# Granular aqueous suspensions with controlled inter-particular friction and
adhesion

Lily Blaiset  Université Paris Cité, CNRS, Matière et Systèmes Complexes UMR
7057, F-75013, Paris, France.  Laboratoire SIMM, ESPCI Paris, PSL University,
CNRS, Sorbonne Université, 75231 Cedex 05Paris, France.  Bruno Bresson
Laboratoire SIMM, ESPCI Paris, PSL University, CNRS, Sorbonne Université,
75231 Cedex 05Paris, France.  Ludovic Olanier  Laboratoire SIMM, ESPCI Paris,
PSL University, CNRS, Sorbonne Université, 75231 Cedex 05Paris, France.
Élisabeth Guazzelli  Université Paris Cité, CNRS, Matière et Systèmes
Complexes UMR 7057, F-75013, Paris, France.  Matthieu Roché  Université Paris
Cité, CNRS, Matière et Systèmes Complexes UMR 7057, F-75013, Paris, France.
Nicolas Sanson  Laboratoire SIMM, ESPCI Paris, PSL University, CNRS, Sorbonne
Université, 75231 Cedex 05Paris, France<EMAIL_ADDRESS>
###### Abstract

We present a simple route to obtain large quantities of suspensions of non-
Brownian particles with stimuli-responsive surface properties to study the
relation between their flow and interparticle interactions. We perform an
alkaline hydrolysis reaction on poly(methyl methacrylate) (PMMA) particles to
obtain poly(sodium methacrylate) (PMAA-Na) particles. We characterize the
quasi-static macroscopic frictional response of their aqueous suspensions
using a rotating drum. The suspensions are frictionless when the particles are
dispersed in pure water. We relate this state to the presence of electrosteric
repulsion between the charged surfaces of the ionized PMAA-Na particles in
water. Then we add monovalent and multivalent ions (Na+, Ca2+, La3+) and we
observe that the suspensions become frictional whatever the valency. For
divalent and trivalent ions, the quasi-static avalanche angle $\theta_{c}$ at
large ionic strength is greater than that of frictional PMMA particles in
water, suggesting the presence of adhesion. Finally, a decrease in the pH of
the suspending solution leads to a transition between a frictionless plateau
and a frictional one. We perform Atomic Force Microscopy (AFM) to relate our
macroscopic observations to the surface features of the particles. In
particular, we show that the increase in friction in the presence of
multivalent ions or under acidic conditions is driven by a nanoscopic phase
separation and the bundling of polyelectrolyte chains at the surface of the
particle. Our results highlight the importance of surface interactions in the
rheology of granular suspensions. Our particles provide a simple, yet flexible
platform to study frictional suspension flows.

##  I Introduction

Suspensions of particles dispersed in a liquid play a significant role in
natural and industrial contexts. Examples of suspensions include sediments,
biological fluids (e.g. blood), and industrial products such as paints and
inks. Understanding their mechanical behavior is crucial for efficient
handling and reduction of environmental impact in processes such as water
treatment, mineral processing, and material synthesis.

The history of suspension rheology dates back to the early 20th century with
the seminal work of Einstein Einstein (1906, 1911) on the effective suspension
viscosity in the dilute regime. Subsequently, significant progress has been
made in the semi-dilute regime.Batchelor and Green (1972); Guazzelli and
Morris (2012) However, only recently has a description of concentrated
suspension flows emerged that is in line with experimental observations. This
conceptual advance acknowledges the growing importance of direct interparticle
contacts over hydrodynamic interactions as the volume fraction of particles
$\phi$ increases,Guazzelli and Pouliquen (2018) leading to a frictional
description of the flow of suspensions inspired by that of dry granular media,
which relates the pressure $P$ applied to the flowing suspension to the shear
stress $\tau$ through a macroscopic friction coefficient $\mu=\tau/P$.
Forterre and Pouliquen (2008) Experiments support this approach. Boyer et al.
(2011); Guy et al. (2015); Clavaud et al. (2017); Tapia et al. (2019);
Etcheverry et al. (2023) The importance of the contact contribution is
especially seen close to the jamming transition, where the viscosity diverges
at a critical volume fraction $\phi_{c}$.Lemaire et al. (2023); Gallier et al.
(2014)

In this framework, the ability to tailor the details of interparticle
interactions becomes crucial. Emerging research relies on engineering the
surface properties of particles to impart specific rheological characteristics
to the suspension. While many protocols exist for colloidal particles,Ballesta
et al. (2012); Hsu et al. (2018, 2021) efforts in the non-colloidal regime of
present interest are more recent. Control over interparticle interactions can
be achieved by adjusting the roughness and/or friction of the particles on the
microscopic scale Tapia et al. (2019) or by controlling the nature of the
interactions between the particles with adsorbed ions,Clavaud et al. (2017)
surfactants,Richards et al. (2020) or custom polymer coatings.Guy et al.
(2015); Moratille et al. (2022) However, these methods, especially the latter,
produce small amounts of particles and require complex chemical protocols that
are difficult to establish outside of specialized chemistry laboratories. To
the best of our knowledge, a simple and scalable procedure to produce large
quantities of particles with controllable surface properties once immersed in
water with the aim of investigating the frictional behavior of non-Brownian
suspensions is still missing.

