# Diffusiophoretic manipulation of particles in a drop deposited on a hydrogel

François Boulogne  Laboratoire Matière et Systèmes Complexes (MSC), UMR 7057
CNRS, Université Paris Diderot, Bâtiment Condorcet, 10 rue Alice Domon et
Léonie Duquet, Paris, France Department of Mechanical and Aerospace
Engineering, Princeton University, Princeton, NJ 08544, USA Laboratoire de
Physique des Solides, CNRS, Univ. Paris-Sud, Université Paris-Saclay, Orsay
91405, France Sangwoo Shin  Department of Mechanical and Aerospace
Engineering, Princeton University, Princeton, NJ 08544, USA Department of
Mechanical Engineering, University of Hawaii at Manoa, Honolulu, HI 96822, USA
Julien Dervaux  Laboratoire Matière et Systèmes Complexes (MSC), UMR 7057
CNRS, Université Paris Diderot, Bâtiment Condorcet, 10 rue Alice Domon et
Léonie Duquet, Paris, France Laurent Limat  Laboratoire Matière et Systèmes
Complexes (MSC), UMR 7057 CNRS, Université Paris Diderot, Bâtiment Condorcet,
10 rue Alice Domon et Léonie Duquet, Paris, France Howard A. Stone  Department
of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ
08544, USA

###### Abstract

We report an experimental study on the manipulation of colloidal particles in
a drop sitting on a hydrogel. The manipulation is achieved by
diffusiophoresis, which describes a directed motion of particles induced by
solute gradients. By letting the solute concentrations for the drop and the
hydrogel be different, we control the motion of particles in a stable
suspension, which is otherwise difficult to achieve. We show that
diffusiophoresis can cause the particles to move either toward or away from
the liquid-air interface depending on the direction of the solute gradient and
the surface charge of the particles. We measure the particle adsorption
experimentally and rationalize the results with a one-dimensional numerical
model. We show that diffusiophoretic motion is significant at the lengthscale
of a drop deposited on a hydrogel, which suggests a simple method for the
deposition of particles on hydrogels.

##  1 Introduction

The transport of colloidal particles is of crucial interest for
microfabrication, self-assembly and surface coating [1, 2, 3, 4, 5]. A broad
range of methods are available to manipulate nano- to micrometer size
particles. One of the most common mechanisms for colloidal transport relies on
solvent evaporation. Therefore, drying receives significant attention from the
evaporation of dilute colloidal droplets to the consolidation of thin films of
concentrated solutions (see [6, 7] for reviews). Nevertheless, evaporation is
particularly difficult to control both in time and space. To mitigate coating
inhomogeneities, different strategies can be employed such as the Marangoni
effect [8, 9, 10, 11], flow properties modified by solvent volatility [11] or
convection [12].

Ultimately, to have a good control over the coating process, manipulating the
motion of particles via external forces is required. Indeed, the idea to apply
forces on particles is long standing. Phoretic phenomena have been known for
decades to induce particle motion [13, 14, 15, 16] and they recently regained
interest either for particle manipulation [17, 18, 19, 20, 21] or to explain
an exclusion zone adjacent to solids [22] (see [23] for a recent review). In
particular, Derjaguin, Dukhin and Korotkova revealed the role of
diffusiophoresis in the deposition of latex particles for rubber film
formation [13]. This method has been used since the 1920’s [24]. More
specifically, diffusiophoresis relies on solute concentration gradients to
generate a force on colloidal particles, which is the sum of chemiphoresis and
electrophoresis contributions. Chemiphoresis takes its origin in the
concentration gradient of solutes that applies a non-uniform pressure on the
thin interfacial layer (e.g. Debye layer) and electrophoresis arises from a
local electric field created by the difference of diffusivities between anions
and cations. Recently, gels have been used as a steady source of solute over
long time scales to get a long lasting electrophoresis effect [25].

In previous studies, we reported observations and analyzed the absorption of a
drop [26] and the particle deposition [27, 28] on swelling hydrogels.
Particles are transported along with the solvent such that micron size
particles are deposited at the interface of the hydrogel to form a nearly
uniform coating [27]. Furthermore, we have shown that the radial profile of
the number of deposited particles can be controlled by the particle
concentration [28].

In this paper, we combine diffusiophoresis as a particle transport mechanism
and the porous properties of hydrogels as a solute source/sink. More
precisely, we investigate the effect of a difference of solute concentration
between the colloidal drop and the hydrogel. We show that a gradient of solute
concentration enables the manipulation of microspheres in the drop, either to
repel them from the gel surface or to generate adsorption depending on the
surface charge of the particles. Our aim is to show how the well-studied
diffusiophoresis mechanism can be used for the deposition of particles on
solids from dispensed drops, beyond the classic evaporation technique or the
solvent absorption by the substrate.

