ${Bo}\gg 1$. In the frame of reference
defined in figure 1, the shape of the interface in the macroscopic region, has
a negative curvature.

###  II.4 Viscous-capillary-gravity region

We finally turn to the examination of the region for which all the
contributions of the viscous, capillary, and gravity forces must be kept in
(1). In this viscous-capillary-gravity region, the length scale is comparable
neither to $\lambda$ nor to $R_{0}$ and the sole remaining length scale is the
capillary length. Since $x$ and $h\gg\lambda$, the reduced equation then reads

$3\,{Ca}\,\partial_{r}h+\frac{1}{r}\partial_{rr}\left[h^{3}x\partial_{r}\left(\partial_{rr}h+\frac{1}{r}\partial_{r}h-\frac{h}{\ell_{c}^{2}}\right)\right]=0,$
(8)

where we consider quasi-static spreading and use
$\partial_{t}h=U\partial_{r}h$. This assumption means that at any time, the
velocity of the contact line sets the drop shape with no transient involved.
This constraint is enforced by the hypothesis that $Re\ll 1$ [17]. The
capillary number, $Ca=\eta U/\gamma$, then appears as a natural way to build a
dimensionless velocity.

As mentioned above, the capillary length, $\ell_{c}$, is the only length
available. It is the scale for the variation along the horizontal direction,
$r=\ell_{c}\hat{r}$. However, no obvious length scale emerges in the vertical
direction. We thus define $h=h^{\star}\hat{h}$ where $h^{\star}$ is the (still
unknown) relevant scale for the drop height. Using these renormalisations in
(8) yields

$3Ca\partial_{\hat{r}}\hat{h}+\left(\frac{h^{\star}}{\ell_{c}}\right)^{3}\frac{1}{\hat{r}}\partial_{\hat{r}}\left[\hat{h}^{3}\hat{r}\partial_{\hat{r}}\left(\partial_{\hat{r}\hat{r}}h+\frac{1}{\hat{r}}\partial_{\hat{r}}\hat{h}-\hat{h}\right)\right]=0.$
(9)

For the two terms to be of the same order in (9), one must take the height
length scale to be

$h^{\star}\equiv\ell_{c}Ca^{1/3}.$ (10)

This scale represents the typical height separating the viscous-capillary
region governed by the Cox-Voinov law from the viscous-capillary-gravity
region where gravity comes into play in the force balance [12]. In other
words, this transition scale corresponds to the upper bound of the region of
the drop where the Cox-Voinov relation (3) is still valid. Another interesting
interpretation of $h^{\star}$ is to see that it may delineate the change in
surface curvature and thus can be understood as the inflection point of the
drop interface.

Some orders of magnitude can be provided for the present experimental
conditions. The range of $h^{\star}$ is $20\leq h^{\star}\leq 800$
$\mathrm{\SIUnitSymbolMicro m}$ for typically $\ell_{c}\simeq 2$ mm and
$10^{-6}\leq Ca\leq 10^{-2}$. As a consequence, liquids with sub-millimetre
characteristic length scales, such as suspensions of non-Brownian particles,
may show non-trivial $\theta_{app}-Ca$ relations. Indeed, adding density
matched particles should not modify the drop behaviour in the gravity driven-
region while it should enhance dissipation provided the particles can access
the region where viscosity matters, namely the dissipation region for
$h\lesssim h^{\star}$.

##  III Experimental methods

###  III.1 Particles and fluid

The suspending fluid is a Newtonian PEG copolymer, Poly(ethylene glycol-ran-
propylene glycol) monobutyl ether (average $M_{n}\simeq 3900$, Sigma-Aldrich
reference 438189), with a density close to that of polystyrene, $\rho=1056$
kg/m3, at a temperature $25\text{\,}\mathrm{\SIUnitSymbolCelsius}$. Its
dynamic viscosity, $\eta_{f}=2.4\pm 0.1$ Pa.s, is constant over a large range
of shear rate (0.01-10 s-1) at $25\text{\,}\mathrm{\SIUnitSymbolCelsius}$.
Particles are spherical polystyrene beads (Dynoseeds TS, Microbeads, Norway)
which are sieved when necessary to remove small dust particles or to narrow
the size distribution below a tenth of particle diameter. The mean particle
diameters, $d$, used in the experiment are 10, 20, 40, 80, 140 and
$250\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Density matching between the
liquid and the particles is sufficient to prevent buoyancy effects over hours
at the temperature of $25\text{\,}\mathrm{\SIUnitSymbolCelsius}$ of the air-
conditioned room. To prepare the suspension mixture, the suspending fluid is
weighted and poured into a test tube. A desired mass of particles is then
added to reach the target volume fraction, $\phi$. Good mixing while avoiding
entrapping of air is achieved by hand mixing followed by slow mixing on a
rolling device overnight. In the following experiments, the solid volume
fraction is $\phi=0.4$ as we aim at studying the strongest effects expected at
large volume fractions [44]. The particle surface is completely wet by the
suspending fluid, i.e. particles remain suspended in the fluids and they do
not aggregate. We measured the surface tension of the suspension using pendent
drop experiments. We find that the suspension surface tension is equal to the
surface tension of the suspending fluid, $\gamma=\gamma_{f}\simeq$
$35\text{\,}\mathrm{mN}\text{\,}{\mathrm{m}}^{-1}$, a similar result to that
reported in [9].

###  III.2 Bulk suspension rheology

Figure 2: Relative viscosity (i.e. shear viscosity of the suspension relative
to that of the suspending fluid), $\eta_{s}$, of $(a)$ monomodal and $(b)$
bimodal suspensions. $(a)$ Relative viscosity of monomodal suspensions as a
function of particle volume fraction, $\phi$, for particles with diameters of
$10,\,20,\,40,\,80,\,140,$ $250\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
comparison with experimental data for particles with diameters of $10$ and
$140\text{\,}\mathrm{\SIUnitSymbolMicro m}$ from the experiments of [6] and
[27], respectively. The vertical dotted line is an estimate of the maximum
volume fraction, $\phi_{c}\simeq 0.53$, used to compare data with the
empirical correlations of Krieger-Dougherty,
$\eta_{s}=(1-\phi/\phi_{c})^{-[\eta]\phi_{c}}$ [24], and Eiler,
$\eta_{s}=(1+[\eta]/2\phi/(1-\phi/\phi_{c}))^{2}$ [13] ($[\eta]=2.5$ is the
intrinsic viscosity of the suspension). $(b)$ Relative viscosity of bimodal
suspensions at a fixed total solid volume fraction of $\phi=0.4$ as a function
of the fraction of the small particles in the solid phase, $\zeta_{small}$,
for two-size blends (see legend).

