††thanks: These two authors contributed equally††thanks: These two authors
contributed equally

# Realisation of the Brazil-nut effect in charged colloids without external
driving

Marjolein N. van der Linden  Soft Condensed Matter, Debye Institute for
Nanomaterials Science, Princetonplein 1, 3584 CC, Utrecht, The Netherlands
<EMAIL_ADDRESS>Jeffrey C. Everts  Institute of Physical
Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, PL-01-224 Warsaw,
Poland  Institute of Theoretical Physics, Faculty of Physics, University of
Warsaw, Pasteura 5, 02-093 Warsaw, Poland<EMAIL_ADDRESS>René van
Roij  Institute for Theoretical Physics, Center for Extreme Matter and
Emergent Phenomena, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The
Netherlands<EMAIL_ADDRESS>Alfons van Blaaderen<EMAIL_ADDRESS>Soft
Condensed Matter, Debye Institute for Nanomaterials Science, Princetonplein 1,
3584 CC, Utrecht, The Netherlands

(August 27, 2024)

###### Abstract

Sedimentation is a ubiquitous phenomenon across many fields of science, such
as geology, astrophysics, and soft matter. Sometimes, sedimentation leads to
unusual phenomena, such as the Brazil-nut effect, where heavier (granular)
particles reside on top of lighter particles after shaking. We show
experimentally that a Brazil-nut effect can be realised in a binary colloidal
system of long-range repulsive charged particles driven purely by Brownian
motion and electrostatics without the need for activity. Using theory, we
argue that not only the mass-per-charge for the heavier particles needs to be
smaller than the mass-per-charge for the lighter particles, but that at high
overall density, the system can be trapped in a long-lived metastable state,
which prevents the occurrence of the equilibrium Brazil-nut effect. Therefore,
we envision that our work provides valuable insights into the physics of
strongly interacting systems, such as partially glassy and crystalline
structures. Finally, our theory, which quantitatively agrees with the
experimental data, predicts that the shapes of sedimentation density profiles
of multicomponent charged colloids are greatly altered when the particles are
charge regulating with more than two ion species involved. Hence, we
hypothesise that sedimentation experiments can aid in revealing the type of
ion-adsorption processes that determine the particle charge and possibly the
value of the corresponding equilibrium constants.

## Significance

Understanding the spatial ordering of dispersed particles caused by
gravitational fields is relevant for applications as diverse as the separation
of particles, water purification, the isolation of cells from blood, the
shelf-life of paints and inks, and stabilising emulsions. When external energy
is provided, unusual density distributions have been observed: heavier
particles float on top of lighter particles. We provide a lacking experimental
demonstration that this effect can also occur in Brownian colloidal systems,
without external energy input, in dispersions of charged particles in a low-
polar oil. Additionally, we demonstrate that density-profile shapes in
mixtures of charged colloids, even if out of equilibrium, can give valuable
microscopic information on the equilibrium constants determining the particle
charge.

## Introduction

The effects of gravity on the spatial distribution of matter, such as atoms,
small molecules, proteins, colloids, and granular matter, can lead to a wide
variety of phenomena relevant to science and technology. Understanding such
sedimentation processes leads to new fundamental insights into, for example,
the formation of stars (1), the structure of earth layers in geology (2),
evolutionary biology (3), archaeology (4), and climate science (5). It also
benefits applications such as the characterisation of (macro)molecules in
pharmaceutics (6), water purification (7), and revealing the equation of state
of a system (8). In some cases, sedimentation leads to counter-intuitive
effects, such as in the Brazil-nut effect, where under certain conditions,
larger particles in a granular mixture move to the top, while smaller
particles travel downwards upon shaking the system. Here, we experimentally
demonstrate that a similar effect can occur in suspensions of charged colloids
as an equilibrium phenomenon without needing external energy input.
Furthermore, we theoretically obtained new insights into the Brazil-nut effect
in the case of strongly interacting concentrated colloidal systems.

Fig. 1:  The Brazil-nut effect in granular and charged colloidal systems. (a)
In granular systems, particles can move against the gravitational force
because of the input of external energy (e.g. via shaking). (b) No external
driving is needed for charged colloids: the charged particles exhibit Brownian
motion due to collisions with the surrounding solvent molecules and ions
(inset). There are two different explanations for the occurrence of a Brazil-
nut effect in such systems (c) At low salt concentrations, the entropic cost
for counterions to establish local charge neutrality throughout the sample is
too high, which causes a violation of local charge neutrality at the bottom of
the container where the overall particle density is highest. This charge
build-up self-consistently generates a macroscopic electric field ${\bf E}$
that pushes charged particles against the gravitational force. (d) In an
effective picture of charged particles with an electric double layer (depicted
in purple), a depletion layer is formed around the larger higher-charged
particle at altitudes where there are more small than big particles. The
particle plus depletion layer (dotted circle) has an effective density that is
lower than the surrounding fluid consisting of solvent and small lower-charged
particles, causing an upward buoyant force.

The Brazil-nut effect was named after the large Brazil nuts that rise to the
top of a mixture of nuts of different sizes when it is shaken (9, 10), see
Fig. 1a. In colloidal systems, the Brazil-nut effect has been observed in
systems consisting of active particles, which require continuous external
energy input to maintain the activity (11). The colloidal phenomenon was named
after the granular Brazil-nut effect only because a similar effect occurs:
heavier particles float on top of the lighter ones. Density-functional theory
(12, 13, 14, 15) and computer simulations (12, 15, 16) predict that the same
effect can occur in charged colloidal systems, driven purely by electrostatics
and Brownian motion (Fig. 1b), without the need for activity. Surprisingly,
despite the numerous predictions in theoretical and numerical studies, this
effect has not yet been observed experimentally. We stress that although the
granular and charged colloidal Brazil-nut effects are analogous in the sense
that they both lead to unusual density sorting, the mechanisms behind the
effects are different.

In charged colloids, two theoretical approaches have predicted the Brazil-nut
effect. The first explanation uses the entropic lift effect that has been
experimentally and theoretically studied for monodisperse suspensions (17, 18,
19, 20, 21, 22, 23, 24, 25) and in mixtures of charged colloidal particles
(12, 13, 14, 15, 16). At low ion concentration, it is entropically favourable
for the system to locally violate charge neutrality in a thin layer at the
bottom and top of the container to generate a macroscopic electric field that
pushes charged colloidal particles to higher altitudes (Fig. 1c), defying
gravity. Without this electric field, the high local density of colloidal
particles at the bottom of the container would require a highly inhomogeneous
counterion concentration to maintain local charge neutrality, which is
entropically costly. The entropic cost to maintain local charge neutrality
throughout the sample decreases at high ion concentrations. Therefore, the
macroscopic electric field vanishes and the particles sediment according to a
barometric distribution.

A minimal density functional theory that captures the entropic lift effect, in
which the components of the colloid-ion mixture are treated as massive
colloidal particles and massless ions, found that the colloidal species
separated into layers, such that colloids with the same mass-per-charge
$m_{i}/Z_{i}$ were found at the same height (13). Here $m_{i}$ is the buoyant
mass, and $eZ_{i}$ is the charge of particles of species $i$ with $e$ the
elementary charge. This is equivalent to the same value of $Z_{i}L_{i}$, with
$L_{i}=m_{i}g/(k_{\mathrm{B}}T)$ the gravitational length, where
$k_{\mathrm{B}}$ is the Boltzmann constant, $T$ temperature, and $g$ the
gravitational acceleration. Particles with the lowest $m_{i}/Z_{i}$ are
furthest from the surface onto which the particles sediment. From this
ordering according to $m_{i}/Z_{i}$ it follows that the Brazil-nut effect
occurs for $Z_{\mathrm{L}}/Z_{\mathrm{S}}\gtrsim
m_{\mathrm{L}}/m_{\mathrm{S}}$, or $m_{\mathrm{L}}/Z_{\mathrm{L}}\lesssim
m_{\mathrm{S}}/Z_{\mathrm{S}}$, i.e. the mass-per-charge for the heavier
(large) colloids (L) is smaller than for the lighter (small) colloids (S).
From this condition, it is clear why it is challenging to observe the Brazil-
nut effect in charged colloids: it is non-trivial to introduce large charge
ratios between the two particle species in the system and at the same time
have low ion concentrations (i.e., large Debye screening lengths).