##  2 Experimental details

The substrate is a hydrogel of $3$ mm thickness fully swollen with an ionic
solution of sodium chloride (NaCl) at a concentration $C_{s}^{g}$. A drop
containing fluorescent microspheres in an ionic solution of concentration
$C_{s}^{d}$ of the same solute is deposited on the hydrogel. The system is
placed in a sealed box saturated in water vapor to prevent evaporation (Fig.
1). In the present conditions, since the hydrogel is initially fully swollen,
the drop is not absorbed by the gel in contrast to our previous studies [27,
28].

Figure 1: (a) Schematic of the experimental setup. $C_{s}^{g}$ denotes the
initial solute concentration in the hydrogel and $C_{s}^{d}$ denotes the
initial concentration in the drop. A sealed box with a reservoir of water
prevents the drying of the drop and the hydrogel by saturating the atmosphere.
(b) Picture of a drop deposited on a slab of hydrogel. The drop is colored red
for illustrative purpose.

###  2.1 Hydrogels

The gel is a polydimethyl acrylamide (PDMA) hydrogel [29]. To prepare these
hydrogels, we used a monomer N,N-dimethylacrylamide (DMA), a crosslinker
N,N’-methylene-bis-acrylamide (MBA) and initiators, potasium persulfate (KPS)
and N,N,N’,N’-tetramethylethylenediamine (TMED). All chemicals are purchased
from Sigma-Aldrich, USA. The composition of these hydrogels is the same as
some of those used in our previous study on the deposition by solvent
absorption [27].

In addition, $[\rm{MBA}]$ and $[\rm{mono}]$ are, respectively, molar
concentrations of the crosslinker (N,N’-methylene-bis-acrylamide) and the
monomer. The composition of the gel is set by $m_{\rm DMA}/(m_{\rm
DMA}+m_{w})=0.15$ whereas $m_{\rm DMA}$ and $m_{w}$ are the masses of DMA and
water, respectively. The molar concentration ratio of MBA and DMA is set to
$[\rm{MBA}]/[\rm{DMA}]=0.02$. The quantities of initiators KPS and TMED are
set to a molar ratio of 1% of the monomer quantity. The solution is poured in
a mold made of two glass plates ($75\times 50$ cm2, Dow Corning) separated
with a rubber spacer of $3$ mm thickness (McMaster-Carr). Hydrogels are stored
in their molds in a vapor saturated environment for two days.

After removing the mold, hydrogels are dialysed in deionized water for five
days. During this process, the water bath is changed twice and it results in a
fully swollen gel. These gels are then transferred into NaCl solution at a
concentration of $0.1$ mM or $10$ mM for at least five days. At the end of
this process, we assume that the hydrogels have a solute concentration
$C_{s}^{g}$ equal to the concentration of the NaCl solution.

Prior to each experiment, a gel is removed from the ionic bath and the surface
is gently wiped with a weighing paper (Fisherbrand) to remove the excess
solution. The gel is placed in a hermetic glass cell containing a reservoir of
water to saturate the gas phase and to prevent evaporation of the drop and the
gel. A drop of colloidal suspension is dispensed on the hydrogel with a
micropipette (Research Plus, Eppendorf).

###  2.2 Drop of colloidal suspension

Colloidal suspensions consist of hydrophilic fluorescent particles
(carboxylate-functionalized polystyrene, Lifetechnologies, Orange (540/560))
of diameter $2a=1.1$ $\mu$m and with a density 1.05 g/cm3 purchased as an
aqueous suspension containing 0.02 % thimerosal. Ionic solutions of NaCl are
prepared at concentrations of $C_{s}^{d}=0.1$ mM and $10$ mM. The suspension
is diluted with the ionic solution at a concentration $C_{p}^{0}=8\times
10^{6}$ particles/mL, which corresponds to a volume fraction of $6\times
10^{-6}$. The viscosity of the solvent is $\eta_{s}=10^{-3}$ Pa$\cdot$s. The
zeta potential of the particles is measured as $\zeta=-70$ mV using dynamic
light scattering (Malvern Zetasizer Nano). In all experiments, the drop volume
is $V_{d}=0.8$ $\mu$L, which when placed on the gel results in a drop radius
$R\approx 1.1$ mm and a drop height $h_{0}\approx 0.4$ mm.

##  3 Governing equations

In this section, we present the governing equations describing the motion of
colloidal particles by diffusiophoresis. We define successively the geometry
of the system, the diffusion equation for ions, the diffusiophoretic mobility
and the equation for the particle motion. Note that, in the present work, we
neglect diffusioosmotic flow driven by the surface charge of the hydrogel due
to the small zeta potential [30].