In this paper, we present a study of the surface properties and interactions
of polyelectrolyte particles made of poly(sodium methacrylate) (PMAA-Na),
obtained by complete alkaline hydrolysis of commercial particles made of
poly(methyl methacrylate) (PMMA). PMAA-Na particles can be mass-produced using
this technique, and their dispersion in water is stable. We perform rotating
drum experiments to characterize the quasi-static macroscopic friction
coefficient $\mu_{c}$ of aqueous suspensions of PMAA-Na particles in the dense
regime through a measurement of the quasi-static avalanche angle $\theta_{c}$.
First, we observe that these permeable particles are frictionless in pure
water compared to bare PMMA particles. The addition of monovalent salt in
solution leads to an increase in the friction coefficient. We take advantage
of the polyelectrolytic nature of our particles to modulate their frictional
behavior by tuning the ion valency, pH, or temperature. We discuss our
macroscopic observations in light of a microscopic characterization of the
particle surface using atomic force microscopy.

##  II Material and Methods

###  II.1 Alkaline hydrolysis of PMMA particles

We start with cross-linked monodisperse (3 mol%) poly(methyl methacrylate)
particles with diameter $d_{P60}=$60\pm 2.\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$ (P60) and density
$\rho_{P60}=$1169\text{\,}\mathrm{kg}\text{.}{\mathrm{m}}^{-3}$$ (CA-60
Spheromers, Microbeads®) (see Figure 1a). These particles are synthesized in
the presence of a steric stabilizer to make them water-dispersable. We perform
alkaline hydrolysis on these particles following a protocol inspired by Thanoo
and Jayakrishnan.Thanoo and Jayakrishnan (1990) We disperse 10 g of P60
particles in 240 ml of ethylene glycol (Sigma-Aldrich) containing 25.75 g of
potassium hydroxide (KOH pellets, Sigma-Aldrich, $\geq 85$ wt%). Reactants and
solvents are used as received. The reaction medium is kept at a controlled
temperature of $160\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for 6 hours. P60
particles are then completely hydrolyzed (see Figure 1b). This reaction can be
easily scaled up to 50 g. We find that potassium hydroxide (KOH) and
methacrylate groups (-CH3COOCH3, MMA) must be in a ratio of 4 KOH to 1 MMA
group for the reaction to be optimal. These groups react through an addition-
elimination mechanism,Azevedo and Pinto (2020) resulting in hydrophilic
poly(potassium methacrylate) particles and methanol as a by-product. We purify
the modified particles with a series of 5min-centrifugation at 2880 g: once to
remove the supernatant, then twice in a saturated solution of sodium
hydrogencarbonate (NaHCO3, Sigma-Aldrich, pH = 8.3) to remove excess KOH.
Finally, the particles are washed with ultrapure water (final pH = 7,
conductivity $\sigma\leq$100\text{\,}\mathrm{\SIUnitSymbolMicro
S}\text{.}{\mathrm{cm}}^{-1}$$) using a 106-micron sieve. At the end of the
reaction, we collect hydrolyzed particles made of poly(sodium methacrylate)
(PMAA-Na) (HP60). Particles are freeze-dried overnight and then stored at room
temperature. Figure 1 shows that the profile density functions (P.D.F.)
related to the size distributions of P60 and water-immersed HP60 particles are
narrow, i.e. these particles are monodisperse. Freeze-dried HP60 particles
have a diameter $d_{HP60,dry}=$59\pm 2.\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$ and a density
$\rho_{HP60,dry}=$1373\text{\,}\mathrm{kg}\text{.}{\mathrm{m}}^{-3}$$. They
swell to an average diameter $d_{HP60,wet}=$118\pm
2.\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ when immersed in pure water, with a
density $\rho_{HP60,wet}=$1041\text{\,}\mathrm{kg}\text{.}{\mathrm{m}}^{-3}$$
and a swelling ratio of $d_{HP60,dry}/d_{HP60,wet}\approx$ 2\. Indirect
titration shows that 65 mol% of methylmethacrylate (MMA) groups are converted
into sodium methacrylate (see Figure S1 Supporting Information).

