1

# Universal scaling of shear thickening transitions

Meera Ramaswamy  Department of Physics, Cornell University, Ithaca, New York
14853, USA  Itay Griniasty  Department of Physics, Cornell University, Ithaca,
New York 14853, USA  Danilo B. Liarte  Department of Physics, Cornell
University, Ithaca, New York 14853, USA  Abhishek Shetty  Anton Paar USA,
10215 Timber Ridge Drive, Ashland, Virginia 23005  Eleni Katifori  Department
of Physics, University of Pennsylvania, Philadelphia, Pennsylvania  Emanuela
Del Gado  Department of Physics, Georgetown University, Washington DC, USA
James P Sethna  Department of Physics, Cornell University, Ithaca, New York
14853, USA  Bulbul Chakraborty  Department of Physics, Brandeis University,
Waltham, Massachusetts, USA  Itai Cohen  Department of Physics, Cornell
University, Ithaca, New York 14853, USA

###### Abstract

Nearly all dense suspensions undergo dramatic and abrupt thickening
transitions in their flow behaviour when sheared at high stresses. Such
transitions occur when the dominant interactions between the suspended
particles shift from hydrodynamic to frictional. Here, we interpret abrupt
shear thickening as a precursor to a rigidity transition and give a complete
theory of the viscosity in terms of a universal crossover scaling function
from the frictionless jamming point to a rigidity transition associated with
friction, anisotropy, and shear. Strikingly, we find experimentally that for
two different systems – cornstarch in glycerol and silica spheres in glycerol
– the viscosity can be collapsed onto a single universal curve over a wide
range of stresses and volume fractions. The collapse reveals two separate
scaling regimes, due to a crossover between frictionless isotropic jamming and
frictional shear jamming, with different critical exponents. The material-
specific behaviour due to the microscale particle interactions is incorporated
into a scaling variable governing the proximity to shear jamming that depends
on both stress and volume fraction. This reformulation opens the door to
importing the vast theoretical machinery developed to understand equilibrium
critical phenomena to elucidate fundamental physical aspects of the shear
thickening transition.

††preprint: AIP/123-QED

##  I Introduction

Suspensions of solid particles in a liquid commonly exhibit shear thickening,
an increase in the shear viscosity with the applied stress or strain rate
Brown  _et al._ (2010); Wagner and Brady (2009). This increase in viscosity
can span orders of magnitude and even lead to solidification, as illustrated
by people running atop vats of cornstarch Brown and Jaeger (2014) 111Search
www.youtube.com for walking on cornstarch. Such strong shear thickening
behaviour has been attributed to the change in the nature of particle
interactions – at low stresses, the particle interactions are dominated by the
lubrication forces but at high stresses, the particle surfaces are forced
closer together and frictional contact interactions dominate Lin  _et al._
(2015); Wyart and Cates (2014); Cates and Wyart (2014); Lootens  _et al._
(2005); Seto  _et al._ (2013); Brown and Jaeger (2009, 2014); Wyart and Cates
(2014).

Understanding the mechanism underpinning the thickening transition has led to
numerous studies focused on altering the system properties to modify the shear
thickening behaviour. Commonly, two different strategies have been pursued –
1) changing the interparticle friction and 2) changing the particle
microstructure or suspension packing. Changes to the interparticle friction
can be achieved by altering the particle surface roughness Hsiao  _et al._
(2017); Hsu  _et al._ (2018, 2021); Hoffman (1998); Jamali and Brady (2019);
More and Ardekani (2020); Lootens  _et al._ (2005), interparticle adhesion
Bourrianne  _et al._ (2022), or hydrogen bonding Jaishankar  _et al._ (2015);
James  _et al._ (2018, 2019a). The maximum particle packing in the suspensions
can be altered by tuning the particle shape Brown  _et al._ (2011); James  _et
al._ (2019b); Rathee  _et al._ (2020), size or polydispersity Guy  _et al._
(2020); Bender and Wagner (1996), or applying external fields Ness, Mari, and
Cates (2018); Niu  _et al._ (2020); Lin  _et al._ (2016); Sehgal  _et al._
(2019); Gillissen  _et al._ (2020); Chen  _et al._ (2022); Jackson  _et al._
(2022). In many of these examples, even a small change in the suspension can
result in large changes to the rheology. It has therefore been a major
challenge to develop a unified, suspension-independent framework to predict
shear thickening.

