# Jamming memory into acoustically trained dense suspensions under shear

Edward Y. X. Ong1, Anna R. Barth2, Navneet Singh2, Meera Ramaswamy2, Abhishek
Shetty3, Bulbul Chakraborty4, James P. Sethna2, Itai Cohen2 1 Department of
Applied Engineering and Physics, Cornell University, Ithaca, New York 14850,
USA  2 Department of Physics, Cornell University, Ithaca, New York 14850, USA
3 Department of Rheology, Anton Paar, Ashland, Virginia 23005, USA  4
Department of Physics, Brandeis University, Waltham, Massachusetts 02453, USA

###### Abstract

Systems driven far from equilibrium often retain structural memories of their
processing history. This memory has, in some cases, be shown to dramatically
alter the material response. For example, work hardening in crystalline metals
can alter the hardness, yield strength, and tensile strength to prevent
catastrophic failure. Whether memory of processing history can be similarly
exploited in flowing systems, where significantly larger changes in structure
should be possible, remains poorly understood. Here, we demonstrate a
promising route to embedding such useful memories. We build on work showing
that exposing a sheared dense suspension to acoustic perturbations of
different power allows for dramatically tuning the sheared suspension
viscosity and underlying structure. We find that, for sufficiently dense
suspensions, upon removing the acoustic perturbations, the suspension shear
jams with shear stress contributions from the maximum compressive and maximum
extensive axes that reflect or ”remember” the acoustic training. Because the
contributions from these two orthogonal axes to the total shear stress are
antagonistic, it is possible to tune the resulting suspension response in
surprising ways. For example, we show that differently trained sheared
suspensions exhibit: 1) different susceptibility to the same acoustic
perturbation; 2) orders of magnitude changes in their instantaneous
viscosities upon shear reversal; and 3) even a shear stress that increases in
magnitude upon shear cessation. We work through these examples to explain the
underlying mechanisms governing each behavior. Then, to illustrate the power
of this approach for controlling suspension properties, we demonstrate that
flowing states well below the shear jamming threshold can be shear jammed via
acoustic training. Collectively, our work paves the way for using acoustically
induced memory in dense suspensions to generate rapidly and widely tunable
materials.

##  I Introduction

Dense non-equilibrium systems have a memory of their history and can undergo
large qualitative changes in their material property depending on the protocol
under which they are prepared [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 16, 17, 18]. For instance, the yield stress of a crystalline metal will
increase under repeated deformation (work hardening) [19, 20, 21, 22]. As
another example, a granular material will densify when tapped and become
harder to shear [23, 24, 25, 26, 27, 28, 29]. In suspensions, numerous
investigations have shown that shear induced memory can give rise to
interesting behaviors including return point memory, coupling between
hysterons, and increased yield strains [7, 10, 30, 12, 31, 32]. Memory in
these studies is typically created through applying perturbations such as
small amplitude oscillatory shear, which provides a controlled environment for
studying the physics by limiting the extent to which the microstructure is
disrupted. The downside of this controlled approach, however, is that the
changes induced in the bulk material properties are often relatively modest in
comparison to what should be possible. In particular, theoretical studies have
shown that through reorganization of contact networks (tuning by pruning) it
should be possible to generate up to 13 orders of magnitude change in the
ratio of shear to bulk modulus [33, 34, 35, 36, 37, 34, 35, 38, 39, 40]. These
studies suggest that if we can significantly alter the suspension contact
network in a controlled fashion, we may be able to dial in or select
dramatically altered suspension properties.

Generating and retaining memory of large changes in contact networks in
flowing systems, however, is challenging since the shear flow often
reorganizes the systems and erases any memory of the training protocols. A
promising but relatively unexplored system that could overcome this limitation
is a shear jamming suspension, which is characterized by a rapid transition
from a flowing state to a jammed state when driven beyond a critical applied
shear stress [41, 42, 43, 30, 44, 45, 46, 47, 48, 49, 50]. Here, the
suspension structure could be significantly modified while it is flowing and
signatures of that structure could be remembered by rapidly shear jamming the
suspension. In such systems, it may be possible to create a variety of shear
jammed states with dramatically different material responses through different
training protocols.

Figure 1: Plots of shear stress versus shear rate for silica in dipropylene
glycol at volume fractions of a) 0.565 and b) 0.555. For a), the “X” marker
indicates a shear stress of 380Pa, which most of the experiments in the paper
were performed at. The circle corresponds to a stress of 38Pa, applied for the
protocol in Fig. 6b. The diamond corresponds to a stress of 3.8Pa, applied for
the protocol in Fig. S6 of the supplementary material. For b), the triangle
corresponds to a stress of 380Pa, applied for the protocol in Fig. 6a. c)
Schematic of the customized geometry used to apply both shear and acoustic
perturbation.

We demonstrate that this approach is indeed feasible by applying acoustic
perturbations to shear jamming suspensions. Previous studies have demonstrated
that acoustic perturbations can significantly disrupt a suspension’s
microstructure to dethicken a shear thickened suspension [51, 52]. Memory of
the dethickening, however, is rapidly forgotten upon removal of the
perturbation. By working with shear jamming suspensions, we can retain memory
of this structural reorganization even after both shear and acoustic
perturbations have been removed. Specifically, memory of the training induced
by both shear and acoustic is stored in force networks oriented along both the
maximum compressive and extensional axes. Signatures of this memory is then
preserved when the suspension is jammed by the shear stress. We show that
these antagonistic networks can be tuned to generate unexpected behaviors and
large changes in the material properties of the suspensions.

