# Role of plasticity in the universal scaling of shear thickening dense
suspensions

Sachidananda Barik, Akhil Mohanan  Sayantan Majumdar<EMAIL_ADDRESS>Soft Condensed Matter Group, Raman Research Institute

###### Abstract

Increase in viscosity under increasing shear stress, known as shear thickening
(ST), is one of the most striking properties of dense particulate suspensions.
Under appropriate conditions, they exhibit discontinuous shear thickening
(DST), where the viscosity increases dramatically and can also transform into
a solid-like state due to shear induced jamming (SJ). The microscopic
mechanism giving rise to such interesting phenomena is still a topic of
intense research. A phenomenological model proposed by Wyart and Cates shows
that the proliferation of stress-activated interparticle frictional contacts
can give rise to such striking flow properties. Building on this model, a
recent work proposes and verifies a universal scaling relation for ST systems
where two different power-law regimes with a well-defined crossover point is
obtained. Nonetheless, the difference in the nature of the flow in these two
scaling regimes remains unexplored. Here, using rheology in conjugation with
high-speed optical imaging, we study the flow and local deformations in
various ST systems. We observe that with increasing applied stress, the smooth
flow changes into a spatio-temporally varying flow across the scaling-
crossover. We show that such fluctuating flow is associated with intermittent
dilatancy, shear band plasticity and fracture induced by system spanning
frictional contacts.

††preprint: APS/123-QED

##  I Introduction

A classic example of non-Newtonian flow behavior is the increase in viscosity
of dense particulate suspensions under increasing shear stress/shear-rate, a
phenomenon known as shear thickening (ST) [1, 2]. Depending on the particle
volume fraction ($\phi$), applied stress and nature of interparticle
interactions, dense suspensions show a mild or, drastic shear thickening. The
mild shear-thickening is known as the continuous shear thickening (CST), while
the drastic one is called the discontinuous shear thickening (DST).
Interestingly, as $\phi$ approaches the random close packing $\phi_{rcp}$
limit, many of these systems show a stress-induced transformation to a solid-
like shear jammed (SJ) state. Such transition has attracted significant recent
interest from both fundamental physics as well as, materials design
perspectives [3, 4, 5, 6, 7, 8, 9]. As SJ involves a sudden increase in shear
modulus as the material transforms to a solid-like state from a liquid-like
state, a natural question arises: Is SJ transition a non-equilibrium phase
transition? For granular systems, the increase in dynamic correlation
reflected in growing length and time scales near the jamming transition also
suggest a connection to phase transition and critical phenomenon [10]. In the
system of soft frictionless spheres, the finite size scaling collapse of
elastic moduli and number of contacts further strengthen such idea [11, 12].

For different ST and SJ systems, a very recent work [13] proposes a universal
scaling of viscosity over wide range of stress and volume fraction values.
This study highlights the importance of two fixed points, namely, isotropic
jamming point ($\phi_{0}$) and frictional jamming point ($\phi_{m}$) in
controlling such scaling behavior. Incorporating the concept of stress induced
frictional interaction as proposed in the Wyart-Cates (W-C) model [14, 15,
16], they expressed the viscosity $\eta$ of a shear thickening system as,

$\displaystyle\eta(\phi_{0}-\phi)^{2}\sim\left(\frac{1}{\phi_{0}-\phi_{m}}-\frac{f(\sigma)}{\phi_{0}-\phi}\right)^{-2}$
(1)

where $f(\sigma)=e^{-\sigma^{*}/\sigma}$ denotes the fraction of frictional
contacts, $\sigma^{*}$ is the onset stress for frictional interaction,
$\phi_{0}$ and $\phi_{m}$ represent the jamming volume fraction without and
with the complete frictional interaction, respectively. Such functional form
indicates that irrespective of the value of $\phi$, $\eta(\phi_{0}-\phi)^{2}$
as function of $x_{wc}=\frac{f(\sigma)}{\phi_{0}-\phi}$ should diverge at a
single point $x_{c}=\frac{1}{\phi_{0}-\phi_{m}}$ for $\phi<\phi_{0}$. However,
it is observed that $\eta(\phi_{0}-\phi)^{2}\sim
F\left(\frac{f(\sigma)}{\phi_{0}-\phi}\right)$ shows a clear variation in
diverging point for different $\phi$ values [13]. Incorporating some careful
modifications in Eq. 1, they overcome such discrepancy and derive an universal
scaling relation.

