# Drop Impact Dynamics of Complex Fluids: A Review

Phalguni Shah  Department of Physics and Astronomy, Northwestern University,
Evanston, IL, USA  Michelle M. Driscoll  Department of Physics and Astronomy,
Northwestern University, Evanston, IL, USA

###### Abstract

The impact of fluid drops on solid substrates has widespread interest in ink-
jet printing, agricultural pesticides, and many other industrial coating and
spraying applications. A large body of experimental and theoretical work has
been done to understand the impact dynamics of Newtonian fluids, yet we are
far from understanding the drop impact of complex fluids. In this review, we
outline recent experimental, theoretical, and computational advances in the
study of impact dynamics of complex fluids on solid surfaces. Here, we provide
an overview of this field, with our discussion segmented by different material
constitutions. Throughout, we highlight promising future directions, as well
as ongoing experimental and theoretical challenges in the field.

###### Contents

  1. I Introduction
  2. II Advances in Newtonian drop impact
  3. III Colloidal suspensions
    1. III.1 Colloidal drop impact: shear-thinning regime
    2. III.2 Colloidal Drop impact: shear-thickening regime
  4. IV Granular Suspensions
    1. IV.1 Spreading and receding dynamics
    2. IV.2 Splashing threshold
  5. V Polymeric fluids
    1. V.1 Experimental results
    2. V.2 Numerical studies
  6. VI Outlook
    1. VI.1 Theoretical and computational challenges
    2. VI.2 Open questions and perspectives

##  I Introduction

The impact of fluid drops on solid surfaces is ubiquitous in many natural and
industrial processes such as geological erosion, spraying and coating of
surfaces, ink-jet printing, and agrochemical delivery. The phenomena of drop
impact is a nuanced problem, and the relevant parameter space is vast. In
addition to the impact conditions (e.g. impact velocity, drop size, impact
angle), the dynamics of drop impact is controlled by the properties of the
fluid (e.g. density, surface tension, viscosity), the properties of the
surface (e.g. contact angle with the fluid, surface roughness, and
hydrophobicity), and the properties of the surrounding gas (e.g. density,
pressure, and molecular weight) YarinReview ; JosserandReview . While the
impact dynamics of Newtonian fluids have been studied extensively for over a
century, the body of work done to understand the impact of non-Newtonian or
complex fluid drops has been limited to date. In addition to spreading and
splashing, complex fluids can exhibit a diverse array of behaviors such as
solidification and delayed spreading upon impact [Fig. 1]. The rich variety of
flow behaviors observed in complex fluids make understanding drop impact in
these materials a challenging problem. However, this is an important challenge
to undertake, as many of the industrial applications of drop impact involve
complex fluids. Furthermore, drop impact studies allow us to probe material
behavior at very high strain rates, a regime that is often inaccessible via
conventional rheometry techniques. For example, a fluid drop of diameter
$d_{0}=3$ mm impacting a surface at a velocity of $u=5$ m/s would undergo a
strain rate of $\dot{\gamma}=\frac{u}{d_{0}}=1.67\times 10^{3}$ $s^{-1}$ at
impact; this is approaching the operating limit of most rheometers. Drop
impact experiments can therefore provide a bridge to expand our understanding
of flow properties of complex fluids into the high-strain rate regime.
Furthermore, drop impact allows an exploration of these flow behaviors in a
system with a deformable free surface, rather than the bulk measurements under
confinement one typically encounters in rheometry data. As we will discuss
below, the free surface allows for observation of new and unusual behaviors.

Figure 1: Behavior of impacting drops of Newtonian (a-c) and complex fluids
(d-f). (a) A drop of mercury (a Newtonian fluid) spreading on glass (Adapted
from JosserandReview ), (b) a drop of mercury undergoing a prompt splash when
impacted on superhydrophobized glass (Adapted from JosserandReview ), (c) a
drop of silicone oil exhibiting corona splash on glass, (d) a drop of silica
in water ($\phi=\%$) partially jammed after impacting on glass (Adapted from
AroraDriscoll2020 ), (e) a drop of dense granular suspension after impacting
on glass, making a pile-like shape instead of completely spreading out
(Adapted from Marston ), and (f) A drop of 0.25% Carbopol in water impacting a
wet substrate (Adapted from Blackwell2015 ).

The goal of this review is to provide a summary of the work on the impact of
complex fluid drops. Much of the work summarized here is confined to impact on
dry, solid surfaces — the most straightforward impact scenario to interpret.
Complex fluids exhibit a large variety of rheological behavior, and much of
the literature has focused on rather disparate regimes of this phase space.
Here, we attempt to classify the existing work on complex fluid impact into
physically relevant categories, and draw connections that relate impact
behavior to rheological data. We hope to provide a useful reference point for
future studies and help move towards a more comprehensive understanding of the
physics of complex fluids under high stresses.

The review is structured as follows: Section II provides an overview of the
current state of the Newtonian drop impact field. We have organized the
remaining sections of the review by material composition, grouping studies by
rheological behavior.

The advances in the impact dynamics of colloidal suspensions are discussed in
section III. Section IV outlines the impact dynamics of granular suspensions.
In section V, we summarize both experimental and numerical results for the
impact of polymeric drops. Section VI concludes the review by discussing the
current challenges in the field, and also suggests several directions of
research that would lead to a comprehensive understanding of the impact of
complex fluids on solid substrates.

##  II Advances in Newtonian drop impact

Drop impact of Newtonian fluids has been a topic of study for over a century
Worthington . Despite many key differences between the impact dynamics of
Newtonian and complex fluids, Newtonian drop impact studies have provided the
foundation for their complex fluid counterparts. In this section, we summarize
the state of the Newtonian drop impact field, before we dive into the impact
dynamics of complex fluid drops. We emphasize that this section is not
intended to be a comprehensive review of Newtonian drop impact, as we cannot
possibly capture all aspects of this vast field in this brief overview.
Instead, we aim to give the reader an outline of the major developments in
this field, and to highlight aspects of Newtonian impact which are most
relevant to the impact dynamics of complex fluids. For a more complete
discussion of this field, we refer the interested reader to the following
reviews on the subject YarinReview ; JosserandReview .

The impact behavior of a Newtonian drop can often be predicted by considering
the relative importance of various stresses, namely inertial, viscous, and
surface stresses. As with other fluid dynamics problems, this is often
quantified using dimensionless quantities. The most common dimensionless
numbers used in Newtonian drop impact are constructed using fluid properties
(density, $\rho$, surface tension, $\sigma$, and dynamic viscosity, $\mu$) and
impact parameters (drop diameter, $d_{0}$ and impact velocity, $U$):

  * •

Reynolds number ($Re$): ratio of inertial and viscous stresses, $Re=\frac{\rho
Ud_{0}}{\nu}$. Large $Re$ implies viscous stresses are negligible compared to
inertia.

  * •

Weber number ($We$): ratio of inertial and surface stresses, $We=\frac{\rho
U^{2}d_{0}}{\sigma}$. Large $We$ signifies that surface stresses are
negligible compared to inertia.