Figure 1: Chemical modification of hydrophobic PMMA particles (P60) into
hydrophilic PMAA-Na particles (HP60). Microscopic images and size
distributions of P60 (a) and HP60 (b) particles. (Scale bars 100 µm)

Young moduli values of poly(methyl methacrylate) scale around 3.5 GPa. The
order of magnitude of the Young modulus determined for HP60 particles under
water is reduced to several hundred kPa. This lower Young modulus is in
agreement with the swelling of HP60 particles in water, giving less mechanical
resistance compared to P60 particles.

###  II.2 Suspending liquids

We test the dependence of interparticle interactions in suspension on
physicochemical conditions by dispersing P60 and HP60 particles in aqueous
electrolyte solutions. We investigate the effect of ionic strength
$I=\frac{1}{2}\sum c_{i}z_{i}^{2}$, where $c_{i}$ is the concentration of ion
$i$ and $z_{i}$ its valency, and ion valency with sodium chloride (NaCl,
Sigma-Aldrich), calcium chloride (CaCl2, purchased in its hydrated form
CaCl${}_{2}\cdot$6H2O from Sigma-Aldrich), and lanthanum chloride (LaCl3,
purchased in its hydrated form LaCl${}_{3}\cdot$7H2O from Sigma-Aldrich). We
choose these salts because they are mild Lewis acids and have little effect on
the pH of the solution. We vary the electrolyte concentration between $5\cdot
10^{-5}$ and 0.5 M, with a pH = 6-7. We test the influence of acidity by
varying the pH in acidic solutions of hydrochloric acid (HCl, Sigma-Aldrich).
HP60 particle suspensions are prepared in water, and the pH is adjusted from 7
to 2. All chemicals are used as received.

###  II.3 Rotating drum

Conventional rheological methods typically analyze suspension flows under
constant volume constraints. However, these methods do not offer insight into
the macroscopic friction coefficient of the suspension, a crucial parameter
that we aim to explore in this paper. Recently, alternative approaches have
emerged to characterize dense suspensions, drawing inspiration from pressure-
based experiments conducted with dry granular flows.Boyer et al. (2011);
Clavaud et al. (2017); Etcheverry et al. (2023)

We take inspiration from Clavaud et al.Clavaud et al. (2017, 2018) to build a
rotating drum. This setup is a convenient way to obtain information on the
quasi-static macroscopic friction coefficient of an assembly of
particles.Courrech du Pont et al. (2003) Our drum has a diameter $d$ = 12 mm
and a thickness $h$ = 3 mm. The complete setup used for rotating drum
experiments is sketched in Figure 2. Particles immersed in water at different
salt concentrations or pH are introduced into the drum. After sealing, the
drum is mounted on a precision rotating stage (M-061.PD from Physik
Instrumente). Measurements are performed at room temperature, unless otherwise
specified. In this configuration, the macroscopic friction coefficient $\mu$
of the flowing layer of particles ($\sim 10d$) depends on the tangential
stress $\tau$ and the confining pressure $P$ of the configuration, such as
$\mu=\tau/P$.

Figure 2: Experimental setup used for rotating drum experiments. The
macroscopic friction coefficient $\mu$ of the suspension is defined by
$\mu=\tau/P$, with $\tau$ the tangential shear stress and $P$ the confining
pressure applied on the flowing layer of immersed particles. The quasi-static
macroscopic coefficient $\mu_{c}$ is directly related to the quasi-static
avalanche angle ($\theta_{c}$) of the suspension as
$\mu_{c}\leavevmode\nobreak\ =\leavevmode\nobreak\ \tan{\theta_{c}}$.

After verifying the limit of the quasi-static regime (see Figure S2 in
Supporting Information), a speed of
$\omega=$0.1\text{\,}\mathrm{\SIUnitSymbolDegree}\text{.}{\mathrm{s}}^{-1}$$
is chosen to perform all experiments. Before each measurement, the particles
are resuspended for 5 min at
$90\text{\,}\mathrm{\SIUnitSymbolDegree}\text{.}{\mathrm{s}}^{-1}$, and then
left to sediment for 5 min. To start the measurements, we impose a rotational
speed
$\omega=$0.1\text{\,}\mathrm{\SIUnitSymbolDegree}\text{.}{\mathrm{s}}^{-1}$$
for 1 h. We record images with a Basler camera and a macro lens at a frame
rate ranging from 0.0056 to 0.0083 images.${\mathrm{s}}^{-1}$. Image
processing is done with a custom Python code. We also built a temperature-
controlled drum to perform experiments at various temperatures ranging from 10
to 60 °C (see setup in Figure S3 in Supporting Information). Temperature is
set by a thermostated bath. We wait 20 min for the desired temperature to
stabilize before starting the measurements. We checked that we do not observe
stick-slip avalanches in our experiments. Nowak et al. (2005) By fixing the
rotational speed at
$0.1\text{\,}\mathrm{\SIUnitSymbolDegree}\text{.}{\mathrm{s}}^{-1}$, we ensure
that experiments are carried out under quasi-static conditions, in a
continuous avalanche regime (rolling regime), where $\theta_{m}=\theta_{r}$.
Therefore, we define $\theta_{c}$ as the quasi-static avalanche angle in our
measurements. The quasi-static macroscopic friction coefficient is given by
$\mu_{c}=\tan{\theta_{c}}$. Finally, we compute the particle Stokes
number,Clavaud et al. (2017)

