|  Shear jamming and fragility in fractal suspensions under confinement
---|---
|  Sarika C. K.,a Sayantan Majumdar,a,∗ and A. K. Sood b
|  a Soft Condensed Matter Group, Raman Research Institute, Bengaluru 560080,
India.
|  b Department of Physics, Indian Institute of Science, Bengaluru 560012,
India.
|  (Dated August 28, 2024)
|  Under applied stress, the viscosity of many dense particulate suspensions
increases drastically, a response known as discontinuous shear-thickening
(DST). In some cases, the applied stress can even transform the suspension
into a solid-like shear jammed state. Although shear jamming (SJ) has been
probed for dense suspensions with particles having well-defined shapes, such a
phenomenon for fractal objects has not been explored. Here, using rheology and
in situ optical imaging, we study the flow behaviour of ultra-dilute fractal
suspensions of multi-walled carbon nanotubes (MWCNT) under confinement. We
show a direct transition from flowing to SJ state without a precursory DST in
fractal suspensions at an onset volume fraction, $\phi\sim$ 0.5%,
significantly lower than that of conventional dense suspensions ($\phi\sim$
55%). The ultra-low concentration enables us to demonstrate the fragility and
associated contact dynamics of the SJ state, which remain experimentally
unexplored in suspensions. Furthermore, using a generalized Wyart-Cates model,
we propose a generic phase diagram for fractal suspensions that captures the
possibility of SJ without prior DST over a wide range of shear stress and
volume fractions.

††footnotetext: ∗<EMAIL_ADDRESS>
##

##  1 Introduction

Dense particulate suspensions formed by dispersing inorganic or polymeric
particles in Newtonian fluids show an array of striking non-Newtonian flow
properties at high volume fractions. Among these, shear-induced increase in
viscosity, commonly known as shear thickening, has attracted significant
recent interest from experimental as well as theoretical points of view.
During shear thickening, the increase in viscosity under increasing applied
stress can be mild or abrupt, giving rise to continuous shear thickening (CST)
or discontinuous shear thickening (DST) respectively 1, 2, 3, 4, 5. Many of
the dense suspensions showing DST also exhibit shear jamming (SJ) at high
enough applied stress ($\tau$) and volume fraction ($\phi$) where the system
develops finite yield stress like a solid. Recent studies point out that
beyond a critical onset stress for contact formation, system-spanning contact
networks similar to those observed for dry granular materials can explain the
phenomenon of SJ in dense suspensions 5, 6, 7, 8. Importantly, the SJ state is
reversible: once the applied stress is removed, the system goes back to the
initial fluid-like state in a short span of time 9. Such reversible tuning of
viscosity implies many potential applications, particularly in the field of
designing smart shock-absorbing materials 10, 11.

In general, SJ can be observed only over a limited range of volume fractions:
$\phi_{\mu}<\phi<\phi_{0}$, where $\phi_{\mu}$ and $\phi_{0}$ are the critical
jamming packing fractions with and without friction respectively. For
suspensions in which the friction and other stress-induced short-range
interactions between the particles are negligible, $\phi_{\mu}$ remains close
to $\phi_{0}$ and the range of $\phi$ over which shear jamming is observed
diminishes. With greater friction, $\phi_{\mu}$ reduces and moves away from
$\phi_{0}$ enabling SJ in suspensions over a considerable range of
concentrations. Recent numerical studies 12, 13 show that introducing
constraints on the sliding and rolling motion of the particles by increasing
the inter-particle friction and adhesion can lower $\phi_{\mu}$. Non-spherical
particle shapes can also constrain the rolling motion and consequently reduce
$\phi_{\mu}$ 14, 15. Furthermore, studies show that tuning hydrogen bonding
and solvent molecular weight enhances the particle contact interaction
eliciting shear jamming in conventional suspensions 15, 16, 17, 18. In
colloidal suspensions, it is demonstrated that the surface roughness of
particles enhances DST due to the interlocking of surface asperities 19, 20. A
recent study shows that the critical volume fractions for both CST and DST can
be lowered by tuning the surface chemistry and surface roughness of rigid
particles in suspensions 21. The key aspect emerging from these observations
is the governing role of the contact network structure and the frictional
contact strength between particles in the shear jamming of suspensions. Hence,
a fractal suspension of stiff particles is a model system to probe shear
jamming due to the strong frictional interaction between the particles.
Confinement can facilitate the formation of the characteristic system-spanning
network for shear jamming. With significantly high effective surface roughness
and occupation volume, fractal suspensions are expected to have much lower
$\phi_{\mu}$ compared to the non-fractal suspensions mentioned above.

In this paper, we study the rheological behaviour of very dilute suspensions
of multiwalled carbon nanotubes (MWCNT) dispersed in a Newtonian solvent,
N-methyl-2-pyrrolidone (NMP), which form fractal aggregates (will be referred
as flocs) due to cohesive interparticle interactions. Previously, the colossal
increase in viscosity in this system by almost four orders of magnitude was
thought to be due to DST 22. The present study establishes that this
divergence in viscosity has all the signatures of shear jamming transition.
For dense suspensions of frictional, non-fractal compact grains, as $\phi$
increases from a very low value, the system progressively passes through
Newtonian/shear thinning (flowing), CST, DST, SJ and isotropically jammed (IJ)
phases 18, 23. Equivalently, for dry granular systems the sequence of phases
is: unjammed, fragile, SJ and IJ 24. Interestingly, we observe a direct
transition from flowing to SJ state with increasing $\phi$, completely
bypassing any intermediate CST/DST phases. Moreover, the ultra-low
concentration range ($\phi\geq 0.5\%$) for observing the SJ in the system
allows us to experimentally probe, for the first time, the fragility and the
associated structural reorganization in shear jammed suspensions.