  * •

Ohnesorge number ($Oh$): compares the effect of viscous stresses with the
combined effect of surface stresses and inertia, $Oh=\frac{\sqrt{We}}{Re}$.
This is an ideal dimensional entity for use in studies where inertial,
viscous, and surface stresses may all be relevant (a common scenario for
impacts at a few m/s).

These dimensionless numbers are often modified to describe complex fluid
impacts; we will introduce these modified quantities as well as new
dimensionless numbers as needed.

When a fluid drop impacts a surface, the drop radially expands on the
timescale of a few milliseconds. During this process, fluid inertia is
converted into surface energy, while being opposed by viscous stresses, fluid-
solid surface interactions, and ambient gas effects. Later in the process,
after the drop has reached its maximum expansion, the fluid recedes; the drop
may undergo several oscillations of spreading and receding depending on fluid
and surface parameters. In some impact regimes, the rim of the spreading drop
becomes unstable, leading to ejection of secondary droplets, this is termed
‘splashing’. A large body of work has been done to understand the spreading
dynamics (spreading time and the maximum spreading diameter), receding
dynamics, and the transition from spreading to splashing YarinReview ;
JosserandReview . Below, we summarize findings about the spreading, receding,
bouncing, and splashing of Newtonian drops. We then proceed to outline studies
on small targets, oblique impacts, and impacts on soft substrates.

The maximum spreading diameter of a drop after impact is governed by the
balance between drop inertia, surface tension, and viscous dissipation, and a
sizable amount of work has been done to obtain scaling laws for maximum
spreading. Energy conservation arguments in the low-viscosity, high-inertia
regime (large $Re$, $We$) suggest that the maximum spread should scale as
${We}^{\frac{1}{2}}$. On the other hand, some experimental data
clanet2004maximal  have indicated a contradictory scaling of
${We}^{\frac{1}{4}}$, though measurements were only made for water drops
laan2014maximum . In the high-viscosity, low-inertia regime (small $Re$), the
maximum spread has been observed to follow the scaling of
${Re}^{\frac{1}{5}}$, but forensic studies have been known to use a
${Re}^{\frac{1}{4}}$ scaling laan2014maximum ; clanet2004maximal . To unite
these disparate experimental observations that report different scaling laws
for different fluids, it has been proposed that a broad crossover regime
exists between the inertial and viscous extremes. Recent works have focused on
developing a universal scaling for the maximum spreading diameter,
incorporating the effects of both inertia and viscosity laan2014maximum ;
lee2016universal . Laan et al. laan2014maximum  used the first order Padé
approximation to propose the scaling:

$\beta_{max}={Re}^{\frac{1}{5}}(\frac{P^{\frac{1}{2}}}{A+P^{\frac{1}{2}}}),$

where $P=We{(Re)}^{-\frac{2}{5}}$, as suggested by energy conservation
Eggers2010 ; lagubeau2012spreading . This scaling was able to fit not only
data for glycerol-water mixtures of varying viscosities, but for blood, a
shear-thinning fluid, approximated as a Newtonian fluid with viscosity equal
to its high-shear viscosity. Lee et al. lee2016universal  further modified
this scaling law to include the maximum spread at zero impact velocity by
incorporating the details of surface wettability, and showed that this scaling
holds for a range of fluids. In summary, by incorporating surface wettability
effects as well as fluid $We$ and $Re$, current scaling models have
successfully predicted the maximum spread for a large variety of Newtonian
fluid impacts.

After the spreading phase, the impacted drop may retract depending on the
impact Weber number and the interaction with the substrate. Compared to
maximum spread, studies on the retraction dynamics have been less extensive.
Bartolo et al. bartolo2005retraction  studied the retraction rate of drops
impacting at high $We$ on hydrophobic surfaces. They observed that the
retraction velocity, when normalized by the maximum spread, was independent of
impact velocity. This suggests that the impact velocity governs the drop
spreading phase via drop inertia, but it has a negligible effect on the
dynamics of the receding phase. The authors additionally observed the
transition from inertial to viscous regime at around $Oh\sim 0.05$, which
could be predicted with simple hydrodynamic arguments.

In addition to spreading and receding, liquid drops impacting on hydrophobic
surfaces may completely retract and bounce back off the impact surface. For
low-viscosity and high-inertia (high $Re$ and $We$) impacts, drops have been
observed to bounce almost elastically; this behavior has been successfully
modeled as a simple spring-mass systems Richard2002 . Recently, this model has
been extended to drops of moderate viscosity fluids using a damped spring-mass
equation jha2020viscous . This damped model showed that the contact time (time
interval for which the drop was in contact with the substrate) increased
slightly for higher viscosity fluids. At the same time, the coefficient of
restitution (measuring how elastic the collision was) dropped to zero with
increasing fluid viscosity, owing to higher viscous dissipation during contact
jha2020viscous . Decreasing the contact time during impact is key for the
development of water-repellent surfaces, and the use of macrostructures has
been proposed for such applications Bird2013contact . For a more detailed
discussion of drop impact on hydrophobic surfaces, we refer the reader to this
recent review khojasteh2016droplet .

As both intrinsic fluid properties and the fluid-substrate interaction affect
the impact dynamics, decoupling the two effects is essential for a better
understanding of the impact process. To this end, experiments on small targets
have proven to be an effective way to minimize substrate effects. Such targets
are typically comparable in size to the drop diameter, but are much smaller
than the maximum spread of the drop, eliminating fluid-substrate interactions
during much of the spreading process. As gravity plays a negligible role over
the impact timescale (a few milliseconds), small target studies still produce
a horizontally spreading film. Scaling laws have been fairly successful in
characterizing the spreading over time in both viscous and inertial regimes
smalltargets_newtonian . Recent works have also focused on the role of
different target geometries on controlling droplet spread and splash
splash_control ; spherical_target .

After the spreading phase, a drop undergoing a high-$We$ or $Re$ impact often
ejects small droplets from its unstable rim, leading to ‘splashing’. The
transition to splashing was originally proposed to be governed by the
parameter $K=We({Re})^{\frac{1}{2}}$ stow1981experimental ; MUNDO1995151 .
This characterization of splashing was very limited, as it incorporated only
the fluid and impact properties. More recent studies have demonstrated that
the quantitative value of the splashing threshold additionally depends both on
the details of the substrate RANGE1998  and surprisingly, the surrounding gas
properties Xu2005 . The observation by Xu et al. Xu2005  that lowering the
ambient pressure could completely suppress splashing has inspired numerous
works aimed at understanding the mechanism behind this phenomenon. The role of
gas as well as the microscopic details of the moving liquid-solid contact line
have both been explored Riboux2014 ; Riboux2017 ; deGoede2018 ; latka2018 ;
jian2018 ; Quetzeri2019 . Riboux and Gordillo Riboux2014  derived a splash
criteria for drops impacting on partially wetting solids, which incorporated
the liquid and gas viscosity, density, interfacial tension, drop size, and the
gas mean free path. The ratio of the gas mean free path and lamella thickness
was shown to be a key control parameter in this splashing criteria,
potentially suggesting the mechanism by which ambient gas acts to suppress
splashing. Recently, the authors modified this criteria to incorporate
boundary layer effects in the gas Riboux2017 . This model has had great
success predicting the splash threshold for a range of fluids impacting on
multiple substrates Riboux2017 . Complimentary work by Jian et al. jian2018
numerically studied the effect of gas density and viscosity on splashing,
assuming both the liquid and gas to be incompressible. They observed the
splashing threshold to be primarily controlled by gas viscosity, with a small
correction due to density. As this study explored a different range of Weber
and Reynolds number, their findings are not directly comparable to that of
Riboux and Gordillo Riboux2014 ; Riboux2017 . Such a direct comparison would
be an ideal area for future exploration, as it can affirm the validity of our
current understanding of gas effects on splashing, as well as establish
whether the low-pressure splash suppression effect works via the same
mechanism in all regimes. Incorporating the effect of gas compressibility in
future studies would be valuable, albeit challenging.