A thorough discussion of our results requires a comparison of the bulk
viscosity of the granular suspensions with their apparent viscosity extracted
from drop spreading experiments, see §V. Rheological measurements have been
performed using an ARES G2 rheometer (TA Instruments) with a 25 mm-wide plate-
plate geometry for solid blends of moderate size particles with diameters of
10, 20, 40 and $80\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The thickness of
the gap between the plates is typically $1.5\text{\,}\mathrm{mm}$, i.e. at
least 20 particle diameters, to prevent confinement effects [29]. As wall-slip
effects depend on the gap thickness, the fact that no viscosity variation is
measured while changing the gap thickness from 1 mm to 2 mm indicates that
sliding is negligible for these small particles [43, 22].

Confinement and slip become important when performing rheological measurements
with suspensions of particles with diameters of 140 or
$250\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Slip is avoided using a 25 mm-
wide plate-plate crosshatched geometry with a typical roughness of
$1\text{\,}\mathrm{mm}$. Confinement is a trickier issue since increasing the
gap thickness leads to a larger meniscus and thus to a significant error in
viscosity measurements [3]. We circumvent this issue by using a wide reservoir
(a 5 cm-wide cup) mounted on the lower plate and filled with a 5 mm-thick
layer of suspension ($\gtrsim 20d_{p}$) on which the upper tool is lowered to
touch the free interface [6, 7]. The additional torque exerted by the
exceeding fluid in the reservoir can be computed analytically if the reservoir
width and the suspension thickness are known [42]. We have implemented
numerically this correction and we have been able to reproduce both published
data and experimental results with and without the reservoir for Newtonian
fluids and dense suspensions of 80-$\mathrm{\SIUnitSymbolMicro m}$ particles.

The relative bulk viscosity of monomodal granular suspensions, $\eta_{s}$, is
defined as the viscosity of the whole suspension relative to that of the
viscosity of the suspending fluid. It is independent of the particle size and
shear rate and is solely an increasing function of the particle volume
fraction $\phi$ that diverges at a maximum value, $\phi_{c}$, for which the
suspension ceases to flow as shown in figure 2$(a)$. Note that the value
$\phi_{c}\simeq 0.53$ inferred from curve fitting is provided for the plot but
a more exhaustive study of the viscosity in the dense regime would be required
to confirm this value. The value of $\phi_{c}$ depends on the frictional
contacts between particles [40]. It is important to note for the following
discussion that $\phi_{c}$ also depends on the particle-size distribution for
polydisperse suspensions.

For bimodal suspensions, the viscosity not only depends on $\phi$ but also on
the particle sizes, $d_{1}$ and $d_{2}$ ($d_{1}<d_{2}$), of the two species
and on the fraction of small particles in the solid phase, $\zeta_{small}$,
see figure 2$(b)$. Experimental studies of dense bimodal systems in the
literature indicate that their viscosity is also controlled by the value of
the maximum packing fraction, $\phi_{c}$ which is found to be higher than that
of monomodal suspensions [5, 8]. This increase in $\phi_{c}$ results from the
ability of small particles to fill the holes between the large ones [26]. For
bimodal suspensions with a given size ratio $d_{2}/d_{1}$, the relative bulk
viscosity, $\eta_{s}$, is equal to that obtained in the sole presence of the
large particles, when $\zeta_{small}=0\,\%$. Increasing $\zeta_{small}$ leads
to a decrease of $\eta_{s}$, down to a minimum located between
$\zeta_{small}=25$ and 50%, and then increases up to the value obtained for a
suspension of small particles, $\zeta_{small}=100\,\%$. The minimum in
viscosity is more pronounced with increasing $d_{2}/d_{1}$ see figure 2$(b)$.

###  III.3 Experimental apparatus

Figure 3: Sketch of the experimental apparatus

We create droplets of suspension having a volume $V_{0}=$
$300\text{\,}\mathrm{\SIUnitSymbolMicro L}$ (spherical radius $R_{0}=$
$4.15\text{\,}\mathrm{mm}$) with a syringe pump (flow rate
$10\text{\,}\mathrm{m}\mathrm{L}\mathrm{/}\min$), as shown in figure 3. The
droplets spread over a silicon wafer (Si-Mat) cleaned with ethanol and
distilled water and dried with clean-room wipes (plasma cleaning did not
change the results nor the quality of the data and was therefore considered an
unnecessary precaution). Acquisition is made from side and top views with two
synchronised monochrome cameras (Basler acA2440-35um, 2000$\times$2448 pixels)
on which 1:1 macro objectives are mounted (I2S visions, MC series). High
spatial resolution is required to capture the dynamics of the contact line.
These optics give a resolution of $3.45\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{\,}{\mathrm{px}}^{-1}$. Frame rates from 5 to 10 fps are required for
the side view, especially at the beginning as the drop spreads quickly. Frame
rates of 0.5 fps are sufficient for the top view as most of the interesting
data are extracted at long times when spreading is slow and the slope of the
drop interface is not too steep. A typical set of experiments for a single
suspension batch is made of 10 different runs of spreading drops. These runs
are acquired after 10 unused runs (corresponding to a total amount of $\sim$ 3
ml) to avoid effects coming from the front of the advancing suspension in the
tubing and in the needle. After performing these blank runs, the experiments
are seen to be very reproducible at the desired volume fraction $\phi=0.4$.
Care is also taken to account for temperature and humidity variations. These
two factors mainly impact the suspending fluid viscosity, $\eta_{f}$. To this
end, systematic viscosity measurements are performed during the experiments
using a capillary viscometer.