This scaling framework is tested numerically for a ST system of bidisperse
particles where a good agreement has been obtained [17]. Further, the scaling
variable can also be modified to incorporate the effect of orthogonal shear
perturbations [18]. Similar scaling relation has also been used for fractal
suspensions showing a direct transition to SJ from a flowing state under
increasing stress [19]. Such universal scaling theory seems to be promising
approach in combining all the theoretical, numerical and experimental works.
The change in power-law slope in the observed scaling curve [13] is attributed
to the crossover between two fixed jamming points, $\phi_{0}$ and $\phi_{m}$.
This suggest that the frictionless and frictional region belongs to two
different universality classes and the change in slope reflects a transition
from frictionless to frictional regime. Although such picture of stress-
induced transition from a frictionless to a frictional regime is interesting,
a direct experimental manifestation of such transition on the flow behavior is
yet to be explored.

Here we study the steady state flow behavior of two well characterized dense
suspensions using shear rheology and in-situ high resolution boundary imaging.
Using particle imaging velocimetry (PIV) technique, we map out spatio-temporal
flow and deformation of the system across the crossover of the universal
scaling curve. We observe that plasticity plays a major role in changing the
slope of the scaling curve.

Figure 1:  (a) and (c) Variation of viscosity $\eta$ with shear stress
$\sigma$ shows the ST properties of cornstarch particles dispersed in glycerol
and polystyrene particles dispersed in PEG 400 respectively for different
volume fraction $\phi$ as shown in the legends. Slope 1 is indicated by the
solid black line. The corresponding particle image is represented in the
inset. (b) and (d) Variation of WC scaling function $F$ with WC variable
$x_{WC}$ for different $\phi$. Inset (magnified plot): $F(x_{WC})$ does not
diverge at a single point for different $\phi$. Dashed lines represent the
expected point of divergence $x_{c}=7.5$ and $17.24$ for cornstarch and
polystyrene systems respectively. Figure 2:  (a) Variation of WC scaling
function $F$ with modified WC variable $x$ for cornstarch and polystyrene
systems at different $\phi$ as shown in legends. Two dashed lines represent
the divergence points $x_{c}$ for the corresponding system. (b) and (c)
Modified scaling function $H=\eta(g(\sigma,\phi))^{2}$ as function of new
scaling variable $\left|\frac{1}{x_{c}}-\frac{1}{x}\right|$ for cornstarch and
polystyrene system respectively at different $\phi$. The crossover between the
solid and dashed line (marked by the vertical dotted line) represents the
change in the magnitude of slope ($\beta$) from $2$ to $1.5$ and the arrow
indicates the crossover point. Figure 3:  Sample boundary images for dense
suspension of (a) CS at $\phi=0.54$ and (b) PS ($d=8\pm 4$ $\mu m$) at
$\phi=0.61$ with increasing $\sigma$ during the flow curve. The corresponding
$\sigma$ values are mentioned inside the images. Figure 4:  Variation of
normal stress $\sigma_{N}$ with shear stress $\sigma$ during the flow curve
for (a) CS system at $\phi=0.54$ and (b) PS system $\phi=0.61$. The
corresponding crossover point stress $\sigma_{c}$ is marked by the vertical
dashed line. Figure 5:  Velocity profiles at four different stress values,
during the flow curve, for dense suspensions of (a) CS particles at
$\phi=0.54$ and (b) PS particles at $\phi=0.61$. Each panel represents the
velocity profiles for one second, as shown in the color bars, and the
corresponding stress values are mentioned inside the panels.  Figure 6:
Space-time plot (in the y-t plane) of dimensionless velocity gradient (
magnitude of the velocity gradient normalized by the maximum value of gradient
magnitude as indicated by the color bar) across the shear gap at four
different stress values, as mentioned near each panel, for dense suspensions
of (a) CS particles at $\phi=0.54$ and (b) PS particles at $\phi=0.61$.