$St=\frac{\sqrt{\rho_{p}\Delta\rho gd^{3}}}{18\eta_{f}},$ (1)

and the particle-to-fluid density ratio $r=(\rho_{p}/\rho_{f})^{1/2}$,Courrech
du Pont et al. (2003) where $\Delta\rho_{p}=\rho_{p}-\rho_{f}$, $g$ being the
gravity, $d$ the particle diameter, and $\eta_{f}$ the fluid viscosity. In
both P60 and HP60 suspensions, $r\approx 1.1$ and $St\approx 10^{-2}$, meaning
that inertial effects are negligible.

###  II.4 AFM surface scanning

We use atomic force microscopy (AFM) to characterize the surface of our
particles. Dry particles (P60 and HP60) are deposited on a glass substrate
covered with a thin layer of Cyanolit® glue. The sample is dried overnight,
then carefully rinsed with Milli-Q water to remove particles that are not
glued to the substrate. A few water drops are placed on the substrate before
putting the sample under the AFM apparatus. A Bruker AFM (Dimension Icon®)
configured in Peak Force Quantitative Nanomechanical (QNM) tapping mode is
used to image the surface at the particle apex in aqueous media. Each image
pixel contains a force curve that we can relate to topographical features. We
obtain the latter with ScanAsyst-Fluid probes (Bruker, tip radius $r\approx$
20 nm, frequency $f\leavevmode\nobreak\ \approx$ 150 kHz, spring constant
$k\approx$ $0.7\text{\,}\mathrm{N}\text{.}{\mathrm{m}}^{-1}$). We perform a
force-distance curve on a glass plate to measure the beam deflection at the
diode output in volts.

####  II.4.1 Topography.

To probe surface roughness, we change the scan size to 10x5 or 5x2.5
${\mathrm{\SIUnitSymbolMicro m}}^{2}$, while the scanning rate, the normal
force and the peak force amplitude are set at 0.3 Hz, 350 pN, and 300 nm,
respectively. The application of a low normal force is crucial to prevent the
cantilever from damaging the particle surface, and the high amplitude prevents
the cantilever from damaging or deforming the surface.

We acquire a series of images of the apex of the same particle to observe the
effects of ionic strength $I$ with NaCl and ion valency with CaCl2 and LaCl3.
We increase the salt concentration in-situ, i.e. during an experiment by
pumping a volume of suspending fluid out with a syringe and replacing it with
a few drops of the new salt solution at the target concentration. With this
protocol, the sample always remains immersed. We repeat this process three to
four times (10 to 15 min) until the sample reaches equilibrium, i.e. when the
diameter of the particles on the substrate decreases until it reaches a
constant value. We apply the same protocol for pH variation with HCl
solutions.

We process the scans with the NanoScope Analysis software. We apply a second-
order flattening to the topography image and estimate the surface roughness as
the root mean square average $R_{q}$ of vertical displacements from the mean
image data plane:

$R_{q}=\sqrt{\frac{\sum(Z_{i})^{2}}{N}},$ (2)

where $Z_{i}$ is the local displacement and N is the number of points within
the selected region of interest. The root mean square $R_{q}$ is commonly
reported to calculate surface roughness, due to its high sensitivity to tip-
sample distortion.

####  II.4.2 Young modulus measurements.

The Young modulus of HP60 particles in pure water is determined from images
with a scan size ranging between 0.25 and
$25\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$, with a normal force from 1.5
to 2.5 nN. We extract an estimate of the Young modulus using a standard
procedure of the Nanoscope Analysis software, which automatically fits the
force-distance curves of the image with a Derjaguin-Muller-Toropov (DMT)
model. We obtain an average Young modulus $E=$500\text{\,}\mathrm{kPa}$$.
However, because we focused on surface topography, the exact tip stiffness
usually measured by acquiring a thermal noise spectrum has not been accurately
measured. The procedure gives us an order of magnitude of the Young modulus of
HP60 particles.

