federated setting, as well as the experimental details.

###  J.1 Partitioned networks in FL

In order to apply partitioned networks to the federated setting, we randomly
choose a partition for each client such that the marginal distribution of
partitions follows a pre-determined ratio. A given chunk $\mathcal{D}_{k}$
therefore corresponds to the union of several clients’ datasets. Analogous to
how “partitioned training” is discussed in the main text and Appendix I, we
desire each partition ${\bm{w}}_{k}$ to be updated on chunks
$\mathcal{D}_{1:k}$. Equation 3 in the main text explains which data chunks
are used to compute gradients wrt. parameter partition ${\bm{w}}_{k}$. An
analogous perspective to this objective is visualized by the exemplary
algorithm in Figure 1 and asks which partitions are influenced (i,e., updated)
by data from chunk $\mathcal{D}_{k}$: A data chunk $\mathcal{D}_{k}$ is used
to compute gradients wrt. partitions ${\bm{w}}_{k:C}$ through subnetworks
${\bm{w}}_{s}^{(k)}$ to ${\bm{w}}_{s}^{(C)}$ respectively. Consequently, a
client whose dataset is assigned to chunk $\mathcal{D}_{k}$ can compute
gradients for all partitions ${\bm{w}}_{k:C}$.

##### Updating network partitions

Due to the weight-partitioned construction of the partitioned neural networks,
it is not possible to compute gradients with respect to all partitions in a
single batched forward-pass through the network. Additionally, a change to the
partition parameters ${\bm{w}}_{k}$ directly influences subnetworks
${\bm{w}}_{s}^{(k+1)}$ to ${\bm{w}}_{s}^{(C)}$. In order to avoid the choice
of ordering indices $k$ to $C$ for the client’s local update computation, we
update each partition independently while keeping all other partitions
initialised to the server-provided values that the client received in that
round $t$: Denote $D_{i,k}$ as the dataset of client $i$ where we keep index
$k$ to emphasize the client’s assignment to chunk $k$. Further denote
${\bm{w}}_{j,i}^{t+1}$ as the partition ${\bm{w}}_{j}^{t}$ after having been
updated by client $i$ on dataset $D_{i,k}$.

$\displaystyle{\bm{w}}_{j,i}^{t+1}=\operatorname*{arg\,max}_{{\bm{w}}_{j}}\log
p\left(D_{i,k}|({\bm{w}}_{1}^{t},\dots,{\bm{w}}_{j}^{t},\hat{{\bm{w}}}_{j+1}^{t},\dots,\hat{{\bm{w}}}_{j+C}^{t}),\psi\right)\quad\forall
j\in[k,C],$ (25)

where the details of optimization are explained in the following section. We
leave an exploration for different sequential updating schemes to future work.
The final update communicated by a client to the server consists of the
concatenation of all updated parameter partitions
${\bm{w}}_{.,i}^{t+1}=\text{concat}({\bm{w}}_{k,i}^{t+1},\dots,{\bm{w}}_{C,i}^{t+1})$.
Note that partitions $({\bm{w}}_{1}^{t},\dots,{\bm{w}}_{k-1}^{t})$ have not
been modified and need not be communicated to the server. The resulting
communication reductions make partitioned networks especially attractive to FL
as data upload from client to server poses a significant bottleneck. In
practice, we expect the benefits of these communication reductions to outweigh
the additional computation burden of sequentially computing gradients wrt., to
multiple partitions.

The server receives ${\bm{w}}_{.,i}^{t+1}$ from all clients that participates
in round $t$, computes the delta’s with the global model and proceeds to
average them to compute the server-side gradient in the typical federated
learning fashion [Reddi et al., 2020].