##  II Materials and Methods

The experiments are carried out for two standard shear thickening (ST) dense
suspensions,(i) Cornstarch (CS) particles (Sigma-Aldrich) with mean diameter
$d=15\pm 5$ $\mu m$ dispersed in glycerol and (ii) Polystyrene (PS) particles
dispersed in polyethylene glycol(PEG) 400. We synthesize PS particles of two
different size distributions ($d=8\pm 4$ $\mu m$ and $d=2.65\pm 0.13$ $\mu m$)
using the dispersion polymerization technique [5, 20]. The samples of
different volume fractions ($\phi$) are prepared by gradually adding the
required amount of dry particles into the solvent and thoroughly mixed at each
step using a spatula. The CS suspensions are then immediately used for
rheological measurement. In the case of of PS particles, the suspensions are
desiccated overnight to remove air bubbles trapped inside. To make sure an
uniform dispersion, the samples are ultrasonicated for 5 minutes (at room
temperature) just before loading for rheological measurements. The volume
fraction range used for CS system is $0.40\leq\phi\leq 0.54$ and that for PS
system is $0.50\leq\phi\leq 0.61$.

All the rheological measurements are performed on a stress control Rheometer
(MCR-702, Anton Paar, Austria) in twin drive 50-50 counter movement mode. For
the CS particle suspension the measurements are carried out using parallel
plate geometry of plate diameter $25$ $mm$ with $1$ $mm$ gap between the
plates and for PS particle suspensions cone and plate geometry with the cone
angle of $2$°and cone/plate of $25$ $mm$ diameter is used. In all cases, the
surface of the geometries is sandblasted to minimize the wall slippage. In
order to remove the loading history, if any, we conduct an oscillatory
amplitude sweep measurement at a frequency of 1 $Hz$ just after loading the
sample. For this, we first increase the strain amplitude ($\gamma$)
logarithmically from a lower to a maximum value and then the amplitude is
again gradually decreased to the initial value. For steady state flow curve
measurements, we increase the applied shear stress ($\sigma$) from a smaller
to maximum value logarithmically with waiting time per data point varying
between $20$ $s$ to $1$ $s$ between minimum and maximum stress, respectively,
and then perform a reverse run.

Using a high-speed in-situ imaging set up we obtain the sample boundary images
during the flow curve measurement. Since the samples are optically opaque, the
in-situ imaging of the boundary is possible only under reflection mode [21].
The sample boundary is illuminated using a LED light source (Dolan-Jenner
Industries) and image the flow gradient plane using a high-speed monochrome
CMOS camera (Phantom Miro C210) with $5X$ and $10X$ long working distance
objective (Mitutoyo). All the images are captured at a frame rate of $100$
$Hz$ with resolution $1280\times 1024$ and $1280\times 720$ pixels for
parallel plate and cone-plate geometries, respectively.

From the sample boundary images, we map out the flow profiles using particle
imaging velocimetry (PIV) technique. The PIV analysis is performed using
custom-written PIV codes developed in MATLAB software.

##  III  RESULTS AND DISCUSSION

The flow curves are obtained for dense suspensions of cornstarch and
polystyrene at different volume fractions ($\phi$) as shown in Fig.1(a) and
1(c) respectively. We observe an increase in viscosity ($\eta$) with
increasing shear stress ($\sigma$) beyond an onset stress indicating shear
thickening. The degree of shear thickening enhances with increasing $\phi$.
The black solid lines in Fig.1(a) and 1(c) having a power-law slope of 1
indicate that CS and PS systems show DST for $\phi\geq 0.48$ and $\phi\geq
0.58$, respectively. The respective onset stress values for CS and PS systems
are close to 1 Pa and 10 Pa. From the flow curves, using Krieger-Dougherty
(KD) equation and WC model we determine $\phi_{0}$, $\phi_{m}$, and
$\sigma^{*}$ for both of these systems to estimate the WC scaling variable and
scaling function mentioned in [13] (details are given in Appendix C1). Here,
we note that the $\sigma^{*}$ value obtained from W-C Model is higher than the
onset stress for shear thickening [5]. Fig.1(b) and 1(d) show the variation of
WC scaling function $F=\eta(\phi_{0}-\phi)^{2}$ with WC variable
$x_{WC}=\frac{f(\sigma)}{\phi_{0}-\phi}$. It shows that for different $\phi$,
$\eta(\phi_{0}-\phi)^{2}\sim F\left(\frac{f(\sigma)}{\phi_{0}-\phi}\right)$
diverges at different points (inset of Fig.1(b) and Fig.1(d)), whereas, from
equation 1, it is expected to diverge at $x_{c}=\frac{1}{\phi_{0}-\phi_{m}}$,
as marked by the dashed lines. In order to avoid this discrepancy, following
[13], Eq. 1 can be modified as,

