# Rheo-acoustic gels: Tuning mechanical and flow properties
of colloidal gels with ultrasonic vibrations

Thomas Gibaud  Corresponding author<EMAIL_ADDRESS>Univ Lyon, Ens
de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, 69342 Lyon,
France  Noémie Dagès  Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS,
Laboratoire de Physique, 69342 Lyon, France  Pierre Lidon  Univ Lyon, Ens de
Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, 69342 Lyon, France
Guillaume Jung  Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire
de Physique, 69342 Lyon, France  L. Christian Ahouré  Univ Lyon, Ens de Lyon,
Univ Claude Bernard, CNRS, Laboratoire de Physique, 69342 Lyon, France
Michael Sztucki  ESRF – The European Synchrotron, 38043 Grenoble, France
Arnaud Poulesquen  CEA, DEN, Univ. Montpellier, DE2D, SEAD, LCBC, 30207
Bagnols-sur-Cèze, France  Nicolas Hengl  Univ. Grenoble Alpes, CNRS, Grenoble
INP, LRP, 38000 Grenoble, France  Frédéric Pignon  Univ. Grenoble Alpes, CNRS,
Grenoble INP, LRP, 38000 Grenoble, France  Sébastien Manneville  Univ Lyon,
Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69342 Lyon,
France  MultiScale Material Science for Energy and Environment, UMI 3466,
CNRS-MIT, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA

###### Abstract

Colloidal gels, where nanoscale particles aggregate into an elastic yet
fragile network, are at the heart of materials that combine specific optical,
electrical and mechanical properties. Tailoring the viscoelastic features of
colloidal gels in real-time thanks to an external stimulus currently appears
as a major challenge in the design of “smart” soft materials. Here we
introduce “rheo-acoustic” gels, a class of materials that are sensitive to
ultrasonic vibrations. By using a combination of rheological and structural
characterization, we evidence and quantify a strong softening in three widely
different colloidal gels submitted to ultrasonic vibrations (with submicron
amplitude and frequency 20–500 kHz). This softening is attributed to micron-
sized cracks within the gel network that may or may not fully heal once
vibrations are turned off depending on the acoustic intensity. Ultrasonic
vibrations are further shown to dramatically decrease the gel yield stress and
accelerate shear-induced fluidization. Ultrasound-assisted fluidization
dynamics appear to be governed by an effective temperature that depends on the
acoustic intensity. Our work opens the way to a full control of elastic and
flow properties by ultrasonic vibrations as well as to future theoretical and
numerical modeling of such rheo-acoustic gels.

###### pacs:

Valid PACS appear here

##  I Introduction

Colloidal gels constitute a class of soft materials with a huge spectrum of
applications, ranging from paints, oil extraction and construction to
pharmaceuticals and food products Larson (1999); Mewis and Wagner (2012).
These gels are typically formed from the aggregation of attractive nanoscale
particles that arrange into a space-spanning network which strength provides
the system with elastic properties at rest. The particle network is, however,
fragile enough that rather weak external forces disrupt the gel structure and
induce flow above a critical yield strain or stress Bonn  _et al._ (2017).
Such unique mechanical and flow properties are key to applications that
require an interplay between solid and fluid behaviour, including cement
placement, ink-jet printing or flow-cell batteries. In this context, decades
of research have investigated the influence of physico-chemical parameters
such as temperature, pH and ionic strength on the aggregation of attractive
particles into colloidal gels Cipelletti  _et al._ (2000); Romer  _et al._
(2000); Trappe  _et al._ (2001); Cardinaux  _et al._ (2007); Zaccarelli
(2007); Lu  _et al._ (2008). While electrostatic repulsion, van der Waals
attraction and steric interaction, combined in the well-known DLVO
interparticle potential, generically lead to the formation of fractal networks
through diffusion- or reaction-limited cluster aggregation, additional
chemical effects such as hydration layers or particle bridging generate a
wealth of more complex gel microstructures Israelachvili (2011). In the quest
of smart, responsive materials, tuning the gel architecture in real-time and
reversibly is a key challenge. However, playing on the above physico-chemical
parameters turns out to be difficult, if not impossible when the chemistry of
the material is strongly constrained. Electro- or magneto-rheological
materials allow for such tuning through the use of electrical and magnetic
fields Hao (2002); Kogan and Solomon (2009); Swan  _et al._ (2012); Helal  _et
al._ (2016a). Yet, applications are obviously restricted to materials with
specific compositions Coulter  _et al._ (1993); Tao (2001); Wereley (2013).

The exquisite sensitivity of colloidal gels to external mechanical
perturbations provides an alternative route to interact with their
microstructure. The effects of steady or low-frequency oscillatory flows on
the gel network are commonly probed through rheological measurements Mewis and
Wagner (2012). Besides viscoelasticity and aging dynamics at rest Zaccarelli
(2007), the signature of the solidlike nature of colloidal gel networks is the
existence of a yield stress: when the applied shear stress is large enough to
break interparticle bonds, the gel network gets fragmented into a dispersion
of aggregates. The yield stress typically shows a power-law increase with the
particle volume fraction Buscall  _et al._ (1988); Chen and Russel (1991) that
has been interpreted in the framework of fractal networks Potanin and Russel
(1996); Shih  _et al._ (1999). Beyond yielding, the dispersion flows and
further shows shear-thinning as the typical aggregate size decreases with the
shear rate Wessel and Ball (1992). Superposition rheology, which consists in
applying small-amplitude oscillatory shear on top of a steady shear flow, in a
direction either parallel or orthogonal to the main shear flow Vermant  _et
al._ (1998); Dhont and Wagner (2001), provides an interesting way to probe the
viscoelastic properties of colloidal gels under nonlinear deformation Colombo
_et al._ (2017); Sung  _et al._ (2018). Furthermore, to get microscopic
insight into yielding, rheological measurements have been combined to
structural characterization Poon and Haw (1997); Vermant and Solomon (2005).
For instance, confocal microscopy has allowed for direct visualization of
shear-induced breakup and relaxation after shear cessation Bonn and Denn
(2009); Koumakis  _et al._ (2015), for tracking of the erosion of rigid
clusters by shear Hsiao  _et al._ (2012) and for detailed investigation of
aggregate morphologies in two dimensional gels Masschaele  _et al._ (2011).
Small-angle light and x-ray scattering has been extensively used to study
shear-induced anisotropy and restructuring, e.g., in silica gels Verduin  _et
al._ (1996); Varadan and Solomon (2001), polystyrene spheres Mohraz and
Solomon (2005) and clay suspensions Pignon  _et al._ (1997). Small-angle
neutron scattering coupled to rheology Eberle and Porcar (2012) was also
instrumental in revealing the shear-induced microstructure of colloidal gels
both under steady shear Eberle  _et al._ (2014); Hipp  _et al._ (2019) and
under oscillatory flow Kim  _et al._ (2014). Finally, recent advances in
numerical simulation schemes for colloidal particles under shear, including
Brownian dynamics or Stokesian dynamics Zia and Brady (2010); Swan and Brady
(2010); Seto  _et al._ (2012), have allowed fruitful comparisons with
experiments on shear-induced structuring of attractive particulate systems
Koumakis  _et al._ (2015); Landrum  _et al._ (2016); Moghimi  _et al._ (2017);
Varga  _et al._ (2019).

All of this previous work resulted not only in a better fundamental
understanding of the complex interplay between flow and gel structure but also
in the optimization of both manufacturing processes and final product
properties of an overwhelming number of colloid-based materials Bergna and
Roberts (2005); Mezzenga  _et al._ (2005); Guvendiren  _et al._ (2012). In
practice, however, applying a mechanical stress in order to fluidize the gel
at will and in situ remains challenging as it involves moving parts such as
pumps, motors or other rotating tools that may not be compatible with the
application. Here, we introduce ultrasonic vibrations, i.e. high-frequency
mechanical vibrations, as a way to control the elastic modulus and/or the
yield stress of a colloidal gel. High-power ultrasound has long been known to
disrupt particulate aggregates with applications to resuspension, filtration
and cleaning Kyllönen  _et al._ (2005); Hengl  _et al._ (2014), optimization
of food products Knorr  _et al._ (2004); Awad  _et al._ (2012); Chandrapala
_et al._ (2012) and drug delivery Mitragotri (2005); Huebsch  _et al._ (2014).
Conversely, intense acoustic waves may induce gelation of some organic
compounds and colloidal systems Cravotto and Cintas (2009); Bardelang (2009).
Yet, in spite of such empirical know-how, no systematic study has been
performed on the effect of ultrasonic vibrations on colloidal gels and it
remains unclear how to distinguish between physico-chemical effects (e.g.
chemical reactions triggered by cavitation and local temperature rise) Suslick
and Price (1999) and mechanical effects (e.g. material deformation due to
acoustic radiation pressure or large-scale flow due to acoustic streaming)
Hamilton and Blackstock (1998); Dalecki (2004). Moreover, besides applications
to material design, the interplay between ultrasound and the complex structure
of colloidal gels raises a fundamental challenge due to the large span of
time- and length-scales involved in the problem.

In the present work, we probe the effects of ultrasonic vibrations on
colloidal gels thanks to two different experimental setups designed to measure
the gel mechanical and structural properties under ultrasound. Vibrations are
generated by piezo-electric transducers driven at frequencies ranging from 20
to 500 kHz in direct contact with the gels. The corresponding acoustic
wavelength ranges from 3 mm to several centimeters, which is always larger
than the thickness of the samples under study. Therefore, in a first
approximation, the gels should be considered as vibrated homogeneously at a
high frequency rather than submitted to propagating ultrasonic waves or to a
stationary ultrasonic field with pressure nodes and antinodes which position
would depend on the boundary conditions. Our approach differs from the
superposition techniques mentioned above in that we probe the low-frequency
rheological response to vibrations with frequencies that are inaccessible to
standard rheometry. In other words, there is a very large separation between
the time scale at which the system is forced and that at which it is probed.

We focus on three different colloidal gels respectively composed of
nanoparticles of calcite, silica and carbon black dispersed at low volume
fractions in a background solvent. As described in more details in Appendix
A.1, all systems aggregate into elastic space-spanning networks thanks to
attractive interparticle forces. At rest, once gelation is complete, all gels
are solid and display an elastic modulus $G^{\prime}$ and a yield stress
$\sigma_{\text{y}}$, the stress above which the gel starts to flow. These gels
are representative of the colloidal gels commonly encountered in soft matter,
featuring elastic moduli $G^{\prime}$ from about 100 Pa to 10 kPa and a wide
range of yielding behaviours from “simple” yielding with little time
dependence for the calcite gel to “delayed yielding” for the carbon black gel
Gibaud  _et al._ (2010); Bonn  _et al._ (2017).

The paper is organized as follows. In Sect. II, we first show that the elastic
modulus of our colloidal gels decreases dramatically under ultrasonic
vibrations. Such ultrasound-induced softening defines “rheo-acoustic” gels,
i.e. gels which elastic properties are strongly dependent on ultrasonic
vibrations. In Sect. III, we couple x-ray scattering to ultrasonic vibrations
and provide evidence for structural modifications hinting at ultrasound-
induced micron-sized cracks within the gel network. Section IV further
explores the way ultrasonic vibrations may modify the flow and yielding
properties of colloidal gels. Ultrasonic vibrations are shown to facilitate
yielding and a tentative interpretation is proposed in terms of an effective
temperature. Section V discusses the various original questions opened by our
experiments with emphasis on the need for a better microscopic understanding
of the coupling between ultrasonic vibrations and the gel network. We close
this article by discussing the implications of our results for future
theoretical and numerical studies.

##  II Gel softening under high-power ultrasonic vibrations

##### Experimental setup.

The mechanical effects of ultrasonic vibrations on colloidal gels are
investigated thanks to a rotational rheometer (Anton Paar MCR 301) equipped
with a parallel-plate geometry where the upper plate is made of sandblasted
Plexiglas. The bottom plate is constituted of a piezoelectric transducer
working either at frequency $f=45$ kHz (Sofranel, diameter of 29 mm) or 500
kHz (Imasonic, diameter of 50 mm). The piezoelectric transducer is fed with an
oscillating voltage of amplitude up to 90 V through a broadband power
amplifier (Amplifier Research 75A250A) driven by a function generator (Agilent
33522A). A thermocouple (National Instruments USB-TC01) monitors the
temperature at the surface of the transducer. A solvent trap covers the
geometry in order to minimize evaporation. The gap width is set to 1 mm for
all experiments except in Fig. 12 in Appendix A.8 where the influence of the
gap width is tested.

Figure 1: Ultrasound-induced softening of colloidal gels. Elastic modulus
$G^{\prime}$ as a function of time $t$ for (a) a 10 % vol. calcite gel, (b) a
5 % vol. silica gel and (c) a 3 % vol. carbon black gel. Ultrasonic vibrations
with frequency 45 kHz are turned on at time $t=0$ and switched off at $t=30$
s. Each curve corresponds to a given acoustic intensity, with darker colors
corresponding to larger intensities. (d) Relative amplitude of the softening
effect $\Delta G^{\prime}/G^{\prime}_{0}$ as a function of the acoustic
pressure $P$ for the calcite, silica and carbon black gels (from top to
bottom). $\Delta G^{\prime}=G^{\prime}_{0}-\min(G^{\prime})$ is the drop in
elastic modulus induced by ultrasonic vibrations, with $G^{\prime}_{0}=G(t=0)$
the initial value of the elastic modulus. The dotted lines emphasize the
threshold at $P_{\text{c}}\simeq 50$ kPa for the silica and carbon black gels
respectively through linear and quadratic behaviours in $P-P_{\text{c}}$. (e)
Relative softening amplitude $\Delta G^{\prime}/G^{\prime}_{0}$ as in (d)
replotted as a function of the dimensionless group
$\Upsilon=\gamma^{\text{s}}_{\text{US}}/\gamma_{\text{NL}}$ that compares the
shear strain induced by ultrasound to the onset of nonlinear response at 1 Hz.
The dotted line shows $\Upsilon=1$.