Figure 2: (a) Training (left) and probing (right) protocols plotted as shear
stress (top), acoustic power (middle), and viscosity (bottom) versus time.
During training, different acoustic powers ($P_{train}$) are applied during
the first 50 s (grey patch), as indicated by different colors. During probing,
a constant acoustic power $P_{probe}$=4 W was applied for all $P_{train}$ from
60-90 s (yellow patch). A constant shear stress $\sigma=380$ Pa was applied
for both training and probing. (b) Phase diagram of shear stress versus volume
fraction ($\sigma-\phi$) based on our model framework. The fixed experimental
stress, $\sigma_{app}=380$ Pa and volume fraction, $\phi_{v}=0.565$ are
indicated by the vertical and horizontal grey lines. The isotropic jamming
volume fraction is labelled as $\phi_{0}$. During training, larger $P_{train}$
shifts the shear jamming boundary farther away from the $P_{train}$=0 W shear
jamming boundary (thick black solid line), as indicated by the hypothetical
dashed curves which color follow the legend in (a). (c) Plot of the average
probe viscosity $\eta_{probe}$ against $P_{train}$ during the probing period
(red dashed region in (a). (d) Zoomed in views of the phase diagram. For
$P_{train}<P_{probe}$ (left), the suspension either shear jams at the
appropriately shifted shear jamming boundary (translucent black, blue crosses)
or at $\sigma_{app}$ (brown cross) after training. During probing, the shear
jamming line is shifted above $\sigma_{app}$, allowing states that were shear
jammed below $\sigma_{app}$ to rearrange their microstructure and flow towards
$\sigma_{app}$ (white arrows). For $P_{train}>P_{probe}$ (right), we expect
$\sigma_{app}$ to limit how deeply the system can probe the shear jammed
regime (translucent cross). Instead, we observe a much larger $\eta_{probe}$
and even jamming. indicating that the shear jammed state lies above the 4 W
($P_{probe}$) shear jamming boundary and well above $\sigma_{app}$ (opaque
cross). Thus, acoustic training must embed some sort of structural memory
which is not accounted for in a simple manner by the applied shear stress. We
emphasize that the dashed SJ boundary and arrows in b) and d) are not
measurements but illustrations based on our hypothesis. e),f) Plots of
$\eta_{probe}$ against e) $P_{train}$ and against f) $P_{probe}$ for a
cornstarch in water suspension with mass fraction $\phi_{m}=$0.57.

##  II Experimental Methods

We carry out the measurements on charge-stabilized silica (Angstrom Sphere)
with a diameter of 2 µm suspended in dipropylene glycol (Sigma Aldrich), with
flow curves for volume fractions of $\phi_{v}=0.565$ and $\phi_{v}^{-}=0.555$
shown in Figs. 1a and 1b respectively. Unless otherwise stated, the volume
fraction used is $\phi_{v}=0.565$ and the applied shear stress is
$\sigma_{app}=380$ Pa (indicated by the “X” marker in Fig. 1a), considerably
lower than the incipient stress for slip but still higher than the critical
shear jamming stress of $\sim 200$ Pa. The sedimentation timescale of the
suspension is $\sim 1000$ s, which is far longer than any experimental time
window in the work presented here. Additional information on the suspension
can be found in the supplementary material.

Measurements were performed using a modified Anton Paar MCR-702 stress-
controlled rheometer with a piezoelectric disk bonded to the lower plate, as
illustrated in Fig. 1c. The piezoelectric disk was purchased from APC
International (material 841, 21mm diameter and 1.80mm thickness) and bonded
via epoxy to an aluminum (6061-T6) bottom plate (19mm diameter and 8.57 mm
thickness). The acoustic perturbations are generated by exciting the
piezoelectric disk using a sine waveform with a frequency of 1.16MHz, over a
peak to peak voltage range of 0V to 160V corresponding to an expected
amplitude range of 0 to 24nm, which is much smaller than the gap size of
0.64mm. To minimize slip, we used sandblasted plates as the shearing surfaces.
The rheometer has a measurement noise floor of $\sim 0.01$ Pa and a strain
rate resolution on the order of $10^{-5}$ s-1. The suspension is considered
shear jammed when the shear rate oscillates around zero at a rate comparable
to the instrument sampling frequency. The transition to the shear jammed state
is visually striking on a logarithmic plot of viscosity versus time where
smooth viscosity curves transition to abrupt large amplitude spikes as seen in
Fig. 2a.

Careful sample preparation, loading, and a consistent preshear protocol prior
to and between measurements yielded highly reproducible results (see S.I. for
more details on sample preparation and equipment characterization). The data
shown in this paper was collected over 6 different samples with volume
fraction $\phi_{v}=0.565$ for Figs. 2,3,4 and over 2 different samples with
volume fractions 0.555 and 0.565 for Figs. 6a and 6b respectively. We note
that the results shown here are highly reproducible and can be observed at a
different volume fraction (Fig. S7), using a different geometry (Fig. S8) and
even with a different suspension (Figs. S9, S10, and S11), as discussed in the
supplementary material.