$\displaystyle\eta(\phi_{0}-\phi)^{2}\sim\left(\frac{1}{\phi_{0}-\phi_{m}}-\frac{g(\sigma,\phi)}{\phi_{0}-\phi}\right)^{-2}$
(2)

where $g(\sigma,\phi)=C(\phi)f(\sigma)$ and $C(\phi)$ is a volume fraction
dependent parameter called the anisotropy factor. Now plotting
$\eta(\phi_{0}-\phi)^{2}$ as function of
$x=\frac{g(\sigma,\phi)}{\phi_{0}-\phi}$ gives an excellent collapse at the
diverging point $x_{c}=\frac{1}{\phi_{0}-\phi_{m}}$ [Fig.2(a)]. We would like
to point out that here we incorporate only the essential modification required
to collapse the diverging points [13].

Eq.2 suggests that, $\eta(\phi_{0}-\phi)^{2}\sim
F\left(\frac{g(\sigma,\phi)}{\phi_{0}-\phi}\right)$ should diverge with
exponent -2. For better visualization of the exponent, Eq.2 can be recast as
(see Appendix A),

$\displaystyle\eta(g(\sigma,\phi))^{2}\sim(\phi_{0}-\phi_{m})^{2}\left(\frac{1}{x}-\frac{1}{x_{c}}\right)^{-2}$
(3) $\displaystyle\eta(g(\sigma,\phi))^{2}\sim
H\left(\bigg{|}\frac{1}{x_{c}}-\frac{1}{x}\bigg{|}\right)$ (4)

Plotting the scaling function $H=\eta(g(\sigma,\phi))^{2}$ as a function of
scaling variable $\left|\frac{1}{x_{c}}-\frac{1}{x}\right|$, we can see that
the viscosity $\eta$ over a wide range of $\sigma$ and $\phi$ collapse into a
single curve, called universal scaling curve as shown in Fig.2(b) and 2(c) for
CS and PS systems, respectively. Interestingly, we observe that magnitude of
the slope of the scaling curve ($\beta$) that represents the exponent of
viscosity divergence, does not remain the same: $\beta$ decreases from 2 below
a certain value of scaling variable and approaches 1.5. Here we note that the
smaller value of the scaling variable
$\left|\frac{1}{x_{c}}-\frac{1}{x}\right|$ corresponds to the larger value of
$\sigma$ and vice versa. This suggests the change in the slope of the
universal scaling curve happens beyond a certain value of the $\sigma$ called
the crossover point stress ($\sigma_{c}$). The crossover points, where the
slope change starts, are marked by the red and green arrows and the
$\sigma_{c}$ for the CS system ($\phi=0.54$) and PS system ($\phi=0.61$) are
$11$ Pa and $67$ Pa, respectively. The change in magnitude of the power-law
slope from 2 to 1.5 is particularly evident for higher $\phi$ values (Fig.2(b)
and 2(c)), for which the flow curves can be obtained for larger stress values
before the sample yields, as shown in Fig.1(a) and 1(c).

In order to understand the physical origin of the change in slope in the
universal scaling curve, we use in-situ sample boundary imaging during the
rheological measurements [22, 23, 24, 20]. Fig.3 represents the boundary
images of the sample at different applied $\sigma$ (marked inside each panel)
for CS (Fig.3(a)) and PS (Fig.3(b)) system for $\phi=0.54$ and $\phi=0.61$,
respectively. During the flow curve measurement, we observe no change in
surface intensity till $\sigma<\sigma_{c}$ (panel (i) of Fig.3(a) and 3(b)).
However, for $\sigma>\sigma_{c}$ significant increase in the surface intensity
is observed (panel (ii) and (iii) of Fig.3(a) and 3(b)). Such change in
intensity arises due to the shear induced protrusion of the particle at the
suspension-air interface due to a mechanism called frustrated dilation [25,
26]. Shear-induced dilation in dense granular suspensions is related to the
shear-induced proliferation of interparticle frictional contacts spanning the
system [25, 26, 27, 20, 28, 29, 30]. The dilation starts at
$\sigma\approx\sigma_{c}$ (panel (ii) of Fig.3(a) and 3(b), Appendix movie 2
and 3). Similar dilation effect has been reported earlier for granular systems
in the shear thickening regime [25, 26]. We find from the imaging experiments
that for $\sigma\approx\sigma_{c}$ the dilation is intermittent in nature
[Appendix movie 1]. However, as $\sigma$ goes beyond $\sigma_{c}$ the dilation
becomes stable and system-spanning. On further increase in $\sigma$, we
observe the development of failures in the system in the form of boundary
fracture or shear-band plasticity (panel (iv) of Fig.3(a) and 3(b)).

