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# New insights on carbon black suspension rheology - anisotropic thixotropy
and anti-thixotropy

Y. Wang  Department of Mechanical Science and Engineering, University of
Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA  Beckman Institute
for Advanced Science and Technology, University of Illinois at Urbana-
Champaign, Urbana, Illinois 61801, USA  Joint Center for Energy Storage
Research, Argonne National Laboratory, Lemont, Illinois 60439, USA  R. H.
Ewoldt  [  Department of Mechanical Science and Engineering, University of
Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA  Beckman Institute
for Advanced Science and Technology, University of Illinois at Urbana-
Champaign, Urbana, Illinois 61801, USA  Joint Center for Energy Storage
Research, Argonne National Laboratory, Lemont, Illinois 60439, USA  Materials
Research Laboratory, University of Illinois at Urbana-Champaign, Urbana,
Illinois 61801, USA

###### Abstract

Abstract: We report a detailed experimental study of peculiar thixotropic
dynamics of carbon black (CB, Vulcan XC-72) suspensions in mineral oil,
specifically the observation of sequential stress increase then decrease at a
fixed shear rate in a step-down test. We verify that such dynamics, though
peculiar, come from a true material response rather than experimental
artifacts. We also reveal how this long-time stress decay is associated with
anti-thixotropy, rather than viscoelasticity, by using orthogonal
superposition (OSP) rheometry to probe viscoelastic moduli during the step-
down tests. The orthogonal storage and loss modulus are present, showing this
two-timescale recovery then decay response, which demonstrates that this
response is anti-thixotropic, and it involves shear-induced structuring. We
further show a mechanical anisotropy in the CB suspension under shear using
OSP. Based on the rheological results, a microstructural schematic is
proposed, considering qualitatively thixotropic structure build-up, anti-
thixotropic densification, and anisotropic structure evolution. Our
observation for these CB suspensions is outside the standard paradigm of
thixotropic structure-parameter models, and the elastic response provides us
with new insight into the transient dynamics of CB suspensions.

##  I INTRODUCTION

Understanding and predicting the rheological properties of carbon black (CB)
suspensions are desired in many applications such as inks [1], paints [2], and
semi-solid flow batteries [3, 4, 5, 6, 7, 8]. However, this is a challenging
task because the structure of CB evolves in a complicated manner under flow.
The fundamental building blocks of CB, the primary particles, are strongly
fused by covalent bonds into fractal aggregates, and the individual aggregates
join together by van der Waals forces to form agglomerates [9]. At high shear
rates, large CB agglomerates break into smaller sizes when the applied shear
force is adequate to overcome the attractive force between them [9]. This
results in a lower hydrodynamic volume, and therefore, a lower viscosity [10].
At low shear rates, large but loosely-connected agglomerates are formed; for
suspensions with volume fraction higher than the percolation threshold, this
leads to a space-filling network [11, 12, 13, 14] which greatly increases the
flow resistance. Under certain shear conditions, these large, open, and
loosely-connected agglomerates can self-organize and inter-penetrate [15], due
to the fractal nature of CB [16]. Such densification decreases the viscosity
of suspensions and has been reported in various flocculated systems [17, 18,
19]. At rest, the elastic modulus of CB suspensions can stiffen with the
applied stress prior to yielding [20], due to the local rearrangement of CB
aggregates [21]. Under shear, CB suspensions can also form anisotropic
structures. For example, shear-induced log-rolling structures aligned in the
vorticity direction at low dimensionless shear rates have been observed with
CB [22, 23, 24, 25]. The structure of CB also shows shear history dependence,
therefore, the suspensions can have a tunable yield stress [26], elastic
modulus [20], conductivity [27], and impedance [28] under different shear
histories.

Many attempts have been made to characterize the structure of CB suspensions
under flow for a better understanding of its rheology. Mewis and Schoukens
[29] compared the mechanical spectra of CB suspensions under different shear
rates and found that larger shear rate results in a smaller structural size in
the CB network. Other characterization techniques that have been used include
orthogonal superposition [30], ultrasonic [31], dielectric [14], and electric
measurement [27] coupled with rheology. Some direct measurements of the CB
structure have been done using neutron scattering [32], light scattering [33],
and optical microscopy [34], but most of these did not measure the transient
response of CB suspensions under flow. Understanding the structure evolution
of CB suspensions under flow through its transient response remains a
complicated and unresolved problem, which is directly relevant to many flow-
related applications.

Furthermore, our understanding of the transient mechanical responses and
structure changes under shear is greatly inhibited due to a lack of clear
differentiation between thixotropy, anti-thixotropy, and viscoelasticity,
while all of them have been reported on CB suspensions before [16, 35, 22].
Thixotropy is defined as “the continuous decrease of viscosity with time when
flow is applied to a sample that has been previously at rest and the
subsequent recovery of viscosity in time when the flow is discontinued” [36,
37, 38, 39]. While this definition of thixotropy refers to a purely viscous
phenomenon [40, 41], time-dependent structure breakdown or buildup generally
impacts viscous, elastic, and plastic stresses. This may be called thixotropy,
or structuration [42], or aging and rejuvenation [40, 43, 38]. Thixotropy and
viscoelasticity are usually differentiated using step-up in shear rate (also
called break-down flow) and step-down in shear rate (also called build-up
flow) tests [44, 35]. Fig. 1 shows the possible shear stress evolutions of
thixotropic, anti-thixotropic, and viscoelastic materials, in response to a
step-down in shear rate. An ideal thixotropic fluid [38] would show an initial
sudden decrease in stress, followed by a gradual increase until steady state
is reached, when the applied shear rate is decreased from $\dot{\gamma}_{i}$
to $\dot{\gamma}_{f}$, because of structure build-up. For a viscoelastic
fluid, on the other hand, the stress response is a monotonic decrease towards
steady state [38], because of elastic stress relaxation. The transient
dynamics of CB suspensions are dominated by thixotropy, because the CB
structure build-up and break-down processes mentioned earlier happen over
timescales typically longer than the viscoelastic relaxation time [38, 36].
The thixotropy of CB suspensions is well-studied using step-up and step-down
in shear rate tests [36, 35], creep [45, 42], and flow sweeps [46]. For
example, a suspension of one particular carbon black (FW2 from Degussa) in
naftenic oil (Shell S6141) [36] was observed to show thixotropic stress
recovery with a step-down in shear rate test, as shown in the last column, "CB
suspensions", in Fig. 1, where the stress increases with time in regime I.

Prior works have also claimed CB suspensions to show anti-thixotropy [26]
(also called rheopexy, or negative thixotropy), as the shear stress decreases
with time in a step-down in shear rate test [15] and increases with time in a
step-up in shear rate test [16], as shown in Fig. 1, the transient stress
response of "CB suspensions" in regime II. The thixotropy and anti-thixotropy
of CB suspensions are therefore contradictory results for the transient stress
dynamics under step shear rate flow. However, the claims of CB being anti-
thixotropic are based on observations that can be due to either anti-
thixotropy or viscoelasticity. As shown in Fig. 1, both anti-thixotropic and
viscoelastic materials would show a monotonic decrease in shear stress to the
steady state value. Therefore, a step in shear rate test cannot differentiate
anti-thixotropy and viscoelasticity, although it is commonly used to
distinguish between thixotropy and viscoelasticity. Fig. 1 only shows the case
of step-down in shear rate, and the same is true for the step-up test. This
results in an indistinguishable stress response for thixotropic, anti-
thixotropic and viscoelastic materials [47].