While it is clear that the dynamic contact angle plays a role in splashing
dynamics latka2018drop ; Quetzeri2019 , it is difficult to understand the
physics of this process as it remains a challenge to measure the microscopic
dynamic contact angle experimentally. The dynamic contact angle is set by
fluid-substrate interactions, and differs during the spreading and receding
phases snoeijer2013 . Recent work by Quetzeri-Santiago et al. Quetzeri2019
experimentally investigated the effect of contact angle on splashing dynamics,
and proposed that the advancing dynamic contact angle is a more suitable
parameter than the static contact angle for predicting the splashing
threshold. Their predictions agree with past studies on hydrophilic surfaces,
and potentially extend our understanding of splashing to hydrophobic surfaces.

Although most drop impact studies have focused on normal impact and hard
substrates, real-life impacts often happen at an angle, or on compliant
surfaces. The best example of this is the spraying of pesticides onto leaves,
a process where both of these modifications come into play. Studies on
splashing onto oblique and translating surfaces bird2009 ; Hao2019 ;
Garcia2020  have found that in some cases splashing can be suppressed in the
‘uphill’ direction, while still occurring in the ‘downhill’ direction due to
the asymmetry of the impact. Models for maximum spreading diameter have been
shown to be applicable to inclined systems, considering the normal component
of impact velocity and short axis of spreading laan2014maximum . Inclined
impacts can also be simulated by imparting a tangential velocity to the
substrate. Bird et al. bird2009  analyzed such impacts on moving surfaces,
also observing asymmetric splash.

Despite the connection with many industrial and agricultural process, the
impact of Newtonian drops onto soft elastic surfaces has not been as well-
explored as that on rigid surfaces. Pepper et al. explored the splash
threshold as a function of impact velocity and surface compliance, finding
that a more compliant surface lowered the impact velocity at which splashing
occurred pepper2008 . More recent work similarly observed suppressed splashing
as the Young’s modulus of the substrate was decreased, and linked it to the
increased energy dissipation occurring due to the softer impact substrate
SoftSplash . Further studies on these lines, for Newtonian as well as complex
fluids, would open up novel yet facile ways to control splashing and secondary
droplet ejection in a variety of applications.

The relevant parameter space for Newtonian drop impact is large, as it
includes properties of the liquid, the substrate, as well as the surrounding
gas. Though much progress has been made Riboux2014 ; Chubynsky2020 ,
simulation and theory in this area are particularly challenging due to the
large dynamic range of both timescales and lengthscales involved
JosserandReview . Additionally, most experimental data are at intermediate
values of Reynolds numbers, a regime that is the most challenging to study
theoretically as neither inertial nor viscous stresses can be neglected. In
addition to the need for more theoretical studies, there are multiple
experimental avenues that deserve more exploration. While it has been well-
established that substrate properties play a role in setting the splashing
threshold tsai2010 ; marengo2011 , likely by controlling the dynamic contact
angle Quetzeri2019 ; latka2018drop , more experiments studying the effect of
micro-patterned substrates (with both regular and random patterns) on spread,
splash, and bounce would be beneficial to a number of industrial applications.

Although complex fluid drops can behave quite differently due to their non-
Newtonian rheology, many parallels can be drawn with the Newtonian impact
field. In the following sections, we make these connections between the two
fields as we summarize impact studies of complex fluids. We additionally point
out phenomena that need more exploration in order to build a more holistic
picture of the similarities and differences between the physics of Newtonian
and non-Newtonian drop impact.

##  III Colloidal suspensions

Colloidal suspensions are typically defined as micron-sized or smaller
particles suspended in a Newtonian fluids such as water. Colloidal suspensions
are often referred to as ‘Brownian’ suspensions due to the ability of
particles in this size range suspended in water-like fluids to exhibit
Brownian motion at room temperature. Suspensions made of particles too large
to exhibit Brownian motion are often termed non-Brownian or granular
suspensions. However, the particle size range that separates colloidal
suspensions from granular ones is not well-established, and can vary based on
flow velocities. Furthermore, drop impact occurs in the regime of high Peclét
number (the ratio of shear rate and particle diffusion rate), that is, the
high-velocity flows which occur during impact and spreading dominate any
diffusive motion. Thus, for the purposes of this review, we have chosen to
divide these two categories based on whether the individual particle inertia
plays a role in the physics of impact. This section discusses studies where
impact behavior can be connected to the bulk rheology of the complex fluid,
whereas Section IV discusses studies where the inertia of individual particles
may governs impact outcomes.

A drop impacting on a solid substrate experiences large shear rates that vary
spatially throughout the drop as well as over time. This variation is
especially relevant to the problem of impacting complex fluids, as their
rheological properties can change dramatically as a function of shear rate.
Below we first summarize the current understanding of shear-thinning and
shear-thickening, the two primary rheological phenomena relevant to the impact
of colloidal drops. We then highlight impact studies for each of these
regimes, and connect them to insights from rheology. We conclude by
elucidating the open problems in the field and highlighting the questions most
pertinent to enhance our understanding to how colloidal suspensions behave
under impact conditions.

Many colloidal suspensions show non-Newtonian behavior under flow, that is the
viscosity of these suspensions is not a material parameter, but depends on the
local stress (viscosity is constant for Newtonian fluids). Typically,
suspension viscosity is a function of both the concentration of colloidal
particles and the value of applied stress mewis2012colloidal ;
DennMortonSoftMatter . At low particle concentrations, a typical colloidal
suspension will behave in a similar manner to a Newtonian fluid (constant
viscosity), albeit with a higher viscosity than the suspending liquid, see the
$\phi$ = 0.09, 0.18 curves in Figure 2a. At moderate particle concentrations,
most colloidal suspensions exhibit shear-thinning, or a decrease in bulk
viscosity with increasing applied stress. Shear thinning liquids have a
relatively high viscosity, the ‘zero-shear viscosity’, that decreases with
increasing shear rates and then plateaus at the ‘infinite-shear viscosity’
value. This is illustrated for the $\phi$ = 0.28 and $\phi$ = 0.34 curves in
Figure 2a. The average of the zero-shear and infinite-shear viscosity, known
as the ‘consistency coefficient’, can be used to characterize the behavior of
these liquids. At high particle concentrations, shear thickening often
emerges; these suspensions show an increasing viscosity and hence a higher
resistance to flow as applied stress is increased, as illustrated by the
curves at $\phi$ = 0.43 and above in 2a. As shown in 2a, the same fluid can
exhibit both shear-thinning and thickening behavior depending on the value of
the applied stress; dense colloidal suspensions often exhibit shear thinning
at low shear stresses and then begin to shear thicken as the shear stress is
increased.