The displacement of the transducer surface remains always much smaller than
the gap size. At the highest achievable acoustic intensity, its maximum
amplitude $a$ reaches $0.5~{}\mu$m for the 45-kHz transducer (as calibrated
with a laser vibrometer, Polytec OVF-505) and up to $0.03~{}\mu$m for the
500-kHz transducer (as derived from the manufacturer calibration). Assuming
spatially homogeneous oscillations, vibrations with amplitude $a$ generate an
oscillatory extensional strain (along the direction of the vibrations i.e.
along the vertical direction) of the order of
$\gamma^{\text{e}}_{\text{US}}\sim a/h$ and a shear strain (along the
horizontal direction) of the order of $\gamma^{\text{s}}_{\text{US}}\sim
Ra/2h^{2}$, with $R$ the radius of the parallel-plate geometry and $h$ the gap
width. Since $h\ll R$,
$\gamma^{\text{e}}_{\text{US}}\ll\gamma^{\text{s}}_{\text{US}}$ and the
maximum strain amplitude is $\gamma^{\text{s}}_{\text{US}}\simeq 0.4$ % at 45
kHz and 0.04 % at 500 kHz. The former order of magnitude is comparable to the
onset of nonlinear viscoelasticity at low frequency
($\gamma_{\text{NL}}=0.1$–1%) and always below the yield strain of the gels
($\gamma_{\text{y}}=1$–10%) as obtained from the standard oscillatory shear
measurements at 1 Hz displayed in Fig. 5 in Appendix A.1. Moreover, the
amplitude of the effective strain rate associated with ultrasonic vibrations,
$\dot{\gamma}^{\text{s}}_{\text{US}}=\omega\gamma^{\text{s}}_{\text{US}}$,
reaches about 200 s-1 for both ultrasonic frequencies. We emphasize that the
high-frequency nonlinear rheology of our gels is unknown so that these
estimates remain difficult to interpret. Still, based on these significant
values for strain and strain rates, one may expect that ultrasonic vibrations
should interact nonlinearly with the samples and have a strong impact on their
mechanical properties.

Rather than in terms of displacement, the acoustic amplitude is usually
reported in terms of the velocity $v$ or pressure $P=\rho cv=\rho c\omega a$,
where $\rho$ is the gel density, $c$ is the speed of sound in the gel and
$\omega=2\pi f$ is the angular frequency. In our experiments, $P$ ranges up to
150 kPa for both transducers. This corresponds to a maximum acoustic power of
about $\mathcal{P}=P^{2}/\rho c\simeq 1.5$ W cm-2.

##### Results.

We start by simply monitoring the viscoelastic moduli $G^{\prime}$ and
$G^{\prime\prime}$ under ultrasonic vibrations following the protocol detailed
in Appendix A.2.1. Our first important result is that the elastic modulus
$G^{\prime}$ of a colloidal gel may dramatically drop under ultrasonic
vibrations. As shown in Fig. 1, this softening effect increases with acoustic
pressure $P$, up to drops in gel elasticity by a factor of about five at the
highest pressures, indicative of ultrasound-induced collapse (see also Fig. 6
in Appendix A.3). The softening depends on the type of gel both qualitatively
and quantitatively. Although the calcite gel in Fig. 1(a) has the largest
elastic modulus at rest, it is the most fragile system and therefore turns out
to be the most sensitive to ultrasonic vibrations by showing a rapid
saturation of the relative softening amplitude with the acoustic pressure
[Fig. 1(d)]. By contrast, the silica gel in Fig. 1(b) and the carbon black gel
in Fig. 1(c), which respectively show strong and weak physical aging at rest
and yield strains larger than the calcite gel (see Fig. 5 in Appendix A.1),
require that ultrasonic vibrations reach some threshold amplitude in order to
induce softening.

In order to rationalize the above observations, we introduce the dimensionless
group $\Upsilon=\gamma^{\text{s}}_{\text{US}}/\gamma_{\text{NL}}$ that
compares the amplitude of the shear strain induced by ultrasound to the strain
amplitude at onset of nonlinear response at 1 Hz inferred from Fig. 5 in
Appendix A.1. As shown in Fig. 1(e), the softening amplitudes for all three
gels appear to fall onto a single curve when the acoustic excitation is
rescaled in terms of $\Upsilon$. The collapse is especially good for the
carbon black and silica gels and points to a critical value of
$\Upsilon_{\text{c}}\simeq 0.1$ below which ultrasonic vibrations have no
measurable effect on the elastic modulus. Although more data at higher
$\Upsilon$ values would be needed for both these gels, the calcite rescaled
data indicate that the softening effect saturates for $\Upsilon\gtrsim 1$,
which highlights the relevance of this dimensionless number. Indeed, we tried
to rescale the same data using other dimensionless groups based on the gel
yield stress $\sigma_{\text{y}}$, including $P/\sigma_{\text{y}}$ and
$\eta_{\text{f}}P/\rho cd\sigma_{\text{y}}$, with $\eta_{\text{f}}$ the
viscosity of the suspending fluid and $d$ the diameter of the colloidal
particles (see also discussion in Section V) but all these definitions led to
scattered data sets for the three gels under study and/or to irrelevant values
of the dimensionless parameter, either much larger or much smaller than 1 over
the whole range of acoustic amplitudes.

Remarkably, after ultrasound is turned off, all gels recover most of their
elasticity over time [see $t>30$ s in Figs. 1(a–c)]. This suggests that the
effect of ultrasound on the colloid network is essentially reversible. For the
silica and carbon black gels, some softening persists for large acoustic
pressures while the calcite gel shows the reverse trend with full recovery for
the largest pressures. Such a dependence with the gel microstructure is not
surprising since a complex interplay between the ultrasonic vibration and the
colloid network is expected at the microscale. This issue is investigated in
more detail below in the case of the carbon black gel.

Since one may also invoke surface effects, such as partial detachment of the
gel at the walls, to explain the softening reported in Fig. 1, it is crucial
to demonstrate that ultrasonic vibrations actually induce _bulk_ modifications
of the gel microstructure. This is what we proceed to show in the next section
where time-resolved structural measurements are performed through ultra small-
angle x-ray scattering (USAXS) as ultrasonic vibrations are being applied to a
4-mm slab of carbon black gel.

##  III Gel structure under high-power ultrasonic vibrations

##### Experimental setup.

Our setup for time-resolved USAXS measurements under ultrasound is described
in full details in Ref. Jin  _et al._ (2014). Briefly, a titanium vibrating
blade of width 2 mm connected to a piezoelectric transducer working at 20 kHz
(SODEVA TDS, France) is immersed in a channel of width 4 mm, depth 8 mm and
length 100 mm. Originally designed for cross-flow ultrafiltration, this setup
is used here in the absence of any flow except that used for filling and
preshearing the sample (see protocol in Appendix A.2.2). USAXS measurements
are carried out with the TRUSAXS instrument at the ID02 High Brilliance
beamline of the European Synchrotron Radiation Facility (ESRF, Grenoble,
France) Narayanan  _et al._ (2018). The incident x-ray beam of wavelength 0.1
nm is collimated to a vertical size of 80 $\mu$m and a horizontal size of 150
$\mu$m. A sample-to-detector distance of 31 m is used and provides access to
scattering magnitudes $q$ of the scattering wave vector from 0.0008 to 0.155
nm-1.

USAXS measurements are performed on the carbon black gel at a distance of
about 1 mm from the vibrating blade through a small window of thickness 0.3 mm
machined in the polycarbonate channel wall. The background scattering from the
channel filled with the mineral oil used in the carbon black gel is
systematically subtracted to the two-dimensional scattering patterns. The
resulting scattering intensity is isotropic so that only radially averaged
spectra are presented. Similar measurements on calcite and silica gels were
not possible due to multiple scattering by the 4-mm thick samples.

Figure 2: Structural measurements under ultrasonic vibrations. (a) Time-
resolved USAXS intensity spectra $I(q)$ recorded in a 3 % vol. carbon black
gel under ultrasonic vibrations with frequency 20 kHz for two different
acoustic pressures $P=64$ kPa (light colors) and $P=240$ kPa (dark colors).
Vibrations are turned on at $t=0$ and switched off at $t=30$ s. Three families
of spectra can be distinguished as a function of time before (red), during
(blue) and just after application of ultrasound (green). The times at which
the spectra are shown correspond to the various colored symbols in (b,c). The
black dash-dotted curve is the form factor used for the data analysis
presented in Appendix A.4. The dotted line indicates the magnitude $q$ of the
scattering wave vector that corresponds to 300 nm, the effective diameter of
the carbon black particles. (b) Exponent $p$ of the best power-law fit
$I(q)\sim q^{-p}$ at low $q$ as a function of time for $P=64$ kPa. The
structure fully recovers after ultrasonic vibrations are turned off [see
orange dot and corresponding spectrum in (a)]. (c) Same as (b) for $P=240$
kPa. The structure remains affected by ultrasound over at least one minute.
Bottom panels: qualitative views of the gel microstructure before (left),
during (middle) and after application of ultrasound (right) as inferred from
USAXS data. The yellow circle highlights a carbon black particle of effective
diameter 300 nm (scale bar).

##### Results.

As displayed in Fig. 2(a), the scattering intensity $I(q)$ is unaffected by
ultrasonic vibrations for scattering wave vector magnitudes $q\gtrsim 10^{-2}$
nm-1, i.e., the structure of the carbon black gel does not change for length
scales smaller than about 500 nm (see also Figs. 7 and 8 in Appendix A.4 for a
more detailed analysis of the USAXS data). However, for $q\lesssim 10^{-2}$
nm-1, the scattering intensity $I$ strongly varies upon application of
ultrasound. To quantify this effect, we measure the power-law exponent $p$ of
the scattering intensity, $I(q)\sim q^{-p}$ at low $q$ values. Starting from
$p\simeq 2.7$ at rest, the exponent reaches values as large as 3.5 under
medium-amplitude ultrasonic vibrations ($P=64$ kPa), before undershooting to
$p\simeq 2.3$ and finally slowly relaxing to its initial value once vibrations
are turned off [Fig. 2(b)]. Under a much higher acoustic pressure ($P=240$
kPa), series of bursts with peak values $p\simeq 3.5$ are observed and the
intensity spectrum does not fully recover its initial shape at rest [Fig.
2(c)].

The above results not only allow us to unambiguously identify the effects of
ultrasonic vibrations as bulk effects, they also lead us to associate them
with a qualitative microscopic scenario. Indeed, exponents $p$ smaller than 3
indicate that the bulk structure of the material contributes the most to the
scattering and that the scattered intensity corresponds to a mass fractal with
fractal dimension $d_{\text{f}}=p$. On the other hand, exponents ranging
between 3 and 4 indicate that interfaces dominate the scattering and such
exponents are interpreted in terms of a surface fractal with fractal dimension
$d_{\text{f}}=6-p$, with $p=4$ as the limiting case of a smooth surface
Schmidt (1991). Here, ultrasonic vibrations are observed to induce a
transition from $p\simeq 2.7<3$ to $p\simeq 3.5>3$ for wave numbers which
magnitudes range from the lowest achievable values up to $q\simeq 5\,10^{-3}$
nm-1, i.e. over half a decade at low $q$. This implies that, for length scales
$2\pi/q\gtrsim 1~{}\mu$m, scattering is dominated by a mass fractal, bulk
structure at rest, which turns into a structure with fractal interfaces under
ultrasound. Therefore, as pictured in the bottom panels of Fig. 2, the most
probable scenario is that the gel network gets fractured by ultrasonic
vibrations over length scales larger than typically 1 $\mu$m. The resulting
cracks are filled with oil and the surface roughness of these channels
accounts for the scattering with $p\simeq 3.5$. This fragmentation results in
a drop of the gel elasticity. The system then coalesces back into a mass-
fractal structure looser than the initial one ($p\simeq 2.3$). At high
acoustic pressures, the large temporal fluctuations and the oscillations from
surface to mass fractal hint at spatial heterogeneity and possible large-scale
flow. After vibrations are turned off, the degree of structural recovery
depends on the acoustic pressure consistently with the behaviour of the
elastic modulus reported in Fig. 1(c).

##  IV Gel flow and yield stress under high-power ultrasonic vibrations

In order to deepen our exploration of the interaction between colloidal gels
and ultrasonic vibrations, we now return to the rheo-ultrasonic setup used in
Fig. 1 and address the effect of vibrations on yielding, both through flow
curve measurements (Fig. 3) and through creep experiments (Fig. 4) following
the protocols in Appendix A.2.1.