##  2 Results and Discussion

Fig. 1: a) Flow curve of MWCNT suspension ($\phi$ = 2.5%), plotted as
viscosity versus applied shear stress in log-log scale. The duration of each
measuring point of shear stress is 100 s. Three different states, CJ-state,
flowing-state and SJ-state are marked as I, II and III respectively, in
different colors. b) Variation of yield stress (circle) and shear jamming
stress (square) with respect to $\phi$, along with power law fit (solid
lines).

We measure the flow curve, viscosity ($\eta$) versus shear stress ($\tau$), of
flocculated fractal suspensions of MWCNT under controlled shear stress using
parallel plate geometry of the rheometer. The detailed fractal structure
analysis of MWCNT flocs is described in the Supplementary Information (S.I.)
and in Fig. S1 where we obtain 2-D fractal dimension of the dried clusters
$\sim$ 1.5 - 1.6 indicating that the flocs have a much larger effective volume
compared to the actual volume of the constituent MWCNTs. We consider a range
of volume fractions varying from 0.26 to 5.35% to measure the flow curves. We
show a typical flow curve for $\phi$ = 2.5% in Fig. 1a. The complete set of
flow curves is shown in Fig. S2. The flow curve exhibits three distinct
rheological signatures over the shear stress range of 0.01 - 10 Pa. The
suspension is initially in a jammed (unyielded) state (Region I) due to
cohesive interaction between the flocs until the yield stress ($\tau_{y}$) is
reached. We term this region as the cohesive jammed state (CJ state). Beyond
$\tau_{y}$, the suspension begins to flow and shear thins upon a further
increase in $\tau$ (Region II). The flowing-state continues until the
viscosity shoots up instantaneously at higher shear stress, $\tau^{*}$ (Region
III). The flow curves plotted as shear stress versus shear rate in Fig. S3a
and b show that in both the regions I and III, the shear rate drops down to
zero, reaching the resolution limit of the rheometer and fluctuates around
zero with positive and negative values. This indicates that regions I and III
are solid-like jammed states as the average shear rate remains negligible in
these regions. Typically, for systems showing DST and SJ, the stress
dependence of viscosity in the shear-thickening regime is given by a power-law
behaviour: $\eta\sim\tau^{\delta}$, with $\delta=1$ marking the onset of DST.
At higher stress values, $\delta$ becomes larger than 1 indicating unstable
flows in the system due to shear-induced local jammed regions. In this regime,
when the flow curve is represented by shear rate ($\dot{\gamma}$) versus
stress ($\tau$), the so-called ‘S-shaped’ curves 7 are obtained showing a
negative slope ($\frac{d\,\dot{\gamma}}{d\,\tau}<0$) just beyond the onset
stress for shear thickening. At higher stress values the system becomes solid-
like with a vanishing $\dot{\gamma}$. As $\dot{\gamma}\rightarrow 0$ at a
finite $\tau$, the viscosity diverges at this point. Hence, region III of the
flow curve manifests clear signatures of SJ transition for $\tau\geq\tau^{*}$.
For solid-like CJ and SJ states, $\eta\rightarrow\infty$. However, the
observed finite values of viscosity for the CJ and SJ states (Fig. 1a) is an
artifact originating from the instrumental limitation in measuring very small
shear rates. Thus, rotational rheometry is not sufficient to characterize the
CJ and SJ states and one requires oscillatory rheology for such
quantification. Oscillatory rheology measurements reveal that both the CJ and
SJ states are viscoelastic solid-like in nature (storage modulus $G^{\prime}$
much higher than the loss modulus $G^{\prime\prime}$), with the magnitude of
$G^{\prime}$ for the SJ state much higher compared to that for the CJ state
22. Nonetheless, stress-controlled rotational rheometry reliably quantifies
the flowing state, as well as, the yielding and SJ transitions that remain the
main focus of the present study.

Fig. 2: Stress reversal in CJ state (a, b) and SJ state (c, d). Applied shear
stress versus time (upper panels) and corresponding viscosity profiles (lower
panels) are plotted. e-f) Optical images of SJ state at the time of reversal,
300 s (e) and at 300.6 s (f) showing the contact breaking of MWCNT flocs upon
stress reversal. Arrow indicates the direction of applied shear and red
circles aid to see contact breaking. Scale bar: 200 $\mu$m.

We observe shear jamming for low volume fractions $\phi\geq 0.5\%$.
Remarkably, we obtain a direct transition to SJ state without any DST regime
for the entire range of applied stress and particle volume fractions. To our
knowledge, such behaviour has never been observed for suspensions. We will be
looking into the details of this atypical feature in the following
discussions. In Fig. 1b, we show the variation of threshold stress for the
onset of yielding ($\tau_{y}$) and shear jamming ($\tau^{*}$) as functions of
volume fraction. The variation of yield stress with volume fraction exhibits a
power law scaling, $\tau_{y}\sim\phi^{2.8}$ similar to other flocculated
suspensions 25. Similarly, the onset stress for SJ, $\tau^{*}\sim\phi^{2}$.
For all the concentrations, we find that the SJ states cannot withstand
significant applied stresses. The SJ state typically fails beyond $\sigma\sim$
15 Pa as also reported earlier 22. Such failure at relatively low stress
values indicates that the rigidity of the MWCNT flocs is much lower compared
to the rigid grains (e.g. corn starch, silica particles).