##  III Measurement Protocols and Results

###  III.1 Training protocol

We systematically generate different shear jammed states via a training
protocol as shown in Fig. 2a. The protocol involves shearing the suspensions
at a constant shear stress, $\sigma=380$ Pa (top panel), and applying acoustic
perturbations with different amplitudes controlled by the acoustic power,
$P_{train}$ (middle panel) for 50 s (gray band). We find that for low acoustic
powers, $P_{train}=0$, and 0.25 W (black and blue curves), the suspension
viscosity (bottom panel) increases and eventually jams. As the training power
is increased further, we find that the suspension remains fluid with a
viscosity that stabilizes at a lower and finite value. Once the acoustic
perturbations are switched off, all trained suspensions rapidly shear jam.

Such behavior is reminiscent of a stress-volume fraction phase diagram
commonly used to describe transitions between flow behaviors in sheared dense
suspensions [42, 43, 46, 55], as shown in Fig. 2b. Here, CST refers to
continuous shear thickening, DST to discontinuous shear thickening, SJ to
shear jamming, and J to isotropic jamming. In the classical description, the
suspension is expected to isotropically jam at an infinitesimally small shear
stress beyond a volume fraction $\phi_{0}$ (grey shaded region). Below
$\phi_{0}$, a sufficiently strong shear stress can induce an anistropic jammed
state (red shaded region) for any volume fraction larger than $\phi_{J}$, as
demarcated by the solid black phase boundary. For our protocol, a similar
fluid-to-solid transition is observed, except here the shear stress and volume
fraction are kept fixed and the training power now determines the final state
of the suspension.

We focus on the shear jamming (SJ) boundary (black curve) between the flowing
and rigid suspension states and its dependence on the training power, as shown
in the schematic in Fig. 2b. Since the $P_{train}=0$ W suspension shear jams,
we deduce that the applied stress must be set above the shear jamming
boundary. Since suspensions trained at higher $P_{train}$ flow, we deduce that
the constant applied shear stress of $\sigma_{app}=380$ Pa is no longer
sufficient to shear jam the suspension at the same volume fraction and thus
the shear jamming line must have shifted above $\sigma_{app}$ i.e. closer to
the boundary of the isotropic frictionless jamming phase (J), as indicated by
the dashed lines. We note that similar shifts in the shear jamming phase
boundary have been proposed for sheared dense suspensions under orthogonal
oscillations [53, 54]. Since the suspension for $P_{train}=0.25$ W and 1 W are
respectively jammed and flowing during the application of acoustic
perturbations, we deduce that the applied stress $\sigma_{app}$ for our
suspension volume fraction $\phi_{v}$ must lie between the shifted shear
jamming boundaries for these powers (i.e. between the dashed blue and dashed
brown curves). When the acoustic perturbation is switched off at t$=$50 s, the
suspensions rapidly shear jam, which we interpret as a rapid recovery of the
shear jamming boundary to its original state (black line).

###  III.2 Acoustic probing

To probe how memory of the acoustic training alters the properties of the
shear jammed states, we apply a constant acoustic perturbation with
$P_{probe}$ = 4 W over t $=60-90$ s (yellow band in Fig. 2a). We find that
under this acoustic probing, suspensions trained at $P_{train}<4$ W fluidize
with a viscosity that stabilizes at a finite value. In contrast, suspensions
trained at much higher acoustic powers remain shear jammed. We quantify this
behavior by plotting the stabilized viscosity during probing, $\eta_{probe}$
(averaged over red dashed region in Fig. 2a) versus the training power,
$P_{train}$ in Fig. 2c. Here, the horizontal dashed line indicates the
viscosity of the $P_{train}=0$ W suspension and the vertical line indicates
the power of the acoustic perturbation used to probe all of the suspensions,
$P_{probe}=4$ W. These observations cannot be explained by simple shifting of
the shear jamming boundary and indicate that acoustic training embeds
structural memory in the sheared suspensions.

To understand why, we contextualize the data for the low and high $P_{train}$
regimes in in Fig. 2d based on the framework presented in Fig. 2b. For flowing
states, the suspension microstructure is known to rearrange under shear to
reach a non-equilibrium steady state (defined by time independent
distributions of contact number and stress) determined by the applied shear
stress, $\sigma_{app}$ [48, 55, 56]. When systems are driven into the jammed
state, they form solid structures that balance the stress via nearly elastic
deformations. This transition from a flowing state to a solid shear jammed
state occurs when the applied stress is increased beyond the critical jamming
stress, which is determined by the shear jamming boundary [41, 43, 45]. For
$P_{train}<P_{probe}$ the measurements conform to these expectations. Here,
during training, the suspension either shear jams at the appropriately shifted
shear jamming boundary or reaches the applied stress $\sigma_{app}$
(translucent black, blue, and solid brown crosses in Fig. 2d Left). Upon
probing these systems at $P_{probe}$ =4 W, the shear jamming line shifts above
$\sigma_{app}$, which fluidizes the systems, enabling the suspension to reach
(white arrows) the non-equilibrium steady state determined by $\sigma_{app}$
(gray horizontal line). Similarly, we would expect that for suspensions
trained at higher $P_{train}>P_{probe}$, the constant applied $\sigma_{app}$
would limit how deeply the systems is able to probe the shear jammed regime
(translucent purple cross in Fig. 2d Right), and a similar fluidization should
be observed when $P_{probe}$ =4 W is applied. However, in stark contrast, we
observe a much larger $\eta_{probe}$ and even jamming. Given that
$\sigma_{app}$ and the volume fraction both remained constant, this
inconsistency indicates that some component of the suspension stress must lie
above the 4 W shear jamming boundary and well above $\sigma_{app}$ (opaque
purple cross on the right of Fig. 2d) to drive the suspension into a shear
jammed state post training. The effect of acoustic training is thus to embed
some sort of structural memory that results in a stress balance that enables
such a stress component to exist under a constant $\sigma_{app}$.