##### Flow curves under ultrasonic vibrations.

We focus on flow curves obtained on the calcite gel by measuring the shear
stress $\sigma$ while sweeping down the shear rate $\dot{\gamma}$ for
different acoustic pressures $P$. As seen in Fig. 3(a), all flow curves show
the same general trend. At large $\dot{\gamma}$, the gel flows and the stress
increases as $\sigma\sim\dot{\gamma}^{n}$ with $0<n<1$ the shear-thinning
index. We observe a sharp increase of $n$ with $P$ [Fig. 3(c)] indicating that
the calcite gel becomes less and less shear thinning as the intensity of
ultrasonic vibrations is increased. Since the whole gel is fluidized at large
shear rates and potential surface effects become negligible, this dependence
of $n$ on $P$ further confirms the bulk nature of the effect of vibrations. At
low $\dot{\gamma}$, the stress plateaus to a value that corresponds to the
dynamic yield stress $\sigma_{\text{y}}$ Bonn  _et al._ (2017). In the calcite
gel, $\sigma_{\text{y}}$ is observed to decrease linearly by almost one order
of magnitude over the explored range of acoustic pressures [Fig. 3(b)]
indicating that increasing the acoustic intensity makes it easier to break and
fluidize the gel. The very same trends are obtained from static yield stress
measurements where the shear stress is swept up from the rest state (see Fig.
9 in Appendix A.3). We conclude that, as one increases the acoustic energy
injected into the material, ultrasonic vibrations can be used to turn a
colloidal gel with a strong yield stress behaviour ($\sigma_{\text{y}}\simeq
G_{0}$) into a more and more Newtonian fluid ($\sigma_{\text{y}}\ll G_{0}$ and
$n\rightarrow 1$).

Figure 3: Flow properties under ultrasonic vibrations. (a) Flow curves, shear
stress $\sigma$ vs shear rate $\dot{\gamma}$, measured in a 10 % vol. calcite
gel for different acoustic intensities [increasing from top to bottom, see
color code in (b,c)]. The frequency of ultrasonic vibrations is 45 kHz. The
shear stress is normalized by the elastic modulus $G^{\prime}_{0}$ measured at
rest. (b) Normalized yield stress $\sigma_{\text{y}}/G^{\prime}_{0}$ and (c)
shear-thinning exponent $n$ as a function of the acoustic pressure $P$. Dotted
lines in (b) and (c) respectively show the best linear and parabolic fits of
the data. The vertical arrow in (a) points to the decreasing region of the
flow curve at low $P$.

Furthermore, we note that in the absence of ultrasonic vibrations, the flow
curve displays a decreasing region for $0.5\lesssim\dot{\gamma}\lesssim 5$
s-1. Such an unstable branch is generically observed in attractive colloidal
systems and interpreted as the signature of localized or shear-banded flows
Bonn  _et al._ (2017); Divoux  _et al._ (2013). Interestingly, under
ultrasonic vibrations, this decreasing portion progressively gives way to a
monotonically increasing branch as the amplitude of the vibrations is
increased. Very similar features have been reported in vibrated frictional
grains Dijksman  _et al._ (2011) as well as vibrated granular suspensions
Hanotin  _et al._ (2012, 2015). This peculiar behavior was linked to the
presence of a dynamical critical point at a finite flow rate due to the
competition between flow-induced fluctuations and external vibration Wortel
_et al._ (2016). Very recently, the previous approach developed for frictional
grains was generalized to soft glassy materials through a mesoscale
elastoplastic model accounting for a shear-banding instability and showing
again a transition from flow curves with a minimum to monotonically increasing
flow curves as vibration is increased Le Goff  _et al._ (2019). The results of
Fig. 3 may constitute the first experimental evidence that such a scenario
applies to colloidal gels.

Figure 4: Ultrasound-assisted fluidization. (a) Shear rate responses
$\dot{\gamma}(t)$ measured in a 3 % vol. carbon black gel for different
acoustic intensities [increasing from right to left, see color code in (c,d)]
after a constant shear stress $\sigma=22$ Pa is applied following preshear.
The frequency of ultrasonic vibrations is 45 kHz. Following a creep period,
the gel starts to flow within a well-defined fluidization time
$\tau_{\text{f}}$ (see black $\times$) that sharply decreases with the
acoustic intensity. (b) Fluidization time $\tau_{\text{f}}$ as a function of
the applied stress $\sigma$ under different acoustic intensities (increasing
from top to bottom). Colored solid lines are the best exponential fits,
$\tau_{\text{f}}=\tau_{0}\exp(-\sigma/\sigma_{0})$, for a given acoustic
intensity. The vertical dotted line shows the shear stress $\sigma=22$ Pa used
in (a). (c) Characteristic stress $\sigma_{0}$ and (d) time $\tau_{0}$
extracted from exponential fits of $\tau_{\text{f}}(\sigma)$ as a function of
the acoustic pressure $P$. Dotted lines are polynomial fits to guide to the
eye. (e) $\tau_{0}$ as a function of $1/\sigma_{0}$ in semilogarithmic scales.
The solid line is the best exponential fit
$\tau_{0}=\tilde{\tau}\exp(\tilde{\sigma}/\sigma_{0})$ with
$\tilde{\tau}=57.1$ s and $\tilde{\sigma}=24.9$ Pa. (f) Master curve for
fluidization times $\tau_{\text{f}}/\tilde{\tau}$ as a function of
$(\tilde{\sigma}-\sigma)/\sigma_{0}$. The black line is the exponential
function $y=\exp(x)$.

##### Ultrasound-induced acceleration of fluidization under stress.

In the experiments of Fig. 3, the flow curves could be reasonably interpreted
as steady states thanks to the weak time-dependence of the calcite gel.
However, when made to flow, colloidal gels generically show strong time-
dependence, through e.g. thixotropy or rheopexy, due to the competition
between physical aging and shear-induced rejuvenation Mewis and Wagner (2009);
Radhakrishnan  _et al._ (2017). Carbon black gels enter such a category. More
specifically, carbon black gels are known to be prototypical of the so-called
“delayed yielding” phenomenon: during a creep experiment, when submitted to a
constant stress $\sigma$ higher than, yet close to, the yield stress, the
presheared gel first consolidates through a long-lasting creep regime before
getting fluidized after a time $\tau_{f}$ that follows an exponential decrease
with the applied stress, $\tau_{\text{f}}=\tau_{0}\exp(-\sigma/\sigma_{0})$,
characteristic of activated processes Gibaud  _et al._ (2010); Sprakel  _et
al._ (2011); Ovarlez  _et al._ (2013); Grenard  _et al._ (2014). Figure 4
summarizes the effect of ultrasonic vibrations on such delayed fluidization.
First, for a given applied stress, vibrations are seen to accelerate
fluidization by a factor of about ten at the highest achievable acoustic
pressure [Fig. 4(a)]. Varying the applied stress for a fixed acoustic pressure
$P$ [Fig. 4(b)], we recover the classical exponential behaviour for
$\tau_{\text{f}}$ vs $\sigma$, yet with parameters $\sigma_{0}$ and $\tau_{0}$
that strongly depend on $P$ [Fig. 4(c,d)]. Mean-field approaches of delayed
yielding predict that (i) $\sigma_{0}\sim\rho_{0}k_{\text{B}}T/\delta$, where
$\rho_{0}$ is the initial area density of strands, $k_{\text{B}}$ the
Boltzmann constant, $T$ the temperature and $\delta$ the range of the
interaction potential and (ii) $\tau_{0}\sim\exp(E_{\text{A}}/k_{\text{B}}T)$
where $E_{\text{A}}$ is the depth of the potential well Lindström  _et al._
(2012). Since ultrasonic vibrations are unlikely to affect $\rho_{0}$,
$\delta$ or $E_{\text{A}}$, a way to interpret our results is to invoke an
_effective_ temperature $T_{\text{eff}}\propto\sigma_{0}$ associated with the
vibrational ultrasonic energy, as commonly done for agitated granular
materials Jaeger  _et al._ (1996); D’Anna  _et al._ (2003); Puglisi  _et al._
(2005). As discussed in detail in Appendix A.6, the remarkable Arrhenius-like
scaling of $\tau_{0}$ with $\sigma_{0}$ demonstrated in Fig. 4(e) further
allows us to collapse all the fluidization time data onto a single exponential
master curve in Fig. 4(f) and to extract an estimate of the interaction energy
$E_{\text{A}}\simeq 20.8\,k_{\text{B}}T$ that is consistent with previous work
Trappe  _et al._ (2001); Prasad  _et al._ (2003); Trappe  _et al._ (2007).
This fully confirms that the characteristic stress $\sigma_{0}$ measures the
effective temperature of the system, which increases by a factor of about 4
over the range of acoustic intensities explored here. Therefore, injecting
acoustic energy into the system amounts to increasing its temperature and thus
facilitates the fluidization of gels that display delayed yielding.

##  V Discussion and conclusion

To summarize, the three colloidal gels investigated here can be called “rheo-
acoustic” gels as their mechanical and flow properties display great
sensitivity to ultrasonic vibrations. We have shown that ultrasonic vibrations
transiently soften these fragile materials and facilitate their flow. This
concept is all the more promising that the gels we chose to prove our point,
calcite, silica or carbon black gels, span a large range of yielding behaviors
usually observed in soft solids. We have quantified these effects thanks to
setups that couple a built-in ultrasonic transducer to rheology and USAXS
measurements and using protocols for gels at rest, under shear flow and
submitted to a constant stress. These effects are robust and persist through
changes in the ultrasound frequency (Figs. 10 and 11 in Appendix A.7) and in
the cell geometry (Fig. 12 in Appendix A.8). Therefore, they should swiftly
become key to applications in flow enhancement, unclogging, extrusion or
3D-printing based on acoustically-tunable colloid-based materials.

The present results also entail a number of new fundamental questions that
should further pave the way for future theoretical and numerical work. At a
mean-field level, our fluidization results strongly suggest that some of the
mechanical effects observed here can be rationalized through an effective
temperature. This corroborates the general idea that external stress or
perturbations can be seen as a “mechanical noise” that adds up to thermal
noise. Experimentally, one could deepen the analogy with real temperature by
performing fluidization experiments as in Fig. 4 over a wide range of
temperatures in the absence of ultrasonic excitation and by checking whether
the acceleration of the dynamics follows the same trend as under ultrasound.
As for theory, the mesoscopic approach proposed in Ref. Le Goff  _et al._
(2019) is based on the assumption that vibrations induce a “fluidizing noise”
that leads to additional activated plastic events of the same nature as those
induced by shear Picard  _et al._ (2005) but with a “vibration rate”
$k_{\text{vib}}$ that quantifies the influence of vibrations. Following this
kind of approach to predict viscoelastic properties of soft glassy materials
under vibrations or to study their effect on yielding dynamics looks very
promising for future comparison with the present experiments. Moreover,
extending classical theoretical approaches such as the Soft Glassy Rheology
framework Sollich  _et al._ (1997); Fielding  _et al._ (2000), that relies on
a “noise temperature,” or Shear-Transformation Zone models Falk and Langer
(1998, 2011), which involve an effective “disorder temperature,” to include
the effect of mechanical vibrations would be most insightful.

Besides the modelling of the global signature of ultrasonic vibrations on
mechanical properties, our experiments raise important physical questions
about the interaction mechanism between colloidal gels and ultrasonic
vibrations at a microscopic level. First, a few ultrasound–gel interaction
mechanisms can be disregarded. Indeed, at the rather low acoustic powers
involved in our study (${\mathcal{P}}\lesssim 2$ W cm-2), cavitation bubbles
are very unlikely to occur Jin  _et al._ (2014), especially in the oil-based
carbon black gel, and any influence of sonochemistry can be excluded
Chandrapala  _et al._ (2012). The acoustic wavelength (3–60 mm) being always
several orders of magnitude larger than the microstructure characteristic
sizes (20 nm–1 $\mu$m), radiation pressure through scattering of ultrasound by
colloidal clusters may also be neglected Hamilton and Blackstock (1998).

Second, the scaling in terms of the parameter
$\Upsilon=\gamma^{\text{s}}_{\text{US}}/\gamma_{\text{NL}}$ reported in Fig.
1(e) suggests that at least the softening effect is controlled by the strain
induced by ultrasonic vibrations. Indeed, it seems natural to think of
ultrasonic vibrations as a source of oscillatory mechanical stress analogous
to that of oscillatory shear although not through simple shear flow.
Repetitively applying the extensional and shear strains
$\gamma^{\text{e}}_{\text{US}}$ and $\gamma^{\text{s}}_{\text{US}}$ introduced
in Sect. II at high frequency could induce fatigue in the gel and therefore
its softening or even its fluidization. Such fatigue could explain that the
effects of ultrasonic vibrations become noticeable as soon as $\Upsilon\gtrsim
0.1$, far below the onset of nonlinearity in the rheological response. Note,
however, that ultrasonic vibrations impose oscillations along the vorticity
direction at a high frequency ($>20$ kHz) whereas the rheometer imposes a
constant or slowly varying shear (typically $<20$ Hz). Such a wide gap in
frequencies makes any direct analogy with the low-frequency response
hypothetical and it is not clear how standard rheological observables such as
the elastic modulus or the yield stress extrapolate to the ultrasonic
frequency domain.

Moreover, while the above argument based on strain-induced fatigue seems
relevant for explaining the softening effect at rest, it is unlikely to also
hold under flow. This raises the question of the existence of a single
dimensionless number that would govern all the effects of ultrasonic
vibrations on colloidal gels. Such a dimensionless group should compare
vibration-induced forces to internal forces within the gel. Still, depending
on whether one considers gels at rest or under shear, one may propose to base
vibration-induced forces on elastic forces or on viscous friction, either at
the single particle level or at the gel network level. As for internal forces,
these obviously result from interparticle attraction but may depend in a
complex manner on the microscopic details of such attraction, on the fractal
geometry of the network, on internal stresses stored during preshear and on
thermal energy.