Fig. 3: Optical images taken at the time of stress reversal and after 0.3 s
for CJ state (a, b) and SJ state (c, d). Scale bar: 1 mm. Arrows indicate the
direction of applied shear stress and the red circles aid to see contact
breaking upon the stress reversal. e and f) Temporal evolution of the
structure of CJ state (e) and SJ state (f), plotted as the variation of light
intensity vector at a fixed line (dashed line) on the images.

One of the main characteristic features that differentiates a shear jammed
state from an isotropically jammed state is the fragility under applied
perturbations 26, 27. To investigate the fragility and associated structural
changes in MWCNT suspensions, we carried out stress reversal experiments
coupled with in situ optical imaging. The latter is possible in the present
case because shear jamming in MWCNT suspensions starts at much lower volume
fractions due to the fractal nature of flocs as compared to the high volume
fraction ($\sim$ 55%) requirement for spherical colloids. This enables us to
track the contact dynamics during the SJ transition using in situ optical
imaging. We first apply a constant shear stress of 0.02 Pa ($<\tau_{y}$) for
300 s to the suspension at rest ($\phi$ = 0.77%). The suspension stays
undisturbed in the CJ state. After 300 s, the direction of the stress is
reversed as shown in Fig. 2a. We observe that the viscosity remains unchanged
during the reversal (Fig. 2b) as the suspension remains jammed, confirming the
non-fragile, isotropic nature of the CJ state. To investigate the fragility of
the SJ state, a constant shear stress of 3 Pa ($>\tau^{*}$) is applied to a
freshly loaded suspension ($\phi$ = 0.77%) as shown in Fig. 2c. Initially, the
suspension flows for $\sim$ 90 s and enters an SJ state (marked as SJ1) which
can support the applied shear stress. The structural transformations
associated with this rheological response, captured using in situ optical
imaging, are shown in Fig. S4. It shows diffuse MWCNT network structure in the
quiescent state, vorticity aligned rolling log-like flocs and dispersed broken
flocs at the flowing-state 28, 29 and interconnected dense floc network
structure at the SJ state 22. After 300 s, the stress direction is reversed
which results in an immediate melting of the SJ1 state characterized by a drop
in viscosity by several orders of magnitude (Fig. 2d). This flowing-state
continues for $\sim$ 20 s and then it enters a re-entrant SJ state (marked as
SJ2) corresponding to the new direction of shear. Optical images captured
before the reversal of stress direction ($t$ = 300 s, SJ1 state) and just
after the reversal ($t$ = 300.6 s) are shown in Fig. 2e and 2f, respectively.
The images clearly show contact breaking between the fractal flocs within a
fraction of a second as the SJ state is melted upon the stress reversal. In
contrast, no change in the contact between the flocs is observed in the case
of CJ state before and after the stress reversal. Such anisotropy in the SJ
state indicates the presence of frictional contacts in the system that can
statically counter the applied stress in one direction but becomes unstable on
stress reversal 30, 26. Although the correlation between fragility and contact
breaking has been studied for SJ in granular matter 6, direct experimental
observation of such correlation remained unexplored as yet in dense
suspensions. In the flowing-state just before shear jamming, the effective
volume fraction of the flocs, $\phi_{eff}\sim 36.3\%$ remains much higher than
the actual volume fraction ($\phi=0.77\%$) (Fig. S4c). We have estimated
$\phi_{eff}$ using optical imaging as described in the S.I.. Such high
effective volume ensures that the system can show SJ even at a very low volume
fraction of MWCNT. Estimation of the 3-D fractal dimension ($d_{f}$) of the
flocs in the flowing-state, where the individual flocs can be identified,
yields a value $d_{f}\sim$ 2.1 (S.I.).

Fig. 4: Temporal evolution of probability distribution functions of pixel
intensity at the stress reversal (a) and across the re-entrant shear jamming
(inset). Color bar represents the time. Inset axis labels are the same as that
of the main graph. b) Computable Information Density (CID) variation with time
at the stress reversal in CJ state and SJ state. Vertical dash lines indicate
the time of reversal (300 s) and that of the re-entrance of SJ state (320 s).
Fig. 5: a) The phase diagram for fractal suspensions in $\tau$ \- $\phi$
parameter space. Stacked columns represent the experimental data and solid
lines denote the phase boundaries extracted from the model for $\alpha=10$.
Different phases marked are cohesively jammed (unyielded), shear
thinning/flowing and shear jammed states. The narrow red region indicates the
DST phase. b) The flow curve of MWCNT suspension ($\phi$ = 2.5%) fitted with
the combined constitutive model of Herschel-Bulkley and Wyart-Cates. Symbols
show the experimental data and the solid line denotes the constitutive model
fit. c-e) Narrowing of the DST phase (shaded area) in the phase diagram with
respect to growth rate, $\alpha$ = 1, 3 and 10 respectively. Boundaries of DST
(red) and shear jamming (green) are shown.

The isotropic nature of the CJ state and the fragile, anisotropic nature of
the SJ state are further confirmed by a large field of view, transmission mode
imaging (Fig. 3). There is no structural change in the CJ state upon stress
reversal (Fig. 3a and 3b) whereas the floc structure in the SJ state
immediately starts to break upon stress reversal (Fig. 3c and 3d). Such
distinct behaviour of CJ and SJ states can be captured by space-time plots
(Fig. 3e and 3f) depicting the time evolution of intensity along the dotted
lines marked in Figs. 3a - 3d. We find that the intensity distribution does
not show any time evolution for the CJ state, but for the SJ state a clear
change is observed during the transition from the SJ1 to the SJ2 state upon
stress reversal (Fig. 3f).