###  III.4 Side-view analysis

Figure 4: Data extraction from a picture of a spreading drop. Reflection on
the wafer helps to detect the advancing contact line. $(a)$ Raw picture and
$(b)$ Sobel filtering. The red rectangle in the top-left panel corresponds to
the blown-up region in panel $(c)$, showing the fitted spline of the drop
profile (green curve), the position of the contact line (yellow dot), and the
drop height $h=$ $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (dash-dotted blue
line). $(d)$ Results extracted from the fit: drop height, $h$, as a function
of the distance to the contact line, $x$ (green), and contact angle computed
from the spline derivation according to
$\theta_{app}(x)=\tan^{-1}\left(\frac{{\rm d}h}{{\rm d}x}\right)$ (orange).

We characterise the dynamics of spreading by measuring the dynamic contact
angle $\theta_{app}$ as a function of the dimensionless contact line velocity
$U$. Angle measurement from the side views has been improved upon the previous
work of Zhao et al. [44] owing to a better optical resolution and the
possibility to perform an automated local angle measurement. We tailor the
background lighting of our system with masks so that the drops captured by the
side camera appear bright on a dark background, as seen in the top panel of
figure 4$(a)$. Drop shape detection is performed with the Sobel filter of the
scikit-image package in Python, shown figure 4$(b)$, and further thresholded
to extract the contact line coordinates and the drop profile, $h(x)$, with
$x=0$ being the position of the contact line, see figure 4$(c)$. The position
of the contact line is defined as the leftmost bright pixel after image
processing, as presented on figure 4$(b),(c)$. The drop profile is then
smoothed with a cubic spline, as shown in figure 4$(c),(d)$. The contact line
velocity, $U$, is obtained by locating the triple contact line while the
apparent dynamic contact angle, $\theta_{app}$, is inferred from the
derivation of this spline. A spline is a piece-wise polynomial functions,
defined by the number of points, or knots, that it passes through. Spline
functions are regular at zero, first, and second order of derivation, as
demonstrated in figure 4$(d)$. A spline is a piece-wise polynomial functions,
defined by the number of points, or knots, that it passes through. To optimise
spline adjustment to the data at all slopes, a small knot-to-knot distance is
chosen at early times and then increased as the drop flattens. This process
prevents meaningless oscillations at small slopes (large distance between
knots). For all pictures, improper fits due to loss of focus or dust are
discarded. The results are found to be similar to those obtained by adjusting
manually a straight line to the air/liquid interface near the contact line in
the range $30^{\circ}\lesssim\theta_{app}\lesssim 85^{\circ}$. The fitted
profile can then be derived once or twice at any point of the interface. The
possibility of changing the measurement height, $h$, of the contact angle is
one of the major benefits of this automated numerical procedure compared to
previous manual measurements, see figure 4$(d)$.

###  III.5 Top-view analysis

Top views are used to visualise the structure of the particle network near the
contact line and also to measure the distance between the particles and the
contact line, $L$. Measurements using the ImageJ FiJi software package [34]
are performed over roughly 30 particles at the front. The precision of these
measurements is set by the resolution of the pictures, 3.45
$\mathrm{\SIUnitSymbolMicro m}\text{\,}{\mathrm{px}}^{-1}$.

##  IV Identifying the region of validity of the Cox-Voinov law

###  IV.1 Simple fluids

Figure 5: Cube of the contact angle, $\theta_{app}^{3}$, versus $(a)$ the
measurement height, $h$, and $(b)$ the horizontal distance to the contact
line, $x$, averaged over 7 experimental runs for 5 capillary numbers,
$Ca=0.0025\,(\lozenge),0.005\,(\circ),0.0075\,(\triangle),0.01\,(\square),0.0125\,(\pentagon)$,
using a Newtonian fluid (PEG copolymer). Red dashed lines: static shape (7)
for $R=8.5,\,8.7,\,8.9,\,9.3$ and $10\text{\,}\mathrm{mm}$ (estimated as to
provide the best fit with the experimental data) and $\ell_{c}=$
$1.82\text{\,}\mathrm{mm}$. $(c)$ Normalised cube of the contact angle,
$\theta_{app}^{3}/\theta_{app}^{3}(h\rightarrow 0)$, versus normalised height,
$h/\ell_{c}$. Black dotted line: threshold for the plateau length at
$\theta^{3}(h^{\star})/\theta^{3}(h\rightarrow 0)=0.9$ . $(d)$ Normalised
transition height, $h^{\star}/\ell_{c}$, versus $Ca$ for 3 different Newtonian
fluids: PEG (black $\lozenge$), Silicone oil V1000 (maroon $\square$)
($\rho=970$ kg/m3, $\gamma=21$ mN/m, $\eta=1.0$ Pa.s, $\ell_{c}=1.46$ mm), and
glycerol (red $\pentagon$) ($\rho=1260$ kg/m3, $\gamma=63$ mN/m, $\eta=1.3$
Pa.s, $\ell_{c}=2.23$ mm) as well as the inflection-point measurements (gold
$\circ$) of [39] with highly viscous Silicone oil. Black solid line:
$h^{\star}/\ell_{c}=0.3\,Ca^{1/3}$.

In § II.4, we have identified the length scale, $h^{\star}$, that
characterises the height of the interface at the transition between the
viscous-capillary regime governed by the Cox-Voinov relation (3) and the
viscous-capillary-gravity regime where gravity starts to prevail and where an
inflection point should exist. We start our experimental characterisation of
$h^{\star}$ by investigating the shape of the interface in the case of simple
fluids.

