# Coexistence of solid and liquid phases in shear jammed colloidal drops

P. Shah1,† S. Arora1,† M.M. Driscoll 1,∗

Department of Physics and Astronomy, Northwestern University

These authors contributed equally to this work.

corresponding author

## Abstract

Complex fluids exhibit a variety of exotic flow behaviours under high
stresses, such as shear thickening and shear jamming. Rheology is a powerful
tool to characterise these flow behaviours over the bulk of the fluid.
However, this technique is limited in its ability to probe fluid behaviour in
a spatially resolved way. Here, we utilise high-speed imaging and the free-
surface geometry in drop impact to study the flow of colloidal suspensions. We
report, for the first time, observations of coexisting solid and liquid phases
due to shear jamming caused by impact. In addition to observing Newtonian-like
spreading and bulk shear jamming, we observe the transition between these
regimes in the form of localised patches of jammed suspension in the spreading
drop. We capture shear jamming as it occurs via a solidification front
travelling from the impact point, and show that the speed of this front is set
by how far the impact conditions are beyond the shear thickening transition.

##  1 Introduction

Complex fluids, such as particulate suspensions[1, 2] and polymer
solutions[3], exhibit a variety of exotic flow behaviours, for instance shear
thickening and solidification via jamming. These behaviours are particularly
relevant to development of smart materials, such as body armours[4] and soft
robots[5]. Rheometry is traditionally used to characterise complex fluids.
However, this technique typically provides measurements averaged over the bulk
of the fluid and obscures the information on local variations in flow. The
free-surface geometry in drop impact systems offers a unique lens to probe
these flow properties, as it provides data on manifestations of non-Newtonian
flow with high spatial and temporal resolution. Here, we use high-speed
imaging to study the drop impact of colloidal suspensions over a large range
of volume fractions and impact velocities, thus sampling impact behaviour from
liquid-like spreading to solid-like jamming. Combined with input from
rheological data, our measurements offer a more holistic picture of complex
fluid flow, especially under dynamic conditions.

An extensive understanding has been developed for the dynamics of a Newtonian
fluid drop impacting a dry solid substrate[6, 7, 8, 9, 10, 11]. However, the
vastly different flow properties of complex fluids substantially modify impact
dynamics. Past studies have explored the spreading and splashing of a variety
of polymeric fluids[12, 13] and particulate suspensions[14, 15, 16, 17, 18,
19]; however each has largely focused on a relatively narrow slice of the vast
parameter space. The role of particulate additives in controlling the
splashing transition has been explored[14, 15], as well as the spreading and
jamming of dense suspension drops[16, 17, 18, 19]. In particular, experiments
on impacting shear thickening fluids have reported solid-like states after
impact[16, 17, 18]. Colloidal suspensions offer a convenient control parameter
— volume fraction — to scan suspension behaviours ranging from Newtonian-like
to shear thickening. Here, we report a systematic study of colloidal
suspension impact spanning a range of volume fractions ($0.09\leq\phi\leq
0.50$) and impact velocities ($0.7$ $m/s\leq u_{0}\leq 4.0$ $m/s$). Our
exploration of this wide parameter space allows us to capture not only
spreading and bulk solidification, but also the transition between these
drastically different flow regimes.

Shear thickening — an increase in viscosity with increasing shear — is one of
the most counterintuitive phenomena exhibited by dense suspensions[1, 2]. The
Peclet number, $Pe=\frac{\text{shear rate}}{\text{rate of diffusion}}$, is a
dimensionless number used to quantify high shear rates, and we expect the
onset of shear thickening at $Pe\gg 1$. The transition to shear thickening
occurs when a suspension with relatively high $\phi$ is subjected to a shear
higher than a critical value[1, 20, 21, 22]. We note that for our experimental
parameters, $Pe>10^{2}$, and thus we expect shear thickening after impact in
the high-$\phi$ limit. Many rheological studies have focused on elucidating
the mechanism of shear thickening, and both lubrication hydrodynamics and
particle interactions have been shown to play a role. Shear thickening has
been proposed as precursor to shear jamming, and the nature of this transition
is an active field of study. For a more detailed discussion, we refer the
reader to the following reviews[2, 20, 21, 22].