Elucidating the link connecting non-Newtonian rheological behavior to material
microstructure has long been a driving goal for the rheological community.
Much progress has been made, but this is a challenging problem and many open
questions remain mewis2012colloidal . A general consensus has emerged that
shear thinning is the result of the microscopic ordering of particles and the
decreasing contribution of entropic forces as the Peclét number increases
ChengShearThinning . However, the physical mechanism for shear thickening has
not been completely elucidated. There are two broad types of shear-thickening
behavior: continuous shear thickening (CST) is defined as a viscosity which
increases continuously with shear rate, while the sudden jump observed in
viscosity at high shear rates is known as discontinuous shear thickening
(DST). Dense particle suspensions often show a transition from CST to DST as
the volume fraction increases. While it has been proposed that the
hydrodynamic rearrangements of particles into clusters can potentially cause
weak shear thickening, there is an emerging consensus that DST arises due to
the transition from lubrication-mediated contacts to friction-mediated
contacts DennMortonSoftMatter ; WyartCates2014 ; Peters2016 . This is
additionally supported by the observation that DST is often a precursor to
‘shear jamming’, beyond which flow in the suspension is not possible without
fracture Brown_2014 .

###  III.1 Colloidal drop impact: shear-thinning regime

Despite the ubiquity of shear thinning fluids, there have been relatively few
studies focused on drop impact in shear-thinning fluids. Surprisingly, the
broad conclusion of these studies is that shear-thinning drops show a
qualitatively similar behavior to Newtonian fluids, and can be well-described
by models used in Newtonian impacts, provided the appropriate viscosity is
used Guemas ; German_2009 .

To study the effect of varying shear viscosity during impact, An and Lee
AnLee2012  compared the quantitative spreading behavior of impacting shear-
thinning (xanthan solutions) and Newtonian (water-glycerol) drops. They
assumed that the average viscosity in the drop oscillates between the fluid’s
zero-shear viscosity and infinite shear viscosity during spreading, receding,
and oscillating phases, as shown schematically in Figure 2b. During spreading,
receding, and subsequent oscillations (if any) of the impacted drop, the fluid
velocity reaches a maximum and goes down to zero, potentially in a cyclical
manner. As the shear rate is linearly proportional to the fluid velocity, it
must also oscillate over the impact process. In their study, An and Lee
AnLee2012  observed that the maximum spreading diameter of the shear thinning
liquid was qualitatively similar to that of Newtonian fluids, with a fluid
viscosity defined by an average of the infinite shear viscosity and the zero-
shear viscosity. Thus, by assuming an average shear viscosity for the shear-
thinning liquid, one can capture its spreading behaviour using an existing
Newtonian model SchellerBousfieldNewtonian .

Figure 2: Rheology and impact behavior of colloidal suspensions. (a) Viscosity
data for colloidal latex suspensions, measured as a function of applied shear
stress, for increasing volume fractions. With increasing shear stress, the
fluid shows shear thinning and the shear-thickening. The non-Newtonian
behavior is more pronounced for more concentrated suspensions. Adapted from
WagnerPhy . (b) Schematic of the hypothesized variation in viscosity during
the impact of shear-thinning drops. Adapted from AnLee2012 . (c) A drop of
dense colloidal suspension (49% by volume) before and after impact on a glass
substrate. After impact, a dimple forms at the top, indicating that the drop
is jammed at the bottom but still liquid-like at the top. Adapted from
AroraDriscoll2020 . (d) Snapshots of cornstarch and polystyrene suspension
drops after impacting on a solid surface. The drop height $h$ was observed to
be independent of impact velocity. Adapted from Boyer .

The impact dynamics of blood (a shear thinning fluid) are of special interest
due to its relevance in forensic analyses. De Goede et al. deGoede2018
studied the impact of blood on different smooth surfaces, and observed that
substrate wettability had little effect on the splashing threshold of blood;
this has ssimilarly been observed in Newtonian fluids Riboux2014 . As with
other studies of shear-thinning fluids, no qualitative differences between the
spreading dynamics of blood and Newtonian fluids were observed, provided the
appropriate viscosity is used.

In future works on shear-thinning drop impact, extensive use of techniques to
measure local stresses in an impacting drop is key. As many fluids exhibit
both shear thinning and shear thickening behaviors depending on the applied
shear rate, it would be interesting to explore drop impact in the range of
shear rates where some fluids transition from shear thinning to shear
thickening; these shear rates are easily experimentally accessible using an
impact height of a few meters. In this region, one might expect to observe
novel instabilities due to the spatio-temporal variation in shear rate. Direct
observations, or numerical work on the spatial variation of shear stresses
during all phases of drop impact could provide a necessary phenomenological
basis for understanding the flow of these liquids at high stresses.

###  III.2 Colloidal Drop impact: shear-thickening regime

Shear thickening fluids typically show an increasing viscosity with with
increasing shear stress, often transitioning to jamming (solid-like behavior)
at the highest stresses. Drop impact studies of such fluids allow one to
access regimes of large shear rates ($\mathcal{O}(10^{3})$ and greater) that
are challenging to access via current rheometery techniques without losing
fluid containment. Despite having great potential to expand our knowledge of
the high stress response of these materials, drop impact studies of shear
thickening fluids have been few and far between, likely due to the
experimental challenges inherent to working with dense colloidal suspensions.

As concentrated suspensions have a high effective viscosity, the drop’s
surface energy typically has a negligible effect on impact dynamics. Therefore
at high impact speeds, inertia and fluid viscosity are the dominate control
parameters. On impact, such dense suspension drops have been observed to
undergo jamming; partially or completely solidifying Boyer ; AroraDriscoll2020
, see Figure 2 c,d. Boyer et al. Boyer  observed that shear-thickening drops
(cornstarch and polystyrene suspensions, $d\sim 5$ to $20\mu m$, volume
fraction $\phi>0.33$) showed a maximum deformation that was independent of the
impact velocity, and the drops stayed frozen at the maximum deformation for
much longer than the characteristic impact timescale [Fig. 2(d)]. Another
study has reported that concentrated colloidal drops($\phi\sim 0.6$) impacted
over hydrophobic PTFE surfaces stayed jammed for days BERTOLA2015412 . This
suggests that the interaction between the drop and the surface plays a key
role in the long-timescale drop dynamics. Investigation of the scaling laws of
these spreading dynamics and comparison with other complex fluids is needed.
Similar experimental work on hydrophilic surfaces exploring the relaxation
time of jammed drops over long timescales would be of great interest, as it
would provide a unique platform to explore the physics of unjamming.