Later, the lift effect was confirmed in simulations that were either based on
the primitive model (18, 21), which includes colloids and ions as separate
species, or used an approach in which colloids interact through an effective
screened Coulomb potential (21), where the presence of the ions is only taken
into account via the Debye screening length. The latter observation might come
as a surprise at first because, in such an effective colloids-only
description, the ions are integrated out, and they can, therefore, not
redistribute such that a macroscopic electric field is generated. This
apparent inconsistency can be understood by comparing the approximations in
both approaches (20).

Within the alternative effective-potential approach, an intuitive explanation
for the multi-component case was proposed in Ref. (26) for the occurrence of a
Brazil-nut effect valid for any system with long-range repulsions: at
altitudes where more small low-charged particles than large high-charged
particles are present, depletion layers are formed around the large particles
because the repulsion between large and small particles is stronger than that
between the small particles (Fig. 1d). The particle with its depletion layer
consisting of solvent and ions is treated as an effective particle with lower
mass density than the surrounding effective fluid composed of small particles,
ions, and solvent. Therefore, due to buoyancy, the particles move to higher
altitudes. At high ion concentrations, the depletion layer is reduced due to
stronger electrostatic screening, which causes the Brazil nut effect to
disappear. In addition, the depletion zone around the more repulsive particles
results in an effective attraction of these particles to the hard bottom wall
of the container, creating a layer of more repulsive particles on the
container wall (27, 26).

The effective-density picture agrees with experimental findings in Ref. (28)
and, more recently, Ref. (29) on sedimentation in charged colloidal mixtures.
However, in both experimental works, no Brazil-nut effect was observed,
presumably because the system parameters were not sufficiently extreme to be
in the Brazil-nut regime. Interestingly, the description based on the
macroscopic electric field already works on the level of point particles,
excluding the possible formation of depletion layers.

In this article, we overcome the challenges of observing the colloidal Brazil-
nut effect by using an experimental system of charged colloids in a low-polar
solvent at low salt concentrations. Our findings support the prediction that
for the Brazil-nut effect to occur, the mass-per-charge for the large colloids
should be smaller than for the small colloids
($m_{\text{L}}/Z_{\text{L}}\lesssim m_{\text{S}}/Z_{\text{S}}$) (12, 13, 15,
16). Furthermore, we quantitatively explain the experimental density profiles
using a minimal dynamical density functional theory where the colloidal
particles are modelled as constant-charge spheres. Finally, we discuss how
theoretical predictions change when charge regulation –where the particle
charge is determined by an equilibrium constant– is added as an extra
ingredient to the model, and we make predictions for future experiments.

Fig. 2:  Colloidal Brazil-nut effect in a binary suspension of large (green)
and small (red) particles (system 2 in Table I of the SI; overall volume
fraction $\bar{\eta}=0.02$). Gravity ($g$) points downwards, but the direction
of sedimentation (s) is upwards, as the mass density of the PMMA particles is
lower than that of the solvent CHB. (a) $xz$ confocal image showing that the
large (green) particles stay underneath the small (red) particles. (b)–(f)
Sequence of $xy$ confocal images taken from top to bottom (spaced 12 $\mu$m
apart; first one taken 8 $\mu$m from the top). Scale bars indicate 10 $\mu$m.
(g) Schematic overview of the sample. $x$, $y$ and $z$ directions are
indicated, as well as the direction of gravity ($g$). In reality, particles
were present across the capillary’s entire width ($y$), but for clarity, the
particles are shown here only against the back wall ($xz$).

## Results

### Experimental observation of the colloidal Brazil nut effect

To arrive at the rather “extreme” conditions for the Brazil-nut effect,
particles are needed with large charge ratios compared to the ratio of their
buoyant masses, at low ion concentrations (large Debye screening lengths). The
latter is established by using a low-polar solvent (cyclohexyl bromide, CHB)
as dispersing medium, which naturally contains a low concentration of ions
(30). We prepared four binary systems of micrometre-sized and sterically
stabilised poly(methyl methacrylate) (PMMA) particles in CHB (see Materials
and Methods) with different buoyant mass ratios
$m_{\text{L}}/m_{\text{S}}=\sigma_{\text{L}}^{3}/\sigma_{\text{S}}^{3}$, where
$\sigma_{\text{L}}$ and $\sigma_{\text{S}}$ are the diameters of the large and
small particles, respectively, neglecting slight polydispersity. We also
determined the charge (number) ratios
$Z_{\text{L}}e/(Z_{\text{S}}e)=Z_{\text{L}}/Z_{\text{S}}$, with
$Z_{\text{L}}e$ and $Z_{\text{S}}e$ the respective charges of the large and
small particles. In choosing the particles for the binary mixtures, we
exploited our observation that the covalent locking of the steric stabiliser
(PHSA-PMMA) molecules to other PMMA chains forming the core of the particle
(see Materials and Methods) has a profound effect on the particle charge (31).
The phase behaviour and electrophoresis measurements indicate that locked
particles carry a higher charge than the same particles in the unlocked state.

First, we test the theoretical prediction of the occurrence of the Brazil-nut
effect, namely when $Z_{\text{L}}/Z_{\text{S}}\gtrsim
m_{\text{L}}/m_{\text{S}}$. We define the Brazil-nut effect by the condition
$h_{\mathrm{L}}>h_{\mathrm{S}}$, with $h_{i}$ the mean height of species $i$;
see Materials and Methods. For cases where the ratio of buoyant masses is too
large or when the charge ratio is too low, no Brazil-nut effect was found; see
the SI Appendix for details.

Fig. 3:  Density profiles for large (green) and small (red) particles in two
systems (corresponding to system 2 in Table I in the SI) with overall volume
fractions (a) $\bar{\eta}=0.02$ and (b) $\bar{\eta}=0.07$. The data points are
an average of 7 stacks. The arrows indicate the mean heights $h_{\text{L}}$
and $h_{\text{S}}$ of the large and small particles (see Materials and
Methods, Eq. (1)). The dotted lines are the best theoretical predictions using
equilibrium theory, whereas the full and dashed lines are based on out-of-
equilibrium theory. In (b), the difference between the full and dashed line is
that for the full lines, an external wall potential is added to mimic the
extra repulsion due to the adsorption of particles on the top wall. See Fig.
2g for an overview of the capillary, the wall onto which the particles
sediment is located at $z=0$. Insets show the same data but on a log-linear
scale. Used parameters of the theory are discussed in the main text. (c)
Graphical depiction of the $\bar{\eta}=0.02$ and (d) the $\bar{\eta}=0.07$
sample.

Next, we focus on a system where $Z_{\text{L}}/Z_{\text{S}}$ varies between
1.9 and 2.8 ($Z_{\text{S}}\approx 240$), depending on the sample, and
$m_{\mathrm{L}}/m_{\mathrm{S}}=2.0$, meaning that the condition for the
Brazil-nut effect to occur is satisfied. $Z_{\text{L}}/Z_{\text{S}}$ was
estimated by comparing one-component systems, which we expect to give a
reasonable estimate for a 1:1 mixture, based on binary cell model calculations
(see SI Appendix). Fig. 2 shows confocal images that were taken for a sample
(system 2) with overall volume fraction $\bar{\eta}=0.02$, which indeed
exhibited the Brazil-nut effect.