A major step towards developing such a framework was proposed by Wyart and
Cates Wyart and Cates (2014); Cates and Wyart (2014), who modelled the
viscosity as a function of the distance to a stress-dependent jamming volume
fraction. As the stress increases, the contacts between the particles become
frictional and the jamming volume fraction decreases, resulting in an increase
in the viscosity. This theory has been used to fit experimental and simulation
data with varying degrees of success Singh  _et al._ (2018); Guy  _et al._
(2020); Rathee  _et al._ (2020); Royer, Blair, and Hudson (2016); Lee  _et
al._ (2020). Others have proposed similar models to describe shear thickening
Singh  _et al._ (2018); Gillissen  _et al._ (2019, 2020), and more recently,
the extent of shear thickening has also been shown to be related to the
jamming volume fraction Pradeep, Jacob, and Hsiao (2020), indicating that
jamming underpins the shear thickening transition. Despite these close
connections between shear thickening, and jamming – a well-established
equilibrium phase transition – the relationship between the shear thickening
and statistical scaling frameworks remains unexplored. More specifically, it
is yet unclear if shear thickening can be described by the proximity to the
jamming critical point via universal scaling functions and if jamming with and
without shear are associated with the same universality class and scaling
exponents.

In this work, we adopt the idea of scaling to analyze the experimentally
observed viscosity of two different shear thickening suspensions and establish
the relationship between shear thickening and the associated critical points.
Since the advent of the ideas of universality and scaling in equilibrium phase
transitions Cardy (1996), it has been known that scaling analysis provides a
litmus test for the presence of critical points in complex phase diagrams.
This approach led to the discovery of non-classical exponents associated with
phase transitions, the identification of universality classes, and the
development of the renormalization group Cardy (1996). In contrast to fitting
data to functional forms, scaling involves collapsing data over a broad range
of multiple control parameters. The advantage of this approach is that it is
model-independent and often reveals the underlying physical governing
principles, the relevant variables controlling the distance to critical points
and the associated scaling functions. Here we propose to use the same
machinery to investigate the critical points associated with the shear
thickening transition.

More specifically, we pursue the idea that thickening is governed by two
different critical points and that a crossover scaling framework can be
utilized to characterize this transition Cardy (1996). Crossover scaling was
originally introduced to describe transitions between thermodynamic critical
points (e.g., Heisenberg magnets with small uniaxial anisotropy behaving like
Ising models), and has become invaluable for describing finite-temperature
behaviour induced by quantum critical points Sachdev (2011, 1997, 1999),
crossovers between universality classes in random matrix theory Adam  _et al._
(2002), and fracture and depinning transitions Chen, Zapperi, and Sethna
(2015). Our analysis shows that the same framework provides an excellent
unified description of thickening transitions.

##  II  Recasting the Wyart and Cates model

We begin by recasting the Wyart and Cates Wyart and Cates (2014) model, the
current state-of-the-art model used to describe shear thickening, into the
framework of crossover scaling. Briefly, the model expresses the viscosity of
the suspension in terms of a distance to a stress dependent jamming volume
fraction, $\phi_{J}(\sigma)$,

$\eta\sim(\phi_{J}(\sigma)-\phi)^{-2}$ (1)

As the stress increases, the nature of the contact forces between the
particles in the suspension changes from hydrodynamic to frictional, and the
jamming volume fraction decreases as:

$\phi_{J}(\sigma)=\phi_{0}(1-f(\sigma))+\phi_{\mu}f(\sigma).$ (2)

Here, $f(\sigma)$ is the fraction of frictional contacts in the system,
$\phi_{0}$ is the jamming volume fraction in the absence of friction, and
$\phi_{\mu}$ is the jamming volume fraction when all the interactions between
the particles are frictional ($\phi_{\mu}<\phi_{0}$). $f(\sigma)$ is a
sigmoidal function of the applied stress, with limits of zero at low stress
and one at high stress, and Eq. 2 defines a line of critical points at which
the viscosity diverges leading to shear-induced jamming at finite $\sigma$.
Thus, increasing the stress activates frictional contacts which lowers the
jamming volume fraction and increases the viscosity. Despite its simplicity,
this model does a remarkable job capturing essential features of the flow
behaviours including continuous shear thickening, discontinuous shear
thickening, and shear jamming.

