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# Quantifying the destructuring of a thixotropic colloidal suspension using
falling ball viscometry

Rajkumar Biswas  [  Debasish Saha  [  Ranjini Bandyopadhyay  [  Raman Research
Institute, Bangalore - 560080, India

###### Abstract

The settling dynamics of falling spheres inside a Laponite suspension is
studied. Laponite is a colloidal synthetic clay that shows physical aging in
aqueous suspension due to the spontaneous evolution of inter-particle
electrostatic interactions. In our experiments, millimeter-sized steel balls
are dropped in aqueous Laponite suspensions of different ages (i.e., time
elapsed since sample preparation). The motion of the falling balls are
captured using a high-speed camera and the velocities of their centroids are
estimated from the images. Interestingly, we observe that balls of larger
diameters fail to achieve terminal velocity over the entire duration of the
experiment. We propose a mathematical model that accounts for rapid structural
changes (expected to be induced by the falling ball) in Laponite suspensions
whose aging time scales are much slower than the time of fall of the ball. For
a range of ball sizes and Laponite suspension ages, our model correctly
predicts the time-dependence of the ball velocity. Furthermore, fits to our
model allow us to estimate the rates of destructuring of the thixotropic
suspensions due to the passage of the falling ball.

††preprint: AIP/123-QED

##  I Introduction

The settling dynamics of objects in a non-Newtonian aging suspension leads to
very interesting modifications of Stokesian flows. It is well known that for
an object falling through a Newtonian medium under gravity, a terminal
velocity, $V_{ter}$, arises due to a balance between the surface drag on the
object and its buoyant weight. For low Reynolds numbers, the flow around the
surface of the object remains laminar. In this regime, the drag force is
linearly proportional to the velocity. The relation between terminal velocity
and drag coefficient of an object falling through a viscous medium at low
Reynolds numbers $(Re<1)$ was first determined by Stokes in 1851 stokes1851 .
The differential equation for a falling ball inside a Newtonian fluid can be
written as, $m\dot{V}=-\kappa V+F_{b}$, where $\dot{V}$ = time derivative of
the instantaneous velocity $V$ of the ball, $\kappa$ = drag coefficient,
$\eta$ = viscosity of the fluid, $D$ = diameter of the ball, $m$ = mass of the
ball, $F_{b}$ = buoyant force. Solving this equation, we can write the
expression for velocity, $V$:

$V=V_{ter}+(V_{0}-V_{ter})\exp(-t/\tau)$ (1)

where $V_{0}$ = initial velocity and $V_{ter}=(\rho_{s}-\rho)gD^{2}/18\eta$,
$\tau=m/\kappa$, where $\kappa$ = $3\pi\eta D$, $\rho_{s}$ = density of the
ball, $\rho$ = density of the fluid, $g$ = acceleration due to gravity. For
high Reynolds numbers $(10-500)$, the flow does not remain laminar around the
ball even though it attains a steady state. Eqn.1 can be used for a range of
Reynolds numbers by including the appropriate correction factor for
dragCoulsonRichardson . The dependence of the drag coefficient on the Reynolds
number has been investigated extensively for Newtonian fluidsbrown2003 ;
goossens2019 ; khan1987 ; little1976 . For intermediate Reynolds number, an
empirical form of correction factor was proposed by Schiller and
NaumannCoulsonRichardson ; goossens2019 . It has also been reported that fluid
motion around a ball remains symmetric in Newtonian fluids, while the symmetry
is broken when the ball falls through an aging viscoelastic fluid with a
finite yield stressgueslin2006 ; Freund .

Novel flow properties such as shear thinning, shear thickening, non-zero
normal stresses and yield stresses emerge in viscoelastic materials such as
polymers, gels and suspensions Guvendiren2012 ; Bonn2010 ; Fong1996 . The bulk
rheological response of a non-Newtonian complex fluid can be correlated with
its structural details at microscopic length scales. Bio-materials like human
bloodGiannokostas , collagenClark , silk fibreLiu  exhibit thixotropic
behaviour. These complex bio-materials are used in disease detection and
treatment, cell carrier therapy, bio-printing etc. The quality of common
everyday materials such as toothpaste, cosmetics, paint, mayonnaise, clays
etc., can also be improved by understanding their non-Newtonian flow Chhabra .
The recovery of oil from soil, which consists of different clay compounds,
sand, organic matter, oil and water in varying proportions, can be enhanced
with the knowledge of the flow of particles in non-Newtonian fluids Zhuang2019
.