Fig. 2a is an $xz$ cross-section of the capillary with the scale bars
indicating 10 $\mu$m. The large green particles at the top of the image are
adsorbed to the top wall of the capillary; the bottom wall is not visible in
the image. Gravity points downwards; the particles, however, sediment upwards,
as they have a lower mass density ($d_{\text{PMMA}}=$ 1.19 g cm-3) than the
solvent ($d_{\text{CHB}}=$ 1.336 g cm-3). Because of the choice of the
coordinate system (Fig. 2g), we can still use the intuition developed for
systems with a positive buoyant mass (Fig. 1). We see that the large green
particles ($L_{\text{L}}=0.71$ $\mu$m) are on average, further from the top
wall than the small red particles ($L_{\text{S}}=1.39$ $\mu$m). Theory (12,
13) predicts that charged particles are ordered according to their mass-per-
charge $m_{i}/Z_{i}$, with particles with the lowest mass-per-charge furthest
from the wall towards which the particles sediment (here: the top wall). This
agrees with the buoyant masses and electrophoresis results (SI Appendix).

Figs. 2b–f are $xy$ confocal images taken parallel to the top wall of the
capillary, spaced 12 $\mu$m apart, starting 8 $\mu$m from the top (Fig. 2b).
Again, it is clear that the small red particles are closer to the top wall
than the large green particles. Furthermore, the small particles have a
smaller interparticle spacing than the large particles, consistent with a
higher charge on the large particles. See also Supplementary Movie 1. Note,
furthermore, the existence of depletion layers around the green particles
(Fig. 2b) that were theoretically predicted (26).

### Density profiles and comparison with theoretical model

Fig. 3 shows two experimentally determined density profiles for two samples of
the system of Fig. 2 (filled dots), at two different overall volume fractions,
$\bar{\eta}=0.02$ (Fig. 3a) and $\bar{\eta}=0.07$ (Fig. 3b), with graphical
depictions in Figs. 3c and d, respectively. The density profiles were
calculated through $\eta_{i}(z)=\pi\sigma_{i}^{3}\rho_{i}(z)/6$, where
$\eta_{i}(z)$ and $\rho_{i}(z)$ are the volume fraction and the number density
as a function of the distance from the wall $z$. Images were taken 1-2 days
after sample preparation. We excluded from the analysis the particles that
were adsorbed to the wall (Fig. 2a), either via electrostatic interactions
between the positively charged particles and negatively charged wall or via a
boundary layering effect (26).

For the $\bar{\eta}=0.02$ sample we found mean heights of $h_{\text{L}}=31$
$\mu$m and $h_{\text{S}}=20$ $\mu$m (Fig. 3a); for the $\bar{\eta}=0.07$
sample we found $h_{\text{L}}=30$ $\mu$m and $h_{\text{S}}=32$ $\mu$m (Fig.
3B). The gravitational lengths for the two species were $L_{\text{L}}=0.71$
$\mu$m and $L_{\text{S}}=1.39$ $\mu$m. For uncharged colloids in the dilute
limit, a barometric profile is expected, with $h_{i}=L_{i}$ (13); at higher
densities, hard-core interactions would yield a more extended profile, but one
would expect $h_{i}$ to be on the order of $L_{i}$ (15, 16). For our systems,
the profiles were far more extended, with $h_{i}\gg L_{i}$, as would be
expected for charged particles (15, 16).

The $\bar{\eta}=0.02$ sample clearly displays the colloidal Brazil-nut effect
with $h_{\text{L}}>h_{\text{S}}$. The $\bar{\eta}=0.07$ sample has
$h_{\text{L}}\approx h_{\text{S}}$: this sample is approximately at the
transition between Brazil-nut effect and no Brazil-nut effect. In
Supplementary Movie 2, we show a stack of confocal $xy$ images of this sample
for various values of $z$.

To describe the experimental density profiles and explain the absence of a
Brazil-nut effect for the sample with higher overall density, we employ a
dynamical density functional theory for the colloid-ion mixture, where we
include the ideal-gas entropy of colloidal particles and ions, mean-field
electrostatics, and hard-core effects of the colloidal particles (see
Materials and Methods). We assume that the ions react instantaneously to the
colloid density field. The resulting Poisson-(Boltzmann)-Nernst-Planck
equation gives for late times $t\rightarrow\infty$ (i.e. equilibrium) the same
theory as discussed in Ref. (12, 13). We chose this approach rather than an
effective potential approach as in Ref. (26) because our approach is simpler
in the numerical calculation, allows easier implementation of charge
regulation, and is known to perform better at low ion concentration (21). The
trade-off of not using the effective potential approach is that depletion
layers (see Fig. 1d dashed lines) are not taken into account, which is
essential to describe particle adsorption on external walls by disruption of
the otherwise spherical depletion layer (26). To overcome this specific
shortcoming of the point-charge approach, we added external particle-wall
potentials, $V_{i}(z)=\chi_{i}\exp(-\lambda z)$ with contact values $\chi_{i}$
($i=\mathrm{L,S}$). Note that $\lambda^{-1}$ is a length scale associated with
the size of the depletion layer, which depends on the ionic strength and
particle charges that determine the inter-particle repulsion strengths between
the same and different types of particles.

The experimental parameters with the highest uncertainty are the effective
particle charges $Z_{\mathrm{L}}^{0}$ and $Z_{\mathrm{S}}^{0}$ of the
homogeneous mixture and the reservoir Debye screening length $\kappa^{-1}$,
and we systematically tune them to get the best match with the experimental
data. Throughout, we set $\kappa^{-1}=1\ \mu\mathrm{m}$ to ensure that the
dilute barometric regime within the sample cell is attained. This value is
smaller than the experimental estimate based on conductivity measurements on
the purified solvent CHB ($\kappa^{-1}=6\ \mu\mathrm{m}$). The Debye screening
length being smaller than the experimental value can be rationalised from the
increase in ionic strength in our systems due to a slight decomposition of CHB
(see Ref. (31)). To fix the particle charges, we used the data from
electrophoresis measurements for one-component systems and estimated the
particle charges in a 1:1 mixture (see SI Appendix) using a binary cell model
(32). We fixed the ratio to be $Z_{\mathrm{L}}^{0}/Z_{\mathrm{S}}^{0}=3$. The
only remaining parameters that are changed to describe the experimental data
are then the particle charges at a fixed charge ratio. Furthermore, due to the
point-charge-like nature of the colloidal particles in the model, the charges
should be treated as effective charges that are often lower than the
experimental values (33). The electrostatic potential in our calculation
should therefore be viewed as a local Donnan potential. In a real system, the
particle charges couple to the surface potentials rather than to the Donnan
potential. A mapping between effective and experimental particle charges is
possible using, for example, cell-model approaches (see SI Appendix). Other
parameters are taken from the experiments (see Methods).

Surprisingly, the theory in equilibrium ($t\rightarrow\infty$) never
quantitatively describes the density profiles, not even when the parameters in
our theory are adjusted (Fig. 3, dotted lines); the theory predicts in
equilibrium stronger particle segregation than was experimentally observed.
However, our theory can give a quantitative agreement with the experimental
density profiles (full lines, Fig. 3), provided that the density profiles are
taken on an intermediate time $t\approx 200\tau_{\mathrm{D}}$ for the low-
density sample (Fig. 3a) and $t\approx 80\tau_{\mathrm{D}}$ for the high-
density sample (Fig. 3b). This observation suggests that equilibrium has not
been reached in our experiments. Here,
$\tau_{\mathrm{D}}=\sigma_{\mathrm{L}}^{2}/(4D)$ is the diffusion time with
$D$ the diffusivity of the colloidal particles taken equal and density
independent. For this system and our model, the equal diffusion constant
approximation is not an unreasonable assumption, see SI Appendix. Given that
hydrodynamic effects are also not explicitly accounted for we feel that trying
to (only) correct the different diffusivities of the different species,
without taking into account other effects such as concentration dependencies,
will not yield additional insights. In this work, we are not after a
quantitative mapping of the time scale.