###  3.1 Drop shape

We define the capillary length $\ell_{c}=\sqrt{\gamma/(\rho g)}$ where
$\gamma$ is the surface tension, $\rho$ the liquid density and $g$ the
gravitational acceleration. For a drop radius $R<\ell_{c}$, to a good
approximation the drop shape is a spherical cap. Thus, at each time, the drop
profile $h(r,t)$ represented in Fig. 1(a) is described by a portion of a
sphere,

$h(r,t)=\sqrt{\left(\frac{R^{2}+h_{0}(t)^{2}}{2h_{0}(t)}\right)^{2}-r^{2}}-\frac{(R^{2}-h_{0}(t)^{2})}{2h_{0}(t)},$
(1)

with $h_{0}(t)=h(0,t)$. We assume that the drop height is much smaller than
the drop radius, $h_{0}\ll R$, which corresponds to small contact angles.
Thus, equation (1) becomes

$h(r)\simeq h_{0}\,\left(1-\frac{r^{2}}{R^{2}}\right).$ (2)

###  3.2 Ionic diffusion

As the drop and the gel have different initial NaCl concentrations, ions
diffuse through the gel-drop interface. In cylindrical coordinates, the
concentration field $c(r,z,t)$ is a solution of the time-dependent diffusion
equation

$\frac{\partial c}{\partial t}={\cal D}_{s}\left(\frac{\partial^{2}c}{\partial
r^{2}}+\frac{1}{r}\frac{\partial c}{\partial r}+\frac{\partial^{2}c}{\partial
z^{2}}\right),$ (3)

where ${\cal D}_{s}$ is the ambipolar diffusion coefficient for the solute
defined as ${\cal D}_{s}=2{\cal D}_{+}{\cal D}_{-}/({\cal D}_{+}+{\cal
D}_{-})$, where ${\cal D}_{+}$ and ${\cal D}_{-}$ are the diffusivities of the
cation and anion, respectively. For NaCl, the diffusion coefficient of cations
is ${\cal D}_{+}=1.33\times 10^{-9}$ m2/s and for anions, ${\cal
D}_{-}=2.03\times 10^{-9}$ m2/s [31]. Hence, ${\cal D}_{s}=1.6\times 10^{-9}$
m2/s. The initial conditions are $c(r,z>0,t=0)=C_{s}^{d}$ and
$c(r,z<0,t=0)=C_{s}^{g}$.

###  3.3 Diffusiophoretic mobility

The diffusiophoretic mobility ${\cal D}_{dp}$ of the particles relates the
particle velocity $\mathbf{v}_{p}$ due to diffusiophoresis and the solute
gradient as [15, 32]

$\mathbf{v}_{p}={\cal D}_{dp}\mathbf{\nabla}\ln c,$ (4)

where $c$ is the solute concentration. Under the thin Debye layer
approximation, the diffusiophoretic mobility is given by [32]

${\cal
D}_{dp}=\frac{\epsilon}{\eta_{s}}\left(\frac{k_{B}T}{Ze}\right)^{2}\left[\frac{Ze\zeta\beta}{k_{B}T}+4\ln\cosh\left(\frac{Ze\zeta}{4k_{B}T}\right)\right],$
(5)

where $\epsilon$ is the permittivity of the medium, $k_{B}$ is the Boltzmann
constant, $T$ is the temperature, $Z$ is the valence of the solute, $\zeta$ is
the zeta potential, and $\beta=({\cal D}_{+}-{\cal D}_{-})/({\cal D}_{+}+{\cal
D}_{-})=-0.21$ for NaCl.

The diffusiophoretic mobility consists of electrophoretic and chemiphoretic
terms, which are, respectively, the first and the second terms in the bracket
on the right-hand side of equation (5). Electrophoresis is driven by the
diffusion potential that is set by the difference in the diffusivities between
cation and anion whereas chemiphoresis originates from the osmotic pressure
difference within the Debye layer due to a solute gradient.

The diffusiophoretic mobility is determined from the zeta potential of the
colloidal particle, which is measured as $-70$ mV. Using equation (5), the
calculated diffusiophoretic mobility is ${\cal D}_{dp}=7.44\times 10^{-10}$
m2/s, where for the conditions of our experiments the electrophoretic
contribution is $2.95\times 10^{-10}$ m2/s and the chemiphoretic contribution
is $4.49\times 10^{-10}$ m2/s.

The particle motion is generated by the diffusiophoretic velocity given by
equation (4) and the Brownian motion associated with the particle diffusivity
${\cal D}_{p}$. The diffusion coefficient of particles is obtained from the
Stokes-Einstein relation ${\cal D}_{p}=\frac{k_{B}T}{6\pi\eta_{s}a}=4.0\times
10^{-13}$ m2/s. Therefore, the particle concentration $C_{p}(r,z,t)$ in the
drop is given by

$\frac{\partial C_{p}}{\partial
t}+\mathbf{\nabla}\cdot(\mathbf{v}_{p}C_{p})={\cal D}_{p}\nabla^{2}C_{p},$ (6)

where the initial condition is $C_{p}(r,z,t=0)=C_{p}^{0}$.