To further characterize the flow regimes, we study the variation of normal
stress in the system under shear. For smaller applied stress values below
$\sigma_{c}$ the normal stress ($\sigma_{N}$) remains negligible, however,
after the crossover point ($\sigma\geq\sigma_{c}$) a significant positive
$\sigma_{N}$ starts to develop that increases with increasing $\sigma$
(Fig.4(a) and 4(b)). Such behavior of normal stress, together with the
appearance of dilation confirms the frictional nature of the flow above
$\sigma_{c}$ [31, 25, 26, 20]. Here we observe that the onset of dilation and
positive normal stress coincides with $\sigma_{c}$. On the other hand, dense
suspensions of smaller polystyrene particles (closer to colloidal regime)
require very high stresses $\sigma>>\sigma_{c}$ for the observation of
dilation and positive $\sigma_{N}$ (Appendix C2). This is due to the high
values of confining stresses (proportional to the inverse of particle size)
[32].

Next, we quantify the flow dynamics in the different scaling regimes using
Particle Image Velocimetry (PIV) technique. For a given applied stress
($\sigma$) we measure the average value of the velocity components ($v_{x}$)
parallel to the plate motion. The variation of $v_{x}$ across the gap ($y$)
between the shearing plates gives the velocity profiles in the flow-gradient
plane. Here, as per our convention for the twin drive 50-50 counter movement
mode, the $v_{x}$ close to the top plate ($y$ = 0) is positive, and the bottom
plate is negative (Appendix C3). Fig.5 represents the evolution of the
velocity profile with increasing $\sigma$ for both corn starch (CS) and
polystyrene (PS) suspensions (Fig.5(a) and 5.(b)) for $\phi=0.54$ and
$\phi=0.61$, respectively. For $\sigma<\sigma_{c}$, as shown in panel (i) of
Fig.5(a) and 5(b), the velocity profiles remain stationary in time and the
nature remains close to a linear profile. The position of the zero velocity
plane ($v_{x}=0$) appears near the middle of the gap between the two plates.
Interestingly, when $\sigma$ approaches $\sigma_{c}$ the velocity profiles
lose the stationary behavior showing random shifting in time (Panel (ii) in
Fig.5(a) and 5(b)). Here, the zero velocity plane starts shifting towards one
of the plates. Similar to the dilation event at $\sigma\approx\sigma_{c}$, the
velocity profiles are also spatio-temporally fluctuating in this stress
regime. However, they become more stable with increasing $\sigma$. At
$\sigma>>\sigma_{c}$ (Panel (iii) and (iv) in Fig.5(a) and 5(b)) we observe
shear band plasticity and fracture in the system (Appendix Movies 2 and 3). We
find that the nature of shear banding may differ from system to system. For
the CS system (Panel (iii) and (iv) in Fig.5(a)), the entire sample moves in
the direction opposite to the direction of motion of the top plate, whereas,
the PS system shows two band structures moving opposite to each other (Panel
(iii) and (iv) in Fig.5(b)). The density matching of the system (CS is not
density matched with the solvent but, PS is) might play a role in creating
such differences.