We note that the key aspects of acoustic training and probing observed for the
silica suspension can be reproduced using a suspension of cornstarch in water
at a mass fraction of 0.57 (see Figs. S9 and S10 in the Supplementary
Information for more suspension and experimental information), suggesting that
acoustic training can be used as a general method to tune the behavior of
different types of suspensions. The response of the cornstarch suspension to
two different probe powers, $P_{probe}=$2.25W and $P_{probe}=9$ W is shown in
Fig. 2e. For both $P_{probe}$ values, we find that the average probe viscosity
is larger for larger training power, with suspensions trained at sufficiently
high $P_{train}$ remaining shear jammed after the probing power is switched
on. These findings are consistent with what was observed for the silica
suspension. In addition, we find that the threshold $P_{train}$ required for
the suspension to remain shear jammed is larger for higher $P_{probe}$, and
the probe viscosity for unjammed states is lower for higher $P_{probe}$. These
observations were again reproduced when a sweep across $P_{probe}$ was
conducted while keeping $P_{train}$ fixed, as shown in Fig. 2f.

Our results demonstrate that the shear jammed state remembers, or contains
signatures of the acoustic training. In particular, the choice of training
power determines whether the suspension will flow or shear jam under later
acoustic probing at some $P_{probe}$. Since both the volume fraction and
applied stress remained constant across the training and probing protocols, we
hypothesize that the acoustic training changes the stress balance of the
suspension state such that for the shear jammed states, a stress component
lies above both $\sigma_{app}$ and the shifted shear jamming boundary at
$P_{probe}$.

Figure 3: (a) Protocol for the shear reversal experiment plotted as stress
(top) and acoustic power (bottom) against time. The first 60 s corresponds to
training and the reversal occurs after 60 s. Colored plots correspond to
different $P_{train}$. (b) Plot of the viscosity versus shear strain after
reversal. Larger $P_{train}$ results in larger viscosities immediately after
reversal ($\eta_{min}$) and shear jamming at lower critical strains
($\gamma_{J}$). As an example, $\eta_{min}$ and $\gamma_{J}$ are indicated by
the dashed horizontal and dotted vertical line for $P_{train}=16$ W. (c) Plot
of $\eta_{min}$ versus the training power. (d) Plot of $\gamma_{J}$ versus the
training power. (e) Illustration of the deduced microstructures before and
after reversal for low (top) and high (bottom) $P_{train}$. Yellow spheres
indicate particles in primary force chains while green spheres indicate those
in secondary force chains. For large $P_{train}$, significant secondary force
chains exist prior to reversal which can resist the reversed shear flow and
require little rearrangement to shear jam in the reversal direction.

###  III.3 Shear reversal

To gain further insights into our trained shear jammed states, the contact
stress anisotropy of the suspension was characterized via a stress reversal
protocol. Previous studies of continuously shear thickening and shear jamming
suspensions undergoing shear reversal have demonstrated that the contact
stresses responsible for shear thickening and jamming go to zero upon shear
reversal as the particles which were initially compressed by the shear are now
pulled apart [59, 57, 58, 60, 61]. As such, the viscosity after reversal,
$\eta_{min}$, was observed to become significantly smaller (or finite) when
compared to the shear thickened (or shear jammed) viscosity. With the contact
stresses disappearing upon reversal, $\eta_{min}$ has been shown to be
representative of the remaining hydrodynamic stresses and has been observed to
remain nearly constant across different applied stresses before and after
reversal.

The stress reversal protocol is shown in Fig. 3a. Here, after training, we
apply a shear stress with the same magnitude as the training stress,
$\sigma_{app}=380$ Pa, but in the reverse direction. The evolution of the
suspension viscosity with strain after reversal is shown in Fig. 3b, where
only a subset of the curves were included for clarity. The figure shows that
immediately after reversal, the suspension is unjammed and is characterized by
a finite viscosity, $\eta_{min}$, that depends on $P_{train}$ (e.g. dashed
grey line in Fig. 3b indicating $\eta_{min}\sim 10^{3}$ Pa.s for
$P_{train}=16$ W). Once a sufficiently large strain is built up, the
suspension viscosity increases dramatically and eventually shear jams in the
reverse direction at a critical strain, $\gamma_{J}$ (e.g. vertical grey
dashed line for $P_{train}=16$ W). We find that both $\eta_{min}$ and
$\gamma_{J}$ retain signatures of the acoustic training.