##  4 Experimental observations

To describe the different solute concentration gradients, we define the solute
concentration ratio as

$\Gamma=C_{s}^{g}/C_{s}^{d},$ (7)

where $C_{s}^{g}$ and $C_{s}^{d}$ are, respectively, the initial
concentrations in the gel and in the drop. For different solute concentration
ratios $\Gamma$, we analyze the particle motion in the drop. To achieve this
analysis, we combined two techniques. First, confocal microscopy is used to
visualize the particle positions in the cross-section of the drop. Then, we
observed the particles deposited on the gel interface with fluorescence
microscopy.

###  4.1 Particle distribution in the drop cross-section

Figure 2: Position of particles in the drop about 4 minutes after the
dispensing. The solid line is the drop interface given by equation (2). (a)
$C_{s}^{g}=0.1$ mM, $C_{s}^{d}=10$ mM ($\Gamma=0.01<1$) (b) $C_{s}^{g}=10$ mM,
$C_{s}^{d}=0.1$ mM ($\Gamma=100>1$).

To measure the particle position, we use a confocal microscope (Leica TCS SP5)
with a 10$\times$ dry objective (PL Fluotar). At this magnification, we image
a quarter of a 0.8 $\mu\ell$ drop. Images in the horizontal $(x,y)$-frame are
taken with a $z$-step of 10 $\mu$m with a pinhole size set to $70$ $\mu$m. The
image resolution is $1024\times 1024$ pixels and we used a scanning rate of
$200$ Hz. To visualize the full cross-section of the drop, we choose a
$z$-range larger than the height of the droplet, which is typically
$h_{0}\approx 0.4$ mm. Images are recorded from the fluorescence and the
transmission signals. We start the acquisition about 2 minutes after the drop
deposition and the duration of the acquisition is about 3 minutes, such that
early time dynamics is not accessible. The initial position of the scanner is
located near the surface of the gel and the scanner moves upward during the
acquisition.

Figure 3: (a-b) Confocal images captured at the level of the hydrogel
interface for (a) $\Gamma=0.01$ and (b) $\Gamma=100$. We observe the interface
reflection outside the drop. The dashed lines show the quarter of the drop
used in the analysis. (c) Particle concentration profile for three solute
concentration ratios $\Gamma$ captured typically between 2 to 5 minutes after
deposition. Green points correspond to the data presented in Fig. 2(a),
$\Gamma=0.01$, and blue points to Fig. 2(b), $\Gamma=100$. Orange points are
for $C_{s}^{g}=C_{s}^{d}=0.1$ mM, i.e. $\Gamma=1$.

From a $z$-stack, we start by detecting the gel interface and the contact line
of the drop. From a region where the gel-vapor interface is visible on
pictures, we measure the average fluorescence intensity as a function of the
$z$ position. With the interface reflectivity, the peak of intensity
corresponds to the position of the gel interface. On the corresponding image
in transmission mode, the position of the contact line is detected by a
circular Hough transform [33]. As the gel and the drop are now located, we
focus on the detection of the particles.

A particle can be seen on several images of the $z$-stack with a displacement
in the $(x,y)$ frame due to Brownian motion occurring during the acquisition.
To determine an average position, we use the Python library Trackpy [34] along
with the scientific libraries numpy, scipy, pandas [35, 36, 37, 38]. On the
fluorescence images, the particle $(x,y)$ positions are tracked in the
$z$-stack. From the resulting particle trajectories, we calculate the average
$x$, $y$ and $z$ positions for each particle. The total number of particles in
the entire image corresponds to $C_{p}V_{d}/4$ to a precision of $15$%.

Figure 4: (a) Radial profiles of the particle deposited on the hydrogel
($\Gamma=100$) for times spanning from 1 to 21 min obtained from fluorescence
microscopy. The profiles are non-dimensionalized by the total number particles
$N_{{\rm tot}}$ initially present in the drop. (b) Number of adsorbed
particles $N_{{\rm ads}}$, non-dimensionalized by the total number of
particles $N_{{\rm tot}}$. Green points are for $C_{s}^{g}=0.1$ mM and
$C_{s}^{d}=10$ mM ($\Gamma=0.01$) and blue points are for $C_{s}^{g}=10$ mM
and $C_{s}^{d}=0.1$ mM ($\Gamma=100$). The solid black line is the numerical
solution with a height $h_{0}^{(n)}$ adjusted at 1.2 mm as a fitted parameter.