High field of view imaging and sampling the dynamics of a large number of
flocs enable us to probe the statistics of the stress-induced structural
evolution of the SJ state. In Fig. 4a, we show the time evolution of the
probability distribution of pixel intensity $P(I)$ recorded in the
transmission mode during the stress reversal. Before reversal, $P(I)$ shows a
bimodal nature with two distinct peaks at low (the dark region where MWCNT
flocs are present) and high (void space) intensities. Immediately after the
reversal ($t>$ 300.1), the two peaks start to merge giving rise to a single
peak at an intermediate intensity closer to the low intensity peak. In the
transmission mode imaging, the intensity at each pixel of the image is a
direct measure of transmitted light intensity through the sample across the
gap between the shearing plates: $I=I_{0}\leavevmode\nobreak\ e^{-\lambda d}$,
where $I_{0}$ is the incident light intensity, $\lambda$ is the absorption
coefficient and $d$ is the thickness of MWCNT flocs across the shear gap.
Under the assumption that the value of $\lambda$ remains constant for the
flocs, the increase in the intensity at any pixel indicates a reduction in the
thickness of MWCNT flocs across the gap between shear plates at the
corresponding position. Hence, the merging of two intensity bands reflects the
dissociation of fractal flocs and redistribution of disjointed flocs in the
lateral direction. We observe an exact opposite trend during the recurrence of
shear jamming as shown in the inset of Fig. 4a: a single peak of $P(I)$ in the
flowing-state splits into two distinct peaks in the re-entrant SJ state,
signifying the building up of system spanning contacts.

To further quantify the complex structural transition associated with SJ, we
measure the computable information density (CID) that can capture such
transitions in a wide range of equilibrium and non-equilibrium systems 31. We
show the variation of CID obtained from the ratio of the compressed
($L^{\prime}$) to the uncompressed ($L$) image size as a function of time
during the stress reversal in Fig. 4b. We observe that in the SJ state CID
remains almost constant. Interestingly, CID shows a sharp drop after the
stress reversal as the system starts to flow. As the re-entrant SJ state sets
in, CID increases again and reaches almost the same value as in the initial SJ
state. The strong correlation between the floc structure and SJ is remarkable
and novel. We conjecture that the lower value of CID in the flowing-state is
due to the uniform distribution of broken flocs as compared to the SJ state
which shows larger and distinct fractal structures. No such time-variation is
observed under stress reversal for the CJ state as seen in Fig. 4b.

Now we model the rheological behaviour of our system which is strikingly
different from the conventional dense suspensions. We require a constitutive
relation that incorporates a cohesive jammed (CJ) state for $\tau<\tau_{y}$,
shear thinning for $\tau_{y}<\tau<\tau^{*}$ and shear jamming for
$\tau>\tau^{*}$. For our system, yielding from the CJ state to shear thinning
state is well described by Herschel-Bulkley (HB) equation,
$\tau_{{}_{\text{HB}}}=\tau_{y}+k\dot{\gamma}^{n}$ ($k$: consistency index;
$n$: power-law exponent). Since the stress reversal experiments show the
existence of frictional contacts in our system, we consider the Wyart-Cates
(WC) model 7 which is widely used for capturing DST and SJ in suspensions of
frictional particles. In WC model the stress dependence of jamming
concentration is given by
$\phi_{J}(\tau)=\phi_{\mu}f(\tau)+\phi_{0}\left(1-f(\tau)\right)$. Here, the
function, $f(\tau)\in[0,1]$, indicates the fraction of frictional contacts in
the system. At low shear stress, when almost all the contacts are lubricated,
$f(\tau)\sim 0$. At higher applied stress, the lubrication layers between the
particles are breached to form frictional contacts. When a sufficient number
of the contacts are transformed into frictional ones at very high stress,
$f(\tau)$ approaches 1. For MWCNT suspensions, the values of $\phi_{0}$ and
$\phi_{\mu}$ are 5.4% and 0.51%, respectively. Such stress-induced increase of
$f(\tau)$ is empirically modeled by an exponential function 30, 32. The
strength of frictional contact at a microscopic scale also affects $f(\tau)$.
The sharp rise to the SJ state in our system indicates a rapid increase in the
sufficient number of contacts between the flocs. We describe this behaviour
with a stretched exponential function 33, 34,
$f(\tau)=1-e^{-(\frac{\tau-\tau_{y}}{\tau^{*}-\tau_{y}})^{\alpha}}$ for
$\tau>\tau_{y}$ and $f(\tau)=0$ for $\tau\leq\tau_{y}$, where the exponent
$\alpha$ determines the growth rate of $f$ with respect to $\tau$ (see Fig.
S5). It ensures that all the contacts are lubricated till the yielding and the
frictional contacts start to form at higher stress values. For non-fractal
frictional particles, $\alpha\sim 1$. With $\alpha>>1$, the rapid increase of
$f(\tau)$ makes $\phi_{J}$ to drop towards $\phi_{\mu}$ giving rise to an
abrupt divergence of viscosity. We find that a constitutive model combining
Herschel-Bulkley and Wyart-Cates relations for viscosity 35,

$\eta(\tau,\phi)=\frac{k^{1/n}\leavevmode\nobreak\
\tau_{y}}{(\tau-\tau_{y})^{1/n}}+\frac{k^{1/n}}{(\tau-\tau_{y})^{1/n-1}}+\eta_{0}\left(1-\frac{\phi}{\phi_{J}}\right)^{-\beta}$
(1)

where $\eta_{0}$ is the solvent viscosity, captures the flow curve very well
in the limit of $\alpha>>1$, as shown in Fig. 5b. Unlike earlier models, the
onset stress ($\tau^{*}$) to overcome the lubrication barrier between
particles is not a constant for our system but increases with $\phi$ as shown
in Fig. 1b. This indicates that the stress-induced frictional contacts are
possible only when the applied stress is high enough to overcome the yield
stress arising from the cohesive interparticle interactions. The values of
parameters $\eta_{0}$, $n$ and $k$ used for the modeling of experimental data
is given in the Supplementary Information.