Fig. 4: Sedimentation dynamics of a binary mixture of charged colloids We
quantify the mean height ratio $h_{\mathrm{L}}/h_{\mathrm{S}}$ and density
profiles, as function of time. (a) We consider a charge ratio that results in
a Brazil-nut effect in equilibrium ($Z_{\mathrm{L}}^{0}/Z_{\mathrm{S}}^{0}=3$,
blue) for the constant-charge case (full lines) and the charge regulation case
with only cation adsorption (dashed lines). We also consider a charge ratio
with no Brazil-nut effect ($Z_{\mathrm{L}}^{0}/Z_{\mathrm{S}}^{0}=1.5$,
purple). The insets show snapshots of the volume fraction profiles on times
indicated by the arrows. (b) Same initial bulk charge ratios
$Z_{\mathrm{L}}^{0}/Z_{\mathrm{S}}^{0}$ as in (a), but for particles where the
large particles can adsorb anions (full blue line, $\alpha_{\mathrm{L}}=0.5$)
suppressing the Brazil-nut effect, or when this only can happen for the small
particle (full purple line, $\alpha_{\mathrm{S}}=0.6$), inducing the Brazil-
nut effect. In both panels
$\bar{\eta}_{\mathrm{L}}=\bar{\eta}_{\mathrm{S}}=0.01$ and
$Z_{\mathrm{S}}^{0}=100$.

For matching the low-density experiment we used $Z_{\mathrm{L}}^{0}=300$ and
$Z_{\mathrm{S}}^{0}=100$, whereas for the high-density experiment we used
$Z_{\mathrm{L}}^{0}=150$ and $Z_{\mathrm{S}}^{0}=50$. Both pairs of effective
charges are expected to correspond well to the measured particle charges, see
SI Appendix. The lower bulk charges needed to describe the high-density
experiment can be explained by a global discharging of particles upon
increasing the overall density due to charge regulation (where the particle
charge is described in terms of an equilibrium constant) (32). For the high-
density sample, we also needed the external particle-wall potentials to
describe the experimental data because here we observed a strong particle
absorption on the top wall. We take
$\chi_{\mathrm{L}}/\chi_{\mathrm{S}}=Z_{\mathrm{L}}^{0}/Z_{\mathrm{S}}^{0}$
with $\chi_{\mathrm{L}}=50$ and $\lambda^{-1}=12.5\ \mu\mathrm{m}$. The
theoretical prediction without such an external potential is shown by the
dashed lines of Fig. 3b.

We obtain a lower bound for $\tau_{\mathrm{D}}$ using the single-particle
diffusion coefficient calculated with the Stokes-Einstein relation. We find
$\tau_{\mathrm{D}}=10$ s meaning that the density profiles in Fig. 3 are
reached within less than an hour, whereas on the experimental scale, the
profiles were reached in a day. More realistically, the single-particle
diffusion coefficient should be replaced by a collective diffusion coefficient
known to depend sensitively on particle density, the strength of electrostatic
interactions, and hydrodynamic interactions (34, 35). In particular, due to
these effects, the collective diffusion coefficient has a lower value than the
single-particle one, explaining at least qualitatively the gap between
experimental and theoretical values.

### Metastable sedimentation profiles

To understand why the high-density sample does not exhibit a Brazil-nut
effect, we study in Fig. 4a the generic sedimentation dynamics within our
theoretical model for
$Z_{\mathrm{L}}^{0}/Z_{\mathrm{S}}^{0}>m_{\mathrm{L}}/m_{\mathrm{S}}$ (blue
line), and
$Z_{\mathrm{L}}^{0}/Z_{\mathrm{S}}^{0}<m_{\mathrm{L}}/m_{\mathrm{S}}$ (purple
line). The blue line is thus a system that is expected to give an equilibrium
Brazil-nut effect, whereas the purple line does not. For both cases, we see a
clear separation of time scales. When we view the
$Z_{\mathrm{L}}^{0}/Z_{\mathrm{S}}^{0}<m_{\mathrm{L}}/m_{\mathrm{S}}$ case,
and probe the ratio of mean heights $h_{\mathrm{L}}/h_{\mathrm{S}}$ as a
function of time, we see that there is never a Brazil-nut effect:
$h_{\mathrm{S}}>h_{\mathrm{L}}$ for all times. One could hypothesise that the
absence of a Brazil-nut effect in the high-density sample is because of too-
low a ratio of the _bulk_ charges between large and small particles. However,
when we look at the density profiles at various times for the purple curve in
Fig. 4a, none match the ones from the high-density experiment. In other words,
the absence of a Brazil-nut effect for the high-density sample cannot be
explained by a bulk charge ratio that is too small.

For $Z_{\mathrm{L}}^{0}/Z_{\mathrm{S}}^{0}>m_{\mathrm{L}}/m_{\mathrm{S}}$
(blue line), there is initially no Brazil-nut effect
($h_{\mathrm{L}}<h_{\mathrm{S}}$) and the density profiles have the same shape
as the one from the high-density experiment. However, for $t\gtrsim
100\tau_{\mathrm{D}}$ a Brazil-nut effect is found
($h_{\mathrm{L}}>h_{\mathrm{S}}$) and the shape of the density profiles
changes to the one that strongly resembles the low-density experimental
measurement. Then after around $t\approx 10^{3}\tau_{\mathrm{D}}$, equilibrium
is reached with clear segregation of particles.

It is tempting to conclude that we never found the equilibrium state in
experiments, which would require longer waiting times. However, as the ionic
strength keeps increasing due to the decomposition of the solvent, the
equilibrium state at a constant Debye length of $\kappa^{-1}=1\
\mu\mathrm{m},$ is experimentally unattainable. Furthermore, the system can
freeze in a first-order phase transition at volume fractions where the
colloids start to interact strongly or get arrested in an out-of-equilibrium
glassy state. In both these limits, the collective diffusion coefficients
become very small. Therefore, the high-density sample is expected to have a
higher value of $\tau_{\text{D}}$ than the low-density sample. This gives a
possible explanation for why there is no Brazil-nut effect in the high-density
sample: after sufficiently long times, the sample is stuck in a metastable
state with $h_{\text{L}}/h_{\text{S}}<1$, whereas the dynamics of the low-
density sample is appreciably faster, such that a state of
$h_{\text{L}}/h_{\text{S}}>1$ can be reached that is unattainable for the
high-density sample, see Fig. 4a blue line. We note that the quantitative
agreement between our highly simplified model and the experimental data could
be fortuitous, for instance due to a cancellation of errors. However,
considering the distinct sedimentation profile shapes generated as a function
of time, we argue that our model captures quite reasonably the absence of a
Brazil-nut effect in the high-density sample, which is the main reason of
constructing the dynamical theory.