For our experiments, we synthesize charge-stabilized silica spheres (diameter
830 $\pm$ 20 nm, Fig. 1a) using the Stöber process[23, 24] and suspend them in
water. Spherical drops of diameter $d_{0}=3.0\pm 0.1$ mm are formed by drawing
a known volume of fluid ($15$ µL) into a micropipette. We set the impact
velocity by changing the height from which the drops are released, and record
the drops impacting on a dry, hydrophilic glass substrate using a high-speed
camera. To minimize the effects of particle sedimentation, all samples are re-
suspended immediately before the experiments using a vortex mixer. All
experiments are performed in a humidity chamber, which additionally mitigates
air currents (see Methods for details).

To connect impact behaviours with rheological properties, a mapping between
impact velocity and rheological parameters such as shear rate or shear stress
is necessary. Precisely quantifying shear rates in drop impact systems is
challenging due to the nonuniformity of shear in both space and time. However,
a simple dimensional argument can be used to estimate the shear rate at
impact. At the instant of impact, the bottom point of the drop comes to rest,
while the apex continues to fall at the impact velocity $u_{0}$, as the shear
caused by impact has not had time to propagate across the drop. Dividing this
difference in speeds, $u_{0}$, by the drop size $d_{0}$ thus provides an
estimate of the maximum shear rate at the moment of impact:
$\dot{\gamma}_{impact}=u_{0}/d_{0}$. With the drop size of $3$ mm, we could
access shear rates in the range $233$ $s^{-1}\leq\dot{\gamma}_{impact}\leq
1333$ $s^{-1}$. Thus, we are able to span a large spectrum of flow behaviours
in these suspensions, and observe how non-Newtonian flow gives rise to a rich
variety of impact phenomena. The results we present here take us closer to an
understanding of the shear jamming transition and the properties of shear
jammed solids.

##  2 Results

Bulk rheometry measurements [Fig. 1b] demonstrate the variety of flow
behaviours exhibited by our suspensions. At low $\phi$ (black and pink lines),
the fluid viscosity is constant, akin to a Newtonian fluid. Shear thinning
(decreasing viscosity) becomes pronounced as $\phi$ is increased (green and
purple curves), and shear thickening (indicated by increasing viscosity)
appears for $\phi\geq$ 0.47 at high shear stresses (shear stress above 100 Pa,
orange, blue, and red curves). We observe fascinating consequences of this
non-Newtonian rheology in our impact experiments. At $\phi=0.47$, where weak
shear thickening appears at high stresses in bulk rheology, we observe patches
of localised solidification during spreading — panel 3 of Fig. 2a shows small
solid-like bumps that protrude from the spreading drop, but vanish in panels 4
and 5 [SI video 1[25]]. At higher $\phi$, we observe partial solidification of
the drop — Panel 2 in Fig. 2b shows that the bottom part of the drop acts as a
solid, while the top part remains fluid and flows over the solidified region
throughout panels 3-5 [SI video 2[25]]. Finally, at $\phi=0.49$ and high
impact velocities, most of drop solidifies as shown in Fig. 2c [SI video
3[25]]. Here, we show that this variety of solidification behaviours is a
direct consequence of shear jamming[26], evidenced by their occurrence much
below the static jamming threshold[27].

We encapsulate this broad range of impact outcomes in a $\phi-u_{0}$ state
diagram [Fig. 3]. Green circles, indicating simple spreading [SI video 4[25]],
dominate the low $\phi$ and low $u_{0}$ region. With increasing $\phi$ or
$u_{0}$, the localised solidification regime appears (orange diamonds),
followed by the bulk solidification regime (blue triangles), where a larger
and larger portion of the drop solidifies upon impact. The transition between
these regimes is a function of both $\phi$ and $u_{0}$, as all regimes can be
accessed by varying either of the parameters while keeping the other constant.
Additionally, we find that the drop behaviour is very sensitive to small
changes in $\phi$, consistent with the transition to shear thickening in
rheological measurements [Fig. 1b].

To quantify this range of impact outcomes, we compute the normalized maximum
diameter of the impacted drops, $\beta=d_{max}/d_{0}$, and plot this metric
against $u_{0}$ [Fig. 4a]. For $\phi\leq$ 0.47, $\beta$ increases with
increasing impact velocity. However, $\beta$ drops to 1 at $\phi\geq$ 0.49 and
high impact velocities. This is because the drop no longer spreads after
impact (lower inset). This result is consistent with recent studies that
observed similar solidification in suspension impact at high $\phi$[16, 18].
Our drops remain solid for a few milliseconds; however, they spread like a
liquid over the timescale of a second [SI video 5[25]]. Thus, the solid-like
state we observe is transient in nature, further evidence that this
solidification is a direct result of shear jamming. A recent result suggests
that the substrate wettability affects this timescale of unjamming[18], but
this problem remains largely unexplored.