Figure 1:  Viscosity versus stress measurements and scaling predictions from
the Wyart and Cates model. ab. The viscosity as a function of applied stress
for a cornstarch suspensions with volume fractions ranging from 0.16 (yellow)
to 0.54 (dark blue) and b silica suspensions with volume fractions ranging
from 0.4 (light pink) to 0.52 (black). These suspensions show a range of shear
thickening behaviour across volume fractions from weakly shear thickening to
discontinuous shear thickening to shear jamming. c, d. Plots of the predicted
Wyart and Cates scaling function $\mathcal{F_{WC}}=\eta(\phi_{0}-\phi)^{2}$
versus the Wyart and Cates scaling variable $x_{WC}=f(\sigma)/(\phi_{0}-\phi)$
for cornstarch suspensions (c) and silica suspensions (d), showing promising
but incomplete scaling collapse. Inset shows the zoom-in of the high $x_{WC}$
region. In the inset of c, the cornstarch scaling function diverges at $\sim$
10 at low volume fractions, $\sim$ 6 at intermediate volume fractions and
$\sim$ 10 at large volume fractions. The silica data in the inset of d
diverges at $\sim$ 10 at low volume fractions and $\sim$ 20 at large volume
fractions.

To recast this model into the crossover scaling framework, we substitute Eq. 2
into Eq. 1 to obtain:

$\eta\sim(\phi_{0}(1-f(\sigma))+\phi_{\mu}f(\sigma)-\phi)^{-2}.$ (3)

Pulling out a factor of $(\phi_{0}-\phi)^{-2}$, we find that the viscosity can
be expressed as a function that only depends on a specific combination of
$f(\sigma)$ and $\phi_{0}-\phi$:

$\eta(\phi_{0}-\phi)^{2}\sim\mathcal{F}_{WC}\left(\frac{f(\sigma)}{\phi_{0}-\phi}\right)$
(4)

where the crossover scaling function $\mathcal{F}_{WC}$, specific to the Wyart
and Cates model is

$\mathcal{F}_{WC}\sim\left(\frac{1}{\phi_{0}-\phi_{\mu}}-\frac{f(\sigma)}{\phi_{0}-\phi}\right)^{-2}.$
(5)

At small values of the scaling variable, $x_{WC}=f(\sigma)/(\phi_{0}-\phi)$,
the scaling function is a constant and the system behaviour is governed by the
frictionless jamming critical point $\eta\sim(\phi_{0}-\phi)^{-2}$. Crucially,
however, the crossover scaling function $\mathcal{F}_{WC}$ has a divergence at
$x_{c}=1/(\phi_{0}-\phi_{\mu})$ indicating that as $x_{WC}/x_{c}\rightarrow 1$
the system is governed by a line of frictional jamming critical points such
that $\eta\sim(x_{c}-x_{WC})^{-2}$. Notably, with this form of the crossover
scaling function, the viscosity diverges with exactly the same exponent of
$-2$ all along the jamming line. This recasting of the Wyart and Cates model
for the thickening transition in terms of crossover scaling clearly indicates
that a major assumption in the model is that frictionless and frictional shear
jamming (at nonzero $x_{WC}$) are controlled by the same fixed point and that
the only effect of friction is to change the location of the critical point.

Practically, this formulation enables us to move beyond merely fitting the
model to viscosity data, and instead attempt a scaling collapse to elucidate
the underlying physics controlling this transition. More specifically, Eq. 4
suggests that plotting $\eta(\phi_{0}-\phi)^{2}$ as a function of
$f(\sigma)/(\phi_{0}-\phi)$ should collapse the viscosity across various
stresses and volume fractions onto a universal curve, revealing the scaling
function and its singularities. This collapse will also allow us to determine
whether thickening is indeed controlled by a unique scaling exponent.