At this stage, it is interesting to compare the present study on colloidal
gels to previous findings on dense suspensions. Recently, shear thickening in
concentrated Brownian and non-Brownian suspensions could be tuned both through
the superposition of an oscillatory flow with frequency up to 500 Hz to steady
shear Lin  _et al._ (2016) and through the application of acoustic
perturbations Sehgal  _et al._ (2019), a process that was shown to be rate-
controlled. Similarly, friction and liquefaction in both dry and wet granular
materials under low-frequency mechanical vibrations appear to be driven by the
velocity amplitude $v=\omega a$ Hanotin  _et al._ (2012); Lastakowski  _et
al._ (2015). These effects have been interpreted in terms of consolidated vs
mobile states depending on whether or not grains are in contact through force
chains Hanotin  _et al._ (2015), and further rationalized in terms of the
lubrication Peclet number, $Pe_{\text{lub}}=\eta_{\text{f}}\omega a/\mu
P_{\text{g}}d$, where $\eta_{\text{f}}$ is the viscosity of the suspending
fluid, $\mu$ the effective friction coefficient, $P_{\text{g}}$ the granular
pressure and $d$ the bead diameter Hanotin  _et al._ (2012, 2015). This
dimensionless number compares the viscous forces induced by vibrations to the
internal friction forces in the granular system. An equivalent number for
colloidal gels is $\Pi=\eta_{f}\omega
a/d\sigma_{\text{y}}=\eta_{\text{f}}P/\rho cd\sigma_{\text{y}}$. Measurements
on the calcite gel at two different ultrasonic frequencies (Fig. 10 in
Appendix A.7) suggest that $v=\omega a=P/\rho c$ could be the relevant control
parameter but the softening data of Fig. 1(d) do not collapse as a function of
$\Pi$. Therefore, future experimental studies should systematically explore
the effects of the vibration frequency on gels with different volume fractions
and for the various protocols. These studies will tell whether a universal
dimensionless group involving $v$ (or equivalently $P$) or $a$ (or
equivalently $\gamma^{\text{s}}_{\text{US}}$) and based on the yield strain or
yield stress can generally describe the effects of acoustic perturbations on
colloidal gels.

Finally, a detailed microscopic scenario remains to be uncovered. USAXS
measurements unambiguously reveal that the gel network gets fractured by
ultrasonic vibrations. Yet, structural measurements at larger length scales,
e.g. through light scattering, are needed to get a full picture of the gel
microstructure under ultrasonic vibrations. Moreover, in the absence of shear,
we could not reach fully fluidized states (characterized by
$G^{\prime\prime}>G^{\prime}$) even though we pushed our transducers to the
maximum achievable acoustic intensities. This suggests that, in spite of the
presence of cracks and of possible large-scale rearrangements, some
connectivity remains in the gel network so that solidlike behavior dominates.
Future experimental work, if possible under even higher acoustic intensities,
will focus on colloidal gels that can be imaged under a microscope in order to
directly follow the breakup of the gel microstructure through high-speed
imaging. At a mesoscopic level, particle imaging velocimetry or ultrasonic
velocimetry will be helpful to investigate the possibility of vibration-
induced wall slip and to confirm that shear-banding instabilities can be tuned
by ultrasonic vibrations Le Goff  _et al._ (2019). Deeper rheological
investigations are also planned to explore the influence of previous shear
history Ovarlez  _et al._ (2013); Helal  _et al._ (2016b); Hipp  _et al._
(2019) on the ultrasound-induced effects evidenced here as well as hysteresis
phenomena in flow curve measurements under ultrasound Divoux  _et al._ (2013);
Radhakrishnan  _et al._ (2017). We also believe that recent simulation methods
developed for colloidal gels Colombo  _et al._ (2013); Varga and Swan (2015);
Landrum  _et al._ (2016); Bouzid  _et al._ (2018), once adapted to account for
ultrasonic excitation, will certainly bring valuable insight into rheo-
acoustic gels by investigating the local impact of ultrasonic vibrations on
the gel microstructure and its relationship with the global mechanical
response.

## Acknowledgements

The authors thank T. Divoux and E. Freyssingeas for insightful discussions on
rheological and USAXS data, W. Chevremont and T. Narayanan for technical help
with the USAXS–ultrasound combined setup, H. Bodiguel and C. de Loubens for
sharing their beam time at ESRF with us, and C. Barentin and T. Liberto for
providing us with calcite. This work was funded by the Région Auvergne-Rhône-
Alpes “Pack Ambition Recherche” Programme and by the European Research Council
under the European Union’s Seventh Framework Programme (grant agreement No.
258803). LRP is part of Labex TEC 21 (Investissements d’Avenir, grant
agreement No. ANR-11-LABX-0030), PolyNat Carnot Institute (Investissements
d’Avenir, grant agreement No. ANR-11-CARN-030-01) and of IDEX UGA program
(ANR-15-IDEX-02).

## Appendix A

###  A.1 Colloidal gels

We focus on three different colloidal suspensions respectively made of
calcite, silica and carbon black particles. All systems aggregate into space-
spanning networks thanks to attractive interparticle forces. However, the
resulting gels present different mechanical properties depending on the
details of the interactions, as illustrated in Fig. 5.

Calcite is the most stable polymorph of calcium carbonate (CaCO3). Calcite
powder (Socal 31 from Solvay, average particle diameter of 60 nm, density of
2710 kg m-3) is suspended in water at a volume fraction of 10 %. The
suspension is homogenized following the protocol described in Ref. Liberto
_et al._ (2017) and spontaneously gels within a few minutes at pH $\simeq
8.8$. The resulting network shows little aging over time yet great sensitivity
to shear. A mild preshear is applied for 10 s at 10 s-1 prior to any
measurement. After a rest period of 60 s, the elastic modulus at rest reaches
a steady value of about 20 kPa.

Figure 5: Elastic modulus $G^{\prime}$ (filled symbols) and viscous modulus
$G^{\prime\prime}$ (open symbols) as a function of strain amplitude $\gamma$
for (a) a 10 % vol. calcite gel, (b) a 5 % vol. silica gel and (c) a 3 % vol.
carbon black gel. After the preshear and rest protocol described for each
system in Appendix A.1, the strain amplitude $\gamma$ is logarithmically swept
up with 10 steps per decade and 10 s per point. The frequency of the
oscillatory strain is 1 Hz. The dotted lines show the strain
$\gamma_{\text{NL}}$ at which the elastic modulus falls below 85 % of its
value in the linear regime and which we take as the lower limit of the
nonlinear regime. The dashed lines show the yield strain $\gamma_{\text{y}}$
defined as the point where $G^{\prime}$ and $G^{\prime\prime}$ cross each
other. Experiments conducted in a parallel-plate geometry of gap 1 mm. The
upper plate is sandblasted in (a) and covered with sandpaper in (b,c).

Silica particles (SiO2, Ludox TM 50 from Sigma, average particle diameter of
22 nm, density of 2350 kg m-3) are made to aggregate at pH $\simeq 9$ by
mixing equal volumes of the 50 % vol. stock Ludox solution and a concentrated
NaCl solution to a final salt concentration of 2 M and final solid volume
fraction of 5 % as described in Ref. Condre  _et al._ (2007). Screening of the
electrostatic repulsion leads to fast gel formation through attractive van der
Waals forces and interparticle siloxane bonds Depasse (1997); Drabarek  _et
al._ (2002). Such a gel shows pronounced physical aging over time Kurokawa
_et al._ (2015). Here, we minimize the effect of aging thanks to a strong
preshear at 500 s-1 applied for 120 s that partially erases the previous
history. The gel is then left to rest for 300 s, after which the elastic
modulus reaches about 500 Pa. As seen in Fig. 6(b), the systematic increase of
the elastic modulus with successive experiments indicates that preshear does
not allow for full rejuvenation of the sample.

Carbon black particles (Cabot Vulcan XC72R, density of 1800 kg m-3) are
fractal aggregates of radius of gyration 100–200 nm and made of fused primary
particles of typical diameter 40 nm Richards  _et al._ (2017). When dispersed
in a mineral oil (density 838 kg m-3, viscosity 25 mPa s, Sigma Aldrich), they
aggregate via van der Waals attraction Hartley and Parfitt (1985) and form
gels that show weak aging over time yet strong time-dependence under shear,
including rheopexy Ovarlez  _et al._ (2013), delayed yielding Gibaud  _et al._
(2010); Grenard  _et al._ (2014) and fatigue Gibaud  _et al._ (2016). It was
also shown that the preshear history can be used to tune the initial
microstructure of carbon black gels Osuji  _et al._ (2008); Ovarlez  _et al._
(2013); Helal  _et al._ (2016b). Here, we focus on a dispersion of carbon
black particles at a volume fraction of 3 % (weight fraction of 6 % w/w).
Because of the fractal nature of the particles, this corresponds to an
effective volume fraction $\phi_{\text{eff}}\simeq 20$ % Trappe  _et al._
(2007). The preshear protocol is made of two successive steps of 200 s each, a
first step at $-1000$ s-1 in the “reverse” direction followed by one step at
$+1000$ s-1 in the “positive” direction that defines the direction of positive
strains and stresses. After preshear, the dispersion recovers an elastic
modulus of typically 1 kPa within a few seconds.

The nonlinear viscoelasticity of the three colloidal gels under study is
probed in the absence of ultrasonic vibrations by imposing an oscillatory
shear with a fixed frequency of 1 Hz and an amplitude that increases from
about 0.01 % to 100 %. As seen in Fig. 5, both the elastic modulus
$G^{\prime}$ and the viscous modulus $G^{\prime\prime}$ first remain
independent of the oscillatory strain amplitude $\gamma$, which corresponds to
linear response. We consider that the linear regime ends at a strain
$\gamma_{\text{NL}}$ when the elastic modulus becomes smaller than 85 % of its
value in the linear regime (see dotted lines) and that the yield point is
reached at the strain $\gamma_{\text{y}}$ when $G^{\prime}=G^{\prime\prime}$
(see dashed lines). In the calcite gel, the linear regime extends only up to
strain amplitudes of $\gamma_{\text{NL}}=0.09$ % and the yield strain is
$\gamma_{\text{y}}=1.5$ % [Fig. 5(a)]. This behaviour was classified as that
of a fragile gel in the “strong link” regime Shih  _et al._ (1990); Liberto
_et al._ (2017). The linear regime of the silica gel extends up to
$\gamma_{\text{NL}}=0.8$ % and the gel yields at $\gamma_{\text{y}}=4.5$ %
[Fig. 5(b)]. Similarly, the response of the carbon black gel remains linear up
to $\gamma_{\text{NL}}=1.3$ % and the gel yields at $\gamma_{\text{y}}=5$ %
[Fig. 5(c)]. Therefore, in spite of a larger elastic modulus, the calcite gel
is significantly more fragile than both the silica and the carbon black gels.

Finally, since the colloidal particles and the suspending fluids considered
here are not density matched, sedimentation may affect the gel structure over
long time scales. We estimate the gravitational Peclet number as defined in
Ref. Kim  _et al._ (2013):

$Pe_{\text{g}}=\frac{4\pi|\Delta\rho|g}{3k_{\text{B}}T}\,r^{4}\phi^{(d_{\text{f}}+1)/(d_{\text{f}}-3)}\,,$
(1)

where $\Delta\rho$ is the density difference between the particles and the
suspending fluid, $g$ is the gravitational acceleration, $r$ is the particle
radius and $\phi$ is the volume fraction. For the calcite gel, using a fractal
dimension $d_{\text{f}}=2.23$ as determined in Ref. Liberto  _et al._ (2017)
leads to $Pe_{\text{g}}\simeq 0.2\lesssim 1$. In the case of the silica gel,
assuming that aggregation is driven by a DLCA process for which
$d_{\text{f}}\simeq 1.9$ is expected Condre  _et al._ (2007), we estimate
$Pe_{\text{g}}\simeq 5.10^{-4}\ll 1$. For the carbon black gel, giving an
estimate of $Pe_{\text{g}}$ is more difficult since carbon black particles are
themselves fractal aggregates and thus far from ideal spheres. Adapting the
approach of Ref. Kim  _et al._ (2013) to the case of fractal aggregates of
radius $r$ and made of primary particles with radius $r_{\text{p}}$ with a
fractal dimension $d_{\text{p}}$, one finds:

$Pe_{\text{g}}=\frac{4\pi|\Delta\rho|g}{3k_{\text{B}}T}\,r_{\text{p}}^{3-d_{\text{p}}}r^{d_{\text{p}}+1}\phi_{\text{eff}}^{(d_{\text{f}}+1)/(d_{\text{f}}-3)}\,,$
(2)

where $\phi_{\text{eff}}$ is the effective (hydrodynamic) volume fraction of
the fractal aggregates. Taking $r_{\text{p}}=20$ nm and $d_{\text{p}}=2.7$ as
reported in Ref. Richards  _et al._ (2017) for Vulcan carbon black and $r=150$
nm and $d_{\text{f}}\simeq 2.6$ as inferred from USAXS measurements, one gets
$Pe_{\text{g}}\simeq 5.10^{3}\gg 1$ for which gravitationally facilitated
phase separation is expected. All our measurements are taken only 300 s after
preshear (see protocols below in Appendix A.2.1) so that our data are most
likely not affected by gravity whatever the gel. Yet, we note that
$Pe_{\text{g}}$ spans many orders of magnitude covering both $Pe_{\text{g}}\ll
1$ and $Pe_{\text{g}}\gg 1$. This emphasizes the robustness of the effects
such as the ultrasound-induced softening evidenced in our work while calling
for a future detailed study of long-term gravitational effects under
ultrasonic vibrations. Such a study could indeed provide a potential
explanation for some of the discrepancies observed in Fig. 1 between the
various gels.