At $\phi\leq$ 0.47, the drops spread in a manner qualitatively similar to
Newtonian fluids [SI video 4[25]]. Previous experiments[28] with Newtonian
fluids have shown that $\beta$ scales as the dimensionless parameter
$ReWe^{1/2}$, where $We$ is the impact Weber number, $\rho
u_{0}^{2}d_{0}/\sigma$, and $Re$ is the Reynolds number, $Re=\rho
u_{0}d_{0}/\eta$. Here, $\sigma$ is the surface tension of the suspending
fluid (for this case water, $\sigma=72$ mN/m), $\rho$ is the fluid density,
calculated as:

$\rho=\rho_{silica}\phi+\rho_{water}(1-\phi)$ (1)

with $\rho_{silica}=2000$ kg/m3 and $\rho_{water}=1000$ kg/m3, and $\eta$ is
the suspension viscosity. For impacting colloidal drops, the calculation for
the Weber number remains identical to Newtonian fluids. Estimating the
Reynolds number, however, is less straightforward due to the non-constant
fluid viscosity of complex fluids. Immediately after impact, the shear rate
experienced by the drop is at its maximum
($\dot{\gamma}_{impact}=u_{0}/d_{0}$). As the fluid spreads and slows down,
the shear rate continuously drops to zero. We use the average of these two
extremes, $\dot{\gamma}_{avg}=u_{0}/2d_{0}$, as the most straightforward
estimate of shear rate throughout the spreading process. We then use the
viscosity value corresponding to $\dot{\gamma}_{avg}$ from our bulk
rheological data to calculate the effective Reynolds number,
$Re_{\textit{\scriptsize{\mbox{eff}}}}$ at each $\phi$ and $u_{0}$. For our
experimental conditions, the range of dimensionless numbers was $20<We<1000$
and $50<Re_{\textit{\scriptsize{\mbox{eff}}}}<1600$. In Fig. 4b, we plot
$\beta$ in the spreading regime against
$Re_{\textit{\scriptsize{\mbox{eff}}}}We^{1/2}$; the dashed black line
indicates the power-law fit:

$\beta={{\color[rgb]{0,0,0}(0.81\pm
0.02)}}(Re_{\textit{\scriptsize{\mbox{eff}}}}We^{1/2})^{{{\color[rgb]{0,0,0}0.164\pm
0.002}}}.$ (2)

The exponent of this fit agrees with that reported by Scheller et al.[28] for
viscous Newtonian fluids, $\beta=0.61(ReWe^{1/2})^{0.166}$. Thus, our simple
estimate of shear rate and in turn effective viscosity, provides a useful
framework to quantify the maximum spread of impacting colloidal drops. Given
that the fluids considered in this study are highly shear thinning at higher
$\phi$, the agreement we report here with Newtonian models is surprising. We
note that more recent works[8, 9] have modified the scaling by Scheller et
al.[28] to additionally account for surface wettability. However,
incorporating these effects for impacting colloidal fluids is highly non-
trivial, and would likely modify only the low-$We$ spreading behaviour.

In the localised solidification regime [orange diamonds in Fig. 3, SI video
1[25]] the bulk of the drop still spreads like a Newtonian fluid [Fig. 4b],
but shear thickening is apparent via solidified patches that appear and then
disappear. These patches appear during the spreading phase, around 1
millisecond after impact. However, indicators of jamming are present earlier,
in the form of nonuniformity in the spreading rim of the drop [see for
example, panel 2 in Fig. 2a]. In most cases, these patches outlive the
spreading phase and disappear during the receding phase, over tens of
milliseconds. Our observation of this regime coincides with the onset of weak
shear thickening in the bulk rheology data [orange curve in Fig. 1b].
Moreover, the transient nature of these patches is strong evidence that
regions of high viscosity are embedded in a lower-viscosity fluid phase. We
note that we can only observe these patches on the drop surface in high-speed
imaging data, and there is a large variance in the spatial and temporal
distribution of these patches. This limits our ability to extract quantitative
information about localised jamming. For higher $\phi$, where shear thickening
is pronounced, the drop exhibits drastically different behaviour, and does not
spread at all.