Inspired by the form of the Cox-Voinov law (3), we measure, for capillary
numbers in the range $0.0025\leq Ca\leq 0.0125$, how the cube of the angle
between the interface and the horizontal, $\theta_{app}^{3}$, depends on the
interface height, $h$, and on the horizontal distance to the contact line,
$x$, see figure 5$(a),(b)$, respectively. The data come from the average over
7 experimental runs. We identify two regions. Starting from the contact line,
for any capillary number, $\theta_{app}^{3}$ is first independent of the
height at which it is measured, see figure 5$(a)$. A plateau can thus be
defined for $h$ ranging from 10 to almost
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Note that this plateau is better
seen in semi-log scale in figure 5$(a)$. Discussion of these results in the
light of the Cox-Voinov law would lead us to expect that $\theta_{app}^{3}$ is
an increasing function of $h$. However, two experimental facts may prevent us
from seeing this increase. First, an inflection point can be seen for some
experiments, see for instance figure 4$(d)$, but the process of averaging over
several runs likely smears out the change of curvature. Second, the length
scales that we can probe are at least three orders of magnitude larger than
the nanometric cut-off scale, $\lambda$. We thus expect the logarithmic term
to increase slowly with distance, leading to difficulties in distinguishing
the shape of the interface from a straight line. Similar conclusions regarding
the local slope near the contact line have been reached by [31]. It is worth
mentioning that the plateau value increases with the capillary number as
expected from the Cox-Voinov law (3). Away from the contact line, the angle
decreases, suggesting a growing contribution of gravity to the force balance.
On figure 5$(b)$, the experimental contact angle is plotted and the
theoretical prediction obtained from the balance between gravity and
capillarity is superimposed at largest distances from the contact line. We
find that the agreement of the static solution (7) with the data is in general
very good, and it improves with decreasing $Ca$, provided we leave the radius
of the droplet a fitting parameter as it cannot be inferred in our
experiments.

Comparison of the datasets is made easier if we normalise
$\theta_{app}^{3}(h)$ by its asymptotic value $\theta_{app}^{3}(h\rightarrow
0)$ and $h$ by the capillary length $\ell_{c}$, see figure 5$(c)$. From these
plots, we define the experimental transition height $h^{\star}$ as the height
at which $\theta_{app}^{3}(h)$ departs from $\theta_{app}^{3}(h\rightarrow 0)$
by 10%. Figure 5$(d)$ shows the inferred $h^{\star}$ as a function of $Ca$ for
different simple fluids. The normalised transition height,
$h^{\star}/\ell_{c}$, increases as $Ca^{1/3}$ in agreement with the prediction
(10) of the dimensional analysis in § II.4. We also report in this graph the
measurements of the inflection point of the interface obtained by [39],
although the present interpretation as an upper limit of dissipation was not
mentioned in this work. These data agree well with the present estimates of
$h^{\star}$ as well as with the $Ca^{1/3}$ scaling. If we grant $h^{\star}$
the interpretation of a viscous-capillary cut-off length and refer to figure
5$(d)$, we can thus conclude that measuring contact angles at heights well
below $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ warrants to probe the region
of the droplet where the apparent dynamic contact angle is set by a balance
between viscous dissipation and capillarity only.

###  IV.2 Granular suspensions

We now test the relevance of $h^{\star}$ to the spreading of drops of granular
suspensions. Because they are density-matched to the suspending fluid and do
not modify surface tension, the particles are expected to modify only viscous
dissipation and to leave gravitational and capillary effects unchanged.

Figure 6$(a)$ presents the variation of $\theta_{app}^{3}$ with the
measurement height, $h$, for monomodal granular suspensions of
10-$\mathrm{\SIUnitSymbolMicro m}$ particles and bimodal suspensions of
10-$80\text{\,}\mathrm{\SIUnitSymbolMicro m}$ particles with
$\zeta_{10}=50\,\%$, at a constant capillary number
$Ca_{0}=\eta_{f}U/\gamma_{f}$, where subscript $0$ in the capillary number
emphasises that it is computed using properties of the suspending fluid.
Reference data for the pure suspending fluid at $Ca_{0}$ are also provided for
comparison in figure 6$(a)$. The behaviour of $\theta_{app}^{3}(h)$ for the
suspensions is similar to that seen for the reference fluid, see also figure
5$(a,c)$. A plateau region is again observed close to the contact line, for
$10\lesssim h\lesssim$ $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$, while
there is a decay at larger distances. The addition of the particles leads to
an increase of the plateau value of $\theta_{app}^{3}$, which depends on the
particle mixture components. Provided measurements are undertaken within the
plateau region at constant height across all experiments, we can obtain an
unequivocal apparent contact angle, at a given capillary number. The
dependence between these two quantities is then a priori interpretable in
terms of the Cox-Voinov relation. In the following, the measurement height is
set at $h=$ $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

Figure 6: (a) Cube of the contact angle, $\theta_{app}^{3}$, versus the
measurement height, $h$, for different fluids: pure suspending fluid (grey
$\triangle$), 10-$\mathrm{\SIUnitSymbolMicro m}$ monomodal suspension (red
$\square$), and 10-$80\text{\,}\mathrm{\SIUnitSymbolMicro m}$ bimodal
suspension at $\zeta_{10}=50\,\%$ (orange $\circ$), resulting from the
analysis of 7, 8 and 10 drop-spreading runs, respectively, and at the same
fluid capillary number $Ca_{0}=\eta_{f}U/\gamma_{f}=0.0025$. Inset: normalised
cube of the contact angle, $\theta_{app}^{3}/\theta_{app}^{3}(h\rightarrow
0)$, versus normalised height, $h/\ell_{c}$. Blue dashed-dotted line: selected
measurement height $h=$ $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$. $(b)$
Variation of $\theta_{app}^{3}/9$ as a function of $Ca_{0}$ performed at a
height $h=$ $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ for the different runs
for spreading drops made of pure suspending fluid (grey dots), 10
$80\text{\,}\mathrm{\SIUnitSymbolMicro m}$ monomodal suspension (red dots),
and 10-$80\text{\,}\mathrm{\SIUnitSymbolMicro m}$ bimodal suspensions at
$\zeta_{10}=50\,\%$ (orange dots). The different colour shades represent
different experiments. The dashed lines correspond to the average of the
linear fits of data coming from each run and are used to infer the relative
apparent viscosity, $\eta_{w}$.

Figure 6$(b)$ displays typical variations of $\theta_{app}^{3}/9$ as a
function of the capillary number of the suspending fluid, $Ca_{0}$. Data from
different experimental runs are plotted for the same fluids as those used in
figure 6$(a)$. For a given fluid (suspensions or reference fluid), the tight
collapse of the different $\theta_{app}^{3}(Ca_{0})$ curves coming from the
different runs bears witness of the good reproducibility of the experiments.
All the datasets collapse on straight lines with unity slopes in log-log
representation, i.e. $\theta_{app}^{3}/9\propto Ca_{0}$. For each run, a
linear fit is performed and the average (dashed) line for a given fluid is the
average of the corresponding linear fits.