We now quantify the spatio-temporal fluctuations in the velocity profile with
increasing applied stress, as mentioned above. Fig.6 shows the space-time plot
(STP) of the magnitude of the normalized velocity gradient across the shear
gap. The maximum magnitude of the gradient corresponding to a particular
applied stress ($\sigma$) is considered to be the normalization factor. Thus,
in our case the normalized gradient always remains between 0 and 1. We find
that for the lower value of $\sigma$ ($<\sigma_{c}$), the spatio-temporal
fluctuations of velocity gradient are negligible, as evident from the uniform
colour in panel(i) of Fig.6(a) and 6(b). For $\sigma\approx\sigma_{c}$ when
$\sigma$ approaches 11 Pa and 67 Pa in CS and PS systems, respectively, the
spatio-temporal fluctuations get significantly enhanced as reflected by the
appearance of patterns in the STP (panel(ii) of Fig.6(a) and 6(b)). For
sufficiently high $\sigma$ ($>>\sigma_{c}$), the spatio-temporal fluctuation
increases further leading to the shear band and plasticity and fracture in the
system (panel(iii) and (iv) of Fig.6(a) and 6(b)). These observations point
out that as the frictional contacts proliferate in the system with increasing
$\sigma$, the flow becomes more and more heterogeneous leading to material
failure. As mentioned earlier, the details of the plasticity and failure
depends on the nature of the sample. We find that for CS system, the
enhancement of fluctuations continues above $\sigma_{c}$. However, for PS
system, just above $\sigma_{c}$ we find a clear fracture when the sample
splits into two different parts moving in two opposite directions with the
shearing plates. Each part has negligible gradient as shown in panel (iii) and
(iv) of Fig.6(b). Such differences are more clearly visible in the Appendix
movies 2 and 3.

##  IV  Conclusions

In summary, we study the spatio-temporal flow behavior of two well-studied
shear-thickening granular suspensions, in relation to the recently reported
universal scaling behavior in shear-thickening systems. Similar to the earlier
study, we also observe the power-law scaling in these systems with a cross-
over point marking a stress-induced transition from frictionless to frictional
jamming regime. Remarkably, our in-situ boundary imaging reveals clear
signatures of frictional flow, such as, dilation, positive normal stress,
above the critical applied stress corresponding to the scaling cross-over.
Thus, our study experimentally verifies the transition from lubrication to
frictional regime as predicted by the universal scaling theory for shear-
thickening systems [13, 17]. However, we find that such one to one correlation
exists only for granular suspensions. Close to colloidal regime, significant
dilation takes place for stress values much beyond the onset stress for shear-
thickening due to higher confining stresses in these systems. Thus, from
boundary imaging, visible signature of plasticity and scaling cross-over point
do not match in this case [Appendix C2]. Deciphering the role of plasticity in
controlling the scaling cross-over point for colloidal systems remains an
interesting future direction to explore. Our observation highlights the role
of plasticity in reducing the slope of the scaling curve from -2 towards -1.5.
Such change in slope may be linked to the the plasticity induced softening of
viscosity divergence in shear jammed dense suspensions [5], where the
viscosity-divergence exponent close to jamming is given by a Krieger-Dougherty
type relation. However, the generality of such connection needs be tested for
a wider range of shear-thickening systems.

##  V  Acknowledgments

S.M. thanks SERB (under DST, Govt. of India) for a Ramanujan Fellowship. We
acknowledge Ivo Peters for developing the MATLAB codes used for PIV analysis,
K.M. Yatheendran for helping with the SEM imaging. S.M. thanks A.K. Sood for
helpful discussion.

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## Appendix A

Using Wyart-Cates (W-C) model the viscosity $\eta$ of a shear thickening
system can be expressed as

$\eta(\phi_{0}-\phi)^{2}\sim\left(\frac{1}{\phi_{0}-\phi_{m}}-\frac{f(\sigma)}{\phi_{0}-\phi}\right)^{-2}$
(5)

where $f(\sigma)=e^{-\sigma^{*}/\sigma}$ denotes the fraction of frictional
contacts, $\sigma^{*}$ is the onset stress for frictional interaction,
$\phi_{0}$ is the isotropic jamming point and $\phi_{m}$ is the frictional
jamming point representing the jamming volume fraction without and with the
complete frictional interaction, respectively.

After incorporating careful modifications, such as, multiplying $f(\sigma)$
with
$C=\left(\frac{\frac{1}{\phi_{0}-\phi_{m}}}{\frac{f(\sigma)}{\phi_{0}-\phi}}\right)$,
the equation (A1) becomes:

$\eta\sim(\phi_{0}-\phi)^{-2}\left(\frac{1}{\phi_{0}-\phi_{m}}-\frac{Cf(\sigma)}{\phi_{0}-\phi}\right)^{-2}$
(6)

$\eta\sim\left(\frac{\phi_{0}-\phi}{\phi_{0}-\phi_{m}}-{Cf(\sigma)}\right)^{-2}$

Considering $g(\sigma,\phi)=Cf(\sigma)$

$\eta(g(\sigma,\phi))^{2}\sim\left(\frac{1}{Cf(\sigma)}\frac{\phi_{0}-\phi}{\phi_{0}-\phi_{m}}-1\right)^{-2}$