To better visualize trends in these signatures, we plot $\eta_{min}$ and
$\gamma_{J}$ versus $P_{train}$ in Fig. 3c and Fig 3d respectively. We find
that $\eta_{min}$ increases by orders of magnitude with increasing training
power, in stark contrast to the approximately constant $\eta_{min}$ found from
typical shear reversal experiments [59, 57, 58, 60, 61]. These measurements
indicate that for higher training powers, there are many more secondary force
networks built up in the jammed suspension that are aligned along the maximum
extensional axis (i.e. perpendicular to the compressive axis normally
responsible for shear jamming and DST). Upon reversal, these secondary
networks align with the maximum compressive axis of the reversed shear
direction and inhibit the flow Fig 3c). Consistent with this picture, we find
that after reversal much lower strains are necessary to jam suspensions
trained at higher powers Fig 3d). Our results suggest that larger $P_{train}$
generates shear jammed states that have significantly stronger secondary force
chains along the extensional axis of shear prior to reversal.

These findings are summarized by the illustrations shown in Fig. 3e. Here, the
primary force networks responsible for shear thickening and jamming are
depicted in yellow and are oriented along the shear compressive axis.
Particles making up the secondary force networks along the maximal extension
axis are depicted in green. At low training powers (top half), the suspension
microstructure prior to reversal is strongly asymmetric, consisting mainly of
primary force chains. Upon reversal, such force chains rapidly break up and
the viscosity drops to the suspension viscosity prior to thickening. In
addition, large strains are necessary to rearrange the particles in order to
generate a shear jammed state. For high $P_{train}$ (bottom half), the
suspension microstructure prior to reversal consists of both primary and
secondary force networks (bottom left of figure). Upon reversal, even if the
primary force networks break up, the secondary force networks can remain and
resist the shear. This resistance results in large increases to the viscosity
immediately after reversal. The presence of these secondary force networks
also seeds or promotes an earlier transition to the shear jammed state after
reversal. This picture of the suspension microstructure is strongly
reminiscent of the “fixed principle axes” model, which attributes the ability
of a shear jammed state to bear loads orthogonal to the compressive axis to a
build-up of force chains along these orthogonal axes [41, 62].

In suspensions with a prominent secondary force network, the applied shear
stress is balanced by the combined projection of the stresses from networks
oriented along both the compressive and extensional axes. Importantly, the
primary and secondary force networks contribute to the shear stress in
opposite directions – the primary force networks resist the flow while the
secondary force networks aid the flow. As such, the same applied shear stress
can be accommodated by different combinations of shear stress projections from
the competing force networks along the compressive and extensional axes.
Therefore, to understand the behaviors of the differently trained jammed
suspensions, the individual shear stress contributions from these competing
force networks must be determined.

Figure 4: (a) Protocol for the rate cessation experiment where the shear rate
(top) and acoustic power (bottom) are plotted versus time. The first 60s
corresponds to training and cessation occurs from 60s onwards. Colored curves
correspond to different $P_{train}$ as labeled in (b). (b) Plot of the
instantaneous stress normalized by the applied shear stress versus time after
cessation for different $P_{train}$. Dashed silver lines are fits to
$Ae^{-\tilde{t}/11}+Be^{-\tilde{t}/1000}-Ce^{-\tilde{t}/67}$ (c) A
decomposition of the stress contribution from the $Be^{-\tilde{t}/1000}$ and
$-Ce^{-\tilde{t}/67}$ terms used to fit the $P_{train}$=16W curve. The stress
decays much more rapidly for the secondary force chains ($-Ce^{-\tilde{t}/67}$
term, illustrated in green) compared to the primary force chains
($Be^{-\tilde{t}/1000}$ term, yellow), resulting in the short time decay
dynamics to be dominated by the relaxation of the secondary force chains. (d)
Plot of the normalized stress amplitudes obtained as fit parameters for the
fits in (b) versus $P_{train}$.

###  III.4 Shear cessation

We attempt to measure the shear stress contributions from the primary and
secondary force networks by studying the suspension stress relaxation behavior
via a rate cessation protocol as shown in Fig. 4a. Here, the shear rate
$\dot{\gamma}$ is fixed at zero after training, and the stress acting on the
top plate while the suspension relaxes between its two stationary confining
boundaries is measured. Previous studies of shear thickening and shear jamming
suspensions undergoing shear cessation have found that the shear stress
monotonically decreases after the shear rate has been set to zero [64, 63, 65,
66]. In addition, multiple timescales are required to describe the purportedly
exponential decay behavior for larger volume fractions, and the exact origins
of these timescales remain an area of active research. For our study, we are
agnostic of the exact physical origin of these timescales, but instead use
shear cessation as a technique to back out the individual contributions from
the primary and secondary stress networks, which is possible if the networks
relax at sufficiently different rates.