As expected, the data from the suspending fluid are in excellent agreement
with the Cox-Voinov law (3). The logarithm factor in (3), is found to be
$\approx 11.9$ in good agreement with values in the literature [41]. It is
important to note that the measurements are undertaken at a constant height,
$h$, within the plateau region and not at a constant $x$. However, since $x$
and $h$ are of the same order of magnitude and sufficiently large compared to
$\lambda$, the logarithm factor is not varying significantly and can be
considered as a constant.

For suspensions, data for $\theta_{app}^{3}(Ca_{0})$ on figure 6$(b)$ equally
collapse on a straight line with slope of unity in the log-log representation
(red and orange dashed lines): the spreading of these materials still seems to
follow a Cox-Voinov relation (3). However, since the values of
$\theta_{app}^{3}$ in the plateau are changed by the addition of particles as
shown in figure 6$(a)$, the different $\theta_{app}^{3}(Ca_{0})$ lines are
shifted with respect to each other in figure 6$(b)$, in agreement with
previous experiments [44]. Assuming that the logarithmic factor has the same
order of magnitude for the pure suspending fluid and for suspensions (as we do
not expect the particles to modify the mechanics at the nanometric scale), we
can superimpose all curves and recover the Cox-Voinov law (3) for all liquids
if we adjust the viscosity used in the capillary number, $Ca=\eta U/\gamma$,
with $\eta=\eta_{f}\eta_{w}$ where $\eta_{w}$ is the relative apparent wetting
viscosity of the suspensions. Again, the shift and consequently the apparent
wetting viscosity strongly depends on the particle mixture components. Here,
we see that the suspension of 10-$\mathrm{\SIUnitSymbolMicro m}$ particles has
a larger apparent wetting viscosity than the 10-80-$\mathrm{\SIUnitSymbolMicro
m}$ bimodal suspension. The behaviour of the apparent viscosity, $\eta_{w}$,
is examined in detail in the following § V for suspensions consisting of
different particle combinations.

##  V Probing dissipation with particles

###  V.1 Suspensions with particles having a large difference in size

Figure 7: Experimental results for suspension blends with fixed $d_{1}=$
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and $d_{2}=$
$80\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and varying $\zeta_{10}$ at
$\phi=0.4$. $(a)$ Relative wetting viscosity, $\eta_{w}$, as a function of the
fraction of the small particles in the solid phase, $\zeta_{10}$. Inset: bulk
viscosity, $\eta_{s}$, of the corresponding suspensions versus $\zeta_{10}$.
The dotted lines are guides for the eyes. $(b)$ Top-view pictures taken when
$\theta_{app}=35^{\circ}$ at $h=$ $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$
for $\zeta_{10}=0,\,25,\,50,\,75,\,100\,\%$ from top to bottom, respectively.

We focus first on suspensions consisting of particles having a large
difference in diameter, $d_{1}=$ $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$
and $d_{2}=$ $80\text{\,}\mathrm{\SIUnitSymbolMicro m}$ ($d_{2}/d_{1}=8$).
Figure 7$(a)$ shows that the relative wetting viscosity of these suspensions,
$\eta_{w}$, increases with increasing fraction of the small particles,
$\zeta_{10}$, with a steeper growth beyond $\zeta_{10}\approx 50\,\%$. This
behaviour is in stark contrast with the $\zeta_{10}$-dependence of the
relative bulk viscosity of the same mixtures shown in figure 2$(b)$ and
plotted again in the inset in figure 7$(a)$. The relative bulk viscosity of
the blends presents a minimum at $\zeta_{10}=35\,\%$, i.e. a mixture
containing roughly a third of small particles. This minimum viscosity is 40 %
smaller than the values obtained in the monomodal cases at $\zeta_{10}=0\,\%$
and 100 % (consisting of monomodal suspensions of 80 and
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$, respectively). It is worth noting
that, even in the monomodal case with the small particles, $\eta_{w}$ is
smaller (by a factor of order 2) than the corresponding value of its bulk
viscosity, $\eta_{s}$.

Top-view pictures of the suspension near the moving contact line provide
information about the origin of the discrepancy between the apparent viscosity
extracted from bulk rheology and its counterpart extracted from the dynamics
of spreading, see figure 7$(b)$. As previously shown in the monomodal case
[44], there is a pure-fluid region devoid of particles near the contact line.
We observe a similar region in the monomodal and bimodal cases depicted in
figure 7$(b)$. The extent of this region is larger in the case of the
80-$\mathrm{\SIUnitSymbolMicro m}$ particles, $\zeta_{10}=0$% (top picture),
than in the 10-$\mathrm{\SIUnitSymbolMicro m}$ case, $\zeta_{10}=100$% (bottom
picture). For intermediate values of $\zeta_{10}$, the
10-$\mathrm{\SIUnitSymbolMicro m}$ particles are able to move in-between the
80-$\mathrm{\SIUnitSymbolMicro m}$ particles. The particles closest to the
contact line arrange in an orderly, crystalline structure. This ordering is
particularly marked for the 80-$\mathrm{\SIUnitSymbolMicro m}$ particles, seen
on the top pictures of figure 7$(b)$. The large 80-$\mathrm{\SIUnitSymbolMicro
m}$ particles seem to be maintained at the same distance from the contact line
as small 10-$\mathrm{\SIUnitSymbolMicro m}$ particles are added, i.e. as
$\zeta_{10}$ is increased, but their linear density decreases.