##  III Results and discussion

###  III.1 Hydrolyzed particles are frictionless in pure water

As shown in Figure 3a-b (left), steady avalanches of P60 particles reach an
angle $\theta_{c}=$27.3(0.6)\text{\,}\mathrm{\SIUnitSymbolDegree}$$, giving a
quasi-static macroscopic friction coefficient $\mu_{c}=0.52$, which are
typical values expected for standard frictional granular materials and
suspensions.Courrech du Pont et al. (2003); Clavaud et al. (2017); Carrigy
(1970) However, this value is relatively high compared to what has been
observed with other experimental setups, such as pressure-imposed rheometers.
In this type of configuration, the quasi-static macroscopic friction
coefficient $\mu_{c}$ of a frictional suspension is generally lower, with
$\mu_{c}\approx$ 0.3-0.4. Tapia et al. (2019); Boyer et al. (2011); Etcheverry
et al. (2023) We explain this difference by the different confining pressures
applied to the particles. In pressure-imposed rheometers, particles are
subjected to pressure typically on the order of a few tens of Pascals, while
in the rotating drum configuration, the confining pressure generated by the
fluid imposed on the flowing particles can be calculated as:Perrin et al.
(2019)

$P=\phi\Delta\rho gh\cos(\theta_{c}),$ (3)

with $\phi$ the volume fraction of the suspension,
$\Delta\rho=\rho_{HP60}-\rho_{f}$ the density difference between the particles
and the suspending fluid at $25\text{\,}\mathrm{\SIUnitSymbolCelsius}$, $g$
the acceleration of gravity, $\theta_{c}$ the avalanche angle of the
suspension, and $h$ the particle layer flowing in the steady-state regime. We
estimate the thickness of the flowing particle layer of the particles to be
$h\approx 10d$, where $d$ is the particle diameter. In the case of P60
particles, the pressure acting on the flowing layer of the particles is
$P\sim$0.5\text{\,}\mathrm{Pa}$$. With HP60 particles,
$P\sim$0.3\text{\,}\mathrm{Pa}$$. This low pressure applied to the particles
implies that the avalanche angles are very sensitive to particle contacts and
interactions. For P60 particles, the relatively high value of $\mu_{c}$ may
also be due to the presence of the steric stabilizer used during the
manufacturing process of the particles which may induce some adhesion effects,
which can only be detectable under the low pressure imposed on the particles.

Figure 3: Frictional versus frictionless behavior of P60 (left) and HP60
(right) particles suspended in pure water (pH = 7). (a) Rotating drum images
(b) Average quasi-static avalanche angle $\theta_{c}$ for P60 and HP60
particles. Inset: Avalanche angle $\theta_{c}$ versus time. (c) AFM images,
roughness profile height, and corresponding Gaussian distribution. (d)
Schematics of P60 show particle-particle contact and HP60 shows lubrication at
the interface due to electrosteric repulsion in pure water.

Much lower values of the avalanche angle are obtained for hydrolyzed HP60
particles (Figure 3 a-b),
$\theta_{c}=$6.2(0.8)\text{\,}\mathrm{\SIUnitSymbolDegree}$$. This value
corresponds to a quasi-static macroscopic friction coefficient $\mu_{c}=0.11$,
typical of a frictionless suspension.Clavaud et al. (2017); Peyneau and Roux
(2008) We may then wonder if and how the hydrolysis reaction affects the
surface roughness $R_{q}$ of the particle. We characterize the surface
topography of P60 and HP60 particles with AFM scans from which we extract the
surface height distribution (Figure 3c). We see that the surface height
distribution of both types of particles is well-fitted with a Gaussian
function. The height distribution of HP60 particles (Figure 3 d) is broader
than that of P60 particles. HP60 particles have a roughness $R_{q}$ = 70 $\pm$
16 nm, larger than $R_{q}$ = 26 $\pm$ 6 nm for P60 particles. The greater
surface contrast and roughness we measure characterize the swollen PMAA-Na
network immersed in water. If the particles had no affinity for the solvent
and were in direct contact,Pohlman et al. (2006) HP60 particles should display
a larger $\theta_{c}$ than their non-hydrolyzed counterparts, in contrast to
our results showing the opposite. We think that the smaller quasi-static
friction coefficient $\mu_{c}$ of HP60 particles compared to P60 particles
results from the fact that the former are made of PMAA, a polyelectrolyte that
will induce electrosteric repulsion between the particles. Kagata and Gong
(2001); Ohsedo et al. (2004) This interaction, which is in general a
combination of an electrostatic repulsion and a steric one due to the
impossibility of the polymer networks of two particles to interpenetrate,
screens roughness effects and permits lubrication of the contact (Figure 3d).
In contrast, P60 particles show no electrosteric repulsion and are in direct
contact. The nature of this repulsion opens vast possibilities for controlling
it with various physicochemical stimuli that we explore in the rest of the
paper.