Fig. 4b shows a plot of the normalized instantaneous shear stress,
$\sigma/\sigma_{app}$, versus the time after cessation, $\tilde{t}$, where
$\sigma_{app}=$ 380Pa corresponds to the constant stress applied during
training that is fixed across all cessation experiments. For P${}_{train}\leq
1$ W, the stress decay is monotonic as expected. Strikingly, for larger
$P_{train}$, the stress increases during cessation. Motivated by our stress
reversal results, we model the relaxation of the normalized stress as a
competition between the decay of the force networks formed along the
compressive and extensional axes. We find that we are able to reasonably fit
all the data by modifying the typical sum of two exponential terms used in
literature to include an additional negative exponential component that
represents the relaxation of the antagonistic secondary force chain:

$\sigma(\tilde{t})/\sigma_{app}=Ae^{-\tilde{t}/11}+Be^{-\tilde{t}/1000}-Ce^{-\tilde{t}/67}$
(1)

with the constraint:

$A+B-C=1$ (2)

and A,B,C $>$ 0\. Here, the stress contribution from the primary force
networks are captured by the first two terms [63, 64, 65, 66], and the stress
contribution of the secondary force networks in the negative direction are
captured by the third term. We note that there is no reason a priori to assume
that the timescales for the decay should remain constant, and the decision to
fix these timescales in our paper was mainly to minimize the total degree of
freedom required to achieve a reasonably good fit to all the cessation curves.
Our fit results reveal that for large $P_{train}$, where significant increases
in the stress is observed after cessation, A $\approx 0$ and the stress
dynamics can be captured using only the B and C terms. To illustrate how the
competition between force network relaxations can give rise to an increasing
stress upon cessation, we plot the stress contributions from the
$Be^{-\tilde{t}/1000}$ and $Ce^{-\tilde{t}/67}$ terms for $P_{train}=16$ W in
Fig. 4c. Since the stress decay dynamics are dominated by the faster
relaxation of the secondary stress network (green and lower schematics), we
observe a net gain in total stress (purple), even as the primary stress
network is also decaying yellow and upper schematics). Consistent with this
picture, we observe that at very long times the stress decays to zero.

Figure 5: (a) Shear stress projections from the primary and secondary force
chains calculated from Fig.4d. The primary stress projection. $\sigma_{P}$, is
represented by the “$+$” marker; the secondary stress projection,
$\sigma_{S}$, by the “$-$” marker; and the (constant) applied stress,
$\sigma_{app}$, by the “X” marker. The solid arrows are vectors indicating how
the primary and secondary shear stress projections compete and sum to result
in the applied shear stress for $P_{train}=16$ W. (b) The same vector
representation is shown on the $\sigma-\phi$ phase diagram, illustrating how
$\sigma_{P}$ (purple $+$ marker) can be above the 4W shear jamming boundary
(yellow dashed line) even when the applied shear stress (horizontal grey line)
is below the boundary at a given volume fraction (vertical grey line). The
jammed structure along the primary direction and thus the suspension can
remain shear jammed during probing under the 4W perturbation even though the
applied shear stress remained unchanged. We note that the dashed SJ boundary
and arrows in b) are not drawn to scale and are qualitative illustrations
based on our hypothesis.

The values for the stress amplitudes $A$, $B$ and $C$ for all $P_{train}$ are
potted in Fig. 4d. We find that the coefficient $B$ that describes the main
contribution of the primary force networks increases with training power and
dominates the other terms. The coefficient $C$, is initially negligible but
rapidly increases with $P_{train}$. This trend is consistent with the results
in Fig. 3c-e, which indicate an increased contribution from the secondary
force networks in suspensions trained at larger $P_{train}$. Collectively,
these cessation measurements and the accompanying analysis enable us to
decompose the total stress into contributions from the primary and secondary
force networks to explain the discrepancies described in Fig. 2d.

###  III.5 Shear stress projections

Using the fits from the cessation measurements, we extract the stress
contributions from primary and secondary force networks projected onto the
shear axis, as shown in Fig. 5a. The projection from the primary stress onto
the shear direction, $\sigma_{P}=(A+B)\sigma_{app}$, is represented by the “+”
marker; the projection from the secondary stress onto the shear direction,
$\sigma_{S}=C\sigma_{app}$, by the “$-$” marker; and the (constant) applied
stress, $\sigma_{app}=\sigma_{P}+\sigma_{S}$, by the “X” marker. We find that
for low $P_{train}$, the stress contribution from the secondary network is
negligible, $\sigma_{S}\sim 0$, so that $\sigma_{P}\sim\sigma_{app}$. For
large $P_{train}$, the contribution from the secondary network in the negative
direction becomes significant. Consequently, the stress contribution from the
primary force network must be greater than the applied stress,
$\sigma_{P}>\sigma_{app}$. The antagonistic relationship between the stress
contributions $\sigma_{P}$ and $\sigma_{S}$, and how they sum up to
$\sigma_{app}$ is represented vectorially (black arrows) for $P_{train}$=16W
in Fig. 5a. These results indicate that for systems with sufficiently
developed secondary force networks, the contribution from the primary stress
networks can be substantially larger than the applied stress.

These large differences between $\sigma_{P}$ and $\sigma_{app}$ account for
the behavior of suspensions trained at high $P_{train}$. We schematically
illustrate the contributions of the primary and secondary force networks to
the stress during the training and probing sequences in Fig. 5b. During
training the shear jamming boundary (solid black curve) shifts as illustrated
by the purple dashed curve. Here, the training gives rise to a significant
contribution from the secondary force network. To accommodate the applied
stress, the contribution from the primary force network must be higher than
$\sigma_{app}$ so that the sum of the contributions from the primary and
secondary force networks match the applied stress (summation of purple
arrows). Once the acoustic perturbations are removed, the suspension jams,
preserving a memory of the force networks formed. During the probing period,
if $\sigma_{P}$ is above the shear jamming boundary associated with
$P_{test}=4$ W (dashed yellow line), the primary force networks remain shear
jammed.