For $\phi\geq$ 0.49, a large fraction of the drop solidifies upon impact. To
quantify the dynamics of this partially solidified state, we measure the
height of the drop apex as a function of time [Fig. 5a]. Consistent with
another study of impacting shear-thickening drops[16], we observe two regimes
in the $h$ vs. $t$ curve — a free-fall regime and a plateau regime.
Immediately after impact, $h$ decreases at a rate identical to the impact
velocity (free-fall regime) [Fig. 5b], and then plateaus at a constant value,
$h_{min}$ (plateau regime). This is strong evidence that any shear from the
impact event has not yet propagated to the top portion of the drop, and hence
the top portion must still remain a liquid. Studies of impacting Newtonian
drops have also observed a similar ‘free-fall’ regime where the drop apex
moves at the impact velocity[29, 10, 30]. It is worth noting that in contrast
to Newtonian fluids, where a broad transition regime was observed between the
free-fall and the plateau regimes, we observe a sudden transition from the
free-fall to plateau regime [Fig. 5a], a direct indication of a shear jammed
drop.

We quantify the spatial extent of solidification by plotting the normalized
minimum height, $h_{min}/d_{0}$ against $u_{0}$ [Fig. 5c]. The increase in
$h_{min}/d_{0}$ with $u_{0}$ indicates that a larger and larger volume of the
drop is solidified as the impact velocity is increased. Interestingly, at high
impact velocities, $h_{min}/d_{0}$ plateaus to a value smaller than $1$,
indicating that the solidified drop also undergoes deformation along the
impact direction, along with slight bulging in the plane transverse to impact
[Fig. 2c]. Furthermore, the high temporal resolution (100,000 fps) of our
imaging enables us to capture the details of this solidification as it occurs.

Immediately after impact, we observe a disturbance travelling upward along the
drop surface over hundreds of microseconds [orange and green arrows in Fig.
6a]. To better visualise this front, we subtract successive frames of the
image sequence, so that only the parts that change between frames are
highlighted [right panel of Fig. 6a, SI video 6[25]]. The location of the
front is given by the lower end of the bright edge [Fig. 6b]. As this front
travels upward, the portion of the drop above the front still maintains its
pre-impact curvature [red circles in Fig. 6a], indicating that it is
unaffected by the impact event until the front reaches it (consistent with
$u^{\ast}=u_{0}$, Fig. 5b). The angular location of this front plotted against
time reveals that the front travels at a constant speed,
$u_{\textit{\scriptsize{\mbox{front}}}}$ [slope of the line in Fig. 6c].
$u_{\textit{\scriptsize{\mbox{front}}}}$ increases with increasing $u_{0}$,
and its value is several times larger than $u_{0}$ [Fig. 6d]. As evident from
the rheology, the suspension thickens when the applied shear surpasses a
critical value. Indicated by the dotted lines in Fig. 6e, the critical shear
rate where thickening is observed, $\dot{\gamma}_{c}$, is much lower for
$\phi$ = 0.50 than for $\phi$ = 0.49. We plot
$u_{\textit{\scriptsize{\mbox{front}}}}$ against the excess shear rate over
this critical value, $\dot{\gamma}_{impact}-\dot{\gamma}_{c}$, and the data
indeed collapses on a single curve for both $\phi$ [Fig. 6f] . This suggests
that the speed of this disturbance is set by this excess shear rate.

As the impact velocity is increased, a larger and larger volume of the drop
solidifies upon impact. At $\phi=0.50$ and $u_{0}=4$ $m/s$, we observe that
the drop bounces off the substrate, with the coefficient of restitution
$\epsilon=0.1$ [SI video 7[25]]. This rebound behaviour is especially striking
given the hydrophilic nature of the substrate. By coupling this coefficient of
restitution with the drop’s deformation along the impact axis, we can semi-
empirically estimate the elastic modulus of the solidified drop. The drop
impacts the substrate with an initial velocity $u_{0}$, remains in contact
with the substrate for time $\Delta t=200$ µs , and then rebounds with the
final velocity $\epsilon u_{0}$. While in contact with the substrate, we
measure that the drop is deformed in the direction of impact by the amount
$\Delta x=0.24$ mm. We calculate the force experienced by the drop upon impact
using momentum conservation:

$F=\frac{m\Delta u}{\Delta t}=\frac{m(1+\epsilon)u_{0}}{\Delta t},$ (3)

To convert the force to a stress, we divide by the contact area for a Hertzian
contact[31], $\pi a^{2}=\pi d_{0}\Delta x/2$:

$\sigma=\frac{F}{\pi d_{0}\Delta x/2}=\frac{2m(1+\epsilon)u_{0}}{\pi
d_{0}\Delta x\Delta t}.$ (4)

The strain experienced by the drop is $\gamma=\Delta x/d_{0}$. Thus, the
elastic modulus of the rebounding drop can be computed as

$E=\frac{\sigma}{\gamma}=\frac{2m(1+\epsilon)u_{0}}{\pi{(\Delta x)}^{2}\Delta
t}.$ (5)

using $m=2.25\times 10^{-5}$ kg, we find $E=5$ MPa. A more thorough estimate
using Hertz’s equations[31] for two colliding elastic bodies leads to a
similar estimate of $E$. Calculating the elastic modulus in this way for other
impact conditions is challenging, as measuring the contact time in the absence
of rebound is nontrivial.