Figure 2:  Universal scaling of the suspension viscosity. a. The scaling
function $\mathcal{F}=\eta(\phi_{0}-\phi)^{2}$ as a function of the scaling
variable $x=e^{-\sigma^{*}/\sigma}C(\phi)/(\phi_{0}-\phi)$ for all the
cornstarch (squares) and silica (diamond) suspensions data. We find that all
the data collapse onto a single universal curve that diverges at $x=x_{c}$. b.
The scaling function $\mathcal{H}=\eta g^{2}(\sigma,\phi)$ versus
$|1/x_{c}-1/x|$ for all the cornstarch and silica suspensions data. This way
of scaling the data clearly illustrates two distinct regimes - a regime
characterized by a power law of -2 at small $x$ and a power law of $\sim-3/2$
at $x\approx x_{c}$. The solid black lines indicate power laws of $-2$ and
$-1.5$ respectively. c The non-linear scaling variable, $g(\sigma,\phi)$ as a
function of the stress for a range of volume fractions of the cornstarch
suspensions. We note that this parameter has the same sigmoidal shape, now
well established for the fraction of frictional contacts, while $C(\phi)$
dramatically affects the overall scale. d. The anisotropy factor $C(\phi)$ as
a function of the volume fraction for both silica and cornstarch. We note that
$C(\phi)$ is a smooth analytic function as required by theories for scaling
variables such as $g(\sigma,\phi)=f(\sigma)C(\phi)$ far from the critical
point.

##  III  Experimental Methods

We test this scaling theory on two different non-inertial, low Reynolds number
systems - a mixture of cornstarch in glycerol and a mixture of hard sphere
silica particles in glycerol both of which show shear thickening behaviour but
over different ranges of volume fractions. The samples were prepared by
weighing out the solutes - cornstarch (Argo) and silica (2 $\mu$m charge
stabilized spheres from Angstrom Sphere) and the solvent - glycerol (Sigma-
Aldrich). We use glycerol as the solvent in each of these cases because of its
low vapour pressure and the ease with which these glycerol-based suspensions
can be loaded onto the rheometer. The cornstarch suspensions were used
immediately after preparation and the silica suspensions were sonicated for 60
minutes prior to use. The viscosity of the suspension is measured using a
stress-controlled method and a parallel plate geometry, with a set gap of 1
mm, on an Anton Paar MCR 702 rheometer. The sample was presheared at a
constant stress of 1 Pa for five minutes. The suspension viscosity was then
measured by performing a descending stress ramp.

The viscosity of the cornstarch and silica suspensions of different volume
fractions as a function of the stress are shown in Figs. 1a and 1b
respectively. The low volume fraction data (yellow and light pink data in
Figs. 1a and 1b respectively) have smaller viscosities and mild shear
thickening, or continuous shear thickening (CST). The intermediate volume
fraction data (teal and purple data in Figs. 1a and 1b respectively) have
larger viscosities and show discontinuous shear thickening (DST), where
$d\log(\eta)/d\log(\sigma)\geq 1$.

##  IV  Results

To collapse the viscosity using Eq. 5, we need to determine the isotropic
jamming fraction, $\phi_{0}$, and the fraction of frictional contacts,
$f(\sigma)$. We determine $\phi_{0}$ from the divergence of the low stress
viscosity (see SI for details). We use the form
$f(\sigma)=e^{-\sigma^{*}/\sigma}$ for the fraction of frictional contacts,
which is consistent with fits of the Wyart and Cates model in prior literature
Guy, Hermes, and Poon (2015, 2015); Lee  _et al._ (2020). We have explored a
limited set of different expressions for $f(\sigma)$, none of which
qualitatively change the results reported here (see SI for results with a
different functional form of $f$). By fitting the flow curves in Fig. 1a,b to
the Wyart and Cates model, we determine $\sigma^{*}$. Using the calculated
values of $f(\sigma)$, and $\phi_{0}$, we plot
$\mathcal{F}_{WC}=\eta(\phi_{0}-\phi)^{2}$ as a function of
$x_{WC}=e^{-\sigma^{*}/\sigma}/(\phi_{0}-\phi)$ in Figs. 1c and d. While we
find that the collapse is promising, at higher volume fractions, the data
diverge at different values of $x_{WC}$ (Fig. 1 insets). Since this
reformulation of the Wyart and Cates model in the language of crossover
scaling assumes a very particular form of the scaling function,
$\mathcal{F}_{WC}$, and the scaling variable, $x_{WC}$, we naturally ask
whether relaxing these assumptions and using the full machinery of crossover
scaling leads to better data collapse and a more accurate description of the
observed thickening transitions.