### Effects of local charge regulation on sedimentation profiles

We included the effects of charge regulation by adjusting the bulk charges;
however, we neglected that charges can locally change due to an increase or
decrease of the local densities compared to the bulk densities. One could
argue that the absence of the Brazil-nut effect in Fig. 3b is because of such
local discharging processes. To test this hypothesis, we extend our theory to
include charge-regulation effects on the local level. First, we assume only
charging of colloids due to cation adsorption, for which we specify the same
bulk charges as the constant-charge case by fixing the value of the
equilibrium constant appropriately (see Materials and Methods). Local
discharging of particles at altitudes where the local density is larger
changes the density profiles at low altitudes only quantitatively (see SI
Appendix), following what is known in one-component systems (19).
Qualitatively, no change was observed, as is seen from the
$h_{\mathrm{L}}/h_{\mathrm{S}}$ curve as a function of time; in the dashed
line of Fig. 4a

When a particle can also adsorb anions, the discharging tendency is more
significant, which causes more extreme local charge ratios. In our theory, we
describe anion adsorption by the parameters $\alpha_{i}$ that represent the
anion adsorption affinity for particle species $i=\mathrm{L,S}$, which is a
free parameter even when we fix the bulk charges (see Materials and Methods).
A high value of $\alpha_{i}$ means a large tendency for anions to adsorb;
therefore the local charge ratio $Z_{\mathrm{L}}(z)/Z_{\mathrm{S}}(z)$
increases for $\alpha_{\text{S}}>0$ at low $z$, whereas it decreases for
$\alpha_{\text{L}}>0$. Therefore, for all $t$ $\alpha_{\text{L}}>0$ suppresses
a Brazil-nut effect that one expects to occur based on the bulk charges (Fig.
4b, blue line), whereas at late times $\alpha_{\text{S}}>0$ can induce a
Brazil-nut effect that would be absent based on bulk charges (Fig. 4b, purple
line). Besides inducing or suppressing a Brazil-nut effect, anion adsorption
in these cases also drastically changes the shape of the density profiles, see
the time snapshots in Fig. 4b, to be compared with Fig. 4a (where
$\alpha_{\mathrm{L,S}}=0$). The change in the density profiles’ qualitative
features suggests that the colloidal particles’ charging mechanism can be
inferred from the shape of the density profiles in a sedimentation setup.
Comparing the theoretical density profiles in Fig. 4 to the experimental
density profiles (Fig. 3), we conclude that anion adsorption on the large
particle does not play a prominent role in our experiments and that local
charge regulation does not explain the absence of the Brazil-nut effect in
Fig. 3b.

## Discussion

Our combined experimental and theoretical approach demonstrates the
realisation of a colloidal Brazil-nut effect in a non-active system driven
purely by Brownian motion and electrostatics. Furthermore, the drastic change
in shape when the charge regulation involves more than one ion species
suggests that one can use sedimentation as a valuable tool to obtain
microscopic information on the charging mechanisms of colloidal particles and
possibly the value of the corresponding equilibrium constants. Finally, we
also show that the occurrence of the Brazil-nut effect depends crucially on
the overall density of the system, arguing that at higher density, slow
dynamics prevented the system from reaching an equilibrium state that displays
the Brazil-nut result. Our work, therefore, leads to new fundamental insights
in partially glassy and crystalline systems (36) and provides inspiration for
future work.

First, it would be interesting to experimentally and theoretically investigate
the sedimentation dynamics of various charge-regulating particles with
different charging mechanisms and see what kind of distinct time-resolved
density profiles one can obtain as a function of salt concentration and
overall density. Concretely, it would be interesting to consider an
experimental system in which either the large or the small particles can also
adsorb anions and compare the sedimentation profiles to a system where only
one species of ions can adsorb to the particles. However, the present work
demonstrates that the parameter space for observing the colloidal Brazil-nut
effect may be limited in systems where charge regulation is essential, as is
almost always the case in low-polar solvents. Namely, it was shown in Ref.
(32) that if the charge density difference between the two colloidal species
is too large, the charge on the higher-charged particle may induce oppositely
charged patches on the other particle species, resulting in string formation
and other phenomena that would frustrate the Brazil-nut effect (32). Adding
salt would provide better control of the screening length and make it possible
to test the impact of the screening length on the density profiles in a binary
system. Furthermore, it would be interesting to see how the density profiles’
shape depends on how the sample has been prepared (initial conditions).

Second, from the theory viewpoint, it is surprising that our relatively simple
mean-field dynamical density functional theory can quantitatively describe the
experimental density profiles. In future work, it is interesting to include
effects we have neglected here, such as hydrodynamic interactions, a
renormalisation of collective diffusion coefficients due to direct
interactions, and memory effects, such as caging. The latter is expected to
play a prominent role: from Supplementary Movies 1 and 2, the particles are
not freely diffusing. We could envisage that a more elaborate approach, for
example, by using the many-body Smoluchowski equation (35), can shed further
light on why the metastable states from the experiments are observed and
stable for such a long time and how the dynamics of a more complicated theory
compares with our findings. Here, confocal microscopy techniques can be a
valuable asset, seeing that the full dynamics of the system can be probed
using such techniques to extract the particles’ self and collective diffusion
coefficients.

## Materials and Methods

### Model system

We used poly(methyl methacrylate) spheres (PMMA; density $d_{\text{PMMA}}=$
1.19 g cm-3; dielectric constant $\epsilon_{\text{r}}=2.6$; refractive index
$n_{\text{D}}^{25}=1.492$ (37, 38), synthesised by dispersion polymerisation
and sterically stabilised by a so-called comb-graft steric stabilising layer
formed by poly(12-hydroxystearic acid) (PHSA) grafted onto a backbone of PMMA
(PHSA-$g$-PMMA) (39, 38). We used four types of particles of different average
diameters and labelled them with either the red fluorescent dye rhodamine
isothiocyanate (RITC) or the green fluorescent dye
7-nitrobenzo-2-oxa-1,3-diazol (NBD). The two smaller particles had average
diameters of 1.30 and 1.58 $\mu$m, polydispersities of 4.0% and 3.5%,
respectively, and were labelled with RITC, the two larger particles had
average diameters of 1.98 and 2.87 $\mu$m, polydispersities of 3.5% and 2.4%,
respectively, and were labelled with NBD. The average diameters and
polydispersities were determined by static light scattering (SLS) for the two
smaller particles and by scanning electron microscopy (SEM) for the two larger
particles. To determine the average diameter and size polydispersity from the
SEM images, we measured $\sim 100$ particles from each batch using the program
iTEM (Olympus Soft Imaging Solutions GmbH). We note that the SLS diameter is
generally a few per cent larger than the SEM diameter due to the swelling of
the particles as a result of solvent uptake. Some batches of particles
underwent a so-called ‘locking’ procedure (40), in which the PMMA backbone of
the steric stabiliser became covalently bonded to the particle surface; in
this article, these particles are referred to as ‘locked’. In the case of
‘unlocked’ particles, the PHSA-$g$-PMMA stabiliser is adsorbed to the particle
surface but not covalently bonded. For more details on the characterisation of
the PHSA-$g$-PMMA stabiliser and the chemistry involved in the locking step,
which involves an additional heating step, see Ref. (38).

The particles were suspended in cyclohexyl bromide (CHB; Sigma-Aldrich;
density $d_{\text{CHB}}=1.336$ g cm${}^{-3}\ $ (37, 30); dielectric constant
$\epsilon_{\text{r}}=7.92$ (41); refractive index $n_{\text{D}}^{25}=1.4935$
(41, 37)), which nearly matched the refractive index of the PMMA particles
($n_{\text{D}}^{25}=1.492$). This solvent is known to slowly and slightly
decompose in time, a process which generates H+ and Br- ions (42). To reduce
the ionic strength, we cleaned the solvent before use (43, 38) by bringing it
into contact first with activated alumina (Al2O3; 58 Å, $\sim$ 150 mesh,
Sigma-Aldrich) and then with molecular sieves (4 Å, 10–18 mesh, Acros
Organics). The conductivity of CHB after the cleaning steps was on the order
of $10$ pS cm-1 (Scientifica 627 conductivity meter).

### Electrophoresis measurements

We measured the particles’ electrophoretic mobility $\mu$ at a volume fraction
$\eta\approx 0.01$–0.02. These measurements are described in Ref. (31). We
used the theoretical work by Carrique et al. (44) (see also Ref. (45)) to
calculate the electrostatic surface potential $\psi_{0}$ and charge number $Z$
from the mobility. In this work, a Kuwabara cell model is used to calculate
$\psi_{0}$ and $Z$ from the measured mobility $\mu$ for any given screening
length $\kappa^{-1}$ and volume fraction $\eta$ by numerically solving the
full Poisson-Boltzmann equation. An estimate for $\kappa^{-1}$ was obtained
from the measured conductivity of the solvent CHB (see below). Details can be
found in Ref. (31).