##  3 Discussion

In sum, our analysis presents the following picture of the drop dynamics. Upon
impact, the drop experiences a large instantaneous shear at the impact point.
At high enough volume fractions and impact velocities, this stress manifests
itself as pockets of localised solidification embedded in the spreading liquid
phase. At even higher volume fractions or shear, a larger and larger fraction
of the drop solidifies after impact, but some volume at the top remains
liquid. Therefore, the shear front must be dissipating as it moves upward, and
the stress falls below the critical stress for shear thickening before the
entirety of the drop is solidified. Moreover, at the highest impact velocity,
the drop rebounds, and the coefficient of restitution allows us to estimate
the elastic modulus of the shear jammed solid, $E=5$ MPa. Thus, our drop
impact experiments provide a unique window to observe shear jamming _as it
occurs_ , and give rise to a number of questions about the nature of both the
shear jamming transition and the resulting jammed solid.

The occurrence of localised solidification coincides with the appearance of
weak shear thickening in our bulk rheology data. The fact that these
solidified patches vanish over tens of milliseconds is strong evidence that
they are regions of high viscosity embedded in a lower-viscosity fluid phase.
Recent rheological studies using boundary stress measurements (BSM) have
reported finite regions of enhanced stress in silica suspensions[32, 33, 34].
In these works, Rathee et al.[32, 33, 34] argued that the transition from
shear thickening to shear jamming is governed by the growing size of such
localised shear jammed regions. Our observations of transient localized
solidification are thus striking visual evidence of such a mechanism. Further
spatially resolved stress measurements performed on impacting drops[11] could
provide more information on the nature of localised solidification in free-
surface systems.

In the bulk solidification regime, the coexistence of liquid and solid regions
is a result of shear traveling upward from the impact point, and
simultaneously dissipating due to the high suspension viscosity. Although
recent studies of low-viscosity Newtonian fluids have established the velocity
and pressure fields within an impacting drop[7, 10], they are not applicable
in case of high-viscosity systems. The highly non-Newtonian nature of
colloidal suspensions adds further complexity, as the fluid viscosity is a
strong function of the applied shear. Numerical work investigating transient
shear might be a useful next step to uncover the mechanism of dissipating
shear fronts. Though challenging, measurements of the flow inside the drop via
methods such as particle tracing, would provide key information about the flow
field in an impacting colloidal drop.

The nature of the upward-travelling front raises a number of interesting
questions. Before the front reaches the top, the speed of the drop apex
$u^{\ast}$ is identical to $u_{0}$ [Fig. 5b], and the curvature of the top
portion is the same as it was before impact. This confirms that the
information of the impact event reaches the top portion only with the front,
thus establishing that it is a solidification front. Why the speed of this
front is constant along the drop surface is an intriguing question. One would
expect a shear front to travel through the bulk of the drop, upward from the
region in contact with the substrate. Given the visual nature of our
measurements on an opaque drop, we can naturally observe this front only on
the surface. The most likely explanation, therefore, is that the front we
measure is this bulk shear front after it interacts with the drop boundary.

Our experiments are especially well-positioned to capture such a front due to
the free-surface conditions here that are absent in other studies of shear
fronts[35, 36, 37, 38, 39]. Past work has established that shear fronts in
dense suspensions are not a result of densification, and their velocity is set
by the external driving speed[38]. We emphasize that the drop impact system
has the unique capability to study complex fluid flow under large and
instantaneous shear. As we demonstrate above, the fluid responds to this shear
over a few hundred microseconds. As typical steady state rheometers apply
stress to the fluid over much longer timescales, shear rates above
$\dot{\gamma}_{c}$ are inaccessible. Thus, the instantaneous nature of shear
in drop impact enables us to access fluid behaviour under these high shear
rates. The dependence of the front speed on
$\dot{\gamma}_{impact}-\dot{\gamma}_{c}$ in our experiments [Fig. 6f] suggests
that the suspension properties near the shear jamming transition are governed
by the distance from the onset of shear thickening. This is consistent with
measurements in static jamming, where material properties depend on the
distance from the critical point[40]. The functional form of this dependence
potentially contains insights into the nature of the shear jamming transition.
Numerical work exploring the impact of suspension drops, although incredibly
challenging due to the strong role of hydrodynamics in colloidal systems,
might provide crucial information in this respect. Unfolding the physics of
these fronts will not only extend constitutive models for complex fluid
rheology to much higher stress regimes, but will also help us understand more
about the nature of the shear jamming transition.