Figure 1: Step shear rate tests can distinguish between thixotropy and anti-
thixotropy, but not anti-thixotropy and viscoelasticity. For the step down
input, $\dot{\gamma}$, possible evolutions of shear stress, $\tau(t)$, are
shown: ideal thixotropy, viscoelasticity, and anti-thixotropy. More realistic
stress responses of carbon black suspensions are shown as "CB suspensions",
where some CB suspensions in literature show a stress growth (regime I, as
observed in Ref. [36]), while others show a stress decay (regime II, as
observed in Ref. [16, 15]).

Orthogonal superposition (OSP) rheometry [48] can potentially help to
distinguish between viscoelasticity and anti-thixotropy. This is performed by
applying a step-down in shear rate flow in the primary (rotational) direction,
and simultaneously superposing a small amplitude oscillatory flow in the
orthogonal direction. Therefore, the evolution of orthogonal storage modulus,
$G_{\perp}^{\prime}$, and orthogonal loss modulus, $G_{\perp}^{\prime\prime}$,
under the step change in the main flow direction (the rotational flow) can be
observed. The change in $G_{\perp}^{\prime}$ and $G_{\perp}^{\prime\prime}$
will give insight to distinguish viscoelastic and anti-thixotropic materials.
This is because viscoelasticity involves storage and relaxation of elastic
energy and stress, while anti-thixotropy is a result of change in structure,
such as change in the size of agglomerates, which itself does not store
energy, but affects the viscosity and moduli of suspensions. Due to the
fundamental difference, we conjecture that the moduli under the step-down flow
evolve differently for viscoelasticity and anti-thixotropy: for anti-
thixotropic materials, both $G_{\perp}^{\prime}$ and
$G_{\perp}^{\prime\prime}$ decrease with time, while for viscoelastic
materials, the change of $G_{\perp}^{\prime}$ and $G_{\perp}^{\prime\prime}$
is material and model specific. Therefore, observing $G_{\perp}^{\prime}(t)$
and $G_{\perp}^{\prime\prime}(t)$ under step in shear rate flow provides us
with new insight to distinguish between anti-thixotropy and viscoelasticity,
and it can help us resolve the uncertainty in the origin of stress decay in CB
suspensions.

In this paper, we aim to experimentally study the transient dynamics of one
particular CB, Vulcan XC-72, in heavy mineral oil suspensions under the step-
down in shear rate flow, and resolve the ambiguity between thixotropy, anti-
thixotropy, and viscoelasticity of CB suspensions. We do this by recording the
shear stress, $\tau$, evolving with time during step-down in shear rate tests,
when the applied shear rate is decreased from initial shear rate,
$\dot{\gamma}_{i}$, to various final shear rates, $\dot{\gamma}_{f}$. A short-
time recovery, followed by a long-time decay in $\tau$ is observed. Such
peculiar decay in stress has been reported before [15], but several hypotheses
remain to be tested. Experimental artifacts must be checked, such as wall-
slip, which can potentially lead to apparent stress decrease. The
repeatability and reversibility of such peculiar stress decrease are tested,
and we verify that such recovery-then-decay dynamics is a reversible,
reproducible, and intrinsic material response. Moreover, whether the decrease
in stress comes from anti-thixotropy or viscoelasticity remains unknown. With
the help of OSP, we record the orthogonal moduli, $G_{\perp}^{\prime}$ and
$G_{\perp}^{\prime\prime}$, under step-down tests. This helps us attribute the
long-time decay to shear-induced anti-thixotropic structure rearrangement,
rather than viscoelastic stress relaxation. The anti-thixotropy is also proved
by hysteresis flow sweep. By comparing the results for shear stress and
orthogonal moduli, we show a mechanical anisotropy in the thixotropic
timescales and the magnitude of moduli of CB suspensions under shear. We
further propose a micro-structural schematic, considering qualitatively
thixotropic structure build-up, anti-thixotropic structure rearrangement, and
anisotropic structure evolution. This paper is organized by first showing
transient shear stress results in Sec. III.1 and transient orthogonal moduli
in Sec. III.2, followed by the anisotropy of CB suspensions, by comparing the
responses in both directions, in Sec. III.3. The concentration dependence and
additional evidence from hysteresis are shown in Sec. III.4 and Sec. III.5
respectively. Discussion on the growth of anisotropy and mechanical schematics
are shown in Sec. IV.1 and IV.2.

##  II MATERIALS AND METHODS

###  II.1 Shear rheological measurements

Rheological measurements were performed at $\rm 25^{o}C$ on a rate-controlled,
separated motor-transducer rheometer (ARES-G2 from TA Instruments). A cone-
and-plate geometry with a diameter of 25 mm and a cone angle of 0.1 rad was
used, unless specified otherwise. There was no edge failure or sample
degradation during the tests, verified by repeat measurements and videos of
the sample edge during the tests, and there was no wall slip or shear banding
during the tests, verified by a 25 mm parallel plate geometry at different
measuring gaps [49].

The behavior of CB suspensions is shear history dependent. Therefore, a 10 min
preshear at 300 $\rm s^{-1}$ was applied before every measurement [42], until
the shear stress reached the steady state (within 3$\%$ variation). The
preshear protocol is intended to erase the memory of any mechanical/shear
history of the sample [50] and to ensure consistency across all measurements.
CB samples were subjected to several rheological test procedures, each
proceeded with the standard preshear protocol. These were: (i) step-down in
shear rate tests, also called the structure recovery tests, to record the
transient shear stress response; (ii) repeated step in shear rate tests using
different geometries and gaps to verify the results; and (iii) hysteresis
tests that provide additional evidence for anti-thixotropy. The details and
results of each of these tests are described and discussed in Sec. III.1 and
Sec. III.5.

###  II.2 Orthogonal superposition (OSP)

Orthogonal superposition (OSP), in which a small amplitude oscillation is
superposed in the direction perpendicular to that of the primary shear flow,
is valuable in measuring shear-controlled structures [51, 29, 52, 53], probing
anisotropic structures [54, 53], and studying nonlinear viscoelasticity using
linear concepts [48, 55, 54]. While scattering, microscopy, and other methods
mentioned earlier can characterize the structure of CB under flow, OSP is
useful for studying the transient behavior and investigating the anisotropy in
the mechanical properties of materials.