Though current studies of shear-thickening fluids are sparse, they have
clearly identified novel behaviors, such as partial/complete solidification
upon impact. To use drop impact as platform to study stress-induced jamming,
this work should be expanded, with special attention paid to suspensions in
both the CST and DST limits. To understand the physics of localized shear
jamming, systematic measurements of shear stresses in the impacted drop are
crucial. Overall shear stresses in impacting drops can be estimated and
measured, but localized information is quite challenging obtain
experimentally. This information is key to understanding the observed partial
jamming processes, and development of new experimental techniques to obtain
this information should be prioritized. Promising avenues for obtaining these
measurements include localized measurements of boundary stresses (akin to
traction force microscopy measurements style2014traction ) and the development
of techniques for visualization suspension microstructure with high temporal
resolution.

Another avenue ripe for exploration is the role of particle shape. It is well-
known from bulk rheological measurements that the critical volume fraction for
jamming varies with particle shape. In particular, elongated (large aspect
ratio) particles can exhibit jamming behavior at dramatically lower volume
fractions than their spherical counterparts williams2003 ; james2019c .
However, the role of particle shape on drop impact outcomes has thus far been
unexplored. These studies would have both the potential to probe the role of
shape asymmetry in the high-stress behavior of suspensions as well as to aid
in connection to industrial processes, where the component particles in
slurries and suspensions are often far from spherical.

##  IV Granular Suspensions

Granular suspensions, composed of large (here greater than $\sim$ 100 microns)
particles behave quite differently upon impact than colloidal suspensions. In
these suspensions, inter-particle interactions are typically negligible, and
individual particle inertia can play a dominant role in the physics of impact.
As we will show, especially at high impact velocities, these systems are
better understood in a particle-based manner, rather than in the bulk
rheological context used to understand the impact of colloidal suspension
drops.

Figure 3: Impact of granular suspensions (a) Comparison of impact of a 26 mm
granular drop (left) and a 3.6 mm water drop (right) onto a hydrophobic glass
plate. Adapted from Marston . (b) granular drop impact can be divided into
splashing and non-splashing regimes with respect to the particle-based Weber
number and the Stokes number of the suspending liquid. Adapted from
Schaarsberg . (c) Radial distribution of particles in the spread droplets is
uniform for low Reynolds number of surrounding fluid, and annular for high
$Re$. Adapted from nicolas_2005 .

In the low- and moderate- volume fraction range (approximately $\phi<0.60$,
though this value is set by the details of the particles used), the drop
impact of granular suspensions can be studied in terms of the dimensionless
parameters typically used for Newtonian and colloidal drop impact
($We=\frac{\rho U^{2}d_{o}}{\sigma}$, $Re=\frac{\rho Ud_{0}}{\mu}$, and
$Re_{\textrm{eff}}=\frac{\rho Ud_{o}}{\mu_{\textrm{eff}}}$). However, for very
dense suspension drops impacting at high $We$, granular particles can overcome
the surface forces of the surrounding fluids during impact and individually
escape. Owing to this, the impact dynamics of granular suspensions can show
features that are qualitatively different from that of Newtonian and colloidal
fluids [Fig. 3(a)]. In this regime, spreading and splashing are best
characterized using particle-based dimensionless parameters. We discuss these
developments in further detail below.

###  IV.1 Spreading and receding dynamics

Multiple studies have investigated the spreading dynamics of granular drops at
a range of volume fractions jorgensen2020deformation ; Ok ; nicolas_2005 ;
Boyer ; Raux ; Marston ; Grishaev ; GRISHAEVdist . The maximum drop diameter
$d_{max}$ has been observed to decrease with increasing particle volume
fraction, but was not found to be significantly affected by individual
particle size Grishaev . The drop diameter $d$ during spreading has been shown
to grow as the square root of time, in a manner identical to the spreading of
Newtonian drops Marston  (Figure 3(a)) Marston . For low and moderate volume
fractions (significantly lower than the jamming limit), spreading dynamics has
been quantitatively understood in terms of the steady effective viscosity of
the suspension. Theoretically, the bulk effective viscosity is an extension of
the Einstein viscosity beyond the linear term krieger1959mechanism , though,
in practice the experimental value inferred from the rheological data is often
used. As effective viscosity grows with particle concentration, $d_{max}$
after impact decreases. Jorgensen et al. jorgensen2020deformation  used
rheological measurements for effective viscosity for particle sizes larger
than $145$ $\mu m$, and compared their findings with the spreading of
Newtonian drops of similar viscosity values. The agreement between the two
fluids, for volume fractions up to $0.50$, verifies that the spreading
dynamics for granular suspensions is primarily controlled by the bulk
effective viscosity. In contrast, the maximum apex height of granular
suspension drops has been observed to scale differently with time as compared
to Newtonian fluids. Thus, even at volume fractions below jamming, the
particle based nature of granular suspensions must be considered to understand
the physics of impact.

Studies of the retracting phase have established that, akin to Newtonian
droplets, it is controlled by substrate properties and the viscosity of the
suspending liquid. Several studies have shown that the dynamics of the
retracting phase are the largest factor in determining the final particle
distribution in the impacted drop. Nicolas et al. explored the steady-state
particle distribution as a function of fluid $Re$, and found that long after
impact, the particles were concentrated in an annular region for a high-$Re$
impacts, while they were uniformly distributed for low-$Re$ impacts [Fig.
3(c)]. At low fluid $Re$, spreading/receding oscillations of the drop diameter
are suppressed; this suppression accounts for the $Re$ dependence of the
steady-state particle distribution nicolas_2005 . Additional studies have
shown that substrate wettability can be used as an additional mechanism for
controlling the post-impact particle distributions GRISHAEVdist .

As with Newtonian impacts, target-based experiments allow for minimization of
viscous stresses in the spreading process. Using this technique, Raux et al.
found that the $d_{max}$ and $t_{max}$ were independent of particle size
(varied from $40$ $\mu$m to $140$ $\mu$m Raux ), indicating that even on small
targets, particles seem to only affect the drop spreading dynamics through the
bulk effective viscosity. However, the retraction phase was slowed down by the
presence of particles. Larger particles have been observed to make the film
unstable during this retraction phase, leading to rupture and a decrease of
the film lifetime.

Close to the critical volume fraction $\phi_{c}$ for jamming, this effective
viscosity framework no longer holds. Lubbers et al. Lubbers  studied the high-
inertia impact of dense suspensions ($d=250\mu m,\phi>0.60$) that resulted in
a particle monolayer. The growth of the monolayer diameter was observed to be
different than the spread of a liquid drop on a surface. They explained this
high-$\phi$, inertia-dominated phenomenon in terms of a particle-based Weber
number $We_{p}$ and liquid Stokes number $St$ ($We_{p}\gg 1$, $St\gg 1$ in
this study). The authors proposed a particle-based chain model and a viscous
drag model to characterize the spreading; this emphasizes the quite different
physics of granular impacts vs. colloidal impacts — in granular suspensions,
particle inertia plays a dominant role.