Due to the transient nature of the shear jammed state, characterising the
jammed solid created after impact is challenging. Using the coefficient of
restitution of the rebounding drop, we were able to estimate the elastic
modulus of the solid phase, $E=5$ MPa. As rebound only occurred at one impact
velocity, how the elastic properties of shear jammed drops are controlled by
the impact conditions remains obscure. The use of superhydrophobic substrates
promotes rebound, even in Newtonian liquid drops[41]. Further colloidal drop
impact experiments on superhydrophobic surfaces could extend the parameter
space where drops rebound, and thus provide the information essential to
understand what controls the properties of this elastic state. Numerous other
properties of the shear jammed solid are of interest: When and how would such
a solid fracture? How broad is its linear elastic regime? How do these
properties compare to those of static jammed solids?

In conclusion, we conduct highly time-resolved drop impact experiments and
systematically probe suspension flow ranging from Newtonian-like to shear
jamming. We show that the impact behaviour in the spreading regime can be
quantitatively understood via an effective viscosity framework, and that the
solidification behaviours at high $\phi$ and $u_{0}$ are direct consequences
of shear jamming. The free-surface geometry in our system provides direct
visual information on how the shear jamming transition occurs, both in
parameter space and in time. Shear jamming occurs via a solidification front,
the speed of which is set by how far into the shear thickening regime the
applied shear rate is. Furthermore, we see this transition occur via a
localised solidification regime that cannot be observed via bulk measurements.
We believe that drop impact is a powerful experimental tool to investigate
macroscopic properties of complex fluids, and provides information that
compliments the data from bulk rheometry.

##  4 Methods

Colloidal sample preparation
We fabricated silica spheres in our lab using the Stöber[23, 24] synthesis
method. The particle size was determined by the number of feeds: we performed
14 feeds after the initiation of the reaction, resulting in particles with a
diameter of $830\pm 20$ $nm$. The reaction mixture was centrifuged and re-
suspended in ethanol 3 times; the suspension was then gravity separated to
improve monodispersity. The particles were then imaged on the Hitachi S4800
Scanning Electron Microscope [Fig. 1a]. The particle size was characterized by
measuring the diameter of a representative sample of 100 particles in ImageJ,
and the polydispersity reported is the standard deviation in particle size.

A concentrated stock suspension of the silica spheres was prepared in water
(with no surfactant), and the weight fraction was measured by drying $100$
$\mu L$ of the stock suspension. The density of silica ($2$ $g/cm^{3}$) was
used to convert weight fractions into volume fractions. Dilutions were then
performed to prepare samples of desired volume fractions. The uncertainty in
volume fractions reported is $0.5\%$ $(0.005)$ or less, determined by repeated
measurements. When not in use, all the sample tubes were sealed using Parafilm
and stored in a refrigerator to minimize evaporation and contamination.

###  4.1 Experimental setup

We used Fisherbrand plain glass slides as the hydrophilic impact substrate.
The slides were cleaned using a 2.5M solution of NaOH in ethanol and water to
remove organic impurities. A micropipette was used to form colloidal drops.
The micropipette was mounted on a vertically moving pipette holder to vary
impact velocities. We used $15$ $\mu L$ of fluid to obtain drops of $3.0\pm
0.1$ $mm$ diameter. The setup was enclosed in a humidity chamber with the
relative humidity maintained between 70–80% using a saturated solution of NaCl
in water, and the humidity was monitored in real time during experiments.
Before every trial of the impact experiments, a vortex mixer was used to re-
disperse the sample, ensuring that it was consistently well-mixed.

The impacting drops were backlit using a white LED light, and filmed using two
high-speed cameras. The first camera, a Phantom V2512, captured the side-view
of the impacting drop at 100,000 frames per second. The second camera, a
Phantom V640L, filmed at 20,000 fps. It was tilted at an angle of $15^{\circ}$
to gather information on how the impact affected the top surface of the drop.
The experiment was repeated at least 5 times for each impact condition to
ensure reproducibility.