### Sample preparation and confocal microscopy

We prepared suspensions with an overall volume fraction of $\bar{\eta}=0.02$
or 0.07 and containing only one type of PMMA particle; note that upon
sedimentation, the local volume fraction $\eta$ will be different from the
overall volume fraction. We mixed equal volumes of the two suspensions and
then transferred the resulting binary suspension to a borosilicate glass
capillary (inner dimensions 5 cm $\times$ 1.0 mm $\times$ 0.10 mm ($x\times
y\times z$); VitroCom) by dipping the capillary into the suspension. We also
made samples by dipping the capillary into one of the suspensions and then
into the second suspension, adding an approximately equal volume of the second
suspension; mixing of the two suspensions took place during the filling step
across a distance of a few micrometres and afterwards by particle diffusion.
The suspension occupied approximately two-thirds of the capillary; the
remaining part was left empty (containing only air). The capillary was mounted
on a microscope glass slide, and both ends of the capillary were sealed with
UV-curing optical adhesive (Norland no. 68), which also attached the capillary
to the microscope glass slide. After curing, the sample was turned upside down
(i.e. with the capillary below the microscope glass slide), which caused the
two particle species to mix by forming swirls. In contrast, they sedimented to
the opposite wall of the capillary (see, e.g. Refs. (46, 47, 48). The
capillary was left to equilibrate for 1–2 days in this horizontal position
with the $z$ axis parallel to gravity, as shown in Fig. 2G.

We used confocal microscopy (Nikon C1 or Leica SP2 confocal microscope) with a
$63\times$ NA 1.4 oil immersion objective (Leica) in fluorescence mode with
543 nm (RITC) and 488 nm (NBD) excitation and sequential scanning mode, to
obtain three-dimensional stacks of images (typical stack: 128 $\times$ 64
$\times$ 300 pixels ($x\times y\times z$); pixel size 0.20-0.24 $\mu$m; $\sim$
3-6 frames per second). Fig. 2G is a schematic overview of the capillary with
$x$, $y$ and $z$ directions and the direction of gravity ($g$) indicated.
Stacks were taken at several positions along the length ($x$ direction) of the
capillary.

### Data analysis

We obtained the positions of the particles using an algorithm as described in,
e.g. Refs. (49, 50), which is an extension to 3D of the 2D method described in
Ref. (51).

From the particle coordinates, we calculated the number density profile
$\rho_{i}(z)$ (number density as a function of the distance $z$ from the top
wall of the capillary; see Fig. 2G) for each type of particle $i$. To improve
statistics, we averaged the profiles from seven independent 3D stacks. We
excluded particles that were adsorbed to the glass wall.

To quantify the colloidal Brazil-nut effect, we can define, after Ref. (12,
13), for each density profile $\rho_{i}(z)$ for colloid $i$ a mean height

$h_{i}=\frac{\displaystyle{\int_{0}^{H}}dz\,z\rho_{i}(z)}{\displaystyle{\int_{0}^{H}}dz\,\rho_{i}(z)}\;,\;\;i=\text{L},\text{S}.$
(1)

The colloidal Brazil-nut effect is defined as $h_{\text{L}}>h_{\text{S}}$,
with $h_{\text{L}}$ and $h_{\text{S}}$ the respective mean heights of the
large and small colloids.

### Debye screening length

The conductivity of CHB after the cleaning steps was on the order of $10$ pS
$\text{cm}^{-1}$, which corresponds to an ionic strength of
$c_{\text{s}}=2.4\times 10^{-10}$ mol L-1 (the ionic strength reduces for this
case of a monovalent salt to the salt concentration $c_{\text{s}}$; note that
the total ion concentration is $2c_{\text{s}}$). We used Walden’s rule to
obtain an estimate for the ionic molar conductances for H+ and Br- in CHB,
which were needed to estimate the ionic strength from the measured
conductivity. A Debye screening length $\kappa^{-1}\approx 6$ $\mu$m follows.
This calculation is described in more detail in Ref. (31).

It is possible that $\kappa^{-1}$ was smaller in the samples for which we
observed the Brazil-nut effect. This might be due to an increase of the ionic
strength in time due to a slight decomposition of CHB.

### Electrostatic surface potential and charge number

From the measured electrophoretic mobilities, we obtained an estimate for the
electrostatic surface potential $\psi_{0}$ and charge number per particle $Z$
(see Ref. (31) for details on the procedure); the results are given in Table I
of the SI Appendix. In Ref. (31), we found that the surface potential for
locked particles was higher than for unlocked particles and that the charge
increased quadratically with the particle diameter. In Ref. (45), it was found
for locked PMMA particles (diameter $\approx$ 1 $\mu$m) in a mixture of CHB
and cis-decalin that at higher volume fractions ($\eta\gtrsim 0.04$; depending
on the system) the charge decreased significantly (for $\eta\approx 0.13$ the
charge was a factor of 1.5–2.0 lower than for $\eta\approx 0.02$). The charge
was approximately constant at low volume fractions ($\eta\lesssim 0.04$;
depending on the system).

To calculate the surface potential and the charge number from the measured
electrophoretic mobility, we needed a value for the Debye screening length; we
took $\kappa^{-1}\approx 6$ $\mu$m, which was obtained from the conductivity
of CHB directly before use, as described above. We did not measure the
electrophoretic mobility for the unlocked particles U29 and U13. As the
surface potential seemed approximately independent of the particle diameter
(31), we assumed that the surface potentials of the particles U29 and U13 were
similar to the surface potentials on the particles U20 and U16, namely $\beta
e\psi_{0}\approx 3.6$. From this, we calculated for particles U29 and U13 the
charge numbers $Z_{i}=6.7\times 10^{2}$ and $1.4\times 10^{2}$, respectively.

The charge numbers of the particles in the samples for which we observed the
Brazil-nut effect might have been different from the charge numbers obtained
from electrophoresis due to charge regulation and differences in ionic
strength, volume fraction and/or number ratio between the two species (see
main text).

### Dynamical density functional theory for constant-charge particles

We use a theoretical model based on the dynamical density functional theory to
describe the experiments. Neglecting hydrodynamic interactions and assuming
the system is close to equilibrium, we find for $i=\mathrm{L,S}$,

$\frac{\partial\rho_{i}({\bf r},t)}{\partial
t}=\nabla\cdot\left\\{D_{i}\rho_{i}({\bf
r},t)\nabla\left[\frac{\delta\beta\mathcal{F}}{\delta\rho_{i}({\bf
r},t)}+\beta V_{i}({\bf r})\right]\right\\}.$ (2)

In general, the diffusivities $D_{\mathrm{L}}$ and $D_{\mathrm{S}}$ depend on
the local densities $\rho_{\mathrm{L}}({\bf r})$ and $\rho_{\mathrm{S}}({\bf
r})$, but for simplicity, we take it here as a constant with
$D=D_{\mathrm{L}}=D_{\mathrm{S}}$. The equal-constant approximation is not a
limiting assumption for this system and model, see SI Appendix. As external
potential, we take

$\beta V_{i}(z)=\frac{z}{L_{i}}+\chi_{i}\exp(-\lambda
z),\quad(i=\mathrm{L,S})$ (3)

describing gravitational effects and that of an external wall potential due to
particle adsorption onto the wall at $z=0$ (52). The intrinsic Helmholtz free
energy functional of a binary colloidal dispersion with small (S) and large
(L) charged colloidal particles in a 1:1 electrolyte with the solvent
approximated as a continuum is