Recently, flow patterns around objects falling through Xanthan gum suspensions
have been studiedMrokowska2019 . Falling ball viscometry is a simple but
elegant technique to study non-Newtonian fluids. In this method, the motion of
a ball falling through a medium is tracked to obtain information about the
properties of the latter. Such experiments can effectively probe the complex
flow properties of fluids whose structures continually evolve with time. In a
worm-like micellar solution, the motion of a falling ball can lead to
instabilities jayaraman2003 ; kumar2012 , with larger balls seen to oscillate
in the direction of their fall due to shear induced structural deformation of
the underlying medium during their passage. Oscillatory motion was also
observed for rising bubbles in a worm-like micellar solution Handzy2013 . The
velocity of a spherical ball in Bentonite clay suspensions was found to
increase continuously over a period of time Briscoe1992 . For non-Newtonian
fluids in general, observations like shear dependent viscosity, fore-aft
asymmetry, negative wake and change in yielding behaviour with aging time make
the motion of the falling object more complicated gueslin2006 .

In this work, the motion of a ball falling through aging Laponite suspensions
is studied. Laponite is a synthetic clay comprising disk-like particles of
thickness $1$ nm and diameter $25-30$ nm (Kroon1998, ). In dry form, the
particles exist in 1-D stacks called tactoids that start disintegrating when
dissolved in water. For jammed aqueous suspensions of Laponite, complete
break-up of tactoids into single particles cannot occur due to strong inter-
particle electrostatic repulsions Ali2013 . The viscosity of a Laponite
suspension of concentration between $2-3.5$% $w/v$ is observed to
spontaneously increase due to the evolution of the internal structures in a
physical aging process. Several studies have reported that continuous
variations in inter-particle interactions occur due to Na+ dissociation from
Laponite surfaces in aqueous suspension Tawari2001 ; Joshi2007 . This process
proceeds simultaneously with the break-up of Laponite stacks or tactoids and
gives rise to a continuous evolution of the microscopic structure of the
suspension. Laponite suspensions exhibit a physical aging phenomenon wherein
the suspension viscosity spontaneously increases with time due to the
underlying structural buildupBonn . Microscopic interactions between clay
particles are sensitive to clay concentration, pH of the medium and
temperature Saha2015 ; Jatav2014 . The concentration-dependent phase diagram
of aqueous Laponite suspensions has been studied extensively in the
literature. These studies reveal that Laponite suspensions can exist in sol,
gel, glass and even a nematic phaseruzika2011 ; Cummins2007 . Rearrangements
between clay particles, resulting from inter-particle screened electrostatic
interactions, lead to the observed sequence of phases.

The rheological behaviour of aging Laponite suspensions has a strong
dependence on the preparation protocol. It can vary noticeably between a
freshly prepared sample and one that has been rejuvenated (where aging has
been reinitialized by shearing a spontaneously evolving suspension). A partial
irreversibility in aging behaviour upon rejuvenation is also
observedShahin2010  and understood by considering that the structural build-up
in Laponite suspensions for long aging times cannot be destroyed even by
applying very large shear deformations. A dichotomic aging behaviour,
indicating different aging dynamics in freshly prepared and rejuvenated
Laponite suspensions, was observed using X-ray photon correlation spectroscopy
and dynamic light scatteringAngelini2013 . It has been reported that a ball
dropped in a rejuvenated Laponite suspension gets stuck inside the
fluidtabuteau2007  or oscillates during its fallFazilati2017 . The critical
stress value that has to be exerted by the ball to exhibit these contrasting
behaviours depends on the ball size and density, and the aging time, $t_{w}$,
defined as the time since preparation of the Laponite suspension moller2006 .
It is observed that the medium in the immediate neighborhood of the surface of
the ball fluidizes due to the shear-induced viscous thinning that arises from
the disruption of the micro-structure due to the passage of the ball. Regions
far from the ball are less fluid-like in comparisonPutz2008 . Most falling
ball experiments performed in Laponite suspensions have used rejuvenated
samplestabuteau2007 ; Fazilati2017 . Since falling ball viscometry has not
been performed in spontaneously aging Laponite suspensions to the best of our
knowledge, we have conducted a systematic study to probe the evolution in the
underlying microscopic structures of freshly prepared Laponite suspensions.