The images displayed in figure 7$(b)$ are obtained for the same dynamic
contact angle, $\theta_{app}=35$ $\mathrm{\SIUnitSymbolDegree}$. The real
capillary number of all these experiments (using the effective viscosity of
the suspension, $Ca=\eta_{w}Ca_{0}$) is thus constant, and equal here to
approximately $2\cdot 10^{-3}$, leading to $h^{\star}\sim 100$
$\mathrm{\SIUnitSymbolMicro m}$ according to (10) and using the prefactor
inferred from the figure 5$(d)$, i.e. $h^{\star}\approx
0.3\,\ell_{c}\,Ca^{1/3}$. The discrepancy between the viscosity measured in
the bulk and that estimated from spreading experiments is now clear. The
80-$\mathrm{\SIUnitSymbolMicro m}$ particles experience strong confinement in
the viscous-capillary region since $h^{\star}\sim d_{2}$, and only a few of
them can penetrate this region. In contrast, $h^{\star}$ is 10 times larger
than the diameter of the 10-$\mathrm{\SIUnitSymbolMicro m}$ particles. Even if
these small particles experience confinement close to the contact line, a
significant part of the viscous-capillary region is filled with a dense phase
of small particles akin to a suspension bulk. Thus, the contribution of the
10-$\mathrm{\SIUnitSymbolMicro m}$ particles to dissipation in the Cox-Voinov
region is expected to be larger than that of the
80-$\mathrm{\SIUnitSymbolMicro m}$ particles, as reported in experiments with
monomodal suspensions. In the case of the bimodal suspensions studied in this
section, increasing the small particle fraction, $\zeta_{10}$, is expected to
lead to a continuous increase of $\eta_{w}$, as observed in figure 7$(a)$.

###  V.2 Varying the size of the large particles

Figure 8: Experimental results for $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$
monomodal suspension ($\bullet$ symbols) and bimodal blends with $d_{1}=$
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and varying $d_{2}$ (= 20, 40,
$80\text{\,}\mathrm{\SIUnitSymbolMicro m}$) at fixed $\zeta_{10}=25\,\%$ and
$\phi=0.4$ ($\square$ symbols). $(a)$ Relative wetting viscosity, $\eta_{w}$,
as a function of the size ratio, $d_{2}/d_{1}$. The horizontal dashed line
indicates the viscosity value for the monomodal suspension consisting of the
sole small particles of size $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$.
Inset: bulk viscosity, $\eta_{s}$, of the corresponding suspensions versus
$d_{2}/d_{1}$. The grey dotted lines are guides for the eyes. $(b)$ Distance
of approach of the large particles for the bimodal suspensions with $d_{1}=$
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and $\zeta_{10}=25\,\%$ ($\square$
symbols) and the $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ monomodal
suspension ($\bullet$). The dashed line corresponds to the geometrical
prediction (11). $(c)$ Top-view pictures taken when $\theta_{app}=35^{\circ}$
at $h=$ $50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ for monomodal suspension of
size $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and bimodal suspensions with
$(d_{1},\zeta_{10})=($$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$$,25\,\%)$ and
$d_{2}$ = 20, 40, $80\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (from bottom to
top, respectively).

We now turn to bimodal suspensions consisting of
10-$\mathrm{\SIUnitSymbolMicro m}$ particles but with large particles of
variable diameter (i.e. $d_{2}=20,40,$ $80\text{\,}\mathrm{\SIUnitSymbolMicro
m}$) at $\zeta_{small}=25\,\%$. Bimodal suspensions formulated this way should
have a bulk viscosity close to the minimum observed on the curves of figure
2$(b)$. Figure 8$(a)$ shows that the apparent wetting viscosity of these
suspensions is maximum in the monomodal case and decreases for bimodal
suspensions as the large particle size increases, i.e. with increasing
$d_{2}/d_{1}$. This result again confirms that particles with a diameter much
smaller than $h^{\star}$ increase the dissipation in the Cox-Voinov region.
This trend is similar to that of the bulk viscosity, $\eta_{s}$, with
increasing $d_{2}/d_{1}$ for $\zeta_{small}=25\,\%$, see the inset figure
8$(a)$. However, the magnitude of the size-ratio effect comes from two
distinct physical origins. For example, the value of the apparent wetting
viscosity, $\eta_{w}$, for $d_{2}/d_{1}=8$ is close to that of the suspending
fluid whereas the value of the bulk viscosity, $\eta_{s}$, is much larger. The
decrease of the bulk viscosity, comes from the size ratio affecting the
maximum packing fraction $\phi_{c}$ (larger for large $d_{2}/d_{1}$), while
the wetting viscosity reduction comes from confinement effects near the
contact line as the dissipation is greater for particles which can efficiently
reach the dissipation region near the contact line ($d\ll h^{\star}$). In the
case of figure 8, the small particles are participating less to the
dissipation as $\zeta_{10}$ is small and the overall dissipation is mostly set
by the large particles.

Figure 8(b) shows that the distance of approach of the large particles, $L$,
increases with their diameter. This measured length follows the prediction of
a geometrical model describing the minimal distance of approach of a particle
in a wedge,

$L=\frac{d}{2}\left[\frac{1}{\sin(\theta_{app})}+\frac{1}{\tan(\theta_{app})}-1\right],$
(11)

where $\theta_{app}$ is the apparent contact angle near the contact line. We
find that this distance is not sensibly affected by the presence of the small
particles. The relation (11) holds both in monomodal and bimodal using the
diameter of the corresponding particles. In the monomodal case, it determines
the size of the depleted region [44].

The top-view pictures in figure 8$(c)$ confirm the existence of size
segregation at the contact line as already mentioned in § V.1. The large
particles form ordered rows behind the small particle region as already noted
in § V.1. However, we seem to observe that the 10-$\mathrm{\SIUnitSymbolMicro
m}$ particles do not go through the network formed by the
20-$\mathrm{\SIUnitSymbolMicro m}$ particles, see bottom image of figure 8(c).
We move to this aspect of suspension spreading in the following section with
clearer visualisations with larger particles.

###  V.3 Varying the size of the small particles

Figure 9$(a)$ gathers the wetting viscosity, $\eta_{w}$, of the monomodal
$80\text{\,}\mathrm{\SIUnitSymbolMicro m}$ suspension and bimodal blends
having a fixed $d_{2}=$ $80\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and varying
$d_{1}$ and $\zeta_{small}$. When the solid phase mainly consists of small
particles ($\zeta_{small}=75\,\%$), $\eta_{w}$ decreases with increasing
$d_{1}$ as seen in figure 9$(a)$. This observation confirms again that the
wetting viscosity is set by the possibility for particles having a diameter
much smaller than $h^{\star}$ to approach close to the contact line. However,
a continuous decrease of $\eta_{w}$ with $d_{1}$ is absent when
$\zeta_{small}=25\,\%$ or $\zeta_{small}=50\,\%$, i.e. when large particles
constitute a significant portion of the solid blend. The relative wetting
viscosity, $\eta_{w}$, reaches a maximum around $d_{1}/d_{2}=0.25$ but it
otherwise shows low values, even lower than those obtained for the monomodal
suspensions of 80-$\mathrm{\SIUnitSymbolMicro m}$ particles for
$d_{1}/d_{2}=0.5$. In other words, for $\zeta_{small}=25-50\,\%$, the apparent
wetting viscosity of the 10-$80\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
40-$80\text{\,}\mathrm{\SIUnitSymbolMicro m}$ blends are particularly low
compared to that of the 20-$80\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and even
to that of the monomodal 80-$\mathrm{\SIUnitSymbolMicro m}$ suspension.