In Fig. 2, we show the particle positions for two solute concentration ratios.
Since the particles are detected in one quarter of the drop, the liquid volume
in a slice ${\rm d}r$ is $h(r)\frac{\pi}{2}r{\rm d}r$. Therefore, at a given
position $z$ in the drop, fewer particles are visible as $r\rightarrow 0$ in
this 2D representation. For larger solute concentrations in the drop, i.e.
$\Gamma<1$, and for this choice of salt (NaCl), we observe that the particles
are repelled from the gel interface and they accumulate near the liquid-air
interface (Fig. 2(b)). In contrast, for $\Gamma>1$, the particles migrate
toward the hydrogel (Fig. 2(b)).

The number of particles ${\rm d}n(z,t)$ contained in a horizontal slice of
liquid of thickness ${\rm d}z$ is related to the concentration $C_{p}(z,t)$ by

${\rm d}n(z,t)=C_{p}(z,t)\,\pi\,r(z)^{2}{\rm d}z.$ (8)

In practice, we count the number of particles located in horizontal cross-
sections of constant volume to ensure that a sufficient number of particles
are present in each elementary volume. In Fig. 3(c), we plot the relative
particle concentration profile $C_{p}(z,t)/\bar{C}_{p}$, where $\bar{C}_{p}$
represents the average particle concentration. For $\Gamma=1$ (orange points),
the solute concentration is uniform everywhere, thus no diffusiophoretic
effect is expected and the experimental concentration profile is nearly flat.

For $\Gamma>1$, i.e. a higher solute concentration in the gel than the drop,
the experimental results show that the concentration near the surface of the
hydrogel is about twice the average concentration. For the opposite gradient
$\Gamma<1$, particles are significantly repelled from the gel surface with a
local concentration of about 30 % the initial concentration [39]. Note that
the direction of the particle motion is independent of the sign of the zeta
potential since chemiphoresis dominates over electrophoresis for NaCl solution
within a wide range of zeta potentials [40]. This also implies that the
particle diffusiophoretic motion can be reversed for positively-charged
particles when an electrophoresis-dominant solute is used [41]. In the next
section, we focus our attention on the particle deposition rate.

###  4.2 Dynamics of the particle deposition on the hydrogel interface

To identify the particles adsorbed on the hydrogel, we performed experiments
with an inverted fluorescent microscope (Leica DMI4000 B) with a $5\times$
objective. The microscope is equipped with a translation stage (Prior) and a
Hamamatsu camera (Digital camera ORCA-Flash4.0 C11440) $2048\times 2048$
pixels. The acquisition is automated with the software Micromanager [42].

The microscope is focused on the top surface of the hydrogel. An acquisition
consists in a sequence of 5 images captured in the fluorescence mode followed
by an image in bright field. The drop position and its radius are measured
with a circular Hough transform as before. With the library Trackpy, we
capture the particle trajectories over a set of 5 images. Among these
trajectories, we define a criterion to determine if a particle is adsorbed on
the substrate, i.e. not mobile. For each trajectory consisting of a set of
positions $(r_{i},\theta_{i})$ in the horizontal plane, we define $\bar{r}$,
the average radial position and $\textrm{std}()$, the standard deviation.
Then, we calculate the radial displacement $\delta r=\textrm{std}(r_{i})$ and
the azimuthal displacement
$\bar{r}\delta\theta=\bar{r}\,\textrm{std}(\theta_{i})$. We consider that a
particle is not mobile if its trajectory satisfies $\delta r<4$ $\mu$m and
$\bar{r}\delta\theta<4$ $\mu$m. This procedure is repeated over the sequences.
The diffusion timescale for the particles over $\delta r$ is $(\delta
r)^{2}/{\cal D}_{p}\approx 1$ s. The time step between each picture is $3$ s,
such that the duration of a sequence is 15 s, which is much larger than the
diffusion timescale to ensure a correct detection of adsorbed particles.

We represent the time evolution of the radial particle distribution in Fig.
4(a). At early times ($t\lesssim 100$ s), the radial profile is nearly flat
and as time passes, the density increases more at the center than near the
contact line. Indeed, we expect that the solute gradient is mainly vertical,
such that the main velocity component of the particles is also vertical as
represented by equation (4). Therefore, fewer particles tend to accumulate
near the drop edge and there is a decreasing concentration profile along the
radius. Interestingly, we noticed that the mobile particles in the drop have
an inward radial motion on the first minutes after the drop deposition.

Next, we plot the total number of absorbed particles as a function of time for
$\Gamma=0.01$ and $\Gamma=100$, as shown in Fig. 4(b). For $\Gamma=0.01$, we
observe a weak particle absorption of only a few percent at the end of the
experiment. Therefore, most of the particles are repelled from the gel surface
and the few adsorbed particles might be due to the irreversible adsorption
when particles move near the surface. The good adhesion of these particles on
PDMA hydrogels has been noticed in our previous studies [27, 28] where we
reported that a contact line cannot detach adsorbed particles. For
$\Gamma=100$, the number of adsorbed particles increases non-linearly in time,
with a decreasing absorption rate. In the next section, we solve numerically a
simplified model based on the governing equations presented in section 3 for
the case $\Gamma>1$ to rationalize our data.