In this article, balls of different sizes are dropped into freshly prepared
Laponite suspensions having different initial states of structure that depend
on the suspension age quantified by the waiting time, $t_{w}$, since sample
preparation. The motion of the ball falling in Laponite suspensions is
captured using a high speed camera and is observed to differ qualitatively
from the trajectory of a ball falling through a Newtonian fluid. We note that
while a ball falling through a Newtonian liquid of comparable viscosities
reaches a steady state within the experimental time window, the velocity of a
ball dropped in the Laponite suspension continues to increase, with the
increase being more dramatic for the larger balls used in this study.
Representative data for a 3 mm ball dropped in a Newtonian glycerol-water
mixture and a non-Newtonian Laponite suspension of comparable viscosity are
plotted in Fig.S1 in Supplementary Material. Interestingly, the larger balls
used in this study, when dropped in the Laponite suspensions of ages 1-3 hrs
explored here, do not reach a steady state over the entire duration of the
experiment. We modify an existing rheological model moore1959  for time-
dependent changes in the structure of an aging non-Newtonian fluid to explain
our experimental observations. We show here that when the time of fall of the
ball is much faster than the aging time of the medium, fits of our
experimental ball-drop data to the model can be used to successfully extract
the destructuring time scales of aging Laponite suspensions.

##  II Experimental Methods

###  II.1 Sample preparation

All experiments reported in this work were performed using Laponite XLG.
Laponite, which is in powdered form and hygroscopic in nature, was first
heated in a hot air oven for $16-20$ hr at $120^{\circ}$C. Colloidal
suspensions of Laponite of concentration $2.8$% $w/v$ were prepared by adding
$19.6$ gm dried Laponite powder to $700$ ml Milli-Q water. The concentration
of Laponite is expressed as the weight of Laponite powder in grams that was
added to $100$ ml Milli-Q water. Laponite suspensions were vigorously stirred
in a magnetic stirrer for $1$ hr to disperse the Laponite powder in the
aqueous medium. After $1$ hr, the suspension became optically clear. It was
then filtered in a closed container through a milli-pore membrane sheet (pore
size = $0.45$ $\mu$m) with the help of a vacuum pump to break up any remaining
particle clusters.

Figure 1:  (a) Schematic diagram of the experimental set-up for ball drop
experiments. (b) The cylindrical tube (length $65$ cm and diameter $4$ cm)
filled with Laponite suspension placed inside a water bath at $25^{\circ}$C.
(c) The electro-magnet used to drop steel balls. (d) High speed camera - IDT
Motion Pro Y4-S2. (e) Trajectory of a falling ball in the viscometer displayed
in (b), tracked using Video Spot Tracker (CISMM), is shown by a yellow line.
Figure 2:  The time evolution of viscosity of a Laponite suspension
(concentration 2.8% $w/v$). In the inset, viscosity is plotted with shear rate
for 2.8% $w/v$ Laponite suspensions at different aging times.