In this study, the transient orthogonal storage and loss moduli,
$G_{\perp}^{\prime}(t;\omega,\gamma_{0})$ and
$G_{\perp}^{\prime\prime}(t;\omega,\gamma_{0})$, were monitored as functions
of time, $t$, by superposing a small amplitude oscillatory shear (SAOS) at a
constant frequency, $\omega$, and amplitude, $\gamma_{0}$, in the orthogonal
direction during a step-down in shear rate in the rotational direction. To
quantify the mechanical anisotropy and its growth with shearing time in CB
suspensions, a 2D-small amplitude oscillatory shear (2D-SAOS) [53] was
applied, where an axial deformation $\gamma_{\perp}=\gamma_{0}{\rm sin}(\omega
t)$ was superposed on the conventional angular deformation
$\gamma=\gamma_{0}{\rm sin}(\omega t)$, and the peak amplitude is
$\sqrt{2}\gamma_{0}$. A double gap slotted concentric cylinder OSP geometry on
the ARES-G2 was used to apply simultaneous flows in the primary (rotational)
and orthogonal (axial) directions. The diameters of the internal surface of
the cup, internal surface of the bob, external surface of the bob, and
external surface of the cup are 27.735, 29.403, 32.072, and 33.996 mm
respectively; the geometry can be operated at immersed height between 42.655
to 38.655 mm. The consistency in measurements with different immersed heights
are tested using Carbopol, as shown in the supporting information (Fig. S1).
For CB suspensions, an immersed height of 42.655 mm was used for all
measurements in this study. Similar to shear rheology tests, a 10 min preshear
at 300 $\rm s^{-1}$ in the rotational direction was applied before every
measurement, until the shear stress reached the steady state (within 3$\%$
variation). The details and results of OSP tests are shown in Sec. III.2 and
Sec. IV.2.

###  II.3 Materials

The carbon black used in this study is Vulcan XC-72 (Cabot Corporation), which
has a density of 1,800 $\rm kg/m^{3}$. The Vulcan XC-72 CB is fractal,
exhibiting a hierarchy of morphological features, and is well studied. The
diameter of the primary particles is 20 nm, which are strongly fused by
covalent bonds into aggregates with a fractal dimension $D_{f}=2.7$ [14]. The
CB particles are dispersed in a heavy mineral oil (Fisher Scientific), which
has a density of 1,200 $\rm kg/m^{3}$ and a Newtonian viscosity of 0.14 $\rm
Pa\cdot s$ at 25 $\rm{}^{o}C$, shown in Fig. S2. The suspensions were prepared
using a rotor-stator homogenizer (IKA) by shearing at 4000 rpm for more than
10 min to make a well dispersed and stable CB suspension in mineral oil.

(a)

(b)

Figure 2: Carbon black (CB) suspensions in heavy mineral oil with four
different mass fractions: (a) flow sweep of CB suspensions with four mass
fractions, from the top to bottom are 10, 8, 6, 4 wt$\%$ respectively. Symbols
indicate the maximum transient stress during the step down shear tests, when
the shear rate decreases from 100 $\rm s^{-1}$ to the shear rates of interest;
the stress keeps decreasing after reaching the maximum value, as shown by the
line segment below each symbol; full transient data shown in Fig. 3. The solid
lines show the Herschel-Bulkley fits to the open symbols. (b) The apparent
yield stress, $\tau_{y}$, obtained from the Herschel-Bulkley model fits,
versus the mass fraction; the red line is the power-law fit, which gives a
slope of 3.4; the inset has photos of CB suspensions at four different mass
fractions, with visual evidence of yield stress behavior.

In this study, four mass fractions of CB suspensions were prepared and tested,
which are 4, 6, 8, 10 wt$\%$; the flow curves are shown in Fig. 2(a). With the
increase in CB concentration, the yield stress increases, as suggested by the
photos in Fig. 2(b) (inset). The flow curve in Fig. 2(a) was obtained by
shearing the suspension at 100 $\rm s^{-1}$ until the steady state was
reached, then stepping down to the shear rate of interest. This is to erase
any mechanical history and to ensure that the suspension was at the same
structural state at each shear rate of interest [15], instead of sweeping up
or down shear rates and collecting the steady value at each rate, as is
conventionally done. At higher shear rates, stresses increased with time and
reached steady state very quickly, and could be recorded within 10 s, which is
in agreement with the thixotropic dynamics reported before [15]. At lower
shear rates, however, no steady state was observed, and the stress started
decreasing once it reached its maximum value and it kept decreasing even after
very long times ($\sim$ 3000 s). The absence of steady state has also been
reported by Hipp $et~{}al.$ [15]. In this case, the maximum stress observed
during the transient increase-and-decrease was recorded and plotted as an open
circle in the flow curve, as shown in Fig. 2(a). The vertical solid line
extending below each circle shows the extent of subsequent stress decay in the
time that follows. The full transient stress responses are shown in Fig. 3,
which will be discussed in details later. From Fig. 2(a) we can see that at a
given shear rate, the higher the concentration, the larger the decrease in
stress. For the 4 wt$\%$ CB suspension (shown in blue), the decrease in stress
is unnoticeable at shear rate of 10 $\rm s^{-1}$, while for the CB suspension
of 10 wt$\%$ (shown in red), the stress decreases by around 40 Pa in 600 s at
the same shear rate. For a given concentration, the higher the shear rate, the
smaller the decrease in the stress. For the 10 wt$\%$ suspension, the decrease
becomes unnoticeable when the shear rate is larger than around 80 $\rm
s^{-1}$. Here, we use the maximum stresses during step-down in shear rate
tests to fit the Herschel-Bulkley model [56, 57] for each concentration,

$\tau=\tau_{y}\left[1+\left(\frac{\dot{\gamma}}{\dot{\gamma}_{\rm
crit}}\right)^{n}\right],~{}~{}~{}~{}~{}~{}\rm for~{}|\tau|>\tau_{y},$ (1)

with $\tau$ being the stress and $\dot{\gamma}$ being the shear rate;
$\tau_{y}$, $\dot{\gamma}_{\rm crit}$, and $n$ are fit parameters. The fit
results for each concentration are shown in Tab. 1, and the fitted apparent
yield stress, $\tau_{y}$, is plotted in Fig. 2(b) as a function of
concentration. Consistent with the photos (inset), the yield stress increases
with concentration, and the increase follows a power law of slope 3.4, shown
by the red solid line in Fig. 2(b). The value of the slope is consistent with
what was found in previous literature for fractal CB suspensions, which
reported a slope of 3.5 [42].