##  5 Model

Figure 5: Normalized particle concentration as a function of the rescaled drop
height $z/h_{0}$. Solid lines are the concentration profiles obtained
numerically for an effective drop height $h_{0}^{(n)}=1.2$ mm at different
times. Points are the experimental data presented in Fig. 3(c) for
$\Gamma=100$, rescaled by the drop height $h_{0}=0.4$ mm. The acquisition of
the experimental data started 2 minutes after the drop deposition and the
duration is about 3 minutes.

The derivation of an analytical solution is particularly difficult because of
the finite size effect due to the limited drop height. We make the choice to
simplify the geometry of the system to retain the essential physical behavior.
Thus, we consider a 1D model where the drop is approximated as a thin cylinder
of height $h_{0}^{(n)}$. The governing equations (3), (4) and (6) can be
rewritten for $c(z,t)$ as

$\displaystyle\frac{\partial c}{\partial t}$ $\displaystyle={\cal
D}_{s}\frac{\partial^{2}c}{\partial z^{2}},$ (9a) $\displaystyle v_{p}$
$\displaystyle={\cal D}_{dp}\frac{\partial{\ln c}}{\partial z},$ (9b)
$\displaystyle\frac{\partial C_{p}}{\partial t}$
$\displaystyle=-\frac{\partial(C_{p}v_{p})}{\partial z}+{\cal
D}_{p}\frac{\partial^{2}C_{p}}{\partial z^{2}},$ (9c)

where $C_{p}(z,t)$ is the particle concentration in the drop. We define
diffusion timescales $\tau=h_{0}^{2}/{\cal D}$ across the drop height $h_{0}$
for the three diffusion coefficients in the problem, namely ${\cal D}_{s}$,
${\cal D}_{p}$ and ${\cal D}_{dp}$. For a drop height $h_{0}\approx 0.4$ mm,
we have $\tau_{s}\approx 100$ s, $\tau_{p}\approx 4\times 10^{5}$ s and
$\tau_{dp}\approx 200$ s. The shorter two times indicate that at the timescale
of the experiments, the finite size of the droplet matters. Hence, we next
consider boundary conditions accounting for the finite geometry.

We consider the case $\Gamma>1$ corresponding to the experimental observation
of migration of the particles toward the gel interface. Because the volume of
the drop is small compared to the volume of the gel, we assume that the
migration of solute molecules from the gel toward the drop does not
appreciably change the concentration of solute inside the gel. Furthermore, we
shall assume that colloidal particles are irreversibly adsorbed when they
reach the drop/gel interface, i.e. $n(z=0,t)=0$, such that the interface
behaves as a perfect sink [43]. In addition, neither solutes nor colloidal
particles can cross the drop/air interface and thus, for a 1D drop with height
$h_{0}^{(n)}$, the boundary conditions read:

$c(z=0,t)&=C_{s}^{g},\\\ C_{p}(z=0,t)=0,\\\ \frac{\partial c}{\partial
z}\bigg{\rvert}_{z=h_{0}^{(n)},t}=0,\\\ {\cal D}_{p}\frac{\partial
C_{p}}{\partial
z}\bigg{\rvert}_{z=h_{0}^{(n)},t}=C_{p}\left(h_{0}^{(n)},t\right)v_{p}\left(h_{0}^{(n)},t\right),$
(10)

together with the initial conditions:

$c(z,t=0)&=C_{s}^{d},\\\ C_{p}(z,t=0)=C_{p}^{0}.$ (11)

We then solved numerically equations 9(a-c), subjected to boundary and initial
conditions (10) and (11). In Fig. 4(b), since the model assumes a simpler
geometry, we fit our data using the thickness $h_{0}$ of the cylindrical drop
as a free parameter. The best fit is obtained for $h_{0}^{(n)}=1.2$ mm, which
is of the same order of magnitude as the experimental value $h_{0}\approx 0.4$
mm. Given the highly simplified geometry used in this model, this estimate is
reasonable. Similar agreement for the time-dependent dynamics can be obtained
analytically with further simplifications (see Appendix).

In Fig. 5, we show the numerical prediction of the particle concentration
profiles at different times ranging from 0 to 1400 s. The curves indicate a
depletion effect near the interface as a result of the perfect sink boundary
condition that removes the particles from the liquid phase as they
irreversibly adsorb. This depleted zone has a maximum characteristic thickness
of $0.03\,h_{0}^{(n)}$. Then, the concentration monotonically decreases along
the drop height as the particles migrate toward $z=0$ in time. The data points
represented in Fig. 5 are extracted from Fig. 3(c) and the acquisition has
been performed between $t=120$ s and $t=300$ s. The numerical model predicts
rather accurately the vertical particle distribution, as can be seen in Fig. 5
(green curve). The size of the depleted zone is clearly outside of the
experimental resolution such that it is not surprising that it has not been
observed.