###  II.2 Falling ball viscometry

The schematic diagram of our falling ball viscometer is shown in Fig.1(a). For
all the experiments reported here, the viscometer was inserted in a water bath
whose temperature is maintained at 25∘C by a temperature controller
(Polyscience Digital Inc.). After filtration, a freshly prepared Laponite
suspension (approximately 600 ml) was loaded into the cylindrical viscometer
shown in Fig.1(b). Once sample loading was complete, the top of the
cylindrical tube was covered with parafilm to isolate the Laponite suspension
from the atmosphere. The suspension was left to age, and its micro-structure
was allowed to develop spontaneously due to the physical aging process. The
aging time of the Laponite suspension is assigned the value $t_{w}$ = 0 as
soon as sample loading was completed. Aging time is measured from the moment
the viscometer was loaded till the time the ball was dropped in the
viscometer. The time taken for the ball to fall through the suspension (less
than 1 second) is negligible compared to the aging times of the Laponite
suspensions (1-3 hours) used in this study. The sample age was monitored
continuously using a stopwatch for the entire duration of the experiment. The
parafilm that was used to cover the Laponite suspension immediately after
loading was completed was removed just before the ball was dropped. An
electro-magnet shown in Fig.1(c) was used to drop steel balls along the
vertical axis of the tube and was positioned at a fixed height of 1.5 cm above
the surface of the Laponite suspension. This method of ball-drop was adopted
since Laponite suspensions, particularly of higher ages, were seen to stick to
the surface of the ball when the latter was immersed entirely in the
suspension. The instant at which the falling ball enters the Laponite medium
is recorded as time $t=0$. The corresponding ball velocity is designated as
initial velocity, $V_{0}$, which is a fitting parameter in our model.
Experiments were performed at several aging times corresponding to different
structural states of the Laponite suspensions. After time $t_{w}$, the
parafilm cover was removed from the top and steel balls (density $7815\pm 50$
kg/m3) of different sizes were dropped using the electro-magnet while
minimizing any rotation. The path followed by the ball was recorded with an
IDT Motion Pro Y4-S2 high speed camera (Fig.1(d)) at 100 frames per second for
all balls. The position of the falling ball was tracked from the images
recorded by the camera with Video Spot Tracker (CISMM). Laponite suspensions
were filled in the falling ball viscometer upto heights of 56-57 cm. According
to TannerTanner , end effects are insignificant if the ball is away from the
bottom of the container by a distance that is more than the latter’s radius.
Following this, and since the radius of our viscometer is 2 cm, we have
tracked the falling balls for a maximum of 54 cm from the top of the container
(or 2 cm from the bottom). The yellow line in Fig.1(e) is a representative
ball trajectory and was constructed from the position of the ball centroid in
each frame. Horizontal displacement of the ball is estimated in Supplementary
Materials (Fig.S2) and the horizontal velocity is seen to be negligible
compared to the vertical velocity.The velocity of the falling ball was
calculated from the position $vs.$ time data by time-differentiating the
positions of the ball centroids using a central differential formula Garcia .
The error in velocity was estimated using the formula Feng2011 , $\delta
V=\sqrt{\delta r_{j}^{2}+\delta r_{j+1}^{2}}/\Delta t$, where $j$ is the frame
number and the uncertainty in position ($\delta r=\sqrt{\delta x^{2}+\delta
y^{2}}$) is the fitting error in estimating the mean of the intensity profile
of the ball along the $y$ and $x$ directions (direction of the fall of the
ball and the direction perpendicular to it, i.e., parallel to the plane of the
camera, respectively) using the Gaussian function. The time interval, $\Delta
t$, was calculated from the frame rate of the camera. The data was fitted to a
mathematical model using Jupyter notebook (Python v3.7). Experiments were
performed using Newtonian fluids of viscosities comparable to those of the
Laponite suspensions used here (Fig.S3 and Table.TS1 in Supplementary
Material). The effects of the viscometer wallKehlenbeck ; Chhabra  and
Reynolds number on the motion of the falling ball have been estimated from
these experiments (Fig.S4 and Table.TS2 in Supplementary Material). The
effects of the wall were found to be negligible in these measurements. By
incorporating the Schiller Naumann correction CoulsonRichardson ; goossens2019
, we found a good match between viscosity values obtained in falling ball and
rheometric measurements.

###  II.3 Rheology

For rheological measurements Macosko ; Ewoldt , a stress controlled Anton Paar
MCR-501 rheometer was used. The temperature was fixed at $25^{\circ}$C using a
water circulation system. A double gap geometry (DG - $26.7$, outer diameter =
$13.796$ mm, inner diameter = $11.915$ mm, cell height = $42$ mm, sample
volume = $3.8$ ml) was used for our experiments. Aqueous suspensions of dried
Laponite of concentration 2.8 % $w/v$ were stirred for $1$ hr at $400$ rpm to
break up big particle clusters. For each experimental run, the sample was
loaded slowly into the annular region of the geometry using a syringe.
Silicone oil of viscosity 5 cSt was used as solvent trap oil to minimize
sample evaporation. A shear rate of 500 sec-1 was applied for 5 minutes to
break any remaining particle clusters. This procedure ensures an identical
initial state and is essential for the acquisition of reproducible data. A low
shear (1 sec-1) was applied for subsequent data acquisition and a freshly
prepared sample was allowed to age spontaneously in the rheometer geometry
upto a predetermined $t_{w}$ before the start of each measurement.

##  III Results

###  III.1 Time dependence of viscosity

Figure 3: Time dependent viscosity of 2.8% $w/v$ Laponite suspensions, of
aging times ($t_{w}$) $1$ hr and $3$ hr, subjected to constant shear rates of
$100$ and $180$ sec-1 at $25^{\circ}$C.  Figure 4: (a) Displacement $vs.$ time
of the steel balls (diameter = $2$, $3$, $4$, $5$, $6$ mm) falling through
Laponite suspensions of concentration $2.8\%$ $w/v$ at $1$ hr aging time
($t_{w}$). (b) Velocity $vs.$ time is calculated from the displacement data in
(a). The error bars are calculated using the protocol specified for balls
falling in a medium Feng2011 .