In summary, as with colloidal suspensions, the impact of low- and moderate-
volume fraction drops can be understood using the bulk effective viscosity of
the suspension. However, for high volume fraction drops, the liquid merely
acts as an agent that binds the particles into a drop; individual particle
inertia plays a significant role in the impact dynamics. Furthermore, the
particle-based parameters that describe spreading can also be used to
characterize the splashing threshold for granular suspensions, as described
further in the next section.

###  IV.2 Splashing threshold

Just as with Newtonian drops, granular suspensions can also splash under the
right conditions. Peters et al. Peters  studied the splashing threshold of
dense granular suspensions ($0.59<\phi<0.65$) of particle sizes greater than
$80$ $\mu m$. ‘Splashing’ in this case comprised of individual particles
ejected from the edge of the drop. They found that this splashing threshold
was best characterized in terms of the particle-based Weber number
$We_{p}=\frac{\rho_{p}r_{p}U^{2}}{\sigma}$, rather than the typical fluid
$We$. This suggests that splashing occurs when an individual particle can
overcome the surface energy of the surrounding liquid and escape. As opposed
to Newtonian splashing, they observed the onset of splashing for these
suspensions (at $We_{p}\geq 14$) was independent of the substrate wettability
and roughness.

This $We_{p}$ dependence predicts that larger and denser particles splash at
lower drop velocities due to their high inertia, and it has been verified in
subsequent studies Schaarsberg ; Grishaev . On the other hand, for bimodal
suspensions comprised of two particle sizes, Peters et al. Peters  found that
smaller particles were ejected more easily than larger particles. Although
seemingly contradictory to the $We_{p}$ based predictions, this can be
understood by considering the distribution of velocities in particle-particle
collisions. Due to momentum conservation, smaller particles end up with higher
velocities than the larger ones, enabling them to be ejected earlier. The
bigger picture that emerges is that the splash of dense granular suspension
drops is governed by the collisions of particles with the surface and with
each other, the liquid primarily providing the surface energy to bind the drop
together. Consequently, Schaarsberg et al. Schaarsberg  experimentally
investigated the effect of liquid viscosity on dense ($\phi=0.59$) granular
splashing, showing that the phase space defined by $We_{p}$ and $St$, the
Stokes number, is cleanly divided into splashing and non-splashing regimes
[Fig. 3(b)]. Thus, it seems that the interaction of particle inertia and
viscous interactions between individual particles and the suspending liquid
control the physics of the splashing threshold, as opposed to the bulk
rheological properties of the suspension that govern the spreading dynamics.

Marston et al. Marston  tracked individual particles ejected after splashing
(grain size $\sim 350\mu m$) and found that the maximum particle velocity was
typically twice that of the drop velocity, a much smaller ratio as compared to
droplets ejected during Newtonian splashing. Their image analysis suggested
after touching the substrate, the drop acted as a compressible fluid and
particles rearranged. They observed that particles were ejected when the
volume fraction was close to jamming; they primarily used the observed area of
the drop for this analysis, assuming an axisymmetric drop. For a more
comprehensive understanding of this correlation, the elastic energy of the
jammed network needs to be taken into account, as near the jamming
concentration long-range correlations among the particle structure are likely
to be significant.

As with colloidal suspensions, it would be interesting to explore the effect
of anisotropy in particle shape on the spreading and splashing dynamics in
impacting drops. Particle shape not only affects the critical volume fraction
$\phi_{c}$ williams2003 ; james2019c , but the realignment of particles due to
shear can also affect bulk flow properties of the suspension. Additionally, in
the dense suspension limit where the bulk viscosity framework is inadequate,
using particles that are index-matched with the liquid phase and introducing a
few tracer particles could be a way to visualize the rearrangements within the
drop during impact. This would allow one to obtain highly spatially resolved
data for $\phi$, and enable an exploration of drop microstructure during the
impact process.

##  V Polymeric fluids

Polymer additives are used in a large variety of industrial applications, both
to achieve desired fluid properties and to control the interaction of fluids
with solid surfaces. The two most convenient parameters to experimentally
study polymeric fluids are polymer concentration and the molecular weight (and
in turn, chain length) of the polymer. Changing either of these quantities can
affect multiple rheological properties of the fluid, for example the yield-
stress, the shear-thinning coefficient, and the zero-shear viscosity.
Polymeric liquids exhibit a range of non-Newtonian behaviors; thoroughly
characterizing their rheology is far beyond the scope of this review. Below we
outline the rheological properties that have thus far been invoked to describe
the physics of polymeric drop impact.

Many polymeric fluids exhibit a ‘yield stress’, a critical stress at which a
fluid transitions from solid-like behavior to liquid-like flow properties
BARNES1999133 ; Balmforth2014 ; COUSSOT201431 . Although this framework is
simplified, the concept of a yield-stress is still useful to describe
materials that exhibit a dramatic change in flow behavior at a critical shear
stress. The simplest model for a yield-stress fluid is the two-parameter
Bingham model Bingham1916 ; Bingham1922 ; coussot2017bingham , which features
only a yield stress and an infinite-shear viscosity. The most commonly used
model is the 3-parameter Herschel-Bulkley model Herschel1926 , which adds
power-law shear-thinning, better capturing measured flow curves.

The rheological properties of a polymeric fluid under flow are connected to
the deformation of individual polymers; this stretching has the most
pronounced effect in elongational/extensional flows. Thus, rather than a shear
viscosity, an elongational viscosity is more relevant for characterizing
polymeric flows extensional_PETRIE20061 . As the spreading phase of an
impacting drops is largely an extensional flow, this viscosity coefficient is
often used to characterize impact processes. Polymeric materials often exhibit
viscoelastic behavior which depends on the applied strain rate; these
materials exhibit both elastic and viscous behavior when they are deformed, so
that their response to applied strain is rate-dependant. Furthermore,
polymeric rheology can change with time; many polymeric fluids are known to be
strongly thixotropic, that is their material properties (for example,
viscosity) change as the material ages due to slow restructuring processes
thixotropy_BARNES19971 ; MEWISthixotropy ; ThixotropyLarson . Below, we will
first outline the experimental results in polymeric drop impact, highlighting
connections between material rheology and drop impact outcomes. We will then
describe numerical work in this field, concluding with suggestions for future
directions.