###  III.2 The ionic strength and valence of ions controls friction and
adhesion

The material our particles are made of, PMAA-Na, is a polyelectrolyte in water
that is responsive to ionic strength and ion valency.Li et al. (2022);
Drifford and Delsanti (2001) We investigate these effects using sodium
chloride (NaCl), calcium chloride (CaCl2) and lanthanum chloride (LaCl3).
Molar concentrations of monovalent, divalent, and trivalent salts are
calculated such as the ionic strength varies $I$ from 5.10-5 to 0.5 M. First,
optical microscopy indicates that the particle size decreases as the ionic
strength increases (Figure S4 in Supporting Information). The decrease in
particle size with an increase in ionic strength results from osmotic effects
between the ions confined within the particle network and those in the
surrounding bulk solution.Eichenbaum et al. (1998) Second, we observe that the
suspensions become more frictional as $I$ increases. At low ionic strength
($I\leq$ 5.10-3 M), the quasi-static avalanche angle increases with the ionic
strength, independent of both the nature of the ion and its valency (Figure
4). Hence, at these concentrations, load and decreasing solvation of the
polyelectrolyte chains drive macroscopic friction.Israelachvili (2011)
However, for $I_{c}$ $>$ 5.10-2 M, the avalanche angle $\theta_{c}$ of HP60
particles appears to saturate in the presence of NaCl while it still increases
in the presence of Ca2+ or La3+, even exceeding the value of $\theta_{c}$
obtained for a suspension of frictional P60 particles (Figure 4). The effect
is more pronounced for La3+ than for Ca2+. Thus, not only do multivalent ions
induce a larger apparent friction coefficient than monovalent ones, but
friction coefficients also increase with valency.

Figure 4: Impact of ion valency on the frictional properties of HP60
particles. Quasi-static avalanche angle $\theta_{c}$ of HP60 suspensions with
monovalent cation Na+ ($\bigcirc$), divalent cation Ca2+ ($\triangle$), and
trivalent cation La3+ ($\square$) with increasing ionic strength $I$. The grey
dashed line corresponds to the avalanche angle $\theta_{c}$ of PMMA particles
P60. Figure 5: Surface patterning resulting from phase separation on HP60
particles with increasing ion valency. (a) AFM images of HP60 particles with
CaCl2 = 0.17 M (left) and LaCl3 = 0.11 M (right). (b) Schematic of the
particle surface HP60 in the presence of multivalent cations Ca2+ or La3+.

Direct analysis of the force-displacement images obtained with the AFM
provides interesting clues on the dependence of the particle surface profile
as a function of the valency of the ions. For example, we observe
inhomogeneities on HP60 particles in the presence of multivalent cations, as
shown in Figure 5a. These nodules have a lateral size $l$ = $284\pm
114.\text{\,}\mathrm{nm}$ and $l$ = $61\pm 20.\text{\,}\mathrm{nm}$ for HP60
particles in presence of CaCl2 and LaCl3, respectively. This observation is
reminiscent of the formation of bidentate and tridentate complexes between
charged groups and cations observed on polyelectrolyte brushes in the presence
of multivalent cations.Williams (1993); Yu et al. (2018) In this case, the
formation of nodules is attributed to electrostatic bridging between
polyelectrolyte chains inside the brush, leading to brush collapse. As
illustrated in Figure 5b, polyelectrolyte chains at the periphery of the
particle can adopt a pinned micelle-like structure in the presence of divalent
or trivalent cations. The pinned micelle-like structure seems to be
accentuated in the case of trivalent cations (La3+), since the addition of
trivalent cations emphasizes the lack of polymer solvation.Williams (1993)
This is consistent with the reduced swelling behavior of HP60 particles with
La3+ (see Figure S4 in Supporting Information). By choosing water as a
suspending fluid to study HP60 particles, which is typically a good solvent
for polyelectrolyte chains in the absence of added salt, we can differentiate
between the effects of low solubility and the influence of cation valency on
surface structure. When the chains bundle, they also become neutral and lose
affinity for water. A hydrophobic effect may appear on an interparticular
scale that tends to induce adhesive forces between the particles.Israelachvili
(2011); Yu et al. (2017) These additional adhesive forces in the presence of
multivalent cations could induce greater macroscopic friction in those
suspensions compared to those of monovalent cations.