Rheological studies
Stress-controlled rheological measurements were performed on the colloidal
samples over $0.09\leq\phi\leq 0.50$. The measurements were done on a TA
Instruments Discovery HR-2 rheometer at room temperature ($\sim 21^{\circ}C$)
using the cone-plate geometry with 40 mm diameter and a $1^{\circ}$ cone
angle. The truncation gap was $25$ $\mu m$. We covered the edges of the
samples with a microscope immersion oil to minimize evaporation. The samples
were pre-sheared to remove effects of shear history.

Data analysis
All high-speed videos were background-divided and analysed using ImageJ. The
plots were made using python, and all errors reported are standard deviations
over multiple trials. The maximum drop spread $d_{max}$ was determined by
locating the frame in the impact timeseries where the extent of the spreading
drop was the greatest. The height of the tallest point on the drop relative to
the substrate, $h$, was measured for each frame in the image sequence. The
minimum height $h_{min}$ was defined as the drop height at the crossover point
between the decreasing and the plateau regimes in the $h$ vs. $t$ plot. The
time of first observation of $h_{min}$, measured since the impact event, was
defined as $t^{\ast}$. The slope of the linearly decreasing regime in the $h$
vs. $t$ plot was defined as $u^{\ast}$. To calculate the coefficient of
restitution, the speed of the drop before impact $u_{0}$, and the speed after
rebound, $u_{f}$ were computed using several frames of the image sequence. The
coefficient of restitution was then computed as $\epsilon=u_{f}/u_{0}$.

To calculate the speed of the upward-moving front, the side-view impact videos
recorded at 100,000 fps were used. For every frame of the image sequence, the
pixel-wise difference between consecutive frames was taken in ImageJ, so that
only the elements that changed between consecutive frames (corresponding to
the location of the moving front) were highlighted [SI video 7 [25]]. This
enabled us to locate the jamming front with a time uncertainty of 10 $\mu$s.
The images were then adjusted for brightness and contrast to enhance the
moving front. The vertical height $h_{\textit{\scriptsize{\mbox{front}}}}$ of
the disturbance from the impact substrate was measured for each frame of the
image sequence, until the front was no longer visible. For every high-speed
video, the left and right half of the drop were separately analyzed to obtain
two datasets for $h_{\textit{\scriptsize{\mbox{front}}}}(t)$. In order to
convert $h_{\textit{\scriptsize{\mbox{front}}}}$ to the position along the
drop surface, $r\theta_{\textit{\scriptsize{\mbox{front}}}}(t)$, we
approximated the drop profile as a circle of radius $r=1.5$ $mm$ (disregarding
the slight deviation from spherical shape during front propagation), and used
the relation
$h_{\textit{\scriptsize{\mbox{front}}}}(t)=r(1-\cos\theta_{\textit{\scriptsize{\mbox{front}}}}(t))$,
such that $\theta\textit{\scriptsize{\mbox{front}}}(0)=0$ at the impact point,
to obtain the angle $\theta_{\textit{\scriptsize{\mbox{front}}}}(t)$. A line
was then fit to the $r\theta_{\textit{\scriptsize{\mbox{front}}}}$ vs. time
plots, and the slope, averaged over the two halves of the drop and several
movies for each impact condition [Fig. 6c], was reported as
$u_{\textit{\scriptsize{\mbox{front}}}}$ with error bars indicating the
standard deviation.

We thank Jeff Richards, Sid Nagel, and Xiang Cheng for useful discussions.
This work was supported by the National Science Foundation under award number
DMR-2004176. This work made use of the EPIC facility of Northwestern
University’s NUANCE Center, which has received support from the SHyNE Resource
(NSF ECCS-2025633), the IIN, and Northwestern’s MRSEC program (NSF
DMR-1720139). We thank the Richards Lab at Northwestern University for the use
of their rheometry facilities.

The authors declare that they have no competing financial interests.

Correspondence and requests for materials should be addressed to MMD.

(michelle.driscoll@northwestern.edu).

SA contributed to the conception of the work, experimental design, and data
acquisition and analysis. PS contributed to data interpretation and analysis
and drafted the manuscript. MMD contributed to the conception of the work,
data analysis and interpretation, and drafted the manuscript.