We find that using a scaling variable where the numerator is both a function
of stress and volume fraction; $x=g(\sigma,\phi)/(\phi_{0}-\phi)$ dramatically
improves the scaling collapse (Fig. 2). To simplify the search for the
function, $g(\sigma,\phi)$, we assume a product form,
$g(\sigma,\phi)=C(\phi)f(\sigma)$. Impressively, this single parameter,
$C(\phi)$, for the data set at each volume fraction collapses both the
cornstarch and silica suspension data across all measured volume fractions
onto a single universal scaling function $\mathcal{F}$ as shown in Fig.
2a222We find that the scaling functions for cornstarch silica differ by a
multiplicative factor of $\sim$2, which simply reflects a different solvent
viscosity..

Figure 3:  Phase diagrams for cornstarch and silica suspensions as derived
from the scaling analysis. Three distinct regions are seen in the phase
diagrams for the cornstarch a and silica b systems- continuous shear
thickening (CST) in purple, discontinuous shear thickening (DST) in blue and a
jammed region in red. The shear jamming line (maroon) is determined by
$x=x_{c}$, where $x=e^{-\sigma^{*}/\sigma}C(\phi)/(\phi_{0}-\phi)$ and the DST
line (blue) is determined by the condition $d\log\eta/d\log\sigma=1$. The
vertical dotted lines indicate the values of frictionless jamming point,
$\phi_{0}$, and shear jammed point, $\phi_{\mu}$, and the volume fraction at
which $C(\phi)$ is maximum.

The scaling function has several characteristic features that are consistent
with the expected behaviour for shear thickening suspensions. At small $x$,
$\mathcal{F}$ is a constant, and the divergence in the viscosity is:

$\eta\sim\frac{1}{(\phi_{0}-\phi)^{2}}.$ (6)

which is consistent with a number of previous studies Singh  _et al._ (2018);
Guazzelli and Pouliquen (2018); Morris and Boulay (1999); Guy  _et al._
(2020). As $x$ increases, $\mathcal{F}$ also increases, until it diverges at
$x=x_{c}$, with an exponent $\delta$:

$\mathcal{F}(x)\sim\frac{1}{(x_{c}-x)^{\delta}},$ (7)

with $x_{c}=14.5$. We note that since $C(\phi)$ is a multiplicative factor,
there is an overall scale that can be chosen for $x_{c}$. This divergence
implies that the viscosity also diverges with the exponent $\delta$.
Remarkably, we find that $\delta<2$ and is significantly different from the
exponent observed for frictionless jamming (See SI, Fig. S3). To visualize
this change in exponents, we follow Cardy (Cardy, 1996, Sect. 4.2) and write:

$\eta g^{2}(\sigma,\phi)\sim\mathcal{H}\left(|1/x_{c}-1/x|\right).$ (8)

where $\mathcal{H}$ is a universal scaling function. We then plot $\eta
g^{2}(\sigma,\phi)$ as a function of $|1/x_{c}-1/x|$ in Fig. 2b. We find
excellent collapse over seven orders of magnitude in the scaling variable with
two easily distinguishable regimes each characterized by clearly different
power-law exponents. At small $x$, far from $x_{c}$, the behaviour is governed
by the frictionless jamming point $\phi_{0}$ and
$\mathcal{H}\sim|1/x_{c}-1/x|^{-2}$. As $x$ approaches $x_{c}$, we observe a
clear change in the value of the exponent from -2, with a crossover between
the two regimes at $x/x_{c}\sim$ 0.1. Our best estimate for the new exponent
is -3/2. Importantly, this change in exponent indicates that the crossover
underlying shear thickening is between critical points that belong to
different universality classes. As such, this change in exponent is a
remarkable demonstration of frictional shear-jamming being qualitatively
different from frictionless jamming at zero shear.