As discussed earlier, the time-dependent evolution of inter-particle screened
repulsive forces in Laponite suspensions gives rise to a gradual and
continuous micro-structural build-up. Consequently, Laponite suspensions show
physical aging which manifests in experiments as a spontaneous evolution of
the viscoelastic moduli. In Fig.2, the viscosity of a 2.8% $w/v$ Laponite
suspension, recorded with time under a low shear rate (1 sec-1) for 3 hr, is
displayed. The viscosity is seen to spontaneously increase with time,
signalling the progress of the physical aging phenomenon. The viscosities of
Laponite suspensions of aging times $t_{w}$ = 1,2,3 hours are plotted in the
inset of Fig.2. As expectedgueslin2009 , the viscosity decreases with shear
rate for all sample aging times explored here. The shear stress vs. shear rate
plots for Laponite suspensions at ages $t_{w}=$ 1,2,3 hrs are shown in Fig.S5
of Supplementary Material. Time-dependent changes in the viscosity of Laponite
suspensions are measured for two different shear rates $100$ and $180$ sec-1
and are plotted in Fig.3. It is seen from this figure that the imposition of
constant shear rates results in a decrease in the suspension viscosity at
earlier times. This observation can be attributed to the breakup of sample
microstructures by the high shear rates that are imposed in these experiments.
At longer times, the viscosity shows an increase due to the increased rate of
structural recovery. These competing effects give rise to a non-monotonic
viscosity profile that is sensitive to sample age and the imposed shear rate.

Figure 5: In (a), (b), (c) and (d), velocities of the ball centroids for balls
of diameters 2mm , 3mm , 5mm and 6mm respectively, falling through Laponite
suspensions of different aging times ($t_{w}$), are plotted. Coloured symbols
represent experimental data, coloured vertical bars are the corresponding
error bars Feng2011  and continuous black lines are the fits to Eqn.(7).

###  III.2 Falling ball experiments

The experiments reported here track millimeter-sized steel balls falling
through spontaneously evolving Laponite suspensions. Freshly prepared Laponite
suspensions that are allowed to age for $t_{w}$ =1,2,3 hrs are used. The
viscosity of the suspension increases gradually due to the physical aging
mechanism discussed earlier. A ball that is dropped through an aging
viscoelastic suspension is expected to induce shear thinning of the medium due
to break up of the fragile structures present in its path. For all the
experiments reported here, Laponite suspension concentration is kept fixed at
$2.8\%$ $w/v$.

The extent of shear thinning of the surrounding medium due to passage of the
ball can be evaluated approximately by relating it to the data plotted in
Fig.3. The shear rate imposed by the ball is estimated using the relation
$\dot{\gamma}=V/D$, where $V$ and $D$ are, respectively, the instantaneous
velocity and the diameter of the falling ball. For the values of velocity, V,
measured here and the diameters, D, of all the balls used, the shear rates
imposed by the balls always lie within the range 100-200 sec-1 in our
experiments. A comparison with Fig.3 shows that the aging Laponite samples lie
in the shear-thinning regime within this range of shear rates. A ball falling
through a Laponite suspension would therefore disintegrate the local
structures in its path.

Fig.4 shows representative data when balls are dropped in a Laponite
suspension that has been aged for 1 hr. In Fig.4(a), the displacements of
balls having different diameters (2mm, 3mm, 4mm, 5mm and 6mm ) are plotted.
The overall slopes of the displacement curves tend to decrease with decreasing
ball diameters. The velocities are plotted vs. time for different ball sizes
in Fig.4(b). As expected, the larger balls with higher inertia travel with
greater velocities through the Laponite suspension medium. The ball of
diameter 2mm, released at a high non-zero initial velocity, eventually reaches
a lower terminal velocity. In contrast, the velocities of the balls of
diameters 3-6 mm increase continuously with time throughout our experimental
time window, with the increase becoming more prominent for the larger balls.
Error in velocity due to tracking is calculated Feng2011  and plotted with
vertical bars.

Data for all the ball-drop experiments, acquired for balls of sizes between 2
mm and 6 mm falling through Laponite suspensions of different ages, are
plotted in Fig.5. We observe from Fig.5(a) that the velocity of the 2 mm ball
decays with time and attains a steady state terminal velocity during the
experimental time window for all Laponite suspensions at aging times, $t_{w}$=
1 hr, 1.5 hr, 2.5 hr. For all the Laponite aging times investigated here, the
velocities of the balls of larger diameters (3mm, 5mm and 6mm) continue to
increase during the experimental time window, with the increase being maximum
for the largest ball (Figs.5(b-d)). Such continuous increase in velocity
without attainment of terminal velocity in the experimental time window
accessible in our experiments indicates the prevalence of medium destructuring
over structural recovery under these conditions. As the sample age increases,
the ball velocity decreases in all the experiments. Vertical bars with lighter
shades of the symbol colors are plotted to represent the errors in Fig.5.