###  A.2 Experimental protocols

####  A.2.1 Mechanical measurements under ultrasonic vibrations

After the preshear protocol described above for each gel, the elastic modulus
is monitored in our rheo-ultrasonic setup during 300 s through small-amplitude
oscillatory strain (SAOS) of amplitude 0.06 % and frequency 1 Hz. This allows
us to define the “initial” elastic modulus $G^{\prime}_{0}$ prior to
application of vibrations.

In Fig. 1, ultrasonic vibrations are then applied for 30 s while measuring the
elastic modulus with the same SAOS protocol as before (see also Fig. 6). Once
vibrations are turned off, SAOS is continued for at least 170 s to monitor the
recovery. For the flow curve measurements shown in Fig. 3, ultrasonic
vibrations are applied while the shear rate is logarithmically swept down from
300 to 0.03 s-1 within 80 s. For Fig. 9, the shear stress is linearly swept up
from 0 to 50 Pa in 60 s. Finally, for the creep experiments of Fig. 4, both
the target shear stress $\sigma$ and ultrasonic vibrations are applied just
after preshear. Following Refs. Sprakel  _et al._ (2011); Grenard  _et al._
(2014), the fluidization time $\tau_{\text{f}}$ is defined as the last
inflexion point of the shear rate response $\dot{\gamma}(t)$.

In all cases, the preshear protocol is repeated between successive tests at
different acoustic intensities. We measured that after one minute under
ultrasound, the temperature of the transducer surface (and thus of the sample)
increases by less than 0.1∘C for $P<80$ kPa, by about 0.2 to 1∘C for
$80<P<130$ kPa and by up to 4∘C for $P>130$ kPa. For fluidization experiments
in the carbon black gel, which require to apply ultrasonic vibrations for much
longer times, $T$ increases by 2 to 10∘C and the system is left to relax for
one hour after sonication in order for the temperature to go back to its
initial value. It is checked in Fig. 13 in Appendix A.9 that temperature
changes cannot be invoked to explain the softening of colloidal gels observed
under ultrasonic vibrations. Finally, we check in Fig. 14 in Appendix A.10
that rheological measurements are not perturbed by the application of
ultrasonic vibrations.

####  A.2.2 Microstructural measurements under ultrasonic vibrations

The feed channel of our USAXS–ultrasound combined setup is filled with the gel
under study thanks to a syringe pump (Harvard Apparatus PHD 4400). The gel is
then presheared by pumping a volume of 10 mL at a flow rate of 20 mL min-1 in
both directions. Assuming a Poiseuille flow, this corresponds to a wall shear
rate of about 8 s-1 applied for 60 s. After a rest time of 180 s, ultrasonic
vibrations are applied for 30 s with a pressure amplitude ranging from 20 to
240 kPa. USAXS spectra are recorded every second during 100 s starting 20 s
prior to application of vibrations. Thus, these measurements provide insight
into the microstructure under ultrasonic vibrations for acoustic powers
similar to those of previous rheo-ultrasonic experiments.

###  A.3 Ultrasound-induced softening: full data set

Figure 6: Elastic modulus (thick solid lines) and viscous modulus (thin dashed
lines) normalized by the elastic modulus $G^{\prime}_{0}$ measured just before
ultrasonic vibrations are turned on and plotted as a function of time $t$ for
(a) a 10 % vol. calcite gel, (b) a 5 % vol. silica gel and (c) a 3 % vol.
carbon black gel. Ultrasonic vibrations with frequency 45 kHz are turned on at
time $t=0$ and switched off at $t=30$ s. Each curve corresponds to a given
acoustic intensity, with darker colors corresponding to larger intensities
[see color code in Fig. 1(d)]. The right panels show the initial values
$G^{\prime}_{0}$ of the elastic modulus measured for the successive
experiments performed under different acoustic intensities and used for
normalization in the left panels.

Figure 6 shows the viscoelastic moduli normalized by the elastic modulus
$G^{\prime}_{0}$ measured just before ultrasonic vibrations are turned on. The
elastic (viscous resp.) modulus is plotted in solid lines (dashed lines resp.)
for all available acoustic intensities and for the three colloidal gels under
study in Fig. 1 in the main text. The right panels show that the initial
modulus $G^{\prime}_{0}$ displays a clear increasing trend with successive
experiments only for the silica gel, i.e. for the system with the most
pronounced aging effects. The last two measurements on the calcite gel [black
curves in Fig. 6(a)] may have been affected by evaporation. Remarkably, the
viscous modulus never exceeds the elastic modulus for all systems, even
transiently. This means that although the gels display a spectacular
softening, they remain solidlike and never reach a fully fluidized state under
ultrasonic vibrations at the acoustic intensities explored here for a
frequency of 45 kHz.

###  A.4 Analysis of USAXS measurements

In order to go beyond the simple power-law description of the intensity
spectra $I(q)$ proposed in the main text, we model the USAXS data obtained on
the 3 % vol. carbon black gel with the Beaucage model Beaucage (2012).
Originally designed for polymeric mass fractals Beaucage (1995, 1996), this
model uses a “unified” function that interpolates between a Guinier regime at
low $q$ and a power-law Porod scaling at large $q$. The crossover between the
two regimes is controlled by a characteristic length scale $R_{\text{g}}$. In
order to account for successive structural levels covering different ranges of
scales, experimental data are often fitted to sums of such unified functions.
Here, we describe our USAXS spectra by using two structural levels as
contained in the following equation:

$I(q)=G_{1}\exp\left(-\frac{q^{2}R_{\text{g},1}^{2}}{3}\right)+B_{1}\exp\left(-\frac{q^{2}R_{\text{g},2}^{2}}{3}\right){q_{1}^{\star}}^{p_{1}}\\\
+G_{2}\exp\left(-\frac{q^{2}R_{\text{g},2}^{2}}{3}\right)+B_{2}{q_{2}^{\star}}^{p_{2}}$
(3)

where

$q^{\star}_{i}=\frac{q}{\left\\{\text{erf}\left(\frac{qR_{\text{g},i}}{\sqrt{6}}\right)\right\\}^{3}}$
(4)

and $i=1$ and $i=2$ respectively correspond to the large-scale and to the
small-scale level, i.e. $R_{\text{g},1}\gg R_{\text{g},2}$.

Figure 7: (a) Time-resolved USAXS intensity spectra $I(q)$ recorded in a 3 %
vol. carbon black gel under ultrasonic vibrations with frequency 20 kHz for
$P=64$ kPa. Ultrasonic vibrations are turned on at $t=0$ and switched off at
$t=30$ s. The times at which the spectra are shown correspond to the various
colored symbols in (b,c,d). The inset shows the same data in a Kratky plot,
i.e. by plotting $q^{2}I(q)$ vs $q$. The solid lines are the experimental
spectra while the dotted curves are the fits with the two-level Beaucage model
given by Eq. (3). The black dash-dotted curve corresponds to the form factor
$I_{2}(q)$ discussed in the text. The four free parameters of fits to Eq. (3)
are: (b) the exponent $-p_{1}$, (c) the characteristic size $R_{\text{g},1}$
and (d) the prefactors $G_{1}$ and $B_{1}$. In (b), the fit results for
$-p_{1}$ (in black) are compared to the power-law exponents $p$ at low $q$ (in
grey) replotted from Fig. 2(b). See also Video 8.

We first fix the small-scale level by fitting the high-$q$ range of the time-
averaged spectrum measured in the absence of ultrasound by a single unified
function:
$I_{2}(q)=G_{2}\exp(-q^{2}R_{\text{g},2}^{2}/3)+B_{2}{q_{2}^{\star}}^{p_{2}}$.
We find that $R_{\text{g},2}=150$ nm, $p_{2}=-2.9$, $G_{2}=4.5\,10^{4}$ cm-1
and $B_{2}=0.128$ cm-1 provide a very good description of $I(q)$ for
$q>10^{-2}$ nm-1 [see black dash-dotted curves in Figs. 7(a) and 8(a)]. The
characteristic size of 150 nm is fully consistent with both the radius of
gyration (177 nm) and the hydrodynamic radius (135 nm) reported recently for
the same carbon black particles based on neutron and light scattering Richards
_et al._ (2017). The exponent $-2.9$ also agrees well with the fractal
dimension of 2.7 measured on these particles by the same authors. It is
observed in Figs. 7(a) and 8(a) that the high-$q$ region of the USAXS spectra
remains unaffected by ultrasonic vibrations whatever the acoustic pressure
$P$. This means that ultrasound does not modify the structure at the scale of
the carbon black particles, which is consistent with the fact that these
particles are themselves unbreakable aggregates of permanently fused primary
particles of typical diameter 20 nm. Thus, the small-scale level $I_{2}(q)$
can be taken as the form factor of the carbon black particles and remains
unchanged throughout all experiments.

Figure 8: (a) Time-resolved USAXS intensity spectra $I(q)$ recorded in a 3 %
vol. carbon black gel under ultrasonic vibrations with frequency 20 kHz for
$P=240$ kPa. Ultrasonic vibrations are turned on at $t=0$ and switched off at
$t=30$ s. The times at which the spectra are shown correspond to the various
colored symbols in (b,c,d). The inset shows the same data in a Kratky plot,
i.e. by plotting $q^{2}I(q)$ vs $q$. The solid lines are the experimental
spectra while the dotted curves are the fits with the two-level Beaucage model
given by Eq. (3). The black dash-dotted curve corresponds to the form factor
$I_{2}(q)$ discussed in the text. The four free parameters of fits to Eq. (3)
are: (b) the exponent $-p_{1}$, (c) the characteristic size $R_{\text{g},1}$
and (d) the prefactors $G_{1}$ and $B_{1}$. In (b), the fit results for
$-p_{1}$ (in black) are compared to the power-law exponents $p$ at low $q$ (in
grey) replotted from Fig. 2(c). See also Video 8.

Ultrasound, however, has a strong impact on the larger scales ($q<10^{-2}$
nm-1). To quantify this impact, we fit the USAXS spectra for $q>10^{-3}$ nm-1
to Eq. (3) where $R_{\text{g},2}$, $p_{2}$, $G_{2}$ and $B_{2}$ are fixed to
the previous values and $R_{\text{g},1}$, $p_{1}$, $G_{1}$ and $B_{1}$ are
taken as free fitting parameters. Fit results are shown with dotted lines in
Figs. 7(a) and 8(a) for a few spectra representative of the time-resolved
USAXS data before (red), during (blue) and just after sonication (green). The
full set of fitting parameters are shown as a function of time in the right
panels of Figs. 7 and 8. For both ultrasonic pressures under investigation,
the evolution of the exponent $-p_{1}$ is found to be nicely consistent with
the power-law exponent $p$ [compare black and grey bullets in Figs. 7(b) and
8(b)].

For the lower acoustic intensity ($P=64$ kPa, see Fig. 7 and Video 8), all
fitting parameters show a strong peak upon turning on ultrasonic vibrations.
This peak coincides with that observed in the viscous modulus concomitantly
with the strong decrease of the elastic modulus in Fig. 6(c). The fits with
the two-level Beaucage model show that this peak is associated with a
transient increase in the low-$q$ scattering that turns from that of a mass
fractal ($-p_{1}<3$) to that of a surface fractal ($-p_{1}>3$) with a fractal
dimension $6+p_{1}\simeq 2.5$ Schmidt (1991). Since low enough $q$ values are
not accessible to the present USAXS measurements, the crossover to the Guinier
regime is not fully observed so that $R_{\text{g},1}$ can only be taken as a
rough estimate of a characteristic microstructural size under ultrasonic
vibrations. Still, the time evolution reported here points to the scenario
illustrated in the sketches of Fig. 2 in the main text, where the initial
microstructure, made of a percolated network of micronsized fractal aggregates
of carbon black particles, transiently breaks up into a fragmented assembly of
aggregates separated by patches of solvent with a characteristic size of
several micrometers. After this initial break-up, the microstructure heals
back to a space-spanning network yet with lower fractal dimension. Once
ultrasonic vibrations are turned off, the system further relaxes until it
fully recovers its initial microstructure over the course of about one minute
[compare red and orange spectra in Fig. 7(a)].

For the larger acoustic intensity ($P=240$ kPa, see Fig. 8 and Video 8), the
temporal response consists of a series of peaks where the spectra change from
mass fractal to surface fractal with fit parameters similar to those observed
within the single peak measured at the lower intensity. This suggests that the
microstructure constantly changes from a space-spanning network of clusters to
a fractured network with micronsized cracks filled with solvent. At this
stage, it is not clear whether this unsteadiness results from a global
evolution of the whole sample or from more local effects. Indeed the spectra
are averaged over the size of the x-ray beam ($80~{}\mu$m$\times 150~{}\mu$m)
so that the strong fluctuations in the USAXS data could be caused by large-
scale motion of carbon black clusters entering and leaving the beam. Moreover,
the pressure amplitude investigated here in the USAXS setup is above the range
of pressures that can be achieved with the rheometer setup, which makes it
difficult to directly compare the USAXS data with the evolution of global
features such as the elastic moduli. Once ultrasonic vibrations are stopped,
the characteristic radius $R_{\text{g},1}$ goes back to its initial value
within experimental uncertainty but the Porod contribution $B_{1}$ to the
low-$q$ structural level remains significantly larger than the initial one,
with an exponent $-p_{1}$ ranging between 2 and 2.5 indicative of a
microstructure looser than before application of ultrasound.