###  III.3 Effect of pH and temperature

Figure 6: Quasi-static avalanche angle $\theta_{c}$ and surface roughness
$R_{q}$ of HP60 particles showing pH dependence. Inset: microscopic and drum
images at low pH and high pH (Scale bars 50 µm). At pH $\leq$ 5, HP60
particles are in their protonated form which induces H-bonding and hydrophobic
interactions, resulting in a phase separation of the polymer brushes at the
particle surface. In acidic conditions, the avalanche angle reaches a plateau
at $\theta_{c}$ = $28.2(0.6)\text{\,}\mathrm{\SIUnitSymbolDegree}$. Figure 7:
Evolution of an HP60 particle surface in acidic conditions. Series of AFM
images of a hydrolyzed particle HP60 in water with pH adjusted at 7, 4, and 2.

We also investigate the effect of the pH of the solution on friction by adding
hydrochloric acid (HCl) to the suspending liquid. In pure water (pH = 7), we
observe that HP60 particles are frictionless, with an avalanche angle
$\theta_{c}=$6.2(0.8)\text{\,}\mathrm{\SIUnitSymbolDegree}$$ (Figure 6), in
agreement with the data in Figure 3. A decrease in pH shows a transition
between two plateaus in the swelling and frictional behavior of HP60
particles. Above pH $\approx 5.5$, i.e. in mild basic conditions, the particle
size is not affected by the pH
($d\approx$118\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) and the quasi-static
avalanche angle ($\theta_{c}$) remains at relatively low values
($\theta_{c}\leq$8\text{\,}\mathrm{\SIUnitSymbolDegree}$$). Below pH = 5, the
particles shrink by a factor close to 2, with $d=$70\pm
1.\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. At the same time, the values
$\theta_{c}$ increase sharply, reaching a plateau at
$28.2(0.1)\text{\,}\mathrm{\SIUnitSymbolDegree}$, typical of a macroscopic
frictional behavior. The reduced swelling and increased friction observed for
HP60 particles at low pH can be attributed to the protonation of the
methacrylate groups (-COO-), leading to the formation of methacrylic acid
groups (-COOH). The protonated polymer presents a more hydrophobic character
in comparison to its ionic version.Jha et al. (2014) Adjusting the pH results
in a sharper transition from frictionless to frictional behavior compared to
adding salts. The change in the acid dissociation constant of PMAA (pKa $\sim$
5.1)Merle (1987) indicates a transition from predominantly ionized groups to
predominantly non-ionized groups.

We report in Figure 7 AFM images and surface roughness values $R_{q}$
determined for an apex of a particle at pH 7 (water), 4, and 2. While in pure
water (pH = 7) poly(sodium methacrylate) (PMAA-Na) chains are extended in
water, we observe the formation of nodules of non-ionized poly(methacrylic
acid) (PMAA) chains at the particle surface under acidic conditions (pH = 2).
Here, the nodules have a lateral size of $l$ = $24\pm 6.\text{\,}\mathrm{nm}$.
Due to hydrogen bonding, poly(methacrylic acid) (PMAA) chains collapse and
undergo a conformation change, i.e. a phase separation of PMAA at the particle
surface. This is consistent with previous work on the swelling behavior of
PMAA microgel particles.Saunders et al. (1997) This phase separation may
induce adhesive forces of hydrophobic origin between particles, leading to
high macroscopic friction coefficients in suspension.Hadziioannou et al.
(1986); Israelachvili (2011)

Finally, poly(methacrylic acid) (PMAA) exhibits a lower critical solution
temperature (LCST) behavior under acidic conditions,Robin et al. (2022) i.e.
PMAA chains become insoluble and precipitate out of solution above the LCST.
We assess the effect of temperature on the frictional behavior of protonated
particles by measuring the avalanche angle $\theta_{c}$ of HP60 particles
under acidic conditions (pH = 4) at temperatures of 10, 13, 25, 50, and
$60\text{\,}\mathrm{\SIUnitSymbolCelsius}$ (Figure 8 and video in Supporting
Information). With increasing temperature, phase separation and hydrophobic
interactions due to methyl groups (-CH3) are emphasized in the rotating drum,
resulting in a hindered flow of HP60 particles under acidic conditions at high
temperature ($60\text{\,}\mathrm{\SIUnitSymbolCelsius}$) likely resulting from
strong adhesive forces between particles.

Figure 8: Effect of temperature on hydrolyzed particles (HP60) under acidic
conditions (pH = 4). From left to right: the temperature (T) varies from 10,
13, 25 to 50 °C. The avalanche angle ($\theta_{c}$) increases due to the
accentuation of phase separation and hydrophobic effects.

##  IV Concluding remarks

The major outcome of this work is the design of model non-Brownian suspensions
of spheres with stimuli-responsive surface properties that offer control of
friction and possibly adhesion between the particles. The results we obtain in
the rotating drum show that parameters such as the ionic strength, ion
valency, pH in solution, or temperature are reliable proxies for adjusting the
flow properties of suspensions of our polyelectrolyte particles.