We note that when compared to a shear jamming suspension at the same applied
stress but at a lower volume fraction ($\phi_{v}=0.56$, see S.I. Fig. S7), the
acoustically induced increases in the magnitude of $\sigma_{P}$ and
$\sigma_{S}$ are weaker for the same $P_{train}$. This difference in memory of
the same training manifests as a larger threshold $P_{train}$ required to
observe an increase in stress during shear cessation, alongside smaller
changes in $\eta_{min}$ and $\gamma_{J}$ for a given $P_{train}$ for the less
dense suspension. As such, we expect $\sigma_{P}$ and $\sigma_{S}$ to be
potentially useful predictors of these behavioral changes induced by acoustic
training.

###  III.6 Inducing Solidification in Flowing States

Figure 6:  Viscosity against time plot for suspensions with a) a volume
fraction $\phi_{v}^{-}$ where $\phi_{v}^{-}=\phi_{v}-1\%$, sheared at
$\sigma_{app}$ (triangle marker in Fig. 1b) and b) a volume fraction of
$\phi_{v}$ sheared at a stress $\sigma_{app}^{-}=\sigma_{app}/10$ (circle
marker in Fig. 1a). The suspensions were trained for the first 60s (as in
Fig.2a) and the applied shear stress of $\sigma_{app}$ for a) and
$\sigma_{app}^{-}$ for b) was left on for an additional 60s. These flowing
states shear jam when trained at sufficiently large $P_{train}$.

These results raise the intriguing possibility of training a flowing
suspension that is well bellow the jamming line to produce a jammed state. We
demonstrate that this is indeed possible by working with two suspensions: one
at a 1% lower volume fraction ($\phi_{v}^{-}$) sheared at $\sigma_{app}$
(triangle marker in Fig. 1b) and a second at the same volume fraction sheared
at $\sigma_{app}/10$ ($\sigma_{app}^{-}$) (circle marker in Fig. 1a). Here, we
train the two systems as described previously and probe their state by
shearing with the same applied stress for an additional 60s after training.
The viscosity versus time plots are shown in Fig. 6. In both cases, when the
suspensions are trained at $P_{train}=0$ W (black lines in Figs. 6a,b), they
are characterized by a finite steady viscosity. In other words, they flow. As
$P_{train}$ is increased, we see the typical monotonic decrease in the
viscosity when the acoustic perturbations are applied (gray shaded regions).
When the perturbation is removed however, clear viscosity overshoots are
observed. For weak $P_{train}$, these overshoots quickly decay back to the
$P_{train}=0$ W viscosity. This decay timescale increases with $P_{train}$
until the system enters a shear jammed state as seen for $P_{train}=25$ W data
in Fig. 6a and for the $P_{train}=16$ W data in Fig. 6b. We note that the
order of magnitude lower stress applied in Fig. 6b may have resulted in the
relatively faster relaxation from the jammed state, as seen from the decrease
in viscosity shortly after large fluctuations (signature of the suspension
jamming) are observed. Nonetheless, we note that the transient behavior post
training is consistent with Fig. 6a prior to the viscosity relaxation. To
further clarify the effect of stress on the acoustically induced memory in our
suspensions, the same protocol was repeated for a suspension farther away from
the shear jamming line, with the same volume fraction but at a stress 10 times
smaller than in Fig. 6b (i.e. $\phi_{v}=0.565$, $\sigma=3.8$ Pa, diamond
marker in Fig. 1a), as shown in Fig. S6 in the supplementary material. There,
we find a rapid convergence of the viscosity to the $P_{train}=0$ W curve
after the acoustic perturbation is removed for all $P_{train}$ applied. As
such, acoustic training can clearly embed memory in flowing suspensions
sufficiently near to the shear jamming line, even inducing solidification as
demonstrated here. We find that for the ($\phi_{v}^{-},\sigma_{app}$) state,
the induced shear jamming can be preserved for a period at least $10$ times
longer than the training interval (see Fig. S5 in the supplementary material).
While this effect does dissipate for systems driven sufficiently below the
shear jamming line, the ability to shear jam flowing suspensions is
nevertheless a remarkable demonstration of the impact that acoustic training
can have on flowing suspensions.

##  IV Discussion

Our strategy for incorporating tunability in these networks involves two key
steps: 1) the encoding of memory during the flowing state and 2) the rapid
freezing of these network structures to preserve signatures of the encoding.
Since these steps can be implemented in a variety of shear jamming materials
using different memory encoding and preservation mechanisms, our strategy
opens up a large material design landscape for exploration. For example, in
shear thickening suspensions different shear protocols and perturbations,
which can be multi-directional and/or time dependent, may be utilized to
generate a variety of force networks during the fluid state that can be
preserved via jamming [67, 68, 69, 53, 70, 71, 52, 72, 73, 74]. We expect that
similar protocols would work for preserving memories in magneto-rheological
and electro-rheological fluids. Additional mechanisms to preserve the
suspension memory such as photopolymerization and rapid solidification can
further extend our strategy to less dense suspensions and even to different
systems such as foams and emulsion [75, 76, 77]. Our general strategy for
tuning the fluid state and leveraging a wide range of memory preservation
mechanisms should enable us to generate a wide range of material properties.