Time-resolved USAXS measurements on a 3 % vol. carbon black gel under
ultrasonic vibrations with frequency 20 kHz for $P=64$ kPa. The data
correspond to those analyzed in Fig. 7. The top panel shows the USAXS
intensity spectra $I(q)$ as a function of time. The bottom panel displays the
time evolution of the exponent $p$ obtained from a power-law fit of $I(q)\sim
q^{-p}$ at low $q$. Ultrasonic vibrations are turned on at $t=0$ and switched
off at $t=30$ s. The black dash-dotted curve corresponds to the form factor
$I_{2}(q)$ discussed above. The red dashed line shows the initial spectrum
prior to application of ultrasound. The current measurement is drawn in red,
blue or green when recorded respectively before, during or after application
of ultrasound.

Same as Video 8 for $P=240$ kPa. The data correspond to those analyzed in Fig.
8.

Figure 9: (a) Flow curves, shear stress $\sigma$ vs shear rate $\dot{\gamma}$,
measured in a 10 % vol. calcite gel for different acoustic intensities
[increasing from top to bottom, see color code in (b,c)]. The frequency of
ultrasonic vibrations is 45 kHz. The shear stress is normalized by the elastic
modulus $G^{\prime}_{0}$ measured at rest. (b) Normalized yield stress
$\sigma_{\text{y}}/G^{\prime}_{0}$ and (c) shear-thinning exponent $n$ as a
function of the acoustic pressure $P$. In (b) and (c), the results obtained
from the flow curves shown in (a) and measured by sweeping up the shear stress
are plotted with filled symbols and compared to the results of Fig. 3(b) and
(c) (open symbols) obtained from downward sweeps of the shear rate as
described in Appendix A.2.1.

###  A.5 Flow properties under ultrasonic vibrations: complementary flow curve
measurements

To complement the flow curve measurements of Fig. 3 that characterized the
dynamic yield stress in the main text, Fig. 9 shows the results of increasing
stress ramps in the calcite gel. This protocol allows us to define the static
yield stress as the stress at which the shear rate suddenly jumps from a
vanishingly small value ($\dot{\gamma}<10^{-3}$ s-1) to a finite value
($\dot{\gamma}>0.3$ s-1). Error bars on the static yield stress are estimated
by taking the standard deviation of the stress measured over the plateau that
separates those shear rate values. As for the dynamic yield stress, the static
$\sigma_{\text{y}}/G_{0}$ decreases linearly with the acoustic pressure $P$
[Fig. 9(b)]. Interestingly, the normalized static yield stress remains more
than two times larger than its dynamic counterpart as long as $P\lesssim 70$
kPa while the two yield stresses become indistinguishable at larger acoustic
pressures. For $P\gtrsim 70$ kPa, all flow curves are monotonically increasing
and the shear-thinning index $n$ strongly increases with $P$ [Fig. 9(c)] as
already reported in Fig. 3(c). Here, error bars on $n$ correspond to the
dispersion obtained on the exponent of power-law fits at high shear rate by
varying the fitting interval.

Below the transition at $P\simeq 70$ kPa, the flow curve displays a decreasing
part under an imposed shear rate [Fig. 3(a)] and a true stress plateau under
an imposed stress [Fig. 9(a)]. These two features are characteristic of
attractive systems where flow competes with restructuring dynamics, leading to
mechanical instability and shear banding Bonn  _et al._ (2017). On the other
hand, flow curves that increase monotonically above the yield point are
typical of repulsive soft glassy materials. Therefore, we may conclude that
for the present calcite gel, ultrasonic vibrations induce a transition from an
attraction-dominated system to a repulsive-like yield stress fluid with weaker
and weaker shear-thinning properties as the acoustic intensity is increased.

###  A.6 Ultrasound-assisted fluidization: accounting for ultrasound through
an effective temperature

In the main text, the master curve in Fig. 4(f) shows that, for the carbon
black gel involved in the present study, the fluidization time
$\tau_{\text{f}}$ under a stress $\sigma$ and an acoustic pressure $P$ can be
written as:

$\tau_{\text{f}}(\sigma,P)=\tilde{\tau}\exp\left(\frac{\tilde{\sigma}-\sigma}{\sigma_{0}(P)}\right)\,,$
(5)

where $\sigma_{0}(P)$ is displayed in Fig. 4(c) and the parameters
$\tilde{\tau}=57.1$ s and $\tilde{\sigma}=24.9$ Pa are inferred from Fig.
4(e). We introduce the effective temperature defined as:

$T_{\text{eff}}(P)=\frac{\sigma_{0}(P)}{\sigma_{0}(0)}\,T\,,$ (6)

such that the effective temperature coincides with the thermodynamic
temperature in the absence of ultrasonic vibrations, i.e. for $P=0$. This
allows us to recast Eq. (5) in terms of $T_{\text{eff}}$ and $T$ as:

$\tau_{\text{f}}(\sigma,P)=\tilde{\tau}\exp\left(\frac{(\tilde{\sigma}-\sigma)T}{\sigma_{0}(0)T_{\text{eff}}(P)}\right)\,.$
(7)

In its simplest form, i.e. when taking the “high stress limit” where all bonds
in a strand cross-section break almost simultaneously, the mean-field model
introduced in Refs. Sprakel  _et al._ (2011); Lindström  _et al._ (2012)
predicts a fluidization time:

$\tau_{\text{f}}^{\text{m}}(\sigma)=\frac{1}{\omega_{0}C\sigma}\exp\left(\frac{E_{\text{A}}}{k_{\text{B}}T}-C\sigma\right)\,,$
(8)

where $\omega_{0}$ is the attempt frequency for bond dissociation,
$C=\delta/(n\rho_{0}k_{\text{B}}T)$ with $n$ the average number of bonds per
cross-section of a network strand and, as defined in the main text, $\rho_{0}$
the initial area density of particles, $\delta$ the range of the interaction
potential and $E_{\text{A}}$ the depth of the potential well. $C$ thus
corresponds to the inverse of the characteristic stress $\sigma_{0}$ in the
absence of ultrasonic vibrations. Assuming that the effect of ultrasonic
vibrations with amplitude $P$ can be captured through an effective temperature
$T_{\text{eff}}(P)$ and neglecting the logarithmic correction $1/C\sigma$, we
may rewrite the model prediction in the presence of ultrasonic vibrations as:

$\tau_{\text{f}}^{\text{m}}(\sigma,P)=\tilde{\tau}^{\text{m}}\exp\left(\frac{E_{\text{A}}}{k_{\text{B}}T_{\text{eff}}(P)}-\frac{\delta\sigma}{n\rho_{0}k_{\text{B}}T_{\text{eff}}(P)}\right)\,.$
(9)

Identification of Eq. (7) and Eq. (9) for any $\sigma$ and $P$ leads to:

$\frac{E_{\text{A}}}{k_{\text{B}}T}=\frac{\tilde{\sigma}}{\sigma_{0}(0)}.$
(10)

With the experimentally measured values $\tilde{\sigma}=24.9$ Pa and
$\sigma_{0}(0)=1.2$ Pa, Eq. (10) yields $E_{\text{A}}\simeq
20.8\,k_{\text{B}}T$. This estimate of the interaction energy nicely falls
into the range of values previously published for carbon black, typically
10–$30\,k_{\text{B}}T$, depending on the solvent and on the presence of
dispersant Trappe  _et al._ (2001); Prasad  _et al._ (2003); Trappe  _et al._
(2007). Together with the results shown in Fig. 4, this strongly supports a
mean-field interpretation of ultrasound-assisted fluidization in the framework
of effective temperatures.

###  A.7 Influence of the acoustic frequency

Figure 10: (a) Elastic modulus $G^{\prime}$ as a function of time $t$ for a 10
% vol. calcite gel. Ultrasonic vibrations are applied at $t=0$ for a duration
$T_{\text{US}}$ with a frequency of 45 kHz (brown curve, $T_{\text{US}}=30$ s,
$P=113$ kPa) and 500 kHz (red curve, $T_{\text{US}}=20$ s, $P=115$ kPa) at
about the same acoustic pressure $P$. The elastic modulus is normalized by its
value $G^{\prime}_{0}$ measured just before ultrasonic vibrations are turned
on and time is normalized by $T_{\text{US}}$. (b) Relative amplitude of the
softening effect $\Delta G^{\prime}/G^{\prime}_{0}$ as a function of $P$ at 45
kHz (brown bullets) and 500 kHz (red triangles). The upper plate is covered
with sandpaper. Experiments at 45 kHz were repeated three times for $P\geq
112$ kPa in order to check for reproducibility. Figure 11: Shear rate
responses $\dot{\gamma}(t)$ measured in a 3 % vol. carbon black gel under a
constant shear stress $\sigma=22$ Pa. The two lower curves (in green) are
obtained in the absence of ultrasonic vibrations, respectively before and
after the two upper curves, in order to illustrate reproducibility. The two
blue curves are measured under ultrasound with frequency 500 kHz and pressure
amplitude $P=55$ kPa (middle curve) and $P=108$ kPa (top curve). In order to
avoid damaging the transducer, ultrasonic vibrations are turned on for only
200 s and 120 s respectively. The upper plate is sandblasted and the gap size
is 0.5 mm.

Figure 10 shows that for the same pressure amplitude $P$, the effects of
ultrasonic vibrations on the elastic modulus of the calcite gel almost
superimpose at 45 kHz and at 500 kHz [Fig. 10(a)]. In particular, the relative
softening amplitudes $\Delta G^{\prime}/G^{\prime}_{0}$ follow the same
evolution and even coincide to within experimental dispersion for $P>70$ kPa
[Fig. 10(b)]. Therefore, a ten-fold increase in the acoustic frequency does
not affect the softening effect.

The same conclusion is reached for ultrasound-assisted fluidization. Figure 11
shows delayed yielding experiments on the carbon black gel under ultrasonic
vibrations with frequency 500 kHz. As already observed in the main text in
Fig. 4 for an ultrasonic frequency of 45 kHz, ultrasonic vibrations strongly
accelerate delayed yielding and stress-induced fluidization. Technical
limitations of the 500-kHz transducer, which cannot be used continuously for
more than one minute at the highest intensities, do not allow for a
quantitative comparison. Still, the effects of ultrasonic vibrations on the
fluidization time are very similar for both frequencies.

###  A.8 Influence of the geometry gap

Figure 12: Relative amplitude of the softening effect $\Delta
G^{\prime}/G^{\prime}_{0}$ as a function of $P$ in a 3 % vol. carbon black gel
for different gap sizes as indicated in the legend. The upper plate is
sandblasted and the frequency of ultrasonic vibrations is 500 kHz.

All data presented in the main text were obtained with the same gap size of 1
mm separating the ultrasonic transducer and the upper plate attached to the
rheometer. Figure 12 explores the softening effect discussed in Fig. 1 as a
function of the gap size in the carbon black gel under ultrasonic vibrations
at 500 kHz. While gaps of widths 0.5 and 1 mm yield very similar values of
$\Delta G^{\prime}/G^{\prime}_{0}$, the softening is about two times weaker in
gaps of widths 1.5, 2 and 3 mm. Such a difference could be due to variations
in the structure of the ultrasonic field within the parallel-plate device,
depending on reflections on the upper plates and on interference conditions at
the transducer surface. Indeed, the wavelength at 500 kHz in carbon black is
about 3 mm, which is comparable to the larger gap sizes. A more thorough
investigation of the effects of the geometry is left for future work. Overall,
however, ultrasound-induced softening is robustly observed for all gaps under
study.

###  A.9 Influence of a temperature ramp

Figure 13: (a) Temperature $T$ imposed to a 3 % vol. carbon black gel as a
function of time $t$ for two different ramps: starting from $T=25^{\circ}$C,
the temperature is increased linearly by $0.5^{\circ}$C (solid line) or by
$2^{\circ}$C (dotted line) within 30 s. $T$ is then decreased back to
$25^{\circ}$C over 180 s. (b) Elastic modulus $G^{\prime}$ normalized by its
value $G^{\prime}_{0}$ measured at the start of the temperature increase and
plotted as a function of time during the ramps shown in (a) using
corresponding lines and colors. Experiments conducted in a parallel-plate
geometry of gap 1 mm with a bottom plate equipped with a Peltier element and a
sandblasted upper plate.