Table 1: Herschel-Bulkley model (Eq. 1) fit results for CB suspensions with different concentrations. Concentration | $\tau_{y}$ | $\dot{\gamma}_{\rm crit}$ | $n$
---|---|---|---
wt$\%$ | Pa | $\rm s^{-1}$ |
4 | 6.56 | 6.99 | 0.67
6 | 25.30 | 24.33 | 0.61
8 | 63.61 | 60.76 | 0.59
10 | 150.00 | 122.06 | 0.54

##  III Experimental RESULTS

###  III.1 Verification of a reversible, reproducible, and intrinsic material
response

A peculiar combined short-time recovery and long-time decay is observed in
transient stress responses at certain constant applied final shear rates,
$\dot{\gamma}_{f}$, on 6 wt$\%$ CB suspensions, as shown in Fig. 3. After
preshearing at 300 $\rm s^{-1}$ for 10 min, the sample was sheared at an
initial shear rate, $\dot{\gamma}_{i}=100~{}\rm s^{-1}$ for 2 min, and then
the applied shear rate was stepped down from the initial value to different
final shear rates ranging from 0.5 to 80 $\rm s^{-1}$, at time zero. Fig. 3
shows that two distinct ranges are observed in transient stress responses. In
the high final shear rate range ($\dot{\gamma}_{f}>15\rm~{}s^{-1}$), the
stress shows a typical fast thixotropic recovery: increasing with time and
reaching steady state within 1 s. For the final shear rate of 80 $\rm s^{-1}$,
the recovery is so fast (within motor time limits) and the stress recovered is
so small that the thixotropy is negligible. In the low final shear rate range
($\dot{\gamma}_{f}<15\rm~{}s^{-1}$), the stress shows a peculiar recovery-and-
decay response as discussed before: increasing until reaching its maximum
during the thixotropic recovery, then starting to decrease and never reaching
steady state even after 2400 s. In an extreme case where
$\dot{\gamma}_{f}=1\rm~{}s^{-1}$, the steady state was not observed even after
10,000 s. The grey area at very short time (50 ms) represents the motor
ramping-down time, before which the applied shear rate is unsteady because of
a finite motor response time. The input shear rate time histories during the
step-down tests are shown in Fig S3, where we can see that the motor ramping
time varies for different final shear rates; the lower the final shear rate,
the longer it takes for the motor to reach the steady velocity. The maximum
time for the motor to reach the steady velocity is around 50 ms, guaranteeing
that the transient stress responses outside the grey area are not due to motor
transients.

Figure 3: The transient stress measured during the step-down shear tests with
the initial shear rate of 100 $\rm s^{-1}$ and final shear rates ranging from
80 to 0.5 $\rm s^{-1}$ for the 6 wt$\%$ CB suspension shows a two-region
response. At high shear rates, stress shows a quick thixotropic recovery,
while at low shear rates, a peculiar, combined short-time recovery and long-
time decay dynamics is observed.

The transient recovery-then-decay responses at constant applied shear rates
also appear in CB suspensions at other concentrations (4, 8, 10 wt$\%$), and
the detailed results are shown in Fig. S4. The critical shear rate, at which
the stress response changes from typical thixotropy to the peculiar combined
recovery-and-decay dynamics, shows a concentration dependence. We are unable
to precisely determine the value of the critical shear rate because of the
large spacing between the shear rates applied. But in general, the higher the
concentration, the larger the critical shear rate. This suggests that a larger
hydrodynamic stress is required for more concentrated CB suspensions to
undergo the structural change that is responsible for the two-regions stress
response and the recovery and decay at low shear rates. The structural origins
are discussed in Sec. IV.1.

(a)

(b)

(c)

(d)

Figure 4: Verifying the reversibility and repeatability of the shear stress
response. The step shear as shown in (a) is applied on the 6 wt$\%$ CB
suspension, and the edge of the sample is observed during the test, as shown
in (b). The stress response is measured as (c) and is also plotted in
logarithmic time scale as (d). The edge does not change during the test,
ruling out artifacts such as edge failure. The repeated measurement results
show that the recovery-and-decay dynamics is reversible and repeatable, as the
stress responses for the three steps of time (I, II, and III) overlap.

The reversibility and repeatability of the shear stress response was verified
with a repeated step shear rate scheduling using a 25 $\rm mm$ diameter
parallel plate geometry. Fig. 4(a) shows the input shear rate, where the 6
wt$\%$ CB suspension is sheared at 100 $\rm s^{-1}$ for 120 seconds, then the
rate is stepped down to 1 $\rm s^{-1}$ and maintained for 40 minutes. The
shear rate is stepped up to 100 $\rm s^{-1}$ again and this cycle is repeated
for three times without any time delay. In the meantime, the edge of the CB
suspension sample is visually recorded. The video of the repeated test is
available in the SI, and the photos of the CB sample before and after tests
are shown in Fig. 4(b). From the recording we can see that the sample remains
identical before and after the test, and there is no visual edge failure or
sample loss during the whole process. This rules out experimental artifacts
such as sample loss or edge failure. The stress response over time during the
reversibility tests is plotted in Fig. 4(c). In all three sets of step-down
shear rate tests, the shear stresses show a sudden jump, followed by a decay
at long times, not reaching the steady state even after 40 minutes. We further
coplot the stress responses for each set of the three repeated tests in Fig.
4(d) on a semilog scale, and the stress responses overlap with each other,
which shows that the peculiar combined stress recovery and decay dynamics is
reversible and repeatable.

Figure 5: Verifying the absence of wall-slip and shear-banding for the
combined recovery-and-decay dynamics by measuring the transient stress
response during the step-down in shear rate test at two different gaps using a
25 mm parallel plate geometry. The two stresses do not change with the
measuring gap, showing that there is no slip or shear-banding at the shear
rates tested.

Next, we verify that the dynamics is not due to wall-slip or shear banding, by
confirming the reproducibility of the step-down shear rate tests at two
different operating gaps, 500 and 750 $\rm\mu m$, using the 25 $\rm mm$
diameter parallel plate geometry [49]. Fig. 5 shows the transient shear stress
evolution, when the shear rate is decreased from 100 $\rm s^{-1}$ to 1 $\rm
s^{-1}$, at different gaps. The stress responses do not depend on the
operating gap, proving that there is no wall slip or shear banding occurring
during the step-down shear rate tests for the shear rates tested. This is
consistent with the results in literature, where wall slip is found to occur
only at much lower shear rates (around $0.05~{}\rm s^{-1}$ and lower)[36]. It
should be noted that at short times, the stress signals oscillate around the
mean value. The oscillation is not due to uneven sample loading or
viscoelastic material waves, proved by a mismatch between stress oscillating
frequency, motor rotating frequency, and the viscoelastic wave frequency,
shown in the SI. Another possible reason for such oscillation is the force
balancing transducer response, which we did not investigate here. Although the
reason behind the oscillations is undetermined, the recovery trend at short
times is distinct.

We therefore conclude that the peculiar combined short-time recovery and long-
time decay dynamics is a true material response, instead of being due to
common experimental artifacts such as edge failure, slip, or shear banding,
and it is repeatable and reversible. However, whether the decay at long times
comes from anti-thixotropy or viscoelasticity remains unknown, as both anti-
thixotropic and viscoelastic materials show stress decay in a step-down in
shear test (Fig. 1). In the next section, we will show that OSP combined with
a step-down in shear rate test is needed to clarify the origin of the stress
decay, from which we conclude that the decay is indeed anti-thixotropic and is
related to shear-induced structure rearrangement.