Based on the combined constitutive model, we map out a phase diagram for
fractal suspensions in $\tau-\phi$ parameter space as shown in Fig. 5a. It
demarcates different phases, viz. unyielded (CJ state), shear thinning
(flowing-state), DST and shear jamming. DST occurs when there is a
discontinuous jump in the shear stress at a given shear rate, i.e.,
$\frac{d\,log(\tau)}{d\,log(\dot{\gamma})}=\infty$ and jamming occurs when
$\dot{\gamma}=0$ at a higher stress value. Yielding and shear jamming
boundaries are estimated from Eqn. 1. Interestingly, the growth rate of
$f(\tau)$ with shear stress controls the DST and SJ boundaries: the DST regime
shrinks as the growth rate of $f(\tau)$ increases. Narrowing of the DST phase
in the phase diagram is demonstrated in Fig. 5c-e, for growth rate, $\alpha=$
1, 3 and 10. With high values of $\alpha$ ($>>$1), the DST regime almost
disappears as the DST and SJ boundaries overlap. Consequently, the system can
directly transit from flowing to SJ state, bypassing the narrow DST zone. DST
is a prominent phase for suspensions of compact particles with well-defined
geometries. In contrast, we experimentally observe a direct transition from
flowing to SJ phase in MWCNT fractal suspensions at all volume fractions,
$\phi_{\mu}\leq\phi\leq\phi_{0}$. The absence of DST stems from the rapid
growth of a sufficient number of interlocking flocs forming a network
structure that can easily span the system due to the confinement. Another
notable point from the phase diagram is the rise of the shear jamming boundary
with $\phi$ showing an opposite trend compared to the conventional, non-
fractal systems showing DST and SJ. This is a direct consequence of $\phi$
dependent $\tau_{y}$ due to cohesive interactions in the system. The power-law
exponent in the WC model (Eqn. 1) is taken as $\beta$ = 2, as in the case of
conventional systems. We note that the SJ phase boundary is not very sensitive
to the value of $\beta$. Although the experimental data show a direct
transition from a flowing (yellow region) to SJ state (green region) as
indicated in Fig. 5a, we use a moderate value ($\alpha$ = 10) to indicate a
narrow DST region. This establishes the generality of the model capturing all
the possible phases. However, if we use very high $\alpha$ values
($\alpha\geq$ 50), the DST regime disappears completely.

Fig. 6: a) Scaling function $F(x)=(\eta-\psi(\phi))(\phi_{0}-\phi)^{2}$ vs.
scaling variable $x=\frac{f(\tau)C(\phi)}{(\phi_{0}-\phi)}$ for
0.5%$\leq\phi\leq$ 5.35%. Colors indicate different volume fractions as
depicted in panel (b). b) Variation of $C(\phi)$ with volume fraction.

A very recent study shows that by recasting the WC model, the viscosity of
shear thickening suspensions can be collapsed onto a universal curve 36. The
underlying scaling function $F(x)$ which is related to the anisotropy of the
system, is obtained from the relation,
$(\eta-\psi(\phi))(\phi_{0}-\phi)^{2}=F(x)$ with the scaling variable,
$x=\frac{f(\tau)C(\phi)}{(\phi_{0}-\phi)}$. Here, $\eta-\psi(\phi)$ is the
reduced viscosity where a volume fraction dependent part of the viscosity,
$\psi(\phi)$, ensures the data collapse below the shear-thickening/jamming
onset. Variation of $F(x)$ vs $x$ for a range of $\phi$ values is shown in
Fig. 6a, where we obtain a good scaling of the data. The scaling function
$F(x)$ diverges at $x=x_{c}=(\phi_{0}-\phi_{\mu})^{-1}$. In our case,
$x_{c}\approx 20.4$. We find from Fig. 6b that $C(\phi)$ decreases linearly
with increasing $\phi$ values and eventually vanishes at the isotropic jamming
point, $\phi_{0}$. This linear decrease in $C(\phi)$ is similar to that
observed for the shear-thickening suspensions of silica particles 36. As
mentioned in ref. 36, the function $C(\phi)$ is related to the volume fraction
dependent anisotropy of the SJ state.

##  3 Conclusion

We have presented stress-induced divergence of viscosity in fractal MWCNT
suspensions showing all the signatures of shear jamming at very low
concentrations ($\phi_{\mu}\sim$ 0.5%). By combining stress reversal
experiments along with in situ optical imaging, we bring out the fragility of
the SJ state and associated contact dynamics and structural reorganization.
Such correlations have remained unexplored so far in the experimental studies
of shear jamming in dense suspensions. Interestingly, from in situ optical
imaging, we observe that the effective area coverage of fractal clusters is
significantly reduced in the SJ state as compared to the initial CJ state (see
Fig. 3). This indicates that at stress, $\tau>\tau_{y}$, the flow induces
densification of the flocs making them stiffer which facilitates the contact
network formation required to achieve the SJ state. For dense suspensions of
frictional non-fractal particles, when the phase diagram is plotted in the
parameter plane of stress ($\tau$) and particle volume fraction ($\phi$) 18,
the DST regime narrows down as $\phi\rightarrow\phi_{0}$. However, in our
case, we observe a direct transition from flowing to SJ with no CST/DST regime
for all volume fractions ($\phi$: 0.5 % - 5.4 %) up to the isotropic jamming
point ($\phi_{0}$ = 5.4 %), as shown in Fig. 5a. This signifies that the SJ
phenomenon reported here is quite different from that of compact granular
systems. The very high effective friction related to the stress-induced
interlocking of fractal clusters is responsible for such striking stress
response.