Taking common $\frac{1}{\phi_{0}-\phi_{m}}$

$\eta(g(\sigma,\phi))^{2}\sim(\phi_{0}-\phi_{m})^{2}\left(\frac{\phi_{0}-\phi}{Cf(\sigma)}-(\phi_{0}-\phi_{m})\right)^{-2}$
(7) $\sim(\phi_{0}-\phi_{m})^{2}\left(\frac{1}{x}-\frac{1}{x_{c}}\right)^{-2}$
$\eta(g(\sigma,\phi))^{2}\sim
H\left(\bigg{|}\frac{1}{x_{c}}-\frac{1}{x}\bigg{|}\right)$ (8)

##  Appendix B Movie description

Movie 1 explains the intermittent nature of the dilation around the crossover
point stress ($\sigma_{c}\approx 60$ $Pa$) for polystyrene ($d=8\pm 4$ $\mu
m$) dense suspension at $\phi=0.58$. The dilation phenomena are reflected in
the increase in brightness of the sample boundary image. As the dilation is
intermittent in nature around $\sigma_{c}$, this brightness disappears under
shear and again reforms.

Movies 2 and 3 explain the flow and deformation behavior of cornstarch dense
particulate suspension at $\phi=0.54$ and polystyrene ($d=8\pm 4$ $\mu m$)
dense particulate suspension at $\phi=0.61$ respectively. In movies 2 and 3,
we have shown the sample boundary images and corresponding velocity vector
images simultaneously for three different stress values at three different
regimes of the scaling curve, (i) before the crossover point (ii) during the
crossover point (iii) after the crossover point. For the lower stress, before
the crossover point, the flow behavior of the sample is almost Newtonian as
observed from the velocity vector images, and no change in sample surface
intensity is observed. But during the crossover point stress ($\sigma_{c}$)
the flow profile starts distorting in addition to the visible change in sample
surface intensity due to dilation. At higher stress, beyond the crossover
point, plasticity starts dominating.

All the images are captured using a CMOS camera (Phantom Miro C210) with a
$5X$ long working distance objective (Mitutoyo). The images are captured at a
frame rate of 100Hz with resolution $1280\times 720$ $pixels$ for movies 1 and
3, and $1280\times 1024$ $pixels$ for movie 2.

##  Appendix C Figures

C1:  (a) and (b) Determination of $\phi_{o}$, as marked by vertical dashed
lines, from the Krieger Dougherty (KD) relation fitting for cornstarch (CS)
and polystyrene (PS) system respectively. (c) and (d) Variation of jamming
volume fraction $\phi_{J}$ with $\sigma$ for CS and PS system. The red solid
line indicates the fit to the Wyart-Cates(WC) model to get the parameters
$\phi_{m}$ and $\sigma^{*}$. The top and bottom horizontal dashed lines
indicate the obtained $\phi_{0}$ and $\phi_{m}$ values. For CS system
$\phi_{0}$ =0.64 , $\phi_{m}$=0.51 and $\sigma^{*}=4.5$ $Pa$ ; for PS system
$\phi_{0}=0.65$, $\phi_{m}$ = 0.59 and $\sigma^{*}=59$ $Pa$. C2: (a) Scaling
curves for dense suspension of small polystyrene particles of diameter
$d=2.65\pm 0.13$ $\mu m$ dispersed in PEG 400 at different volume fractions as
shown in the legend. The black solid line and red dashed line represent the
slope $-2$ and $-1.5$ respectively. The crossover point is marked by the green
arrow. (b) Variation of normal stress $\sigma_{N}$ with shear stress $\sigma$
during flow curve for small polystyrene system ($\phi=0.58$). The
corresponding cross-over stress $\sigma_{c}=167Pa$ is marked by the vertical
dashed line. (c) Sample boundary images at four different stress values for
$\phi=0.58$. Panel (iii) and (iv) represents the onset of dilation (at
$\sigma=5657$ $Pa$ ) and fracture (at $\sigma=13129$ $Pa$) respectively which
are found deep inside the regime of slope $-1.5$ as shown in (a). C3: Typical
PIV window for sample boundary image with top and bottom plate moving at
velocity $+v_{x}$ and $-v_{x}$ respectively. Yellow arrows represent the
velocity vector parallel to the plate (obtained from PIV). ‘d’ is the gap
between the two plates.