###  V.1 Experimental results

In coating, spraying, and agricultural pesticide industries, suppressing
bounce and splash is important for the efficient use of chemicals. Bergeron et
al. Bergeron2000  observed that a small amount of polymer additives suppressed
the bounce of liquid droplets impacting on hydrophobic surfaces; this result
was observed to hold for a variety of hydrophobic substrates. Bounce
suppression due to polymer additives has since been confirmed by many other
studies Bartolo2007 ; SmithBertola2010 ; Zang2013 ; CrooksBoger2000 ;
CrooksCooperWhite2001 , and it is has been found to be more pronounced for
higher molecular weight polymers CrooksCooperWhite2001 . Rheological data has
suggested that the fluid shear viscosity was not affected by the presence of
these additives, hence bounce suppression was attributed to the rise in
elongational viscosity Bergeron2000 ; CrooksBoger2000 . Later studies have
disputed this point, arguing that in addition to bounce, both spreading and
receding should be affected by a higher elongational viscosity Bartolo2007 .
Experimental data largely suggests that the spreading phase is unaffected by
the presence of small concentration of polymer additives, but the retraction
is slowed down relative to the Newtonian case. However, in highly concentrated
solutions, suppression in both spreading and receding phases has been observed
GelledFuel . Recent work suggests that, as with Newtonian fluids, contact line
dynamics are crucial to understanding flow modifications due to polymer
additives. Smith and Bertola SmithBertola2010  performed Particle Image
Velocimetry with fluorescent particles, and observed that the contact line
velocity was lower in drops with polymer additives. As a result, they
suggested that dissipation at the contact line could be responsible for
suppressing bounce. While these measurements have the potential to explain the
mechanism of the bounce suppression phenomena, further study is needed. In
particular, studies which focus on altering contact line dissipation through
surface modifications would shed light on this problem.

As with Newtonian fluids, the interaction of the impacting drop with the
surface can be minimized and hence decoupled from fluid processes via
experiments on small targets. Rozhkov et al. Rozhkov2003  observed that the
drop spread normalized with initial drop diameter behaved similarly over time
for water and polymer solutions on small targets. Slower retraction of polymer
drops in other experiments, therefore, has been conjectured to be due to
fluid-surface interaction, in line with the contact-line measurements made by
Smith and Bertola. Additionally, the rupture of ejected films and length of
protruding fingers was suppressed by polymer additives [Fig. 4(c)]. This is
consistent with the stabilizing effect of polymers on splash and bounce
observed in other studies.

While many studies in polymeric fluids have focused on viscous behavior and
dissipation effects, fluid elasticity has been shown to play an important role
in the impact process. Luu and Forterre luu_forterre_2009 ; LuuForterre2013
observed that carbopol (a yield-stress fluid) drops spread to a much larger
extent on rough substrates; the spread was found to be even larger on
hydrophobic surfaces. Their data collapsed well when represented in terms of
the elastic Mach number ($M=\frac{U}{\sqrt{G/\rho}}$, where $U$ is the fluid
velocity and $\sqrt{G/\rho}$ is the elastic velocity of the fluid), indicating
that the fluid elasticity is the dominant mechanism for determining the impact
outcome [Fig. 4(a)].

Figure 4: Impact of polymer drops. (a) Spreading factor of polymeric drops on
partially wetting surfaces depends on the elastic Mach number, as predicted by
Luu and Forterre luu_forterre_2009 . Adapted from Guemas . (b) Phase diagram
of the impact behaviors of polymeric drops impacting on a wet surface, as a
function of four parameters: impact velocity, polymer concentration, drop
diameter, and thickness of the liquid layer on the substrate. Adapted from
Blackwell2015 . (c) Timelapse of a polymeric drop after impact on a small
target. Adapted from Rozhkov2003 .

Even though a number of experiments have focused on impacting drops on dry
surfaces, drops often impact on an already wet surface in industrial
processes. Blackwell et al. Blackwell2015  studied such a system of carbopol
in water, where the surface was already coated with the same liquid. Splashing
was suppressed for higher concentrations of carbopol, in line with with other
studies using polymer additives. Additionally, splash suppression also
occurred when the thickness of the liquid layer was increased; Blackwell et
al. identified the ratio of coating thickness and drop diameter as a key
dimensionless factor to demarcate the splashing regime. Another relevant
parameter was the ratio of inertial and dissipative stresses, including yield
stress and viscosity. In the space of these dimensionless parameters, the data
was separated into sticking and splashing regimes [Fig. 4(b)]. In a further
study, Sen et al. sen_morales_ewoldt_2020  investigated the effect of the
stronger thixotropic aging present in Laponite as compared to Carbopol,
providing further evidence that the impact of these fluids could still be
characterized using the same dimensionless groups. Even though these
dimensionless parameters were useful to interpret the experimental data in
this case, they would have to be modified for systems where surface tension
and elasticity effects are not negligible. More experimental data spanning a
larger range of coating thicknesses is also needed to obtain a complete
picture of polymeric drop impact on wet surfaces.

In summary, the presence of polymer additives has a stabilizing effect on
impacting drops, that manifests through suppression of bounce and splash.
Studies on wet surfaces have also shown that liquid layers on substrates
enhance this stabilizing effect. Many mechanisms have been suggested for
bounce suppression in low-concentration polymeric drops, but a strong
consensus is yet to emerge. Experiments that incorporate direct measurements
of shear, elongational, and normal stresses during spreading would greatly
enhance our understanding of polymeric fluid impact. These measurements would
be quite challenging to do in practice, and thus numerical work has been
useful tool to understand these processes, as we will discuss in detail below.

###  V.2 Numerical studies

In experiments involving polymeric drop impact, the natural choice for the
independent parameter is the concentration of polymer or particulate
additives. The fluid’s yield stress increases with increasing polymer
concentration, but so does the effective viscosity and rate of shear thinning.
This makes it challenging to decouple the effects of yield stress and shear
rheology. Forming spherical drops of fluids with high polymer concentration is
also an experimental challenge. Thus, numerical studies are are crucial to
eliminate artifacts introduced by a non-spherical initial geometry and
understand the effects on yield stress and flow rheology separately.

Kim and Baek KIM201262  performed a numerical study of impacting yield-stress
drops; the fluid rheology was implemented with the Herschel-Bulkley model
Herschel1926 . They separately varied the yield-stress, viscosity, shear-
thinning rate, and surface tension parameters to study the effects on the
spreading and retraction phases. They observed that the spreading phase was
mostly dominated by inertia and effective viscosity, but was not affected by
the yield-stress or surface tension. Retraction, on the other hand, was
inhibited by both yield stress and capillarity. Experimental works indicating
that higher polymer concentration affects both spreading and retraction are
consistent with this data, though a quantitative agreement in
spreading/retraction rates was not found. Failure to get a quantitative
agreement is suspected to be due to the assumptions about dynamic contact
angles that were an input to the calculations, which, just as with Newtonian
fluids, vary during the spreading and retracting phases. The authors observed
locally varying viscosity values in different parts of the drop after impact,
which might be a direct result of local variation in fluid velocity, and in
turn, shear stresses.

Recently, Oishi et al. oishi_thompson_martins_2019  investigated both normal
and oblique impacts numerically, also taking into account thixotropic effects.
Thixotropy was modeled by introducing a delay between applied stresses and the
resulting structural changes in the fluid; their numerical data qualitatively
matches the experimental results of Luu and Forterre luu_forterre_2009 . Oishi
et al. treated the thixotropic timescale as a fitting parameter for their
numerical data for carbopol and kaolin droplets. Their study considered the
phase space of elastic Ohnesorge number $Oh_{e}$, and a parameter $Y$
consisting of the elastic Mach number, the plastic number, and the thixotropic
timescale. This phase space was clearly divided between the sticking,
bouncing, and rolling (in case of inclined planes) behaviors.