At the highest achievable acoustic powers, the ultrasonic transducers used in
this study dissipate a significant amount of heat. Although the elastic
modulus of carbon black gels at rest is known to increase with temperature Won
_et al._ (2005), we need to check that a fast, transient heating of the gel
cannot account for the softening observed under ultrasonic vibrations. To this
aim, the carbon black gel was submitted to a temperature ramp in a parallel-
plate geometry thanks to the standard bottom plate of the rheometer equipped
with a Peltier element. The parameters of the two temperature ramps shown in
Fig. 13(a) are chosen so that they mimic the temperature increases induced by
ultrasonic vibrations during 30 s, respectively at high intensities (1∘C min-1
i.e. $P\simeq 100$ kPa) and very high intensities (4∘C min-1 i.e. $P\simeq
150$ kPa). For the slowest ramp, the elastic modulus decreases by less than 1
% [see light blue solid curve in Fig. 13(b)]. For the fastest ramp, the
evolution of $G^{\prime}/G^{\prime}_{0}$ is more complex, most probably due to
the dilation of the bottom plate, but the effect on the elastic modulus is at
most 4 %, well below the softening amplitudes of 40%–90 % reported under
ultrasonic vibrations at the highest acoustic intensities. This allows us to
rule out any significant effect of the heating due to dissipation within the
ultrasonic transducers on ultrasound-induced softening.

###  A.10 Control experiments

Figure 14: (a) Viscosity $\eta$ of a Newtonian oil (light mineral oil, Sigma-
Aldrich) and (b) elastic modulus $G^{\prime}$ of a 2 % wt. carbopol microgel
as a function of time $t$. Ultrasonic vibrations with frequency 45 kHz and
pressure amplitude $P=110$ kPa are turned on at time $t=0$ and switched off at
$t=20$ s. Experiments conducted in a cone-and-plate geometry of angle 1∘,
diameter 25 mm and truncation of 47 $\mu$m. In (a), the viscosity is measured
under an applied shear rate of 100 s-1. In (b), the elastic modulus is
measured under an oscillatory strain of amplitude 1 % and frequency 1 Hz.

In order to check that ultrasonic vibrations do not interfere with rheological
measurements, we performed control experiments in a Newtonian fluid (a light
mineral oil) and in a simple yield stress fluid (a 2 % wt. carbopol microgel
prepared as described in Ref. Lidon  _et al._ (2017)). Figure 14(a) shows that
viscosity measurements in the Newtonian fluid are unaffected by the
application of large-amplitude ultrasonic vibrations at 45 kHz. Similarly, as
seen in Fig. 14(b), the elastic modulus $G^{\prime}$ of the microgel does not
change by more than 2 % and does not show any systematic trend when ultrasonic
vibrations are applied with the same frequency and pressure amplitude. This is
expected as the carbopol microgel is constituted of a dense assembly of soft,
swollen polymer particles: contrary to the fragile structure of the colloidal
gels investigated in the main text, the microstructure of this soft jammed
material appears to be insensitive to ultrasound. This confirms that neither
the torque transducer nor the optical strain gauge of our stress-imposed
rheometer are perturbed by ultrasonic vibrations applied at the bottom plate
of the geometry.