Importantly, the nonlinear scaling variable that drives the suspension towards
frictional jamming, $g(\sigma,\phi)$, depends on both the stress and the
volume fraction as shown in Fig. 2c. $g(\sigma,\phi)$ is sigmoidal in the
stress, similar to the fraction of frictional contacts, $f(\sigma)$, in
previous works Wyart and Cates (2014); Guy  _et al._ (2020); Royer, Blair, and
Hudson (2016). The volume fraction dependence changes the overall scale and
indicates that the fraction of frictional contacts that contribute to and
determine the shear viscosity varies with $\phi$. The scaling collapse of the
data reveals that this functional dependence, $C(\phi)$, is non-monotonic for
both the silica and cornstarch suspensions (Fig. 2d). Once these material
dependent differences in $C(\phi)$ and $\phi_{0}$ are accounted for, the
collapse is universal.

Since the scaling form (Eq. 8) is valid for both suspensions at all stresses
and volume fractions, we can use it to construct a shear thickening phase
diagram as shown in Fig. 3 333To construct this phase diagram we use an
extended data set with data closer to $x_{c}$. See SI for more details.
Crucially, constructing these phase diagrams is only possible because we have
determined the full functional form of the scaling function over seven orders
of magnitude in the scaling variable, which is not typical for other analyses.
In particular, we plot the shear jamming boundary at $x=x_{c}$ (border of red
region) where the system transitions from a flowing to a jammed state. Since
this border is defined by the scaling variable $x_{c}$, it is independent of
the details of the scaling function $\mathcal{F}$(shape, functional form
etc.). In addition, the boundary for discontinuous shear thickening is
determined from:

$\frac{d\log(\eta)}{d\log\sigma}=1,$ (9)

which does depend on the form of the scaling function. To obtain this boundary
(blue lines), we fit $\mathcal{H}$ to a function consisting of two power law
regimes stitched together by a crossover region (See SI), and use this fit to
compute the derivatives in Eq. 9. This boundary indicates the transition from
continuous to discontinuous shear thickening regimes. Due to the form of the
anisotropy factor $C(\phi)$, and differences in $\phi_{0}$ and $\sigma^{*}$,
the phase diagrams for cornstarch and silica are distinct, and are shown in
Figs. 3a and b respectively. These phase diagrams are qualitatively similar to
those obtained by previous experiments and simulations Peters, Majumdar, and
Jaeger (2016); Seto  _et al._ (2013); Bi  _et al._ (2011) but are generated
directly from the scaling collapse of the data.

##  V  Discussion

###  V.1  Altering the scaling variable to change material properties

A key modification that enabled scaling collapse of the data is the non-linear
scaling variable, $g(\sigma,\phi)=f(\sigma)C(\phi)$, that depends on both the
stress and the volume fraction. Previous studies have interpreted $f(\sigma)$
as the fraction of frictional contacts. If we retain this interpretation, then
the inclusion of $C(\phi)$ in $g(\sigma,\phi)$ suggests that only a portion of
$f(\sigma)$ contributes to the viscosity divergence. Such a modulation of
$f(\sigma)$ is indicative of the role of force network connectivity or
anisotropy in the viscosity divergence consistent with recent simulations
Gameiro  _et al._ (2020); Singh  _et al._ (2020); Gillissen  _et al._ (2020)
and models Gillissen  _et al._ (2019, 2020); Baumgarten and Kamrin (2019).