###  III.2 Verification of anti-thixotropic structure rearrangement using OSP

Orthogonal superposition (OSP) can potentially help to distinguish between
anti-thixotropy and viscoelasticity, as responses from elastic and viscous
components are separated. Fig. 6 shows the time evolution of orthogonal
moduli, $G^{\prime}_{\perp}$ and $G^{\prime\prime}_{\perp}$, for the 6 wt$\%$
CB suspension, where the solid circles in Fig. 6(a) are $G^{\prime}_{\perp}$,
and open circles in Fig. 6(b) are $G^{\prime\prime}_{\perp}$. Similar to the
transient shear stress responses under step shear tests, two distinct ranges
in moduli responses are observed. In the high shear rate range, both
$G^{\prime}_{\perp}$ and $G^{\prime\prime}_{\perp}$ recover from lower values
because of the structure build-up, while in the low shear rate range, a
combined recovery-and-decay dynamics appears in both storage and loss moduli.
When $\dot{\gamma}_{f}$ is high, the recovery is very fast, and the moduli
increase and reach steady states within 3 s. This response time is shorter
than the time needed to collect oscillatory data at 5 rad/s, due to transients
and multiple cycles averaging, and therefore the increase in
$G^{\prime}_{\perp}$ and $G^{\prime\prime}_{\perp}$ cannot be resolved for
times $\lesssim 3\rm~{}s$. With decreasing $\dot{\gamma}_{f}$, the recovery
occurs over a longer timescale; the moduli reaching their respective maximum
values after $\sim$ 100 s for $G^{\prime}_{\perp}$ and $\sim$ 10 s for
$G^{\prime\prime}_{\perp}$. Beyond their respective maximum values, both
$G^{\prime}_{\perp}$ and $G^{\prime\prime}_{\perp}$ start decreasing with no
apparent steady state values even at extremely long times, identical to the
stress transients in the low rate region. The decrease observed in both
storage and loss moduli suggests that the long-time decay is related to
structural rearrangement of CB, which changes both the elastic and viscous
properties. Therefore, the long time decay observed is a result of anti-
thixotropy, instead of viscoelastic stress relaxation. From Fig. 6 we can also
see that the critical shear rate, below which the moduli start to show such
combined dynamics, is around 15 $\rm s^{-1}$, although it cannot be determined
precisely. The value of the critical shear rate is about the same as observed
in the shear stress response in Fig. 3. This suggests that the transient
responses of shear stress and orthogonal moduli have the same structural
origin, which is a result of CB structure rearrangement involving not only
viscous, but also elastic stress contributions.

(a)

(b)

Figure 6: The OSP results prove that the decay at long times comes from anti-
thixotropy. The transient orthogonal (a) storage and (b) loss moduli of 6
wt$\%$ CB suspensions, measured by superposing an orthogonal oscillatory flow
with 0.5$\%$ and 5 rad/s on the step-down shear rate flow in the rotational
direction.

The long-time decay is shown to be shear-induced with the help of OSP,
providing additional evidence for anti-thixotropic structure rearrangements
rather than viscoelastic relaxation. Fig. 7 shows $G^{\prime}_{\perp}(t)$ and
$G^{\prime\prime}_{\perp}(t)$ during the cessation of steady flow at 100 $\rm
s^{-1}$. Without applied shear, the orthogonal moduli only show a thixotropic
recovery, increasing monotonically to their respective steady state values,
and the storage modulus is much larger than the loss modulus during the
cessation. The increase in storage modulus suggests that the suspension
undergoes elastic stiffening at rest, rather than the structure rearrangement
that decreases the viscosity and moduli. The increase in storage modulus at
rest is consistent with the results on fumed silica suspensions [53] and has
been reported on CB suspensions [20], where the value of the storage modulus
depends on the applied shear stress before cessation. The stiffening at rest
also suggests that a low but definite applied shear rate is required for CB
suspensions to undergo the structure rearrangement that decreases the
viscosity and moduli, and the decay in both the viscosity and moduli is
therefore shear-induced. This provides overwhelming evidence that the decrease
is not a result of viscoelastic relaxation, but anti-thixotropic shear-induced
structural change. The ability to measure rheological properties changing with
time after cessation of flow is another advantage of superposition rheometry,
including orthogonal and parallel superposition, which is not possible with
the standard cessation of shear because of no shear applied. The same
technique, using OSP during cessation of steady flow, has been used on fumed
silica suspensions to study the anisotropy of the resulting colloidal gel
[53], but this is the first time OSP has been used on CB suspensions to study
the anti-thixotropic shear-induced structuring of CB particles.

Figure 7: The results of cessation of steady shear flow at 100 $\rm s^{-1}$
show no long-time decay, and therefore such decay as in Fig 6 and Fig. 8 is
shear-induced, which again verifies that the decay is anti-thixotropic. The
transient orthogonal storage modulus (solid circle) and loss modulus (open
circle) of a 6 wt$\%$ CB suspension measured during the flow cessation test,
where the flow is stopped from shearing at the initial shear rate of 100 $\rm
s^{-1}$, while measuring the orthogonal moduli using OSP simultaneously
throughout.

The reversibility and repeatability of the OSP results are verified by
applying the repeated step shear rate scheduling from 100 to 1 $\rm s^{-1}$ as
shown in Fig. 8(a), while measuring $G^{\prime}_{\perp}$ and
$G^{\prime\prime}_{\perp}$ at $\omega=5$ rad/s and $\gamma_{0}=0.5\%$
simultaneously throughout. The corresponding storage modulus as a function of
time during the repeated measurements is shown as Fig. 8(b) and the moduli
responses during the three sets of stepping-down tests are co-plotted on a
logarithmic time scale in Fig. 8(c). The loss modulus is omitted for
simplicity. The response is reversible in repeated tests and the storage
moduli overlap (Fig. 8(c)), all showing the short-time thixotropic recovery
followed by the long-time anti-thixotropic decay. This verifies that the OSP
results are repeatable and reversible, and they come from a true material
response, rather than experimental artifacts.

(a)

(b)

(c)

Figure 8: Verifying the repeatability and reversibility of the orthogonal
moduli response. The step shear rate flow and an oscillatory flow in the
orthogonal direction with 5 rad/s and $0.5\%$ strain, as shown in (a) is
applied on the 6 wt$\%$ CB suspension, and the storage modulus in the
orthogonal direction is recorded, as shown in (b); the modulus response during
the stepping-down for the three sets is also plotted on a log time scale,
which shows the dynamics are repeatable and reversible, as the modulus
responses for the three sets overlap.

It should be noted that the fluctuation of both $G_{\perp}^{\prime}$ and
$G_{\perp}^{\prime\prime}$, especially at the high shear rates, exists in both
storage and loss moduli, which is likely to originate from Taylor vortices due
to high angular rotation rates. Studying the limits of the operating windows
of the OSP, however, is outside the scope of this work. Another caveat of
using OSP is that the orthogonal oscillation can significantly disrupt
particle contacts for some systems [58], and therefore change the properties
in the primary shear direction. To avoid structure disruption by the
orthogonal oscillation, the amplitude used here is as low as possible while
being sufficient to generate a detectable stress signal. For the 6 wt$\%$ CB
suspension, the amplitude is 0.5$\%$ and the frequency is 5 rad/s, which is in
the linear viscoelastic regime of the suspension. The amplitude sweep results
are shown in Fig. S5. We also test the effect of orthogonal oscillation on the
primary shear stress by comparing the shear stress in the rotational direction
during OSP tests with the transient stress response shown in Fig. 3, which
shows the same response at the same timescale, as shown in Fig. S7. This
suggests that the rheological properties, and therefore, the structure of CB
in the primary flow direction are unaffected by the small amplitude orthogonal
oscillation.