###  III.3 Mathematical model

Moore (1959) moore1959  proposed a mathematical model to explain the micro-
structural evolution of thixotropic materials. In this model, the time-
dependent structure of an aging fluid is represented by a dimensionless
structural parameter, $\lambda$. This model incorporates $\lambda$ with the
following physical interpretation: $\lambda=0$ for a fully broken or
destructured state of the fluid (or when the structure is yet to form at the
microscopic level), while $\lambda=1$ for a fully structured fluid state,
i.e., when the structure is not broken at all.

In an aging Laponite suspension that is locally destructured by a ball falling
through it, the time evolution of the structural parameter, $\lambda$, has to
be governed by restructuring/structural recovery and destructuring/structural
breakdown of the suspension:

$\frac{d\lambda}{dt}=F(\lambda)-G(\lambda,\dot{\gamma})$ (2)

In the above equation, $F(\lambda$) and $G(\lambda,\dot{\gamma})$ are
associated with the rates of the restructuring and destructuring processes
that progress simultaneously in a Laponite suspension having an instantaneous
state $\lambda$. Eqn.2 can be simplified by considering a constant
restructuring rate $F=1/\theta$ and a destructuring function
$G=\alpha\lambda\dot{\gamma}$ that varies linearly with local shear
rateferrior2004 ; toorman1997 , i.e.,
$d\lambda/dt=1/\theta-\alpha\lambda\dot{\gamma}$. Here, $\dot{\gamma}$ is the
shear rate imposed by the falling ball and $1/\theta$ and $\alpha$ are
parameters related to the restructuring and destructuring rates of the medium
respectively.

The restructuring rate of the Laponite suspension is decided by its physical
properties such as aging time, concentration and instantaneous state. As the
Laponite suspension ages with time, the structure evolves identically
throughout the medium. In the present scenario, since the structural length
scales of interest are much larger than the size of a single Laponite
particle, the viscosity can be assumed to be equal at all points in the
undisturbed sample. The high shear rates of approximately 100-200 sec-1
experienced by the regions in the immediate neighborhood of the falling ball
are expected to result in shear thinning of the surrounding Laponite medium
due to the breakdown of the fragile microscopic structures. This can be seen
in Fig.3 where we note that structural recovery takes over only after 1000
seconds when suspensions are subjected to these high shear rates. The time of
fall of the ball is less than 1 second and therefore smaller than the time
scale over which the structural parameter is expected to change due to the
physical aging process. The destructuring due to the passage of the ball is
expected to dominate over the restructuring process and $F(\lambda)$ in Eqn.2
can be neglected. To model the destructuring of the medium in our falling ball
experiments, we make the following assumptions:

i) The ball falls inside the sample while shearing only the immediate vicinity
of its surface. During the passage of the ball, it interacts with each point
in the medium that lies in its path. Therefore the time evolution of the
structural parameter of the medium, $d\lambda_{f}/dt$, should be proportional
to the time taken by the ball to travel its diameter, $D/V$.

ii) As higher ball velocity indicates increased disruption in the structure of
the medium surrounding the ball, the time evolution of the structural
parameter is assumed proportional to the instantaneous ball velocity, $V$,
during the fall.

We can therefore simplify Eqn.2 and propose the following modified kinetic
equation for the time-dependent structural evolution in Laponite suspensions,

$\frac{d\lambda_{f}}{dt}\propto-V\frac{D}{V}$ (3)

Table 1: Parameters measured by fitting Eqn.7 to the data in Fig.5. Ball diameter (mm) |  Aging time (hr) |  Initial velocity $V_{0}$ $(m/s)$ |  $W$ $(m/s)$ |  $1/\tau$ $(s^{-1})$ |  Destructuring rate $\beta$ $(s^{-1})$
---|---|---|---|---|---
2 |  1 |  0.653 $\pm$ 0.010 |  0.433 $\pm$ 0.004 |  26.077 $\pm$ 2.255 |  1.07e-17 $\pm$ 0.031
2 |  1.5 |  0.570 $\pm$ 0.019 |  0.392 $\pm$ 0.005 |  33.232 $\pm$ 5.245 |  1.64e-19 $\pm$ 0.037
2 |  2.5 |  0.439 $\pm$ 0.024 |  0.238 $\pm$ 0.003 |  45.801 $\pm$ 6.843 |  7.79e-16 $\pm$ 0.026
3 |  1 |  0.655 $\pm$ 0.011 |  0.747 $\pm$ 0.006 |  16.033 $\pm$ 3.814 |  1.11e-15 $\pm$ 0.011
3 |  3 |  0.436 $\pm$ 0.010 |  0.543 $\pm$ 0.013 |  6.819 $\pm$ 2.001 |  0.017 $\pm$ 0.030
4 |  1 |  0.661 $\pm$ 0.009 |  0.876 $\pm$ 0.013 |  10.882 $\pm$ 1.499 |  0.066 $\pm$ 0.029
5 |  1 |  0.736 $\pm$ 0.007 |  0.982 $\pm$ 0.016 |  9.374 $\pm$ 1.130 |  0.105 $\pm$ 0.031
5 |  2 |  0.713 $\pm$ 0.009 |  0.947 $\pm$ 0.029 |  8.208 $\pm$ 1.719 |  0.103 $\pm$ 0.053
5 |  3 |  0.664 $\pm$ 0.014 |  0.793 $\pm$ 0.015 |  11.757 $\pm$ 3.456 |  0.119 $\pm$ 0.034
6 |  1 |  0.691 $\pm$ 0.010 |  1.001 $\pm$ 0.008 |  16.261 $\pm$ 1.206 |  0.218 $\pm$ 0.018
6 |  3 |  0.627 $\pm$ 0.006 |  0.802 $\pm$ 0.004 |  15.872 $\pm$ 1.242 |  0.232 $\pm$ 0.010