Interpreting $g(\sigma,\phi)$ in this manner enables us to envision how we can
alter the shear thickening phase diagram. In particular, by modifying material
properties corresponding to changes in $\sigma^{*}$, $\phi_{0}$, and $C(\phi)$
we can dramatically shift the borders between the jammed and unjammed regimes
and by extension the discontinuous versus continuous shear thickening. A
change in $\sigma^{*}$, for example, indicates how easily frictional
interactions increase with stress and can be controlled by altering particle
roughness Hsu  _et al._ (2018); Hsiao  _et al._ (2017); Hsu  _et al._ (2021);
Hoffman (1998); Jamali and Brady (2019); More and Ardekani (2020); Lootens
_et al._ (2005), hydrogen bonding Jaishankar  _et al._ (2015); James  _et al._
(2018, 2019a), or solvent-particle interactions van der Naald  _et al._
(2021). Changes to $\phi_{0}$ can be generated by modifying particle roughness
Hsu  _et al._ (2018); Hsiao  _et al._ (2017); Hsu  _et al._ (2021); Hoffman
(1998); Jamali and Brady (2019); More and Ardekani (2020); Townsend and Wilson
(2017), shape Brown  _et al._ (2011); James  _et al._ (2019b); Rathee  _et
al._ (2020), and polydispersity Guy  _et al._ (2020); Bender and Wagner
(1996). Finally, differences in $C(\phi)$ may result from the constraints
governing particle displacements and rotations as well as other perturbations
to the flows Singh  _et al._ (2020); Pradeep, Jacob, and Hsiao (2020);
O’Neill, Royer, and Poon (2019); Sehgal  _et al._ (2019); Lin  _et al._
(2016); Ness, Mari, and Cates (2018); Niu  _et al._ (2020); White,
Chellamuthu, and Rothstein (2010); Han  _et al._ (2018). Thus, differences in
these variables directly inform the types of changes that one can use to
influence the thickening and jamming behaviours.

###  V.2  Exponents related to frictionless and frictional divergence

The scaling analysis in Fig. 2b illustrates two distinct power law regimes,
one with an exponent of -2, associated with frictionless isotropic jamming and
another with an exponent of -3/2 associated with frictional shear jamming.
Measurements of the divergence in the viscosity associated with frictionless
and frictional jamming have been reported previously in a wide range of
systems. Our -2 scaling result for frictionless isotropic jamming is
consistent with a number of previous studies in shear thickening suspensions
Singh  _et al._ (2018); Guazzelli and Pouliquen (2018); Morris and Boulay
(1999); Guy  _et al._ (2020).

We note that in pressure driven suspensions there is a claim that the scaling
exponent in this regime is -2.85. This apparent discrepancy, however, is a
result of an assumption that stress and pressure scale identically close to
jamming, which may not hold for dense suspensions. For the expert reader,
previous works have shown a scaling of
$(\phi_{0}-\phi)\sim\mathcal{J}^{\gamma_{\phi}}$, with $\gamma_{\phi}=0.37$
for frictionless particles and $\gamma_{\phi}=0.7$ for frictional suspensions
DeGiuli  _et al._ (2015); Boyer, Guazzelli, and Pouliquen (2011). Here,
$\mathcal{J}=\eta_{0}\dot{\gamma}/P$ is the viscous number, $\eta_{0}$ is the
solvent viscosity, $\dot{\gamma}$ is the shear rate and $P$ is the applied
pressure DeGiuli  _et al._ (2015); Boyer, Guazzelli, and Pouliquen (2011);
Perrin  _et al._ (2019, 2021). Assuming that $P\sim\sigma$ close to jamming,
we can invert the equation to obtain
$\eta\sim(\phi_{0}-\phi)^{-1/\gamma_{\phi}}=(\phi_{0}-\phi)^{-2.85}$ in the
frictionless regime, giving an apparent discrepancy in the scaling exponent
DeGiuli  _et al._ (2015). The assumption that $P\sim\sigma$ however, is a
conjecture and may not hold for dense suspensions. Indeed, a recent study
extending the analysis presented here has shown that the pressure and the
shear viscosities are associated with different scaling exponents Malbranche
_et al._ (2022); Malbranche, Chakraborty, and Morris (2023). In these
simulations, the shear viscosity exponents are consistent with those presented
here and the pressure viscosity exponents are consistent with those presented
in the pressure driven systems Malbranche  _et al._ (2022).

With respect to the exponent of -3/2 for the shear jamming regime, we note
that others have previously tried to fit exponents to viscosity versus volume
fraction data. Our result is well within the range of previously obtained
exponents Mari  _et al._ (2014); Guy, Hermes, and Poon (2015); Guy  _et al._
(2020); Singh  _et al._ (2018). Moreover, here, we use scaling collapse of the
entire shear thickening transition, using a scaling variable that is a
function of stress and the volume fraction to determine both exponents. This
approach is more rigorous since multiple measurements corresponding to
different combinations of stress and volume fraction are used to determine the
value of the universal function $\mathcal{F}$ at each point $x$.