Figure 9: Experimental results for suspension blends with $d_{2}=$
$80\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and varying $d_{1}$ (= 10, 20,
$40\text{\,}\mathrm{\SIUnitSymbolMicro m}$) at $\zeta_{small}=25\,\%$
($\square$ symbols), $\zeta_{small}=50\,\%$ ($\lozenge$ symbols) and
$\zeta_{small}=75\,\%$ ($\triangle$ symbols). $(a)$ Relative wetting
viscosity, $\eta_{w}$, as a function of $d_{1}/d_{2}$. The dashed line
indicates the viscosity value for the monomodal suspension consisting of the
sole large particles of size $80\text{\,}\mathrm{\SIUnitSymbolMicro m}$.
Inset: bulk viscosity of the corresponding suspensions. $(b)$ and $(c)$ Top-
view pictures taken when $\theta_{app}=35^{\circ}$ at $h=$
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ for bimodal suspensions with
$d_{2}=$ $80\text{\,}\mathrm{\SIUnitSymbolMicro m}$, $\zeta_{small}=25\,\%$
$(b)$ and $\zeta_{small}=75\,\%$ $(c)$. Small particle size is increasing from
top to bottom ($d_{1}=$10, 20, $40\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
monomodal suspension of size $80\text{\,}\mathrm{\SIUnitSymbolMicro m}$).

The top views shown in figure 9$(b)$ provide some clues about the variation of
the apparent wetting viscosity. At $\zeta_{small}=25\,\%$, for small values of
$d_{1}/d_{2}$ (two topmost pictures), small particles flow through the large
particle network and get close to the contact line. In contrast, for
$d_{1}/d_{2}=0.5$ (pictures on the third row from the top), the presence of
40-$\mathrm{\SIUnitSymbolMicro m}$ particles disrupts the ordering of the
80-$\mathrm{\SIUnitSymbolMicro m}$ particles, compared to the other cases. A
similar trend is observed for $\zeta_{small}=50\,\%$. In contrast, for
$\zeta_{small}=75\,\%$, there is a large amount of ordered small particles
near the contact line in front of the large spheres, as seen in figure 9$(c)$.

Figure 10: Sketch of two large particles separated by a small one on a solid
plane.

Geometrical considerations can explain the effect observed for
$d_{1}/d_{2}=0.5$. Two large particles with radius $R_{2}=d_{2}/2$ sitting on
the surface of solid come to contact if the distance, $l$, between them goes
to zero, see figure 10. A small particle of radius $R_{1}$ can pass through
the hole between the two particles and the solid surface if its radius is at
most $R_{1}=R_{2}/4$. Consequently, while the 10- and
20-$\mathrm{\SIUnitSymbolMicro m}$ particles can flow through the interstices
created by the large $80\text{\,}\mathrm{\SIUnitSymbolMicro m}$ particles, the
40-$\mathrm{\SIUnitSymbolMicro m}$ particles cannot. Instead, these particles
induce defects in the 80-$\mathrm{\SIUnitSymbolMicro m}$ particle network,
leading to the distortion of the large-particle matrix when the portion of
small particles is not too large ($\zeta_{small}\lesssim 50\,\%$). This loss
of organisation seems detrimental to the local dissipation. These geometrical
considerations coupled with the wetting viscosity measurements suggest that
the structure near the contact line can be a key element to explain the value
of the wetting viscosity of granular suspensions.

However, the above geometrical arguments cannot explain the lower values of
$\eta_{w}$ for the 10-$80\text{\,}\mathrm{\SIUnitSymbolMicro m}$ blends
compared to those of the 20-$80\text{\,}\mathrm{\SIUnitSymbolMicro m}$ blends
for $\zeta_{small}=25-50\,\%$ in figure 9$(a)$. Instead, this lessening effect
may be attributed to bulk effect impacting on the variation of the wetting
viscosity. The inset of figure 9$(a)$ indicates that the bulk viscosity is
significantly smaller for the 10-$80\text{\,}\mathrm{\SIUnitSymbolMicro m}$
blends than for the 20-$80\text{\,}\mathrm{\SIUnitSymbolMicro m}$ blends for a
given value of $\zeta_{small}$. Moreover, the bulk viscosity for
$\zeta_{small}=25-50\,\%$ is lower than for $\zeta_{small}=75\,\%$ as it
corresponds to the minimum bulk viscosity for bimodal blends, as seen in
figure 2$(b)$. Bulk effects are therefore evidenced in this specific example
and impact the value of $\eta_{w}$ as they happen to overcome size effects.

It is worth noting that, in the monomodal case, variation of the wetting
viscosity can only come from confinement effect near the contact line as the
bulk viscosity does not vary with particle size, as shown in figure 2$(a)$.
The case of fixed $d_{2}$ and increasing $d_{1}$ evidences two competing
effects: confinement effect governing the particle ability to approach the
contact line and the variation of the bulk viscosity with $\zeta_{small}$ and
$d_{1}/d_{2}$. In conclusion, for bimodal suspensions, a high wetting
viscosity results from a complex compromise between a high fraction of small
particles, $\zeta_{small}$, a large size ratio, $d_{2}/d_{1}$, and a high bulk
viscosity, $\eta_{s}$, in addition to size effects already demonstrated.