In Eqn.3, the time dependent structural parameter of the aging Laponite
suspension for falling ball experiments is denoted by $\lambda_{f}$. The
negative sign is incorporated to account for shear-thinning due to disruption
of the fragile suspension structures by the falling ball. Eqn.3 reduces to:

$\displaystyle\frac{d\lambda_{f}}{dt}$ $\displaystyle=-\epsilon D$ (4)
$\displaystyle\ \implies\lambda_{f}$ $\displaystyle=\lambda_{0}(1-\beta t)$
(5)

where $\epsilon$ is a proportionality constant of dimension (length $\times$
time)-1, $\lambda_{f}=\lambda_{0}$ at time $t$ = 0 and the rate of
destructuring is $\beta=\epsilon D/\lambda_{0}$. The viscoelastic, structured
nature of the suspension can be incorporated in the expression for terminal
velocity, $V_{ter}$ obtained from Eqn.1, by considering the functional
formgumulya2014  for viscosity $\eta=\eta_{0}\lambda_{f}^{n}$ where $\eta_{0}$
is the zero-shear viscosity and $n=1$ for a Laponite suspension ferrior2004 .
Viscosity of the suspension at each point changes with its structural
parameter. The expression for the structure dependent terminal velocity,
$V_{\lambda}$, of the ball falling through a thixotropic Laponite suspension
can therefore be written as:

$\displaystyle V_{\lambda}(t)$
$\displaystyle=\frac{(\rho_{s}-\rho)gD^{2}}{18\eta(t)}$
$\displaystyle=\frac{(\rho_{s}-\rho)gD^{2}}{18\eta_{0}\lambda_{0}(1-\beta t)}$
(6)

Here, we have used Eqn.5 and the expression for viscosity of Laponite
suspensions gumulya2014  to obtain the expression for instantaneous viscosity,
$\eta(t)=\eta_{0}\lambda_{0}(1-\beta t)$. Using Eqn.(1) and (6), we rewrite
the equation of motion of a ball falling through a thixotropic suspension with
an initial velocity $V_{0}$:

$\displaystyle V$ $\displaystyle=V_{\lambda}+(V_{0}-V_{\lambda})\exp(-t/\tau)$
$\displaystyle=\frac{W}{(1-\beta t)}+(V_{0}-\frac{W}{(1-\beta
t)})\exp(-t/\tau)$ (7)

where $W=(\rho_{s}-\rho)gD^{2}/18\eta_{0}\lambda_{0}$, $\beta$, $V_{0}$,
$\tau$ are the fitting parameters. Here, $\tau=m/3\pi\lambda_{0}\eta_{0}D$
where $m$ = mass of the ball, $\eta_{0}$ = zero-shear viscosity of the sample,
$D$ = diameter of the ball.