###  V.3 Renormalization group flows

Recasting the non-equilibrium thickening transition as being governed by
crossover scaling between two different critical points suggests a
renormalization group flow diagram with two fixed points. This picture is
directly analogous to that found in equilibrium magnetic systems (e.g.
Heisenberg to Ising crossover scaling). We project the flows, obtained from
the experimentally determined scaling function, onto the $g(\sigma,\phi)$ and
$|\phi_{0}-\phi|$ plane. The nonlinear scaling variable,
$g(\sigma,\phi)=f(\sigma)C(\phi)$, is associated with the relevant direction
at the frictionless jamming point with the nonlinear terms in $g(\sigma,\phi)$
providing analytic corrections to scaling 444just as one defines $u_{h}(P,T)$
as the magnetic-field like perturbation away from the liquid-gas critical
point Cardy (1996). Usually, analytic corrections to scaling are treated
perturbatively near criticality. Here we use them to extend our theory to the
entire shear-thickening regime.. The isotropic jamming fixed point has two
relevant directions with flows away from the fixed point. The first is along
the $|\phi_{0}-\phi|$ axis (black line double arrows) and the second is along
the line of critical points on the shear-jamming line (black line single
arrow) flowing to the shear jamming fixed point. This critical manifold
separates the jammed and the flowing states. The shear jamming fixed point
must also have a relevant variable, and the flow along this line determines
the exponent $\delta$ (drawn schematically in green line double arrows).
Finally, the crossover between the two fixed points can be approximated by the
knee in Fig. 2b and is depicted in the renormalization group flow by the
dashed grey line. This diagram clearly illustrates the deep connection between
the shear thickening transition and renormalization group flows that are the
hallmark of critical phenomena in thermodynamic systems.

Figure 4:  Renormalization group flow diagram governing the shear thickening
transition. We plot the projection of the renormalization group flows onto the
$g(\sigma,\phi)$ and $|\phi_{0}-\phi|$ plane. The two distinct fixed points
are the frictionless isotropic jamming fixed point and the frictional shear
jamming fixed point. The black solid line with single arrows is a line of
critical points, and the dashed line corresponds to the $x$ value for the knee
in Fig. 2b. The flows are outward from the isotropic jamming point, which
corresponds to a critical exponent of -2 along the $|\phi_{0}-\phi|$ axis. The
green line is the outward flow line for the shear jammed fixed point, which
corresponds to a critical exponent of -3/2 for the viscosity.

##  VI  Conclusions

Adopting this statistical mechanics framework to describe shear thickening
opens several novel avenues for future work. For instance, universal scaling
theories and scaling functions such as those presented have previously been
used to predict a number of physical properties in equilibrium systems,
suggesting that similar approaches may be used in shear thickening. In the
well-studied system of Heisenberg and Ising magnets, for example, one can use
the crossover scaling function to determine the specific heat, correlation
functions, and many other system properties Cardy (1996). Similarly, recent
studies have investigated the scaling relations for the jamming transition for
frictionless particles, demonstrating that both static and dynamic
viscoelastic properties such as the shear stress, bulk modulus, and shear
modulus can be predicted by such a scaling formulation Goodrich, Liu, and
Sethna (2016); O’hern  _et al._ (2003); Goodrich  _et al._ (2014); Liarte  _et
al._ (2019, 2021a, 2021b). These studies demonstrate the power of scaling
theories and universal scaling functions and our work suggests that analogous
predictions and theories could be generated for shear thickening and
frictional jamming.

###### Acknowledgements.

We thank Edward Y. X. Ong, Eric M. Schwen, and Stephen J. Thornton for
valuable discussions and Anton Paar for use of the MCR 702 rheometer through
their VIP academic research program. For this work, IC, IG, and MR were
generously supported by NSF CBET award numbers: 2010118, 1804963, 1509308, and
an NSF DMR award number: 1507607. BC was supported by NSF CBET award number
1916877 and NSF DMR award number 2026834. JPS was supported by NSF
CBET-2010118 and DMR-1719490. EK was supported by the NSF Award PHY-1554887,
the University of Pennsylvania Materials Research Science and Engineering
Center (MRSEC) through Award DMR-1720530. EDG was supported by National
Science Foundation (NSF DMR-2026842).

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