Importantly, we present a phase diagram for fractal suspensions based on the
generalized Wyart-Cates model showing a cohesive jammed state, shear thinned
state and shear jammed state. The generality of the phase diagram sets a
framework for the rheology of fractal suspensions. The sharp increase in the
number of frictional contacts over a narrow range of stress is the key reason
to observe a shear jammed state from the flowing state without going through
the DST phase as commonly observed in dense suspensions of spherical or
anisotropic particles. Controlling fractal dimension in these systems by
tuning the inter-particle interactions and observing the resulting flow
behaviour can shed light on the sharp rise in stress-induced contact formation
or contact proliferation. Exploring the stability properties of SJ state in
fractal suspensions using a similar approach used for dry granular materials
37 can be an interesting future direction to explore. We hope that our study
will motivate further experimental and theoretical studies of SJ in
suspensions of fractal particles.

## Acknowledgements

AKS thanks the Department of Science and Technology, India for the support
under Year of Science Professorship and the Nanomission Council. SM thanks
Science and Engineering Research Board, India for the support through
Ramanujan Fellowship.

## Materials and Methods

For the preparation of suspensions, MWCNTs having a diameter of 30-50 nm and
length of a few microns purchased from Sun NanoTech are mixed with an organic
solvent, N-methyl-2-pyrrolidone (NMP), from Sigma Aldrich. Suspensions of
different volume fractions, $\phi=0.26-5.35\%$, are prepared by a continuous
process of mechanical stirring (10 min) and ultra-sonication (10 min) followed
by mechanical stirring again (10 min). This preparation protocol is
implemented to ensure a homogeneous distribution of flocs in the medium. The
volume fraction of MWCNT suspension is calculated from the weight fraction
using density of NMP (1.03 g/cm3) and the true density of MWCNT (1.7 - 2.1
g/cm3).

Rheological measurements are carried out in a stress-controlled rheometer (MCR
102, Anton Paar) at a fixed temperature of 25 oC. Flow curves are obtained
using a sandblasted cone-plate measuring system (diameter: 50 mm, cone-angle:
2o) and a steel bottom plate. Opto-rheological measurements are done on
suspensions ($\phi$ = 0.77$\%$) using a glass parallel plate measuring system
(diameter: 43 mm) along with a glass bottom plate at a fixed shear gap of 60
$\mu$m. Images are captured using a color CCD camera (Luminera, 640 X 480
pixels) fitted with a microscope objective lens. A white light source is used
for illuminating the imaging area. Images are collected at a rate of 50 fps in
transmission mode (Leica lens 5x) and in reflection mode (Mitutoyo lens 5x
with condenser). Optical images of drop cast samples are taken using Olympus
bright-field microscope.