In shear thickening suspensions, such as system studied here, the build up of
antagonistic force networks is a particularly potent mechanism for controlling
the resulting material properties. Specifically, such antagonistic networks
provide an additional degree of freedom for satisfying the applied force
constraint and therefore greater control over the emergent material
properties. Similar strategies are used in a variety of systems ranging from
sophisticated temperature control for regulating flowering behavior in plants
and homeostasis in human [78, 79, 80] to muscular manipulation of limbs for
rapid movement and stabilization [81, 82, 83, 84, 85].

Our results also demonstrate that the applied shear stress $\sigma_{app}$ is
not sufficient to describe the state of the system. This observation is
consistent with simulations and theoretical studies which find that a
tensorial approach is required to fully characterize and predict suspension
properties especially at high volume fractions, where the stress distributions
become more isotropic [86, 87, 88, 55, 89]. The importance of antagonistic
force chains also suggest a way to extend the universal scaling framework [90,
54] to memory-forming suspensions. The lesson learnt is that frictional
contacts along the compressive and dilational directions play very different
roles, and therefore impact the shear-jamming process differently. The results
presented here show how decomposition of the shear stress contributions into
antagonistic components may be required to fully capture the behavior of these
very dense suspensions even under simple shear. Such stress decomposition is
likely to also be necessary to fully capture and understand more complicated
flow behaviors induced by combinations of shear and compression or other
biaxial and even triaxial flow manipulations.

We note that our results may also draw a deep connection to studies on
directed ageing and memory encoding in glasses, gels and granular materials
[11, 91, 7, 10, 92]. In particular, the memory encoding and preservation
processes presented here may be analogous to annealing, quenching and/or
densification. Here, application of larger acoustic power during training
drives the system into a more stable configuration with a larger part of the
network characterized by a longer relaxation timescale. This training process
is reminiscent of annealing, which allows particles to explore their energy
landscape and settle in a more stable energy minima. Our experimental non-
thermal acoustic perturbation protocol, like numerical Wolff cluster moves and
swap Monte Carlo [93, 94, 95, 96], bypasses the usual relaxation barriers and
allows exploration of otherwise challenging or inaccessible materials
morphologies. One notable difference between our approach and these previous
training protocols is that the training in our system is conducted while the
suspension is sheared. This breaking of symmetry enables training of tensorial
components of the stress, which are not as easily accessed in these previous
protocols [11, 91, 92, 10, 32, 12, 14, 31].

While our current study is limited by a lack of imaging data, we emphasize
that this very difficulty in obtaining useful imaging data for very dense
suspensions further highlights the importance of the characterization
protocols used in this paper. There is a growing number of studies that
demonstrate that shear thickening and shear jamming in non-deformable
particles are caused by a redistribution of frictional interactions between
particles without significant rearrangement of the geometric topology [57, 58,
56, 97]. As a result, it remains extremely challenging, if not impossible, for
the spatial changes associated with these changes in frictional interactions
to be resolved optically [56, 98, 99], even with advanced image reconstruction
algorithms that provides nanometer-scale resolution [100, 101]. Given that
direct optical measurements are unlikely to yield results, we have turned to
alternative measurement techniques, such as the protocols used in this study,
to characterize the suspension and through these characterization, deduce
changes in the microstructure. Such deductions are possible because the
structural changes are directly associated with frictional forces being
redistributed [56, 102], and so the fact that we are able to dramatically
change the response of the jammed suspension indicates that there is something
different about the organization of the frictional interaction, which
indicates structural changes. In addition, as new materials and tuning
mechanisms continue to be introduced into the field of dense suspension,
alternative non-imaging based methods to obtain additional information on the
suspension which do not impose additional constraints on the setup or
suspension (e.g. on refractive index matching, transparency of surface etc)
such as the protocols shown here will become increasingly important. As such,
we believe the striking results obtained from these protocols are highly
valuable even without, or perhaps exactly because of the lack of microscopic
data available for suspensions in this very dense regime. We note that a
recently demonstrated experimental protocol [103] that involves carrying out
shear reversal protocols at different angles to the direction of shear may
enable us to resolve the contribution of the acoustic training to the full
stress tensor, and will be pursued in a future study.

##  V Conclusion

Our strategy to jam memory of antagonistic force networks into suspensions
shows how flow protocols can be used to engineer suspensions with dramatically
different material properties. While we have focused on acoustic training and
shear jamming in this study, our strategy is broadly applicable to different
flow tuning and structure preservation mechanisms, which we expect to be
fertile ground for uncovering and engineering unique material properties. In
combination with more sophisticated training, flow, and solidification
protocols, it should finally be possible to generate structured fluids that
can begin to take advantage of the predicted orders of magnitude changes in
materials properties that can emerge from altering network connectivity and
force transmission.

##  VI Acknowledgement

We thank Anton Paar for use of the Twin Drive MCR 702 rheometer through their
VIP academic research program and the Cohen group for their insightful
suggestions. We also thank Cornell Center for Materials Research, an NSF MRSEC
supported by DMR-1719875, for providing access to their facilities. This work
is also supported by NSF CBET award numbers 1804963, 1509308 and 2228681 as
well as DMR-1507607 and DMR-2327094. Anna Barth is supported by NSF
DGE-2139899. Edward Ong is supported by the Agency of Science Technology and
Research, Singapore NSS(PhD) award.

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