##  VI Concluding remarks

The spreading of large drops onto a solid substrate is a rich and complex
phenomenon by itself. Adding particles affects the spreading dynamics in a
non-trivial way as there exists a high degree of inhomogeneity in the
particulate drop going from a random solid bulk phase far from the contact
line ($h\gg d$), to ordered dense monolayers when particles undergo
confinement ($h\gtrsim d$), and finally to a particle-depleted region in the
very close vicinity of the contact line ($h<d$). Using granular suspensions
nonetheless offers an interesting way to investigate dynamical wetting as one
can take advantage of this complex particulate organisation to modify locally
the dissipation near the contact line.

Dimensional analysis of the equation governing the spreading of large drops
($Bo\gtrsim 1$) shows that the profile of the liquid/gas interface between the
droplet and the atmosphere is set by a viscous-capillary balance as long as
its height is smaller than a length-scale $h^{\star}\sim\ell_{c}Ca^{1/3}$
beyond which the drop profile becomes also sensitive to gravity. This scaling
has been validated in the present work for both simple fluids and suspensions
and care was taken to perform measurements below this length-scale (typically
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$) to ensure that gravity is
negligible across the range of capillary numbers that we can probe. Drop of
granular suspensions were then seen to follow the Cox-Voinov law relating the
contact angle to the capillary number in a similar way as found for regular
Newtonian fluids. However, the relative apparent viscosity, $\eta_{w}$,
involved in the capillary number was found to differ from that of the bulk
suspension, $\eta_{s}$.

In the present experiments, the typical magnitude of $h^{\star}\sim$
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ lies in the size range of the non-
Brownian particles used. Therefore, dissipation is affected by particles
during the spreading when $d\lesssim h^{\star}$ while the spreading dynamics
is close to that of the pure fluid when $d\gtrsim h^{\star}$, i.e. when
particles can not reach the viscous-capillary region. These geometrical
considerations help rationalise that $\eta_{w}$ depends on the particle size,
$d$, unlike $\eta_{s}$ that only depends on particle volume fraction, $\phi$.
The relative apparent viscosity, $\eta_{w}$, is found to be maximum for the
smallest particles and decreases to values close to the suspending fluid
viscosity for $d\gtrsim$ $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ even
though $\eta_{s}$ is much larger than $\eta_{w}\simeq\eta_{f}$ for these dense
suspensions ($\phi=0.4$). This size cut-off of the wetting viscosity is thus
roughly of the order of $h^{\star}$. This confirms that particles affect the
wetting dynamics only if their size is small enough for them to reach the
region dominated by viscosity.

It may seem surprising that the Cox-Voinov law applies to such a complex
system. Indeed, the possibility for the particles to occupy the viscous-
capillary region in the case of a spreading droplet evolves with time as
$h^{\star}\propto Ca^{1/3}\propto U^{1/3}$ decreases with increasing time.
However, in the present experiments, the variation of $Ca$ over only one
decade prevents a strong change in $h^{\star}$. Performing measurements in a
much lower $Ca$-range, while rather challenging, may reveal intriguing
effects.

Predicting the value of the apparent wetting viscosity when $d\lesssim
h^{\star}$ remains a difficult task. The value of the bulk viscosity,
$\eta_{s}$, certainly affect the wetting viscosity, $\eta_{w}$. However,
knowing whether this observation comes from long range effects of the bulk
phase far from the contact line is not easy. One must also keep in mind that
$\eta_{s}$ varies with $\phi_{c}-\phi$ and strongly diverges as
$\phi\rightarrow\phi_{c}$. For bimodal suspensions, a lower $\eta_{s}$ is the
signature of a higher $\phi_{c}$ for the corresponding solid phase and
therefore produces a lower $\eta_{w}$, as seen in the present experiments.

Another fundamental ingredient to consider in predicting $\eta_{w}$ is the
possibility for the particles to crystallise in the viscous region. For
monomodal suspensions, a monolayer of several rows of ordered particles is
always observed in the vicinity of the contact line. In the bimodal case,
crystallisation of the two population sizes can be hindered when the small
particles cannot flow through large-particle holes, leading to a significant
diminution of the apparent wetting viscosity. The structure of the particulate
network near the contact line therefore directly impacts the drop spreading
dynamics captured in $\eta_{w}$. It also explains that even for the smallest
particles for which $d<h^{\star}$, the wetting viscosity, $\eta_{w}$, cannot
reach the bulk value, $\eta_{s}$, because of this ordered region which
possesses a lower viscosity.

This ordering very likely comes from the confinement of the particles near the
contact line combined with the pressure of the dense particulate phase farther
out. Rheological measurements combined with direct visualisation of dense
colloidal suspensions ($\phi=0.52$) have clearly established the relation
between a significant viscosity drop and moderate confinement, owing to the
ordering of the particulate phase in sliding layers [30]. Conversely, for much
dilute granular suspensions ($\phi\leq 0.2$), rheological measurements have
exhibited a monotonous increase of dissipation for gap sizes of ten diameters
and less [29]. This apparent disagreement could be explained by the nature of
the solid phase (colloidal/granular) or the dissipation mechanisms
(friction/hydrodynamics) that depends on the solid volume fraction [15].
Indeed, at low volume fraction, dissipation is mainly due to hydrodynamic
interactions while frictional contacts dominate for dense suspensions and may
crucially depend on the local solid structure [18].

With confinement strengthening, complex behaviours become even more apparent,
e.g. viscosity minima when the gap thickness is commensurate with the particle
diameters [30]. Such oscillatory values of the viscosity as function of the
gap have been reported under high confinement and high shear in numerical
simulations of dense granular suspensions [14]. In the present experiments,
confinement effects may however be less easy to study and quantify for the
following reasons. Confinement is changing over time as the drop spreads and
the local fluid thickness varies with the radial position. Moreover, it is set
by a solid surface and a free interface that can deform to relax high stress,
contrary to solid boundaries in a rheometer and most of the simulations.

Finally, the present work may help understand the spreading of suspensions
that are closer to those used in industrial processes and have a wide range of
particle sizes [e.g. from sub-micron to hundreds of microns in cement paste,
see 1, 4]. Bimodal suspensions can be seen as a first step toward polydisperse
solid blends as a smart choice in particle size and fraction in a bidisperse
system can partially mimic an equivalent polydisperse suspension [28].

Declaration of Interests. The authors report no conflict of interest.

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