Further work would be necessary to predict the radial distribution of
deposited particles as shown in Fig. 4(a). This would require to solve the set
of equations (3), (4) and (6) with the geometry of the drop given by equation
(2). In particular, such description would capture the significant finite-size
effect occurring in the wedge formed by the contact line. Also, the solution
for the 3D salt concentration field would probably suggest an outward
diffusive flux in the drop responsible for the radial inward motion of
particles that we observed after the drop deposition.

##  6 Conclusion

We studied particle transport induced by diffusiophoresis in a water drop
deposited on a hydrogel. We show that a difference of solute concentration
between the hydrogel and the drop triggers either a particle motion towards
the gel interface or, on the contrary, toward the liquid-air interface of the
drop, depending on the direction of the solute gradient. This behavior can be
understood qualitatively by inspection of the equations describing the
diffusion of solute and the diffusiophoretic mobility. In the particular
situation of particle deposition, which is our interest, we observed that the
density profile decreases along the radius. This trend is more pronounced as
the deposition by diffusiopheresis proceeds. In addition, we derived a
simplified 1D numerical model that successfully rationalizes our findings for
the time evolution of the total number of adsorbed particles.

We believe that the results reported in this paper are important for at least
two reasons. First, when coating the surface of a hydrogel with colloidal
particles, not only solvent evaporation or absorption can be used;
diffusiophoretic forces allow manipulation of the colloidal particles on
timescales that are comparable to the deposition made by absorption that we
reported previously [27, 28]. Furthermore, the direction of the solute
gradient can be chosen to manipulate the particles either toward the solid
interface or toward the liquid-air interface, depending on the surface charge
of the particles. Our experiments use hydrogels but these conclusions must
remain relevant for various porous substrates. The second message is to bring
attention to solute concentration gradients in situations where colloidal
particles and hydrogels are manipulated simultaneously. Indeed, solute
concentration gradient may appear due to the presence of ions in the colloidal
suspension or in the hydrogel. Whereas we have not observed a particle motion
that can be attributed to diffusiophoresis in our previous experiments [27,
28], the presence of ions must be carefully considered as they can generate
significant effects as we have shown in this paper.

##  7 Acknowledgments

We kindly thank François Ingremeau for useful discussions. F.B. acknowledges
that the research leading to these results received funding from the People
Programme (Marie Curie Actions) of the European Union’s Seventh Framework
Programme (FP7/2007-2013) under REA grant agreement 623541. S.S. and H.A.S.
thank Unilever Research and the Princeton Environmental Institute for partial
support of this research.

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##  8 Appendix: a simple analytical model

We derive in this section a simple model for the case $\Gamma>1$ that can be
used to provide some insights on the dynamics of colloids deposition at the
surface of hydrogels. For the sake of simplicity, let us neglect at first the
finite size of the drop. Under this crude approximation the solute
distribution is self-similar and can be written as [44]:

$\frac{c(z,t)-C_{s}^{d}}{C_{s}^{g}-C_{s}^{d}}={\rm
erfc}\left(\frac{z}{2\sqrt{{\cal D}_{s}t}}\right).$ (12)

As the finite size effects are neglected, this equation is valid only for
$t\ll h_{0}^{2}/{\cal D}_{s}$.

Substituting equation (12) in equation (9b), we obtain the following
expression for the diffusiophoretic velocity:

$v_{p}(z,t)=-\frac{{\cal D}_{dp}}{\sqrt{\pi{\cal
D}_{s}t}}\frac{e^{-\chi^{2}}}{{\rm
erfc}(\chi)},\,\,\,\mbox{where}\,\,\,\chi=\frac{z}{2\sqrt{{\cal D}_{s}t}}.$
(13)

Here, $v_{p}$ is negative as the particles are moving toward the gel interface
for the initial condition we consider. At the gel interface, we have:

$v_{p}(z=0,t)=-\frac{{\cal D}_{dp}}{\sqrt{\pi{\cal D}_{s}t}}.$ (14)

At a vertical position corresponding to the height $h_{0}$ of the drop, we can
estimate the factor $e^{-\chi^{2}}/{\rm erfc}(\chi)$ in equation (13) for
different times that are accessible experimentally:

$v_{p}(z=h_{0},t)\approx-\frac{{\cal D}_{dp}}{\sqrt{\pi{\cal
D}_{s}t}}\left\\{\begin{array}[]{ll}\times 3.0&\mbox{for t=10 s,}\\\ \times
1.6&\mbox{for t=100 s,}\\\ \times 1.2&\mbox{for t=1000 s.}\\\
\end{array}\right.$ (15)