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Figure 1: Rheology of the colloidal suspensions. a SEM image of the colloidal
silica spheres used in our drop impact experiments; the sphere diameter is 830
$\pm$ 20 nm. b Bulk rheological flow curves: the colloidal suspension exhibits
viscous flow, shear thinning, and shear thickening as $\phi$ is increased. The
grey triangle in the bottom right indicates the rate limit of the rheometer.
Figure 2: Exotic impact behaviours of colloidal suspension drops. a Timeseries
of a $\phi=0.47$ colloidal drop expanding after impacting at $u_{0}=$ 3.0 m/s
[SI video 1 [25]]. The spreading drop shows transient pockets of localized
solidification, indicating the onset of shear thickening. b Timeseries of a
$\phi=0.49$ colloidal drop impacting at $u_{0}=$ 2.0 m/s [see also SI video
2]. The bottom half of the drop solidifies, while the still-fluid top portion
flows over it. c Timeseries of a $\phi=0.49$ drop impacting at $u_{0}=$ 3.0
m/s [SI video 3 [25]]. While most of the drop is solidified, the top portion
of the drop is in the liquid phase. All scale bars are 1 mm. Figure 3: State
diagram of colloidal drop impact. $\phi-u_{0}$ state diagram summarizing
impact regimes; representative snapshots corresponding to these regimes are
shown on the right. Green circles denote simple spreading behaviour, which
dominates the low $\phi$, low $u_{0}$ region. Orange diamonds indicate that
transient pockets of localised solidification were observed during spreading.
Blue triangles correspond to the partial/full solidification regime, where the
bottom portion of the drop jams after impact, but a shrinking region at the
top remains fluid. Figure 4: Quantifying maximum drop spreading. a Normalised
maximum diameter, $\beta=d_{max}/d_{0}$, as a function of $u_{0}$ for various
volume fractions $\phi$. For $\phi\geq 0.49$ and high impact velocities,
$\beta$ drops to 1, indicating the drop does not spread. Insets show
representative snapshots of simple spreading (upper), localised solidification
(middle), and bulk solidification (lower). Dotted lines are guides to the eye,
and the dashed black line indicates $\beta=$ 1\. b Normalised maximum
diameter, $\beta$, for $\phi\leq$ 0.47 plotted against the dimensionless
parameter $Re_{\textit{\scriptsize{\mbox{eff}}}}We^{1/2}$. The exponent of the
power law fit (dashed line, $\beta=0.81(ReWe^{1/2})^{0.164}$) is in agreement
with the scaling reported for Newtonian fluids[28]. Figure 5: Characterisation
of the partial solidification regime. a Height of the drop apex from the
impact substrate, plotted against time, for $\phi=0.50$, $u_{0}=3$ $m/s$. $h$
decreases at the speed $u^{\ast}$ until time $t^{\ast}$, then plateaus at the
value $h_{min}$. Inset: post-impact snapshot of a drop at the minimum height
$h_{min}$. b $h$ decreases at a speed identical to the impact velocity,
indicating that over the timescale $t^{\ast}$, the top portion of the drop is
unaffected by the impact event. Dashed line corresponds to $u^{\ast}=u_{0}$. c
$h_{min}/d_{0}$ vs. impact velocity $u_{0}$. $h_{min}/d_{0}$ increases with
increasing impact velocity, and then plateaus at a value less than 1,
indicating finite compression of the drop along impact axis. Dashed line
indicates $h_{min}/d_{0}=1$. Figure 6: Dynamics of the solidification front. a
Timeseries of a $\phi=0.50$ drop impacting at $u_{0}=$ 2.0 m/s [SI video 6
[25]]. Right panels are images obtained by subtracting consecutive frames, so
that the edge of the solidification front is highlighted (shown by arrows).The
red circle indicates the drop profile before impact. Even at 0.19 ms, the
portion of the drop above the front maintains its pre-impact curvature. Scale
bar is 1 mm. b Schematic of a subtracted image of the moving solidification
front, outlining relevant parameters. The height, $h_{front}$, of the edge of
the white outline gives the location of the front, which is then converted to
$r\theta_{front}$ using the spherical geometry. c Example datasets of
$r\theta_{front}$ vs. $t$ for $\phi=0.49$ and $u_{0}=3$ $m/s$. $r\theta$ vs.
$t$ is a straight line, the slope being the front speed along the surface,
$u_{front}$. d $u_{\textit{\scriptsize{\mbox{front}}}}$ plotted against
$\dot{\gamma}_{impact}$. e High- $\phi$ bulk rheological data from Fig.1b re-
plotted as shear stress vs. shear rate. Dotted lines indicate the onset shear
rates $\dot{\gamma}_{c}$ for shear thickening. f The
$u_{\textit{\scriptsize{\mbox{front}}}}$ data for $\phi=0.49$ and $\phi=0.50$,
when plotted against $\dot{\gamma}_{impact}-\dot{\gamma}_{c}$, collapses on a
single curve.