$\displaystyle\beta\mathcal{F}[\rho_{\pm}$
$\displaystyle,\rho_{\mathrm{L}},\rho_{\mathrm{S}}]=\sum_{\alpha=\pm}\int
d{\bf r}\,\rho_{\alpha}({\bf r})\\{\ln[\rho_{\alpha}({\bf
r})\Lambda_{\alpha}^{3}]-1\\}$ $\displaystyle+\sum_{i=\mathrm{L,S}}\int d{\bf
r}\,\rho_{i}({\bf r})\\{\ln[\rho_{i}({\bf r})\mathcal{V}_{i}]-1\\}$
$\displaystyle+\frac{1}{2}\int d{\bf r}\,q({\bf r})\phi({\bf r})+\int d{\bf
r}\,f_{\text{HS}}(\rho_{\mathrm{L}}({\bf r}),\rho_{\mathrm{S}}({\bf r})),$ (4)

with electrostatic potential $\phi({\bf r})/(\beta e)$ constrained by the
Poisson equation

$\nabla^{2}\phi({\bf r})=-4\pi\ell_{\mathrm{B}}q({\bf r}),$ (5)

with the total charge density is $eq({\bf r})=e[\rho_{+}({\bf
r})-\rho_{-}({\bf r})+\sum_{i}Z_{i}\rho_{i}({\bf r})]$ and Bjerrum length
$\ell_{\mathrm{B}}$. The integrations in Eq. (4) are performed in a rectoid
sample cell with height $H$ that is translationally invariant in the $xy$
plane. Finally, $\Lambda_{\pm}^{3}$ and $\mathcal{V}_{\mathrm{L,S}}$, are the
(irrelevant) thermal volumes of ions and colloidal particles, respectively,
and $f_{\mathrm{HS}}({\rho}_{\mathrm{L}},\rho_{\mathrm{S}})$ is the free
energy density of a binary hard-sphere fluid, which in Eq. (4) is taken in the
local density approximation. Such contributions have been considered in
sedimenting charged binary colloids in Ref. (14), although in a different
parameter regime than the experiments of this work.

Explicitly writing out Eq. (2) using Eqs. (3) and (4), we find

$\displaystyle\frac{1}{D}\frac{\partial\rho_{i}(z,t)}{\partial t}=$
$\displaystyle\frac{\partial^{2}\rho_{i}(z,t)}{\partial
z^{2}}+\frac{\partial}{\partial
z}\Bigg{\\{}\rho_{i}(z,t)\frac{\partial}{\partial
z}\Bigg{[}Z_{i}^{0}\phi(z,t)$
$\displaystyle\beta\mu_{i}^{\mathrm{HS}}(\rho_{\mathrm{L}}(z,t),\rho_{\mathrm{S}}(z,t))+\beta
V_{i}(z)\Bigg{]}\Bigg{\\}},$ (6)

where we defined
$\mu_{i}^{\text{HS}}(\rho_{\mathrm{L}},\rho_{\mathrm{S}})=\partial_{\rho_{i}}f(\rho_{\mathrm{L}},\rho_{\mathrm{S}})$
for $i=\mathrm{L,S}$. We will take the Boublik-Mansoori-Carnahan-Starling-
Leland equation of state (53, 54), with chemical potential,

$\displaystyle\beta\mu^{\mathrm{HS}}_{i}=$
$\displaystyle-\left(1+\frac{2\xi_{2}^{3}\sigma_{i}^{3}}{\eta^{3}}-\frac{3\xi_{2}^{2}\sigma_{i}^{2}}{\eta^{2}}\right)\ln(1-\eta)$
$\displaystyle+\frac{3\xi_{2}\sigma_{i}+3\xi_{1}\sigma_{i}^{2}+\xi_{0}\sigma_{i}^{3}}{1-\eta}+\frac{3\xi_{2}^{2}\sigma_{i}^{2}}{\eta(1-\eta)^{2}}$
$\displaystyle+\frac{3\xi_{1}\xi_{2}\sigma_{i}^{3}}{(1-\eta)^{2}}-\xi_{2}^{3}\sigma_{i}^{3}\frac{\eta^{2}-5\eta+2}{\eta^{2}(1-\eta)^{3}},$
(7)

with $\sigma_{i}$ the particle diameters and total volume fraction
$\eta=\eta_{\mathrm{L}}+\eta_{\mathrm{S}}$. Furthermore, we defined the
quantity

$\xi_{k}=\sum_{j=\mathrm{L,S}}\eta_{j}\sigma_{j}^{k-3},\quad k=1,2,3,$ (8)

with volume fraction $\eta_{i}=(\pi/6)\sigma_{i}^{3}\rho_{i}$
($i=\mathrm{L,S}$). The hard-sphere contribution is needed because, locally,
the volume fraction can be larger than 0.1 in the parameter regime we will
consider.

We assume that ions settle instantaneously for every time $t$ given the
particle densities $\rho_{\mathrm{L,S}}({\bf r},t)$, and that they are treated
grand canonically (ion particle number is not conserved) at chemical potential
$\beta\mu_{\pm}=\ln(c_{\mathrm{s}}\Lambda_{\pm}^{3})$, i.e. as if they are in
contact with an ion reservoir at salt concentration $2c_{\mathrm{s}}$. This is
reasonable because CHB acts as an ion reservoir that generates H+ and Br-
ions. Then the ions are Boltzmann distributed, $\rho_{\pm}({\bf
r})=c_{\mathrm{s}}\exp[-\phi({\bf r})]$, combined with the Poisson equation
Eq. (5), results in the modified Poisson-Boltzmann equation

$\nabla^{2}\phi({\bf r},t)=\kappa^{2}\sinh[\phi({\bf
r},t)]-4\pi\ell_{\mathrm{B}}\sum_{i=\mathrm{L,S}}Z_{i}^{0}\rho_{i}({\bf
r},t),$ (9)

with $\kappa^{-1}=(8\pi\ell_{\mathrm{B}}c_{\mathrm{s}})^{-1/2}$ the reservoir
Debye screening length. Eqs. (6) and (9) take the form of modified Poisson-
Boltzmann-Nernst-Planck equations and are solved with colloidal-particle
blocking boundary conditions and imposed global charge neutrality. For the
initial conditions, we use

$\rho_{i}({\bf
r},0)=\frac{\bar{\eta}_{i}}{(\pi/6)\sigma_{i}^{3}},\quad(i=\mathrm{L,S}).$
(10)

From local charge neutrality at $t=0$, we find the initial condition for
$\phi({\bf r},t)$ consistent with Eqs. (10),

$\phi({\bf
r},0)=\phi_{\mathrm{D}}=\mathrm{arsinh}\left\\{\frac{\sum_{i}Z_{i}^{0}\bar{\eta}_{i}/[(\pi/6)\sigma_{i}^{3}]}{2c_{s}}\right\\},$
(11)

with $\phi_{\mathrm{D}}/(\beta e)$ the Donnan potential of the homogeneous
mixture.

In the model, we used experimental input values for the system size $H=100\
\mu\mathrm{m}$, Bjerrum length $\ell_{\mathrm{B}}=7$ nm, gravitational lengths
$L_{\mathrm{L}}=0.71\ \mu\mathrm{m}$ and $L_{\mathrm{S}}=1.39\ \mu\mathrm{m}$,
particle diameters $\sigma_{\mathrm{L}}=1.98\ \mu\mathrm{m}$ and
$\sigma_{\mathrm{S}}=1.58\ \mu\mathrm{m}$, and the overall volume fractions of
the homogeneous mixture $\bar{\eta}_{\mathrm{L}}$ and
$\bar{\eta}_{\mathrm{S}}$. For the low-density experiments we used
$(\bar{\eta}_{\mathrm{L}},\bar{\eta}_{\mathrm{S}})=(0.0125,0.01)$ and for the
high-density experiment
$(\bar{\eta}_{\mathrm{L}},\bar{\eta}_{\mathrm{S}})=(0.035,0.045)$, close to
the experimental values. The values of the Debye screening length and bulk
particle charges are discussed in the main text.