We notice that for the range of timescales and vertical positions that are of
interest, the factor $e^{-\chi^{2}}/{\rm erfc}(\chi)$ is a weak function of
$z$ and $t$. Therefore, from equation (13), we make the following
approximation for the diffusiophoretic particle velocity:

$v_{p}(z=h_{0},t)\approx-\frac{{\cal D}_{dp}}{\sqrt{\pi{\cal D}_{s}t}}.$ (16)

It is important to stress that this approximation is not good for
$t\rightarrow 0$. Now, we consider equation (9c) for the particle motion
which, combined with (16), leads to:

$\frac{\partial C_{p}}{\partial t}-\frac{{\cal D}_{dp}}{\sqrt{\pi{\cal
D}_{s}t}}\frac{\partial C_{p}}{\partial z}={\cal
D}_{p}\frac{\partial^{2}C_{p}}{\partial z^{2}},$ (17)

We define $\xi=z-\frac{2{\cal D}_{dp}}{\pi{\cal D}_{s}}\sqrt{t}$ and rewrite
(17) as:

$\frac{\partial C_{p}}{\partial\xi}+\frac{{\cal D}_{p}\sqrt{\pi{\cal
D}_{s}t}}{{\cal D}_{dp}}\frac{\partial^{2}C_{p}}{\partial\xi^{2}}=0.$ (18)

By integrating this equation twice, we obtain:

$C_{p}(z,t)=A\exp\left(-\frac{{\cal D}_{dp}}{{\cal
D}_{p}}\frac{z}{\sqrt{\pi{\cal D}_{s}t}}\right)+B,$ (19)

where $A$ and $B$ are two constants. We consider the boundary conditions
$C_{p}(z\rightarrow\infty,t)=C_{p}^{\infty}$ and $C_{p}(z=0,t)=0$ because the
particles at the interface are not mobile and the interface behaves as a
perfect sink [43]. Therefore, equation (19) reduces to:

$C_{p}(z,t)=C_{p}^{\infty}\left[1-\exp\left(-\frac{{\cal D}_{dp}}{{\cal
D}_{p}}\frac{z}{\sqrt{\pi{\cal D}_{s}t}}\right)\right].$ (20)

The number of particles $N_{{\rm abs}}$ adsorbed at the interface ($z=0$) is
given by:

$\frac{{\rm d}N_{{\rm abs}}}{{\rm d}t}=-C_{p}(z=0,t)v_{p}+{\cal
D}_{p}\frac{\partial C_{p}}{\partial z},$ (21)

where, by definition of the boundary condition, $C_{p}(z=0,t)=0$. In equation
(21), we substitute the equation (20). To take into account the finite effect
of the drop, we consider that the concentration $n_{\infty}$ varies with the
number of adsorbed particles, i.e. $n_{\infty}\approx(N_{{\rm tot}}-N_{{\rm
abs}})/h_{0}$. Therefore, equation (21) becomes:

$\frac{{\rm d}N_{{\rm abs}}}{{\rm d}t}=\left(N_{{\rm tot}}-N_{{\rm
abs}}\right)\frac{{\cal D}_{dp}}{h_{0}\sqrt{\pi{\cal D}_{s}}t}.$ (22)

We integrate this equation with the initial condition $N_{{\rm abs}}(t=0)=0$
to obtain:

$\frac{N_{{\rm abs}}(t)}{N_{{\rm tot}}}=\left[1-\exp\left(-\frac{{\cal
D}_{dp}\sqrt{t}}{h_{0}\sqrt{\pi{\cal D}_{s}}}\right)\right].$ (23)

As stated for Eq. 12, this solution is valid only for $t\ll h_{0}^{2}/{\cal
D}_{s}$. In our experimental conditions, this assumption is equivalent to
$t\ll h_{0}^{2}{\cal D}_{s}/{\cal D}_{dp}^{2}$. Thus, developing Eq. 23 at the
first order, we obtain

$\frac{N_{{\rm abs}}(t)}{N_{{\rm tot}}}\simeq\frac{{\cal
D}_{dp}}{h_{0}\sqrt{\pi{\cal D}_{s}}}\sqrt{t}.$ (24)

Despite the strong approximation on the particle velocity made at short
timescales with Eq. 16, Eq. 24 correctly describes our observations. Indeed,
the model predicts a prefactor $\frac{{\cal D}_{dp}}{h_{0}\sqrt{\pi{\cal
D}_{s}}}\simeq 0.025$ s-1/2 and a fit of the experimental data provides a
value of $0.017$ s-1/2. However, because of its simplicity, this model cannot,
in contrast to the numerical simulation presented in the main text, accurately
describes the vertical particle distribution.