## Notes and references

  * Barnes 1989 H. Barnes, _Journal of Rheology_ , 1989, 33, 329–366.
  * Hoffman 1972 R. Hoffman, _Transactions of the Society of Rheology_ , 1972, 16, 155–173.
  * Mewis and Wagner 2012 J. Mewis and N. J. Wagner, _Colloidal suspension rheology_ , Cambridge University Press, 2012.
  * Wagner and Brady 2009 N. J. Wagner and J. F. Brady, _Physics Today_ , 2009, 62, 27–32.
  * Brown and Jaeger 2014 E. Brown and H. M. Jaeger, _Reports on Progress in Physics_ , 2014, 77, 046602.
  * Bi _et al._ 2011 D. Bi, J. Zhang, B. Chakraborty and R. P. Behringer, _Nature_ , 2011, 480, 355–358.
  * Wyart and Cates 2014 M. Wyart and M. Cates, _Physical review letters_ , 2014, 112, 098302\.
  * Seto _et al._ 2013 R. Seto, R. Mari, J. F. Morris and M. M. Denn, _Physical review letters_ , 2013, 111, 218301.
  * Barik and Majumdar 2022 S. Barik and S. Majumdar, _Physical Review Letters_ , 2022, 128, 258002\.
  * Lee _et al._ 2003 Y. S. Lee, E. D. Wetzel and N. J. Wagner, _Journal of materials science_ , 2003, 38, 2825–2833.
  * Majumdar _et al._ 2013 A. Majumdar, B. S. Butola and A. Srivastava, _Materials & Design_, 2013, 46, 191–198.
  * Guy _et al._ 2018 B. Guy, J. Richards, D. Hodgson, E. Blanco and W. Poon, _Physical review letters_ , 2018, 121, 128001.
  * Singh _et al._ 2020 A. Singh, C. Ness, R. Seto, J. J. de Pablo and H. M. Jaeger, _Physical Review Letters_ , 2020, 124, 248005.
  * Rathee _et al._ 2020 V. Rathee, S. Arora, D. L. Blair, J. S. Urbach, A. Sood and R. Ganapathy, _Physical Review E_ , 2020, 101, 040601.
  * James _et al._ 2019 N. M. James, H. Xue, M. Goyal and H. M. Jaeger, _Soft matter_ , 2019, 15, 3649–3654.
  * James _et al._ 2018 N. M. James, E. Han, R. A. L. de la Cruz, J. Jureller and H. M. Jaeger, _Nature materials_ , 2018, 17, 965–970.
  * van der Naald _et al._ 2021 M. van der Naald, L. Zhao, G. L. Jackson and H. M. Jaeger, _Soft Matter_ , 2021, 17, 3144–3152.
  * Peters _et al._ 2016 I. R. Peters, S. Majumdar and H. M. Jaeger, _Nature_ , 2016, 532, 214–217.
  * Hsu _et al._ 2018 C.-P. Hsu, S. N. Ramakrishna, M. Zanini, N. D. Spencer and L. Isa, _Proceedings of the National Academy of Sciences_ , 2018, 115, 5117–5122.
  * Hsu _et al._ 2021 C.-P. Hsu, J. Mandal, S. N. Ramakrishna, N. D. Spencer and L. Isa, _Nature communications_ , 2021, 12, 1–10.
  * Bourrianne _et al._ 2022 P. Bourrianne, V. Niggel, G. Polly, T. Divoux and G. H. McKinley, _Physical Review Research_ , 2022, 4, 033062.
  * Majumdar _et al._ 2011 S. Majumdar, R. Krishnaswamy and A. Sood, _Proceedings of the National Academy of Sciences_ , 2011, 108, 8996–9001.
  * Fall _et al._ 2015 A. Fall, F. Bertrand, D. Hautemayou, C. Méziere, P. Moucheront, A. Lemaitre and G. Ovarlez, _Physical review letters_ , 2015, 114, 098301\.
  * Otsuki and Hayakawa 2020 M. Otsuki and H. Hayakawa, _Physical Review E_ , 2020, 101, 032905\.
  * Buscall _et al._ 1987 R. Buscall, P. Mills, R. Stewart, D. Sutton, L. White and G. Yates, _Journal of Non-Newtonian Fluid Mechanics_ , 1987, 24, 183–202.
  * Cates _et al._ 1998 M. Cates, J. Wittmer, J.-P. Bouchaud and P. Claudin, _Physical review letters_ , 1998, 81, 1841.
  * Seto _et al._ 2019 R. Seto, A. Singh, B. Chakraborty, M. M. Denn and J. F. Morris, _Granular Matter_ , 2019, 21, 1–8.
  * Lin-Gibson _et al._ 2004 S. Lin-Gibson, J. A. Pathak, E. Grulke, H. Wang and E. K. Hobbie, _Physical review letters_ , 2004, 92, 048302.
  * Varga _et al._ 2019 Z. Varga, V. Grenard, S. Pecorario, N. Taberlet, V. Dolique, S. Manneville, T. Divoux, G. H. McKinley and J. W. Swan, _Proceedings of the National Academy of Sciences_ , 2019, 116, 12193–12198.
  * Lin _et al._ 2015 N. Y. Lin, B. M. Guy, M. Hermes, C. Ness, J. Sun, W. C. Poon and I. Cohen, _Physical review letters_ , 2015, 115, 228304.
  * Martiniani _et al._ 2019 S. Martiniani, P. M. Chaikin and D. Levine, _Physical Review X_ , 2019, 9, 011031.
  * Mari _et al._ 2014 R. Mari, R. Seto, J. F. Morris and M. M. Denn, _Journal of Rheology_ , 2014, 58, 1693–1724.
  * Guy _et al._ 2015 B. Guy, M. Hermes and W. C. Poon, _Physical review letters_ , 2015, 115, 088304.
  * Dhar _et al._ 2019 S. Dhar, S. Chattopadhyay and S. Majumdar, _Journal of Physics: Condensed Matter_ , 2019, 32, 124002.
  * Singh _et al._ 2019 A. Singh, S. Pednekar, J. Chun, M. M. Denn and J. F. Morris, _Physical Review Letters_ , 2019, 122, 098004.
  * Ramaswamy _et al._ 2021 M. Ramaswamy, I. Griniasty, D. B. Liarte, A. Shetty, E. Katifori, E. Del Gado, J. P. Sethna, B. Chakraborty and I. Cohen, _arXiv preprint arXiv:2107.13338_ , 2021.
  * Zhao _et al._ 2022 Y. Zhao, Y. Zhao, D. Wang, H. Zheng, B. Chakraborty, J. E. Socolar _et al._ , _Physical Review X_ , 2022, 12, 031021.

Supplementary Information

## Fractal structure of MWCNT flocs

For the fractal structure analysis, MWCNT suspension ($\phi$ = 0.03$\%$) is
drop cast onto a glass substrate and dried. Optical microscopic images of the
drop cast samples show the fractal nature of MWCNT flocs. The 2D fractal
dimension ($d_{f}$) is estimated by two methods: Area-Perimeter analysis
($log(A)\propto d_{f}\leavevmode\nobreak\ log(P)$) and box-counting method
($d_{f}=-\lim_{r\to 0}\frac{log(n)}{log(r)}$ where $n$ is the number of filled
boxes and $r$ is the box size). The Area-Perimeter analysis using the image of
discrete flocs (Fig. S1a) yields the value $d_{f}=1.64$ (see Fig. S1b). For
the box-counting method, images of interconnected flocs are collected in which
the fractal arrangement of nanotubes is clear. A typical image and the fractal
dimension estimation ($d_{f}=1.53$) are shown in Fig. S1c and d respectively.
As we increase the concentration for drop casting, such analysis becomes
difficult due to the dense packing of the dried clusters. So, we have used a
very dilute sample ($\phi$ = 0.03$\%$) for such analysis.