The transient orthogonal moduli for CB suspensions of other weight fractions
are shown in Fig. S8-S10, which show that the thixotropic recovery and anti-
thixotropic decay in the low final shear rate region are ubiquitous to all
concentrations tested. OSP provides us with the first definitive evidence of
the distinction between viscoelasticity and anti-thixotropy of CB suspensions,
which is also the first instance of using OSP to resolve the ambiguity between
viscoelasticity and anti-thixotropy. With the help of step-down in shear rate
tests coupled with OSP, we conclude that the time varying mechanical response
of the CB suspension under step shear is a combined short-time thixotropic
recovery and long-time anti-thixotropic decay.

###  III.3 Anisotropy in thixotropic timescales and moduli

We observe different timescales of thixotropic dynamics for different
rheological measures, which indicates anisotropy in CB suspensions. Fig. 9
shows the orthogonal moduli, $G^{\prime}_{\perp}$ and
$G^{\prime\prime}_{\perp}$, and shear viscosity, $\eta$, calculated from the
shear stress, co-plotted for several final shear rates, 0.5, 8, and 40 $\rm
s^{-1}$ (from data in Fig. 3 and Fig. 6). Consistent with earlier plots, the
shear viscosity is plotted as a solid line, the orthogonal storage and loss
moduli are plotted as solid and open circles respectively. It can be seen in
Fig. 9 that at the same shear rate, the thixotropic timescales are different
for shear viscosity in the rotational direction and moduli in the orthogonal
direction. For example, when the final shear rate is $0.5~{}\rm s^{-1}$, as
shown in Fig. 9(a), the shear viscosity reaches its maximum first (at 3 s),
followed by the orthogonal loss modulus (at 15 s), and finally the orthogonal
storage modulus (at 100 s), taking more than one order of magnitude longer
than the shear viscosity. The same trend is observed for
$\dot{\gamma}_{f}=8~{}\rm s^{-1}$ (shown in Fig. 9(b)) and
$\dot{\gamma}_{f}=40~{}\rm s^{-1}$ (Fig. 9(c)). For all the shear rates tested
ranging from 0.5 to 80 $\rm s^{-1}$, there is a separation of thixotropic
timescales between shear stress/viscosity in the rotational direction and the
moduli in the orthogonal direction: the anti-thixotropic structural
rearrangement under shear is delayed in the orthogonal direction. The
separation of thixotropic timescales suggests an anisotropic CB structure in
the rotational and orthogonal directions, $e.g.$, due to the size of CB
agglomerates longer in the orthogonal direction than in the rotational one.

(a)

(b)

(c)

Figure 9: Evidence of anisotropic dynamics from co-plotted parallel transient
shear viscosity and orthogonal storage and viscous moduli during the step–down
shear tests (data from Fig. 3 and Fig. 6). The shear rate is decreased from
100 to (a) 0.5; (b) 8; and (c) 40 $\rm s^{-1}$ respectively.

The mechanical anisotropy also shows in the difference between the amplitude
of moduli, where the storage and loss moduli in the orthogonal direction are
larger than those in the rotational direction by a factor of two. Fig. 10(a)
shows the storage moduli in the rotational direction, $G^{\prime}$, and those
in the orthogonal direction, $G^{\prime}_{\perp}$, at frequencies ranging from
10 to 100 rad/s, measured using a 2D-small amplitude oscillatory shear
(2D-SAOS) at a peak amplitude of $0.5\%$, while Fig. 10(b) shows the loss
moduli, $G^{\prime\prime}$ and $G^{\prime\prime}_{\perp}$. Both storage and
loss moduli are larger in the orthogonal direction than those in the
rotational direction. As a benchmark test, the moduli in both directions are
measured using the same OSP geometry on aqueous Carbopol 940 at 1 wt$\%$,
which is known as a homogeneous, isotropic, model yield stress material, and
our results show that both storage and loss moduli are the same in both
directions within 10$\%$; details in Fig. S11. This validates the anisotropic
moduli measured on CB suspensions to be of a structural origin, and not
experimental artifacts. We can also see from Fig. 10 that all moduli show a
weak dependence on the frequency, and storage moduli are larger than the loss
moduli, which are signatures of soft solid behavior. Similar trends in the
moduli can be observed for CB suspensions at the other concentrations tested,
which suggests a structural anisotropy across all concentrations tested. The
mechanical anisotropy has also been shown using OSP on other attractive
colloidal systems such as fumed silica suspensions, where the storage moduli
in the orthogonal direction are larger[53] than the rotational storage moduli
by a factor of two to a hundred, depending on the shear conditions.

(a)

(b)

Figure 10: Stiffer in the perpendicular direction. The anisotropic (a) elastic
and (b) loss moduli of 6 wt$\%$ CB suspension for frequencies from 10 to 100
rad/s after shearing at 1 $\rm s^{-1}$ for 100 s, measured using OSP 2D
oscillation with peak amplitude of 0.5 $\%$.

The mechanical anisotropy in thixotropic timescales and the magnitude of
linear moduli suggests the formation of anisotropic CB agglomerates under the
applied low shear rate. We hypothesize that this arises from long slender
structures aligned in the orthogonal vorticity direction. Such "log rolling"
anisotropic structures for CB suspensions have been reported before [25]. It
is shear-induced, formed at low shear rates, aligned in the vorticity
(orthogonal, in this case) direction, and separated from each other by a
certain distance to form a striped pattern. The anisotropic log-rolling flocs
have been observed in the same type of CB suspensions of various
concentrations using rheo-optical setups [24, 22]. The striped pattern of the
log-rolling structure is only realized in the range of 1-2 wt$\%$; at the
higher concentrations (2.5-3 wt$\%$) only several "holes" are observable
between the large agglomerates [24]. At concentrations tested here (6-10
wt$\%$), the log-rolling structure is not visible using optical setups, likely
due to the high CB density and an overlap between large flocs. However, the
mechanical anisotropy in both the thixotropic timescale and the moduli
measured using OSP are consistent with the hypothesis of anisotropic
vorticity-aligned structures. The rheological OSP technique is robust and
versatile in detecting the anisotropic structure even without high-resolution
microscopy.