The constant breakdown of the micro-structure as the ball falls through the
suspension leads to shear thinning at a rate parametrized by $\beta$ in Eqn.5.
Naturally, for a Newtonian sample, we expect the value of $\beta$ to be zero.
The data for balls of different sizes falling through Laponite suspensions of
different aging times (Figs.5(a-d)) are fitted to Eqn.7 (solid lines in Fig.5)
and the destructuring rate, $\beta$, is obtained as one of the fitting
parameters. All the fitting parameters are tabulated in Table I. In Fig.6,
$\beta$ is plotted for ball sizes between 2 mm - 6 mm for different aging
times. Only the fitting errors are considered in this plot. It is clear from
the figure that $\beta$ decreases with ball size and is approximately zero for
the smallest ball of diameter 2 mm studied here. When larger balls are dropped
through Laponite suspensions, the destructuring rates increase with ball size
and stay nearly unchanged over the entire range of sample ages. The
approximate insensitivity of $\beta$ to changes in sample age $t_{w}$ arises
from the predominance of destructuring processes over structural recovery in
the freshly prepared Laponite suspensions used here. Our assumption that shear
thinning of the medium dominates over structural recovery in our falling ball
experiments is therefore adequate to accurately compute the ball trajectories.
Furthermore, Eqn.7 explains our experimental data irrespective of the
instantaneous state of the suspension.

Figure 6: Destructuring rate, $\beta$, $vs.$ ball diameter, $D$, is plotted
for different aging times ($t_{w}$) of Laponite suspensions.

##  IV Conclusions

The destructuring of a Laponite suspension due to a ball falling through it is
studied in this work. We show that the motion of solid steel balls through
spontaneously evolving colloidal suspensions of Laponite is very different
from the motion of a solid object in a Newtonian fluid.

Falling ball experiments are performed by dropping millimeter-sized balls in
Laponite suspensions of different ages. In contrast to a ball falling through
a Newtonian fluid, we find that the larger balls do not attain a steady state
while falling through Laponite suspensions during the experimental time
window. By modifying a mathematical model proposed by Moore moore1959  to
account for balls falling over time scales much faster than the suspension
aging times, we successfully extract the destructuring rates of Laponite
suspensions of different ages for different ball sizes. Interestingly, the
gradual increase of velocity within the experimental time window accessible in
these measurements can also be explained using this model. Eventually at later
times and for longer viscometer heights, the ball would have certainly reached
a steady state. The restructuring contribution in Eqn.2 that we have neglected
for the present measurements would then be non-negligible. However, trying to
observe the motion of the ball till it reaches a steady state would require us
to perform experiments with much longer cylinders and enormous sample volumes.
We have therefore restricted our study by exploring the motion of the falling
ball in a regime that can be described by the simplest mathematical model that
quantifies the gradual increase of ball velocity due to shear thinning of the
Lapaonite suspensions. Further, the initial velocity of the ball when it
enters the medium is non-zero as the ball is released from a fixed height
above the fluid surface. Further experiments need to be performed to determine
the effect of initial velocity on the destructuring rate of the suspension.

Our results are fairly general and are applicable in the study of the settling
dynamics of macroscopic objects in a wide variety of thixotropic suspensions.
Balls of different materials can be used to explore microscopic viscosities
and destructuring rates of complex fluids over a large range of shear rates.
We note here that while the viscosity measured by a rheometer reflects the
response of the entire sample, falling ball viscometry explores the viscosity
of the medium only in the neighborhood of the falling ball. We have
parametrized the destructuring rate of the Laponite suspension medium in terms
of a parameter $\beta$. Rheometry and falling ball viscometry probe the system
differently, involve very different shear profiles, and are complementary to
each other. A quantitative relation between $\beta$ and bulk viscosity of a
non-Newtonian fluid obtained in rheological measurements is therefore well
outside the scope of this paper.

To conclude, falling ball viscometry, which involves the macroscopic
measurement of the velocity of a ball falling through a medium, can be
employed to successfully estimate the microscopic dynamics of fragile
structures in shear thinning suspensions. The experiments reported in this
paper provide important insights into the physical aging process in the
presence of externally-imposed forces and contribute to our understanding of
the motion of macroscopic objects through non-Newtonian fluids.

Supplementary Material \- See supplementary material for velocity $vs.$ time
data for millimetre sized balls falling through Newtonian glycerol-water
mixtures and non-Newtonian Laponite suspensions of comparable viscosities,
estimation of radial velocities of the falling balls, estimations of wall
effects and the effects of the intermediate values of Reynolds numbers for
balls dropped in glycerol-water mixtures of comparable viscosities, and the
flow curve of 2.8% $w/v$ Laponite suspensions of ages 1, 2 and 3 hrs.

###### Acknowledgements.

We would like to thank Abhishek Dhar and Rama Govindarajan for useful
discussions. We acknowledge CISMM at UNC-CH, supported by the NIH NIBIB (NIH
5-P41-RR02170) for the tracking software. We also acknowledge Sanjay Kumar
Behera and Yash Rana for their help during the initial stages of the
experiment.

Availability of data \- The data that support the findings of this study are
available from the corresponding author upon reasonable request.

This article has been submitted to Physics of Fluids.

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