When first comparing the P60 particles with their hydrolyzed form HP60 in pure
water, we find macroscopic friction coefficients typical of a frictional
suspension for P60 particles ($\mu_{c}$ = 0.52) in contrast to HP60 particles
which appear to be frictionless ($\mu_{c}$ = 0.11). We think that this
frictionless state stems from an electrosteric repulsion between
polyelectrolyte particles. Polyelectrolytes being known to be stimuli-
responsive, we tune the macroscopic frictional properties of HP60 particles
using physicochemical parameters such as ionic strength, ion valency, pH, and
temperature. By adding monovalent cations Na+ to the medium, osmotic effects
induce the collapse of polymer chains. The absence of electrostatic repulsion
causes direct contact between particles, increasing interparticle friction.
Using multivalent cations (Ca2+ and La3+), we observe a significant increase
in the avalanche angle $\theta_{c}$ above a critical value of the ionic
strength ($I_{c}=10^{-2}$ M). This phenomenon arises from complex formation at
the particle surface, causing phase separation between the polymer network and
the solvent, leading to adhesive forces between particles and increased
frictional properties in suspension. Similarly, protonation of HP60 particles
under acidic conditions induces phase separation and adhesive forces,
particularly at temperatures above 50 °C, where the suspension ceases to flow.

The microscopic surface features of HP60 particles immersed in saline
solutions suggest interesting directions for future studies. First, the
formation of nodules in the presence of multivalent ions and under acidic
conditions point towards the occurrence of phase separation at the surface of
the particles. We already discussed the possible existence of adhesion in our
systems because of this phase separation, suggested by the large value of the
avalanche angles with Ca2+ and La3+ and the formation of lumps at high
temperature under acidic conditions. It would be interesting to explore how
mixtures of monovalent and trivalent ions affect the interaction of particles,
following recent results on polyelectrolyte brushes Yu et al. (2018). In the
latter case, addition of minute amounts of trivalent ions to a solution of
monovalent ions leads to a transition from a frictionless to a frictional
state under load. The existence of such a transition in our systems should
lead to the observation of shear-thickening.

Figure 9: Impact of ionic strength and ion valency on the surface roughness
$R_{q}$ of HP60 particles in the presence monovalent cations Na+ ($\bigcirc$),
divalent cations Ca2+ ($\triangle$), and trivalent cations La3+ ($\square$).
The values were obtained from a series of AFM images performed on the apex of
one HP60 particle per type of salt. The dashed grey line corresponds to a
scaling law with an exponent of $\alpha\approx-0.28$ for Na$+$ and Ca2+.

Second, Figure 9 displays the evolution of the particle roughness $R_{q}$ as a
function of the ionic strength $I$ for mono- and multivalent salts. We see
that the roughness decreases above a threshold in ionic strength following a
power law with an exponent -0.28, at least in the case of NaCl and CaCl2. This
exponent is similar to the one of the power law characterizing the decrease of
the thickness of polymer brushes in the presence of salt.Pincus (1991); Ross
and Pincus (1992) The connection between the roughness of the particles and
the thickness of a polyelectrolyte brush is far from obvious, especially since
there is no reason that the surface layer of our particles be made of a brush.
The similarity in the exponent calls into question the structure of our
particles. It would be interesting to probe in more detail the nature of the
surface layer of our particles, how far it extends in the bulk, and to compare
its features to those of a brush. Also, our protocol makes it possible to
produce core-shell particles where a thin layer of PMAA-Na sits on a PMMA
core, and it would be interesting to have a look at these systems with an AFM
to probe further this resemblance with a brush.

HP60 particles open great perspectives for rheological studies of dense
granular suspensions. Preliminary experiments with a pressure-imposed
rheometer, the Capillarytron,Etcheverry et al. (2023) suggest that suspensions
of these particles remain frictionless over the whole range of shear stresses
and pressures accessible to the apparatus, suggesting that the pressure
applied to the particles never overcomes the repulsion force. Current work in
our group investigates the consequences of these preliminary results.

## Author Contributions

E. G., M. R. and N. S. acquired the funding supporting this research; L. B.,
E. G., M. R. and N. S. conceptualized research; L. B. carried out the
experiments; E. G., M. R. and N. S. supervised the work; L. B., E. G., M. R.
and N. S. analyzed the data; L. B. wrote the initial draft of the paper; L.
B., E. G., M. R. and N. S. reviewed and edited the paper.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

This work was supported by the project 80Prime CNRS “Suspensions modèles de
particules fonctionnalisées” (SUMO). We thank Yoël Forterre and Bloen Metzger
for their kind help during the preliminary measurements with their pressure-
imposed rheometer.

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