###  III.4 Concentration dependence of stresses and moduli

The combined short-time thixotropic recovery and long-time anti-thixotropic
decay in shear stress and orthogonal moduli are ubiquitous across all four
concentrations of CB suspensions tested (4, 6, 8, and 10 wt$\%$). Fig. 11
shows the shear stresses, orthogonal storage moduli, and orthogonal loss
moduli in response to a step down shear from 100 to 1 $\rm s^{-1}$ for CB
suspensions of the four concentrations. The larger the CB concentration, the
higher the shear stress, as well as the orthogonal moduli, as expected. Both
the thixotropic recovery and anti-thixotropic decay are more dramatic for
higher concentration suspensions, showing larger stress and storage moduli
recovered. For the lowest concentration, 4 wt$\%$, the thixotropy and anti-
thixotropy in shear stress and orthogonal moduli are more difficult to
observe.

(a)

(b)

(c)

Figure 11: The combined short-time thixotropic recovery and long-time anti-
thixotropic decay in shear stress and orthogonal moduli is ubiquitous in the
four different concentrations of CB suspensions tested, which are 4, 6, 8, and
10 wt$\%$. The (a) shear stress, (b) orthogonal storage modulus, and (c) the
orthogonal loss modulus for the four concentrations of CB suspensions are
plotted as a function of time during a step-down test, where the shear rate is
decreased from 100 to 1 $\rm s^{-1}$.

There is no obvious difference among timescales for different concentrations.
However, for all concentrations, similar to the 6 wt$\%$ suspension, as shown
in Sec. III.3, the difference in timescales of the shear response and the
orthogonal response is distinct, with thixotropic times for orthogonal moduli
orders of magnitude longer than those for shear stress. The anisotropy in the
thixotropic timescale is consistent with the existence of vorticity aligned
log-rolling structures in CB suspensions over a wide range of concentrations,
as discussed in Sec. III.3.

###  III.5 Additional evidence for anti-thixotropy from hysteresis loops

In this section, we show additional evidence of anti-thixotropy of CB
suspensions using hysteresis tests. There are multiple ways of conducting a
hysteresis test [46, 59, 60, 61]. In this study, the hysteresis flow was
applied on CB suspensions by starting at high shear rate and decreasing it
from 300 to 0.03 $\rm s^{-1}$ through $n=20$ successive logarithmically spaced
steps per decade of $\dot{\gamma}$ with duration $\delta t$ at each step, and
then increasing to 300 $\rm s^{-1}$ again following the same steps and the
same duration $\delta t$ per step. In this way, the total number of shear
rates in a decreasing ramping was $N=81$ steps. Different duration per step,
$\delta t$, were used [46], which were $\delta t=1$, 5, 20, 50, and 150 s.

Fig. 12 shows the first half of the hysteresis loops for 6 wt$\%$ CB
suspension, that is, when the shear rate ramps down from 300 to 0.03 $\rm
s^{-1}$. The hysteresis curve with $\delta t=1~{}\rm s$ is shown in light
green, and the curve with longest duration $\delta t=150~{}\rm s$ is shown in
dark green. For quick ramping, where the duration is $\delta t=1~{}\rm s$, the
total ramping time is 80 s. In this case, the CB structure is not given enough
time to restructure and the suspension behaves like a yield stress fluid with
a Herschel-Bulkley shaped flow curve. With the increase in duration, the CB
structure has more time to rearrange at each step, and the stress decreases
more at each shear rate. For the longest duration, $\delta t=150~{}\rm s$, the
total ramping time is 12,000 s, and the stress is the lowest, with minimal
apparent dynamic yield stress.

Figure 12: Shear stress versus shear rate in first-half of hysteresis loop
(ramping high-to-low) shift down with duration, $\delta t$, measured in a 6
wt$\%$ CB suspension, where shear rate is decreased from 300 to 0.03 $\rm
s^{-1}$ with different duration. Each ramp is performed with $n$ = 20 points
per decade, and a different duration per point $\delta t$ is used for each
experiment. The arrow indicates the direction of ramping (from high to low).

Another feature of the hysteresis tests of CB suspensions is the direction of
the hysteresis loops, which consist of both the downward and upward ramping.
Fig. 13 shows the hysteresis loops with different duration, from top to
bottom, the duration per step increasing from 1 s to 150 s, and the total time
of the hysteresis loop increasing proportionally. The data plotted as circles
represents the downward ramping, labeled with the black arrow and the number
"1" shown in black, while the squares, the red arrow, and the number "2"
represent the upward ramping. From the plot we can see that for quick ramping,
where $\delta t=1~{}\rm s$, the up and down ramps overlap and the area of the
loop is vanishingly small. This is because during the hysteresis test (total
time of 160 s, including both up and down ramping), the CB structure is not
given enough time to restructure and therefore, the rheological properties of
the suspension remain the same in down and up ramping. With the increase in
$\delta t$, the area of the hysteresis loop increases. The increase in the
area of hysteresis loops with duration is consistent with previous studies by
Divoux and coworkers [46, 62], where they found that for thixotropic
materials, the area first increases then decreases with the duration, and the
duration for the maximum area provides a timescale that can be compared
between different materials. However, the previous studies on hysteresis loops
fail to point out the direction of the loops, which is an important feature.
The direction of the hysteresis loop of CB suspensions, if there is any, is
counter-clockwise: the downward ramping stress is higher than the upward
ramping one.

Figure 13: Hysteresis tests indicate anti-thixotropic dynamics with counter-
clockwise loops. Hysteresis loops measured in a 6 wt$\%$ CB suspension, by
decreasing shear rate from 300 to 0.03 $\rm s^{-1}$ and then increasing it
with the same step. Each ramp is performed with $n$ = 20 points per decade,
and a different duration per point $\delta t$ is used for each experiment.

In our recent studies, we show that there are two distinguishing features in
hysteresis loops that can fully differentiate between thixotropy, anti-
thixotropy, and viscoelasticity [63]. The direction of the hysteresis loop for
thixotropic materials is clockwise, while for anti-thixotropic and
viscoelastic materials, they are counter-clockwise. The second feature is that
at quick ramping (small duration $\delta t$), where the down- and upward
ramping flow curves overlap, the stress is a constant for a viscoelastic fluid
that cannot relax, while the flow curve suggests a yield stress fluid for
thixotropic and anti-thixotropic materials. For the CB suspensions tested
here, the direction of hysteresis loops is counter-clockwise, while the stress
rate curve shows a Herschel-Bulkley shape at quick ramping ($\delta t=1~{}\rm
s$). This adds additional evidence that the CB suspension is anti-thixotropic.
It is worth noticing that for the shortest duration ($\delta t=1\rm~{}s$), the
upward ramping stress is slightly higher than the downward ramping stress,
showing a clockwise loop at the low shear rate regime. This is consistent with
the short-time thixotropic dynamics, as shown in the stress recovery in Fig.
3.