Fig. S1: a) Optical image of discrete MWCNT fractal flocs (scale bar: 50
$\mu$m). b) Log(A) versus log(P) of the individual flocs and the linear fit.
c) Optical image of the interconnected fractal flocs (Scale bar: 20 $\mu$m).
d) Log(n) versus log(r) of the interconnected flocs and the linear fit.

The concentration of the drop cast sample is low compared to that of the
sample used for rheology measurements. At those higher concentrations, it
becomes difficult to image the fractal arrangement of nanotubes due to
clustering. To address this issue, we have directly estimated the 3D fractal
dimension of the suspension used for the rheology measurement from the
effective volume fraction: $\phi_{eff}\approx\phi(R/a)^{3-d_{f}}$, where $R$
is the radius of gyration and $a$ is the size of MWCNT 1. Using the image of
dispersed broken flocs in the flowing state (Fig. S4c), the estimated value of
3D fractal dimension is, $d_{f}\approx 2.1$ ($\phi_{eff}=36.3\%$,
$\phi=0.77\%$, $R=100\mu$m and $a=1\mu$m). Here, we have considered only the
flowing state of the sample just beyond yielding, where the isolated clusters
are observed that enables us to estimate the value of $R$. In the CJ and SJ
states, the determination of $R$ is difficult using optical imaging due to the
connectivity of the clusters. For the estimation of effective volume fraction,
the area coverage of the dispersed broken flocs (Fig. S4c) is measured using
ImageJ software. Assuming a uniform distribution of flocs across the shear
gap, $\phi_{eff}$ is estimated.

## Flow Curves of MWCNT suspensions

Flow curves of MWCNT suspensions of different volume fractions ranging from
0.26$\%$ to 5.35$\%$ are plotted as viscosity versus applied shear stress in
Fig. S2. Shear jamming transition is observed for $\phi\geq 0.5\%$. From the
variation of viscosity with shear stress, $\eta\sim\tau^{\delta}$, the onset
of DST is characterized with, $\delta=1$. Here, the divergence of viscosity at
finite shear stress and the corresponding drop in shear rate,
$\dot{\gamma}\sim 0$, imply direct manifestation of shear jamming without
prior DST.

Fig. S2: a - l) Flow curves of MWCNT suspensions of different volume
fractions, 0.26, 0.51, 1.02, 1.52, 2.01, 2.51, 2.99, 3.47, 3.95, 4.42, 4.89,
5.35$\%$ respectively, plotted as viscosity versus shear stress in log-log
scale.

Since MWCNT suspensions belong to the category of frictional suspensions which
exhibit DST or SJ or both, we use S-shaped flow curve representation 2 of the
rheological data in Fig. S3a. In the S-shaped flow curve, SJ is indicated by
arch-shapes with a vanishing $\dot{\gamma}$ at higher stress. A magnified
portion of the flow curve ($\phi$ = 2.5%) is shown in Fig. S3b. It clearly
shows that the shear rate drops down to the resolution limit of the rheometer
and fluctuates around zero in regions I and III. Since the average shear rate
is negligible in these regions, they are solid-like jammed states; namely,
cohesive jammed state and shear jammed state.

Fig. S3: a) S-shaped flow curve of MWCNT suspension ($\phi$ = 2.5%), plotted
as shear stress versus shear rate in log-log scale. The dashed line denotes
the shear rate at the resolution limit of the rheometer. b) Magnified portion
of the flow curve showing the fluctuation of shear rate around zero (vertical
dashed line) in both CJ and SJ states.

## Stress reversal images

In situ optical images of MWCNT suspension ($\phi=0.77\%$) collected in
transmission mode during the stress reversal with a wide-angle lens (Leica,
5x) are shown in Fig. S4. It shows the structural transformations associated
with the initial flow of suspensions when shear stress of 3 Pa is applied. The
structural deformation pathways leading to the shear jamming can be clearly
seen in the images.

Fig. S4: Optical images taken after the application of shear stress (3 Pa). It
shows diffuse MWCNT network structure at the quiescent state (a), rolling log-
like flocs (b) and dispersed broken flocs (c) at the flowing-state and
interconnected dense floc network structure at the SJ state (d). Scale bar: 1
mm.

## Parameters for the modeling

For modeling the rheology data of MWCNT suspensions using the combined
constitutive model of Herschel-Bulkley and Wyart-Cates, we used parameter
values, $n=0.2$, $k=0.1$ and $\eta_{0}=1.65$ mPa s. The critical volume
fraction of jamming without friction, $\phi_{0}$ = 5.4%, is the volume
fraction above which the system remains in the CJ state under the application
of shear stress. The critical volume fraction of jamming with friction,
$\phi_{\mu}$ = 0.51%, is the volume fraction above which the system exhibits
shear jamming.

##  $f(\tau)$ growth at different rates

Growth of fraction of frictional contacts with shear stress,
$f(\tau)=1-e^{-(\frac{\tau-\tau_{y}}{\tau^{*}-\tau_{y}})^{\alpha}}$ for
$\tau>\tau_{y}$ and $f(\tau)=0$ for $\tau\leq\tau_{y}$, for different values
of growth rates, $\alpha$ = 1, 3 and 10, is shown in Fig. S5.

Fig. S5: Growth of $f(\tau)$ with shear stress for different growth rates,
$\alpha$ = 1, 3 and 10.

## Notes and references

  * 1 D. B. Genovese, Advances in Colloid and Interface Science, 2012, 171–172, 1–16.
  * 2 M. Wyart and M. Cates, Physical review letters, 2014, 112, 098302.