### Charge regulation

We extend the theory also to include charge regulation (55, 56). Therefore, we
make the decomposition of the colloid charge density $\rho_{i}({\bf
r})eZ_{i}({\bf r})$ in adsorbed positive ions $Z_{i,+}({\bf r})$ and adsorbed
negative ions $Z_{i,-}({\bf r})$,

$Z_{i}({\bf r})=Z_{i,+}({\bf r})-Z_{i,-}({\bf r}),\quad(i=\mathrm{L,S}).$ (12)

The total local particle charge is then determined by the adsorption of
cations and anions on specific surface sites with total available sites for
cations and anions, $M_{i,\pm}$. In the point-particle limit, the additional
charge-regulation free energy that is added to Eq. (4) is

$\beta\mathcal{F}_{\mathrm{CR}}[\rho_{\mathrm{L}},\rho_{\mathrm{S}},Z_{\pm}]=\sum_{i=\mathrm{L,S}}\int
d{\bf r}\,\rho_{i}({\bf r})\beta g_{i}(Z_{i,\pm}({\bf r})),$ (13)

with “surface” free energy $g_{i}$ for finite size colloidal particles (57)
treated in the point particle limit for $i=\mathrm{L,S}$,

$\displaystyle\beta g_{i}(Z_{i,\pm}({\bf r}))$
$\displaystyle=\sum_{\alpha=\pm}M_{i,\alpha}\Bigg{\\{}\vartheta_{i,\alpha}({\bf
r})\left[\ln\vartheta_{i,\alpha}({\bf
r})+\ln\left(K_{i,\alpha}\Lambda_{\alpha}^{3}\right)\right]$
$\displaystyle+[1-\vartheta_{i,\alpha}({\bf
r})]\ln[1-\vartheta_{i,\alpha}({\bf r})]\Bigg{\\}},$ (14)

describing a binary mixture of occupied and non-occupied sites with single-ion
adsorption free energy $-k_{\mathrm{B}}T\ln(K_{i,\pm}\Lambda_{\pm}^{3})$ and
surface coverages $\vartheta_{i,\pm}$ for $i=\mathrm{L,S}$. Similar models for
charge regulation for mobile charged colloidal particles were considered in
the context of effective screening constants and their density profiles near a
charged wall (58), and the effects on the disjoining pressure between two
charged walls (33).

Using that $Z_{i,\alpha}({\bf r})=M_{i,\alpha}\vartheta_{i,\alpha}({\bf r})$,
we find from using the Euler-Lagrange equation

$\displaystyle\frac{\delta}{\delta\vartheta_{i,\alpha}({\bf r})}$
$\displaystyle\Bigg{[}\mathcal{F+F}_{\mathrm{CR}}$
$\displaystyle-\sum_{j=\mathrm{L,S}}$ $\displaystyle\int d{\bf
r}\,\rho_{j}({\bf
r})\sum_{\lambda=\pm}\mu_{\lambda}M_{j,\lambda}\vartheta_{j,\lambda}({\bf
r})\Bigg{]}=0,$ (15)

the Langmuir form which implies that ions follow equilibrium distributions for
every fixed configuration of colloidal particles at any time $t$ for
$i=\mathrm{L,S}$,

$Z_{i,\alpha}({\bf r},t)=\frac{M_{i,\alpha}}{1+K_{i,\alpha}/\rho_{\alpha}({\bf
r},t)}\approx\frac{M_{i,\alpha}\rho_{\alpha}({\bf r},t)}{K_{i,\alpha}},$ (16)

where in the last step, we made the realistic assumption that
$Z_{i,\alpha}/M_{i,\alpha}\ll 1$, or equivalently for $i=\mathrm{L,S}$,

$Z_{i}({\bf r},t)=\gamma_{i}c_{s}\Big{\\{}\exp[-\phi({\bf
r},t)]-\alpha_{i}\exp[\phi({\bf r},t)]\Big{\\}},$ (17)

with charge regulation input parameters,

$\gamma_{i}=\frac{M_{i,+}}{K_{i,+}},\quad\alpha_{i}=\frac{M_{i,-}K_{i,+}}{M_{i,+}K_{i,-}},\quad(i=\mathrm{L,S}).$
(18)

Note that $\gamma_{i}$ (units of inverse volume) determines the overall
charge, whereas $\alpha_{i}$ characterises the tendency of negative ions to
adsorb compared to the adsorption of positive ions. Also note that the
assumption $Z_{i,\alpha}\ll M_{i,\alpha}$ reduces the number of input
parameters from eight $(M_{i,+},M_{i,-},K_{i,+},K_{i,-})$ to four
$(\gamma_{i},\alpha_{i})$. Eq. (17) is directly used in Eq. (9). Furthermore,
Eq. (6) is altered by the replacement

$Z_{i}^{0}\phi(z,t)\rightarrow-
c_{s}\gamma_{i}\Big{\\{}\exp[-\phi(z,t)]+\alpha_{i}\exp[\phi(z,t)]\Big{\\}}$
(19)

This should be contrasted with the implementation of charge regulation by
Biesheuvel (19) in the context of charge-regulating one-component colloids in
a centrifugal field (23), who used the constant-charge equations and then made
the substitution $Z_{i}\phi({\bf r})\rightarrow Z_{i}({\bf r})\phi({\bf r})$
in Eq. (6), which amounts to the assumption
$\delta\mathcal{F}_{\text{CR}}/\delta\rho_{i}({\bf r})=0$ in our model.
However, we will not adopt this assumption to be consistent with our free
energy. In a few test cases of our numerical calculations, we established that
the qualitative behaviour does not change if we use the Biesheuvel procedure.

Moreover, we set

$Z_{i}({\bf r},0)=Z_{i}^{0},\quad(i=\mathrm{L,S}),$ (20)

where $Z_{\mathrm{L}}^{0}$ and $Z_{\mathrm{S}}^{0}$ are particle charges of a
homogeneous mixture with composition $\bar{\eta}_{\mathrm{L}}$ and
$\bar{\eta}_{\mathrm{S}}$. Given $Z_{i}^{0}$, and $\alpha_{i}$, also
$\gamma_{i}$ is uniquely determined,

$\gamma_{i}=\frac{Z_{i}^{0}}{c_{s}[\exp(-\phi_{\mathrm{D}})-\alpha_{i}\exp(\phi_{\mathrm{D}})]}.$
(21)

Note that equilibrium constants are always positive, hence $\gamma_{i}>0$. All
parameters are fixed by experiment, although the parameters $\kappa^{-1}$,
$Z_{\mathrm{L}}^{0}$, $Z_{\mathrm{S}}^{0}$ are not precisely known. The
quantities $\alpha_{\mathrm{L}}$ and $\alpha_{\mathrm{S}}$ are free
parameters, however, we assume that $Z_{i}^{0}>0$, such that we have the
inequality,

$0<\alpha_{i}<\exp(-2\phi_{\mathrm{D}}).$ (22)

The dynamical equations are solved using COMSOL Multiphysics.

## Author contributions

M.N.v.d.L. performed the experiments and analysed experimental data under the
supervision of A.v.B.; J.C.E. performed the theoretical calculations under the
supervision of R.v.R. All authors contributed to writing and discussing the
manuscript.

## Competing interests

The authors declare no competing interests.

## Data availability

All data and material are included in the article and/or SI Appendix.

## Acknowledgements

We want to thank Johan Stiefelhagen for synthesising the RITC-labelled PMMA
particles, his help with the locking procedure and valuable discussions on
PMMA synthesis, Gulşen Heessels-Gürboğa for synthesising the NBD-labelled PMMA
particles; Niels Boon and Robert Hołyst are thanked for interesting
discussions. J. C. E. acknowledges financial support from the Polish National
Agency for Academic Exchange (NAWA) under the Ulam programme Grant No.
PPN/ULM/2019/1/00257. M.N.v.d.L. acknowledges financial support via a
Toptalent grant from the Dutch Research Council (NWO) funded by the Dutch
Ministry of Education, Culture and Science (OCW). This work is part of the
D-ITP consortium, a program of NWO that is also funded by the Dutch Ministry
of OCW.

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