##  IV DISCUSSION

###  IV.1 Microstructural model to rationalize the anisotropic thixotropy and
anti-thixotropy

We hypothesize a microstructural schematic to rationalize the anisotropic
thixotropic and anti-thixotropic dynamics of CB suspensions under step-down
shear rate flow, as shown in Fig. 14. Our extensive rheological test results
suggest a two-timescales structure rearrangement process, which in short
timescales increases the viscosity and moduli of the suspension, and in long
timescales decreases them. Based on this, we conjecture that at short times,
relatively open, large, and loosely-connected agglomerates form under low
final shear rates due to attractive van der Waals forces between CB
aggregates, resulting in an overall more structured state and a higher
effective hydrodynamic volume fraction. This increases the viscosity of the
suspension and the resulting shear stress. At such low shear rate, log-rolling
structures begin to form, which has been observed and reported for carbon
black suspensions before [25]. Therefore, we hypothesize that the structured
agglomerates are asymmetric in shape, aligning in the vorticity direction and
rotating around the vorticity axis. The anisotropy in structure results in the
difference in the timescales between shear stresses and orthogonal moduli, and
the difference in the magnitude of the moduli in the parallel and orthogonal
directions. At longer times, the observed anti-thixotropic dynamics is due to
these open agglomerates self-organizing and inter-penetrating, condensing the
flocs into a denser, strongly-connected, and closed structure in both shear
and orthogonal directions. This decreases the hydrodynamic volume, and
therefore decreases the shear stress and moduli over long time scale. Such
densification is shear-induced, as evidenced by the lack of moduli evolution
in the cessation flow in Fig. 7, and this can explain the long-time anti-
thixotropic decay.

Figure 14: Possible mechanical schematics of the CB structure under step-down
shear rate flow. The open, large, more structured and loosely-connected
agglomerates (middle panel) are formed during the short-time thixotropic
build-up, which causes shear stress and orthogonal moduli to increase; while
the shear-induced anti-thixotropic densification (right panel) results in
stress decay at long times. The structure is anisotropic, and being longer in
the vorticity direction, and rotates around the vorticity axis under shear. A
concentric cylinder geometry is shown in the schematic for simplicity, whereas
a double-wall concentric cylinder is used in experiments.

At high shear rates, however, the anti-thixotropic decay is negligible in both
the shear stress and the orthogonal moduli, as shown in Fig. 3 and Fig. 6.
This is because the short-term thixotropic structure build-up is small since
the change in the state of structure is small before and after stepping. The
large, open, and loosely-connected anisotropic agglomerates are more dense and
have less void fraction compared with those formed under lower final shear
rates. Therefore, the long-time anti-thixotropic densification is not
significant, resulting in negligible anti-thixotropic decay over long time,
and the stresses and orthogonal moduli reach steady states more quickly. The
dynamics at large shear rates is consistent with the microstructural model
shown in Fig. 14.

###  IV.2 The growth of anisotropy under low shear

The mechanical schematics in Fig. 14 show how anisotropic structure grows
qualitatively; in this section, we will show the growth of anisotropic
structure more quantitatively using 2D-SAOS. Fig. 15 shows the applied shear
input on the CB suspension, where the sample is sheared at $1~{}\rm s^{-1}$
for a given shearing time, after initially reaching steady state at 100 $\rm
s^{-1}$. Then the flow is ceased and 2D-SAOS at 0.5$\%$ peak strain amplitude
and frequency 100 rad/s is imposed to simultaneously measure $G^{\prime}$ and
$G^{\prime}_{\perp}$ as a function of the shearing time. To quantify the
degree of anisotropy, we define an anisotropy parameter,
$G^{\prime}_{\perp}/G^{\prime}$, the ratio of the orthogonal and parallel
storage moduli. The storage moduli and their ratio are plotted as functions of
the shearing time, as shown in Fig. 15(b). Both shear and orthogonal moduli
show thixotropic recovery and anti-thixotropic decay, with moduli first
increasing then decreasing with the shearing time. This is consistent with
what we observed before in Fig. 3 and Fig. 6. However, the ratio of the
orthogonal and shear moduli, $G^{\prime}_{\perp}/G^{\prime}$, keeps rising
with the shearing time, increasing by 33$\%$ in 10 s and reaching a constant
ratio of 1.8. The change in both the elastic moduli and their ratio suggests
that along with the thixotropic structure build-up and anti-thixotropic decay,
the anisotropic structure in CB suspensions under low shear keeps evolving and
increasing over these timescales.

(a)

(b)

Figure 15: The degree of anisotropy increases with the shearing time; the
orthogonal and rotational elastic moduli both show the thixotropic recovery
and anti-thixotropic decay during the step-down shear test (from 100 to 1 $\rm
s^{-1}$), but their ratio keeps increasing.

##  V CONCLUSIONS

In summary, this study resolved the long-existing ambiguity of transient
dynamics of CB suspensions by using OSP during step-down shear flows, and this
is the first time OSP has been used to study the anti-thixotropic shear-
induced structuring of CB suspensions. From step-down flow tests on CB in
mineral oil suspensions at different concentrations from 4 to 10 wt$\%$, a
two-region stress response is identified. In the high shear rate region,
stress shows a negligible thixotropic recovery and reaches steady state
quickly. While in the low shear rate region, it shows a short-time thixotropic
recovery and a long-time decay. The combined recovery-then-decay was verified
to be a reversible, reproducible, and intrinsic material response, and the
long-time decay was attributed to anti-thixotropy, rather than viscoelasticity
with the help of OSP. This demonstrates that the response involves shear-
induced structuring that affects both viscous and elastic properties. We
further showed a mechanical anisotropy in the CB suspensions under shear,
where we found that the decay in the orthogonal direction happens much later
than that in the rotational direction, and the orthogonal moduli are two times
larger than the moduli in the rotational direction. Such mechanical anisotropy
suggests the existence of an anisotropic log-rolling structure aligned in the
vorticity direction during the stepping-down in shear flow. A mechanical
structure schematic was proposed, considering qualitatively thixotropic
structure build-up and anti-thixotropic densification and anisotropic log-
rolling structure. The growth of anisotropy was verified by the moduli
measured in both directions using OSP 2D-SAOS.

It should be noted that a steady state was never observed for the transient
shear stress, or the orthogonal storage and loss moduli of all suspensions
even after 20 minutes of shearing, once they begin to show anti-thixotropic
decay. The orthogonal moduli measured using OSP clarify that such decay is
anti-thixotropic and can possibly be attributed to shear-induced anti-
thixotropic structure densification at long times.

Importantly, the results presented here suggest a combined thixotropic and
anti-thixotropic dynamics that happens at different timescales, which has not
been acknowledged before. This is the first time the thixotropy, anti-
thixotropy, and viscoelasticity of CB suspensions are clearly differentiated,
and using OSP combined with step shear rate tests provides a protocol to
resolve the ambiguity between anti-thixotropy and viscoelasticity. Our
observation for these CB suspensions is outside the standard paradigm of
thixotropic structure-parameter constitutive models, and this means that the
existing kinetic thixotropic models with only one structure parameter to
represent the structure build-up state are not enough to describe the
structure change of CB under even simple shear flow. The microstructural model
of CB we proposed can rationalize the mechanical responses of CB suspensions,
which can be useful in designing flow-related devices, such as semi-solid flow
batteries, where CB is used as a conductive additive.

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