## References

  * Larson (1999) R. G. Larson,  _The Structure and Rheology of Complex Fluids_ (Oxford University Press, 1999).
  * Mewis and Wagner (2012) J. Mewis and N. J. Wagner,  _Colloidal Suspension Rheology_ (Cambridge University Press, 2012).
  * Bonn  _et al._ (2017) D. Bonn, M. M. Denn, L. Berthier, T. Divoux, and S. Manneville, Yield stress materials in soft condensed matter, Reviews of Modern Physics 89, 035005 (2017).
  * Cipelletti  _et al._ (2000) L. Cipelletti, S. Manley, R. C. Ball, and D. A. Weitz, Universal aging features in the restructuring of fractal colloidal gels, Physical Review Letters 84, 2275 (2000).
  * Romer  _et al._ (2000) S. Romer, F. Scheffold, and P. Schurtenberger, Sol-gel transition of concentrated colloidal suspensions, Physical Review Letters 85, 4980 (2000).
  * Trappe  _et al._ (2001) V. Trappe, V. Prasad, L. Cipelletti, P. N. Segre, and D. A. Weitz, Jamming phase diagram for attractive particles, Nature 411, 772 (2001).
  * Cardinaux  _et al._ (2007) F. Cardinaux, T. Gibaud, A. Stradner, and P. Schurtenberger, Interplay between spinodal decomposition and glass formation in proteins exhibiting short-range attractions, Physical Review Letters 99, 118301 (2007).
  * Zaccarelli (2007) E. Zaccarelli, Colloidal gels: equilibrium and non-equilibrium routes, Journal of Physics: Condensed Matter 19, 323101 (2007).
  * Lu  _et al._ (2008) P. Lu, E. Zaccarelli, F. Ciulla, A. B. Schofield, F. Sciortino, and D. A. Weitz, Gelation of particles with short-range attraction, Nature 453, 499 (2008).
  * Israelachvili (2011) J. N. Israelachvili,  _Intermolecular and Surface Forces_ (Academic Press, 2011).
  * Hao (2002) T. Hao, Electrorheological suspensions, Advances in Colloid and Interface Science 97, 1 (2002).
  * Kogan and Solomon (2009) M. Kogan and M. J. Solomon, Electric-field-induced yielding of colloidal gels in microfluidic capillaries, Langmuir 26, 1207 (2009).
  * Swan  _et al._ (2012) J. W. Swan, P. A. Vasquez, P. A. Whitson, E. M. Fincke, K. Wakata, S. H. Magnus, F. De Winne, M. R. Barratt, J. H. Agui, R. D. Green, N. R. Hall, D. Y. Bohman, C. T. Bunnell, A. P. Gast, and E. M. Furst, Multi-scale kinetics of a field-directed colloidal phase transition, Proceedings of the National Academy of Sciences 109, 16023 (2012).
  * Helal  _et al._ (2016a) A. Helal, B. Qian, G. H. McKinley, and A. Hosoi, Yield hardening of electrorheological fluids in channel flow, Physical Review Applied 5, 064011 (2016a).
  * Coulter  _et al._ (1993) J. P. Coulter, K. D. Weiss, and J. D. Carlson, Engineering applications of electrorheological materials, Journal of Intelligent Material Systems and Structures 4, 248 (1993).
  * Tao (2001) R. Tao, Super-strong magnetorheological fluids, Journal of Physics: Condensed Matter 13, R979 (2001).
  * Wereley (2013) N. Wereley,  _Magnetorheology: Advances and Applications_ (Royal Society of Chemistry, 2013).
  * Buscall  _et al._ (1988) R. Buscall, P. D. Mills, J. W. Goodwin, and D. Lawson, Scaling behaviour of the rheology of aggregate networks formed from colloidal particles, Journal of the Chemical Society, Faraday Transactions 1 84, 4249 (1988).
  * Chen and Russel (1991) M. Chen and W. Russel, Characteristics of flocculated silica dispersions, Journal of Colloid and Interface Science 141, 564 (1991).
  * Potanin and Russel (1996) A. Potanin and W. Russel, Fractal model of consolidation of weakly aggregated colloidal dispersions, Physical Review E 53, 3702 (1996).
  * Shih  _et al._ (1999) W. Y. Shih, W.-H. Shih, and I. A. Aksay, Elastic and yield behavior of strongly flocculated colloids, Journal of the American Ceramic Society 82, 616 (1999).
  * Wessel and Ball (1992) R. Wessel and R. Ball, Fractal aggregates and gels in shear flow, Physical Review A 46, R3008 (1992).
  * Vermant  _et al._ (1998) J. Vermant, L. Walker, P. Moldenaers, and J. Mewis, Orthogonal versus parallel superposition measurements, Journal of Non-Newtonian Fluid Mechanics 79, 173 (1998).
  * Dhont and Wagner (2001) J. K. Dhont and N. J. Wagner, Superposition rheology, Physical Review E 63, 021406 (2001).
  * Colombo  _et al._ (2017) G. Colombo, S. Kim, T. Schweizer, B. Schroyen, C. Clasen, J. Mewis, and J. Vermant, Superposition rheology and anisotropy in rheological properties of sheared colloidal gels, Journal of Rheology 61, 1035 (2017).
  * Sung  _et al._ (2018) S. H. Sung, S. Kim, J. Hendricks, C. Clasen, and K. H. Ahn, Orthogonal superposition rheometry of colloidal gels: time-shear rate superposition, Soft matter 14, 8651 (2018).
  * Poon and Haw (1997) W. C. Poon and M. Haw, Mesoscopic structure formation in colloidal aggregation and gelation, Advances in Colloid and Interface Science 73, 71 (1997).
  * Vermant and Solomon (2005) J. Vermant and M. J. Solomon, Flow-induced structure in colloidal suspensions, Journal of Physics: Condensed Matter 17, R187 (2005).
  * Bonn and Denn (2009) D. Bonn and M. M. Denn, Yield stress fluids slowly yield to analysis, Science 324, 1401 (2009).
  * Koumakis  _et al._ (2015) N. Koumakis, E. Moghimi, R. Besseling, W. C. Poon, J. F. Brady, and G. Petekidis, Tuning colloidal gels by shear, Soft Matter 11, 4640 (2015).
  * Hsiao  _et al._ (2012) L. C. Hsiao, R. S. Newman, S. C. Glotzer, and M. J. Solomon, Role of isostaticity and load-bearing microstructure in the elasticity of yielded colloidal gels, Proceedings of the National Academy of Sciences 109, 16029 (2012).
  * Masschaele  _et al._ (2011) K. Masschaele, J. Fransaer, and J. Vermant, Flow-induced structure in colloidal gels: direct visualization of model 2d suspensions, Soft Matter 7, 7717 (2011).
  * Verduin  _et al._ (1996) H. Verduin, B. De Gans, and J. Dhont, Shear induced structural changes in a gel-forming suspension studied by light scattering and rheology, Langmuir 12, 2947 (1996).
  * Varadan and Solomon (2001) P. Varadan and M. J. Solomon, Shear-induced microstructural evolution of a thermoreversible colloidal gel, Langmuir 17, 2918 (2001).
  * Mohraz and Solomon (2005) A. Mohraz and M. J. Solomon, Orientation and rupture of fractal colloidal gels during start-up of steady shear flow, Journal of Rheology 49, 657 (2005).
  * Pignon  _et al._ (1997) F. Pignon, A. Magnin, and J.-M. Piau, Butterfly light scattering pattern and rheology of a sheared thixotropic clay gel, Physical Review Letters 79, 4689 (1997).
  * Eberle and Porcar (2012) A. P. Eberle and L. Porcar, Flow-SANS and rheo-SANS applied to soft matter, Current Opinion in Colloid and Interface Science 17, 33 (2012).
  * Eberle  _et al._ (2014) A. P. Eberle, N. Martys, L. Porcar, S. R. Kline, W. L. George, J. M. Kim, P. D. Butler, and N. J. Wagner, Shear viscosity and structural scalings in model adhesive hard-sphere gels, Physical Review E 89, 050302 (2014).
  * Hipp  _et al._ (2019) J. B. Hipp, J. J. Richards, and N. J. Wagner, Structure-property relationships of sheared carbon black suspensions determined by simultaneous rheological and neutron scattering measurements, Journal of Rheology 63, 423 (2019).
  * Kim  _et al._ (2014) M. J. Kim, A. P. Eberle, A. Kate Gurnon, L. Porcar, and N. J. Wagner, The microstructure and rheology of a model, thixotropic nanoparticle gel under steady shear and large amplitude oscillatory shear (LAOS), Journal of Rheology 58, 1301 (2014).
  * Zia and Brady (2010) R. N. Zia and J. F. Brady, Single-particle motion in colloids: force-induced diffusion, Journal of Fluid Mechanics 658, 188 (2010).
  * Swan and Brady (2010) J. W. Swan and J. F. Brady, Particle motion between parallel walls: hydrodynamics and simulation, Physics of Fluids 22, 103301 (2010).
  * Seto  _et al._ (2012) R. Seto, R. Botet, G. K. Auernhammer, and H. Briesen, Restructuring of colloidal aggregates in shear flow, The European Physical Journal E 35, 128 (2012).
  * Landrum  _et al._ (2016) B. J. Landrum, W. B. Russel, and R. N. Zia, Delayed yield in colloidal gels: creep, flow, and re-entrant solid regimes, Journal of Rheology 60, 783 (2016).
  * Moghimi  _et al._ (2017) E. Moghimi, A. R. Jacob, N. Koumakis, and G. Petekidis, Colloidal gels tuned by oscillatory shear, Soft Matter 13, 2371 (2017).
  * Varga  _et al._ (2019) Z. Varga, V. Grenard, S. Pecorario, N. Taberlet, V. Dolique, S. Manneville, T. Divoux, G. H. McKinley, and J. W. Swan, Hydrodynamics control shear-induced pattern formation in attractive suspensions, Proceedings of the National Academy of Sciences 116, 12193 (2019).
  * Bergna and Roberts (2005) H. E. Bergna and W. O. Roberts,  _Colloidal Silica: Fundamentals and Applications_ (CRC Press, 2005).
  * Mezzenga  _et al._ (2005) R. Mezzenga, P. Schurtenberger, A. Burbidge, and M. Michel, Understanding foods as soft materials, Nature Materials 4, 729 (2005).
  * Guvendiren  _et al._ (2012) M. Guvendiren, H. D. Lu, and J. A. Burdick, Shear-thinning hydrogels for biomedical applications, Soft Matter 8, 260 (2012).
  * Kyllönen  _et al._ (2005) H. Kyllönen, P. Pirkonen, and M. Nyström, Membrane filtration enhanced by ultrasound: a review, Desalination 181, 319 (2005).
  * Hengl  _et al._ (2014) N. Hengl, Y. Jin, F. Pignon, S. Baup, R. Mollard, N. Gondrexon, A. Magnin, L. Michot, and E. Paineau, A new way to apply ultrasound in cross-flow ultrafiltration: application to colloidal suspensions, Ultrasonics Sonochemistry 21, 1018 (2014).
  * Knorr  _et al._ (2004) D. Knorr, M. Zenker, V. Heinz, and D.-U. Lee, Applications and potential of ultrasonics in food processing, Trends in Food Science and Technology 15, 261 (2004).
  * Awad  _et al._ (2012) T. Awad, H. Moharram, O. Shaltout, D. Asker, and M. Youssef, Applications of ultrasound in analysis, processing and quality control of food: a review, Food Research International 48, 410 (2012).
  * Chandrapala  _et al._ (2012) J. Chandrapala, C. Oliver, S. Kentish, and M. Ashokkumar, Ultrasonics in food processing, Ultrasonics Sonochemistry 19, 975 (2012).
  * Mitragotri (2005) S. Mitragotri, Healing sound: the use of ultrasound in drug delivery and other therapeutic applications, Nature Reviews Drug Discovery 4, 255 (2005).
  * Huebsch  _et al._ (2014) N. Huebsch, C. J. Kearney, X. Zhao, J. Kim, C. A. Cezar, Z. Suo, and D. J. Mooney, Ultrasound-triggered disruption and self-healing of reversibly cross-linked hydrogels for drug delivery and enhanced chemotherapy, Proceedings of the National Academy of Sciences 111, 9762 (2014).
  * Cravotto and Cintas (2009) G. Cravotto and P. Cintas, Molecular self-assembly and patterning induced by sound waves. the case of gelation, Chemical Society Reviews 38, 2684 (2009).
  * Bardelang (2009) D. Bardelang, Ultrasound induced gelation: a paradigm shift, Soft Matter 5, 1969 (2009).
  * Suslick and Price (1999) K. S. Suslick and G. J. Price, Applications of ultrasound to materials chemistry, Annual Review of Materials Science 29, 295 (1999).
  * Hamilton and Blackstock (1998) M. F. Hamilton and D. T. Blackstock,  _Nonlinear Acoustics_ (Academic Press, 1998).
  * Dalecki (2004) D. Dalecki, Mechanical bioeffects of ultrasound, Annual Review of Biomedical Engineering 6, 229 (2004).
  * Gibaud  _et al._ (2010) T. Gibaud, D. Frelat, and S. Manneville, Heterogeneous yielding dynamics in a colloidal gel, Soft Matter 6, 3482 (2010).
  * Jin  _et al._ (2014) Y. Jin, N. Hengl, S. Baup, F. Pignon, N. Gondrexon, A. Magnin, M. Sztucki, T. Narayanan, L. Michot, and B. Cabane, Effects of ultrasound on colloidal organization at nanometer length scale during cross-flow ultrafiltration probed by in-situ SAXS, Journal of Membrane Science 453, 624 (2014).
  * Narayanan  _et al._ (2018) T. Narayanan, M. Sztucki, P. Van Vaerenbergh, J. Léonardon, J. Gorini, L. Claustre, F. Sever, J. Morse, and P. Boesecke, A multipurpose instrument for time-resolved ultra-small-angle and coherent x-ray scattering, Journal of Applied Crystallography 51, 1511 (2018).
  * Schmidt (1991) P. W. Schmidt, Small-angle scattering studies of disordered, porous and fractal systems, Journal of Applied Crystallography 24, 414 (1991).
  * Divoux  _et al._ (2013) T. Divoux, V. Grenard, and S. Manneville, Rheological hysteresis in soft glassy materials, Physical Review Letters 110, 018304 (2013).
  * Dijksman  _et al._ (2011) J. Dijksman, G. Wortel, L. van Dellen, O. Dauchot, and M. van Hecke, Jamming, yielding, and rheology of weakly vibrated granular media, Physical Review Letters 107, 108303 (2011).
  * Hanotin  _et al._ (2012) C. Hanotin, S. Kiesgen de Richter, P. Marchal, L. Michot, and C. Baravian, Vibration-induced liquefaction of granular suspensions, Physical Review Letters 108, 198301 (2012).
  * Hanotin  _et al._ (2015) C. Hanotin, S. Kiesgen de Richter, L. Michot, and P. Marchal, Viscoelasticity of vibrated granular suspensions, Journal of Rheology 59, 253 (2015).
  * Wortel  _et al._ (2016) G. Wortel, O. Dauchot, and M. van Hecke, Criticality in vibrated frictional flows at a finite strain rate, Physical Review Letters 117, 198002 (2016).
  * Le Goff  _et al._ (2019) M. Le Goff, E. Bertin, and K. Martens, Criticality at a finite strain rate in fluidized soft glassy materials, Physical Review Letters 123, 108003 (2019).
  * Mewis and Wagner (2009) J. Mewis and N. J. Wagner, Thixotropy, Advances in Colloid and Interface Science 147, 214 (2009).
  * Radhakrishnan  _et al._ (2017) R. Radhakrishnan, T. Divoux, S. Manneville, and S. M. Fielding, Understanding rheological hysteresis in soft glassy materials, Soft Matter 13, 1834 (2017).
  * Sprakel  _et al._ (2011) J. Sprakel, S. B. Lindström, T. E. Kodger, and D. A. Weitz, Stress enhancement in the delayed yielding of colloidal gels, Physical Review Letters 106, 248303 (2011).
  * Ovarlez  _et al._ (2013) G. Ovarlez, L. Tocquer, F. Bertrand, and P. Coussot, Rheopexy and tunable yield stress of carbon black suspensions, Soft Matter 9, 5540 (2013).
  * Grenard  _et al._ (2014) V. Grenard, T. Divoux, N. Taberlet, and S. Manneville, Timescales in creep and yielding of attractive gels, Soft Matter 10, 1555 (2014).
  * Lindström  _et al._ (2012) S. B. Lindström, T. E. Kodger, J. Sprakel, and D. A. Weitz, Structures, stresses, and fluctuations in the delayed failure of colloidal gels, Soft Matter 8, 3657 (2012).
  * Jaeger  _et al._ (1996) H. M. Jaeger, S. R. Nagel, and R. P. Behringer, Granular solids, liquids, and gases, Reviews of Modern Physics 68, 1259 (1996).
  * D’Anna  _et al._ (2003) G. D’Anna, P. Mayor, A. Barrat, V. Loreto, and F. Nori, Observing Brownian motion in vibration-fluidized granular matter, Nature 424, 909 (2003).
  * Puglisi  _et al._ (2005) A. Puglisi, P. Visco, A. Barrat, E. Trizac, and F. van Wijland, Fluctuations of internal energy flow in a vibrated granular gas, Physical Review Letters 95, 110202 (2005).
  * Prasad  _et al._ (2003) V. Prasad, V. Trappe, A. D. Dinsmore, P. N. Segre, L. Cipelletti, and D. A. Weitz, Universal features of the fluid to solid transition for attractive colloidal particles, Faraday Discussions 123, 1 (2003).
  * Trappe  _et al._ (2007) V. Trappe, E. Pitard, L. Ramos, A. Robert, H. Bissig, and L. Cipelletti, Investigation of q-dependent dynamical heterogeneity in a colloidal gel by x-ray photon correlation spectroscopy, Physical Review E 76, 051404 (2007).
  * Picard  _et al._ (2005) G. Picard, A. Ajdari, F. Lequeux, and L. Bocquet, Slow flows of yield stress fluids: complex spatiotemporal behavior within a simple elastoplastic model, Physical Review E 71, 010501(R) (2005).
  * Sollich  _et al._ (1997) P. Sollich, F. Lequeux, P. Hébraud, and M. E. Cates, Rheology of soft glassy materials, Physical Review Letters 78, 2020 (1997).
  * Fielding  _et al._ (2000) S. M. Fielding, P. Sollich, and M. E. Cates, Aging and rheology in soft materials, Journal of Rheology 44, 323 (2000).
  * Falk and Langer (1998) M. L. Falk and J. S. Langer, Dynamics of viscoplastic deformation in amorphous solids, Physical Review E 57, 7192 (1998).
  * Falk and Langer (2011) M. L. Falk and J. Langer, Deformation and failure of amorphous, solidlike materials, Annual Review of Condensed Matter Physics 2, 353 (2011).
  * Lin  _et al._ (2016) N. Y. Lin, C. Ness, M. E. Cates, J. Sun, and I. Cohen, Tunable shear thickening in suspensions, Proceedings of the National Academy of Sciences 113, 10774 (2016).
  * Sehgal  _et al._ (2019) P. Sehgal, M. Ramaswamy, I. Cohen, and B. J. Kirby, Using acoustic perturbations to dynamically tune shear thickening in colloidal suspensions, Physical Review Letters 123, 128001 (2019).
  * Lastakowski  _et al._ (2015) H. Lastakowski, J.-C. Géminard, and V. Vidal, Granular friction: triggering large events with small vibrations, Scientific Reports 5, 13455 (2015).
  * Helal  _et al._ (2016b) A. Helal, T. Divoux, and G. H. McKinley, Simultaneous rheoelectric measurements of strongly conductive complex fluids, Physical Review Applied 6, 064004 (2016b).
  * Colombo  _et al._ (2013) J. Colombo, A. Widmer-Cooper, and E. D. Gado, Microscopic picture of cooperative processes in restructuring gel networks, Physical Review Letters 110, 198301 (2013).
  * Varga and Swan (2015) Z. Varga and J. W. Swan, Linear viscoelasticity of attractive colloidal dispersions, Journal of Rheology 59, 1271 (2015).
  * Bouzid  _et al._ (2018) M. Bouzid, B. Keshavarz, M. Geri, T. Divoux, E. Del Gado, and G. H. McKinley, Computing the linear viscoelastic properties of soft gels using an optimally windowed chirp protocol, Journal of Rheology 62, 1037 (2018).
  * Liberto  _et al._ (2017) T. Liberto, M. Le Merrer, C. Barentin, M. Bellotto, and J. Colombani, Elasticity and yielding of a calcite paste: scaling laws in a dense colloidal suspension, Soft Matter 13, 2014 (2017).
  * Condre  _et al._ (2007) J.-M. Condre, C. Ligoure, and L. Cipelletti, The role of solid friction in the sedimentation of strongly attractive colloidal gels, Journal of Statistical Mechanics: Theory and Experiment 2007, P02010 (2007).
  * Depasse (1997) J. Depasse, Coagulation of colloidal silica by alkaline cations: surface dehydration or interparticle bridging?, Journal of Colloid and Interface Science 194, 260 (1997).
  * Drabarek  _et al._ (2002) E. Drabarek, J. Bartlett, H. Hanley, J. Woolfrey, and C. Muzny, Effect of processing variables on the structural evolution of silica gels, International Journal of Thermophysics 23, 145 (2002).
  * Kurokawa  _et al._ (2015) A. Kurokawa, V. Vidal, K. Kurita, T. Divoux, and S. Manneville, Avalanche-like fluidization of a non-Brownian particle gel, Soft Matter 11, 9026 (2015).
  * Richards  _et al._ (2017) J. J. Richards, J. B. Hipp, J. K. Riley, N. J. Wagner, and P. D. Butler, Clustering and percolation in suspensions of carbon black, Langmuir 33, 12260 (2017).
  * Hartley and Parfitt (1985) P. Hartley and G. Parfitt, Dispersion of powders in liquids. 1. The contribution of the van der Waals force to the cohesiveness of carbon black powders, Langmuir 1, 651 (1985).
  * Gibaud  _et al._ (2016) T. Gibaud, C. Perge, S. B. Lindström, N. Taberlet, and S. Manneville, Multiple yielding processes in a colloidal gel under large amplitude oscillatory stress, Soft Matter 12, 1701 (2016).
  * Osuji  _et al._ (2008) C. O. Osuji, C. Kim, and D. A. Weitz, Shear thickening and scaling of the elastic modulus in a fractal colloidal system with attractive interactions, Physical Review E 77, 060402(R) (2008).
  * Shih  _et al._ (1990) W.-H. Shih, W. Y. Shih, S.-I. Kum, J. Liu, and I. A. Aksay, Scaling behavior of the elastic properties of colloidal gels, Physical Review A 42, 4772 (1990).
  * Kim  _et al._ (2013) J. M. Kim, J. Fang, A. P. R. Eberle, R. Castañeda Priego, and N. J. Wagner, Gel transition in adhesive hard-sphere colloidal dispersions: the role of gravitational effects, Physical Review Letters 110, 208302 (2013).
  * Beaucage (2012) G. Beaucage, Polymer science: a comprehensive reference. (Elsevier, 2012) Chap. 2.14. Combined small-angle scattering for characterization of hierarchically structured polymer systems over nano-to-micron meter: Part II Theory, pp. 399–409.
  * Beaucage (1995) G. Beaucage, Approximations leading to a unified exponential/power-law approach to small-angle scattering, Journal of Applied Crystallography 28, 717 (1995).
  * Beaucage (1996) G. Beaucage, Small-angle scattering from polymeric mass fractals of arbitrary mass-fractal dimension, Journal of Applied Crystallography 29, 134 (1996).
  * Won  _et al._ (2005) Y.-Y. Won, S. P. Meeker, V. Trappe, and D. A. Weitz, Effect of temperature on carbon-black agglomeration in hydrocarbon liquid with adsorbed dispersant, Langmuir 21, 924 (2005).
  * Lidon  _et al._ (2017) P. Lidon, L. Villa, and S. Manneville, Power-law creep and residual stresses in a carbopol microgel, Rheol. Acta 56, 307 (2017).