Existing numerical studies have not only enabled us to study the effect of
fluid parameters separately, they additionally shed light on the dynamics of
drop impact on a microscopic scale, such as varying structural parameters and
viscosities across the spreading/retracting drop. More numerical studies,
especially those focused on wet or thinly coated surfaces, would be helpful to
bridge experimental data with physical insights of how macroscopic polymeric
drop impact behaviors (sticking, bouncing, etc.) are coupled to the
interactions between the drop and the substrate. To understand their
microscopic origins, fitting parameters such as the thixotropic timescale
during impact need more theoretical attention. The suggested mechanisms behind
experimentally observed bounce suppression should be explicitly tested in
future numerical work. Additionally, the link between non-uniform stresses in
the drop and spreading/receding behaviors is likely best explored in a
numerical context.

##  VI Outlook

In this review, we have summarized current work on the drop impact of complex
fluids on solid substrates. Drop impact studies at high velocities enable us
to study fluid properties at very high stresses, often beyond the range of
typical rheometers, and in the absence of boundary conditions. This work thus
has strong potential to enhance not only our understanding of drop impact, but
additionally to lead to a significant expansion in our understanding of the
rheology of complex fluids under high stresses.

This is a relatively young field, and the studies so far have followed very
disparate avenues. Nevertheless, we have attempted to classify results based
on different material compositions. The impact of colloidal suspensions that
show shear-thinning and shear-thickening behavior can in many cases be
characterized in terms of bulk effective viscosity of the fluid, as long as
the concentration is not too close to the critical volume fraction
($\phi_{c}$) for jamming. Shear thickening fluids have also exhibited
fascinating behaviour such as dilatancy of the particle phase (in granular
suspensions) and partially-jammed states after impact with high velocities (in
colloidal suspensions). Dense granular suspensions have been observed to
splash in a fundamentally different manner than Newtonian fluids; their splash
is controlled by the particle-based Weber number and the Stokes number with
the surrounding fluid. Polymer additives seem to suppress bounce of impacting
drops, the effect being more pronounced for both higher molecular weight and
higher concentration of polymers. The physical origin of this bounce
suppression is an active field of investigation.

The experimental studies so far have shown a wide range of behaviors of
impacting complex fluid drops, and scaling laws have been fairly successful in
predicting fluid behaviors in specific regimes. However, a unified description
correlating the available rheological data with impact experiments is still
far out of our reach. More numerical and theoretical studies are necessary to
understand the role of each fluid property in a decoupled way. Additionally,
more studies with local measurements of fluid velocities and stresses after
impact need are needed in order to understand the microscopic origins of the
rich array of observed fluid behaviors. Here, we outline the key theoretical
challenges and suggest a number of avenues for future experimental studies
that will take us closer to a more holistic description of complex fluid drop
impact.

###  VI.1 Theoretical and computational challenges

Capturing processes which occur on over a large dynamic range in both length
and time is the biggest challenge in modeling Newtonian drop impact dynamics.
This problem is only augmented further in case of complex fluids, since
particle size and the interaction timescales of the solid and liquid phases
compete with the already existing scales such as drop size and surface
roughness. Many dimensionless parameters, for Newtonian as well as complex
fluids, have been identified to characterize drop impact dynamics. Thus far,
just as with Newtonian drop impact, these parameters have only been effective
to characterize behavior in specific impact regimes; a generalized picture
still eludes us. Given ever-improving computational capabilities, as well as
the renewed trend in parallel computing, more detailed simulation of drop
impact processes, guided by existing models are on the horizon.

Our understanding of the behavior of impacting complex fluid drops is
inevitably built on the current understanding of the rheology of these fluids.
Although a large amount of rheological data is available and new data is being
added over a wide range of timescales and shear rates, developing constitutive
models for rheological behavior is a highly active avenue of research
StickelRheology . Thus, the empirical nature of much of our current rheology
knowledge is another significant roadblock in developing theoretical
frameworks for complex fluid drop impact. However, we hope that more
systematic drop impact experiments could potentially inform the work on
constitutive models in rheology, allowing the two disciplines to co-evolve in
the near future.

###  VI.2 Open questions and perspectives

The directions followed in the study of impacting complex fluids have so far
been very disparate. Future experiments in the field can draw from our
knowledge of Newtonian impact, as many complex fluids in the dilute limit can
be approximated as Newtonian. While some experiments have looked into the
spreading and receding phases, more systematic studies spanning a range of
additive concentrations and Weber numbers need to be performed. The hope is
that a large amount of systematic experimental data combined with detailed
numerics would give rise to more theoretical models and universal scaling
arguments, in turn providing us with novel ways to control droplet deposition,
bounce, and splashing. Experiments with additive concentrations in the highly
non-Newtonian regime might uncover additional exotic instabilities and help
connect impact behavior with rheological findings.

Shear thickening fluids have recently showed some fascinating behavior, such
as localized or complete jamming, when impacted on a surface AroraDriscoll2020
; Boyer ; jorgensen2020deformation . In almost all studies on colloidal and
granular drops, spherical particles have been used, despite the fact that
anisotropic particle suspensions are much more common in industrial processes.
Exploring the effect of particle shape on impact dynamics as well as the
jamming threshold would contribute to our understanding of jamming in complex
fluids. Furthermore, the use of anisotropic particles adds another parameter
to the microscopic dynamics inside the drop, namely the particle orientation.
Rheological studies of non-spherical particle suspensions have indicated that
they jam at a much lower volume fraction williams2003 ; james2019c .
Statistical studies of the rearrangement of rods inside the impacting drop
could enhance our knowledge about the interaction between the solid and liquid
phases.

While there is some data available about the effect of substrate properties on
the drop impact of complex fluids, more systematic studies are necessary to
form a full picture. In particular, the effect of surface roughness, both
patterned and random, deserves more attention, as it is a pragmatic way of
controlling droplet splashing and deposition. The effect on the advancing and
receding contact angles and of droplet properties such as particle shape,
size, and effective surface tension also deserves further attention, as it
will also guide computational studies in the right direction. The effect of
competing lengthscales of surface roughness and particle sizes on effective
contact angles is a further level of complexity which needs to be examined.

Though much of the work in this field is dominated by the study of drop impact
on dry substrates, impact on wet surfaces is becoming the focus of more
studies. Impact on a surface which is already wet with a coating of the
complex fluid is a more realistic scenario in industrial spraying and coating
processes. A larger range of the liquid layer thicknesses should be explored,
as the physics of complex drop impact could vary substantially with the ratio
of liquid layer thickness and drop size.

While the variety of competing lengthscales and timescales enriches the drop
impact problem, it also makes it extremely challenging. The availability of a
number of microscopy techniques, and advances in high-speed imaging and data
storage capacity make this an exciting time to attack this problem. There have
already been some soft matter studies that combine these tools; for example
refractive-index matching, fluorescent or dyed particle tagging, and particle-
image velocimetry further enrich our ability to extract more localized
information from drop impact studies. The time is ripe for widespread use of
these techniques to obtain more spatio-temporally resolved data, that in turn
will provide direction to further theoretical and computational studies.

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