##### Updating hyperparameters

The computation of gradients on a client $i$ wrt. $\psi$ is a straight-forward
extension of equation 4 and the exemplary algorithm of Figure 1:

$\displaystyle\nabla_{\psi}\mathcal{L}_{\mathrm{ML}}\left(D_{i,k},\psi\right)\approx\nabla_{\psi}\log
p\left(D_{i,k}|{\bm{w}}_{s,i}^{(t+1),(k-1)},\psi\right),$ (26)

where $D_{i,k}$ corresponds to client $i$’s local dataset which is assigned to
chunk $k$ and ${\bm{w}}_{s}^{(t+1),(k-1)}$ corresponds to the $(k-1)$’th
subnetwork after incorporating all updated partitions
${\bm{w}}_{s,i}^{(t+1),(k-1)}=\text{concat}({\bm{w}}_{1}^{t},\dots,{\bm{w}}_{k-1}^{t},{\bm{w}}_{k,i}^{t+1},\dots,{\bm{w}}_{C,i}^{t+1})$.
Note that we compute a full-batch update to $\psi$ in MNIST experiments and
use a batch-size equal to the batch-size for the partitioned parameter updates
for CIFAR10.

Upon receiving these gradients from all clients in this round, the server
averages them to form a server-side gradient. Conceptually, this approach to
updating $\psi$ corresponds to federated SGD.

###  J.2 Federated setup

##### Non-i.i.d. partitioning

For our federated experiments, we split the $50k$ MNIST and $45k$ CIFAR10
training data-points across $100$ clients in a non-i.i.d. way to create the
typical challenge to federated learning experiments. In order to simulate
label-skew, we follow the recipe proposed in Reddi et al. [2020] with
$\alpha=1.0$ for CIFAR10 and $\alpha=0.1$ for MNIST. Note that with
$\alpha=0.1$, most clients have data corresponding to only a single digit. For
our experiments on rotated versions of CIFAR10 and MNIST, we sample a degree
of rotation per data-point and keep it fixed during training. In order to
create a non-i.i.d partitioning across the clients, we bin data-points
according to their degree of rotation into $10$ bins and sample using the same
technique as for label-skew with $\alpha=0.1$ for both datasets. Learning
curves are computed using the $10k$ MNIST and $5k$ CIFAR10 validation data-
points respectively. For the rotated dataset experiments, we rotate the
validation set in the same manner as the training set.

##### Architectures and experimental setup

We use the convolutional network provided at Schwöbel et al. [2021] for MNIST
and the ResNet-9 [Dys, ] model for CIFAR10 but with group normalization [Wu
and He, 2018] instead of batch normalization. We include (learnable) dropout
using the continuous relaxation proposed at Maddison et al. [2016] between
layers for both architectures. We select $3$ chunks for MNIST with a
$[0.7,0.2,0.1]$ ratio for both, client-assignments and parameter-partition
sizes. For CIFAR10, we found a $[0.9,0.1]$ split across $2$ sub-networks to be
beneficial. In addition to dropout logits, $\psi$ encompasses parameters for
affine transformations, i.e., shear, translation, scale and rotation. We
report results after $2k$ and $5k$ rounds, respectively, and the expected
communication costs as a percentage of the non-partitioned baseline.

##### Shared setting

In order to elaborate on the details to reproduce our results, we first focus
on the settings that apply across all federated experiments. We randomly
sample the corresponding subset of $1.25k$, $5k$ data-points from the full
training set and keep that selection fixed across experiments (i,e., baselines
and partitioned networks) as well as seeds. The subsequent partitioning across
clients as detailed in the previous paragraph is equally kept fixed across
experiments and seeds. Each client computes updates for one epoch of its local
dataset, which, for the low data regimes of $1.25k$ data-points globally,
results in single update per client using the entire local dataset. We
averaged over $10$ augmentation samples for the forward pass in both training
and inference.

##### MNIST & RotMNIST

For $5k$ data-points and correspondingly $50$ data-points on average per
client, most clients perform a single update step. A small selection of
clients with more than $64$ data-points performs two updates per round. For
the experiments using the full dataset and a mini-batch size of $64$, each
client performs multiple updates per round. After initial exploration on the
baseline FedAvg task, we select a local learning-rate of $5e-2$ and apply
standard SGD. The server performs Adam Reddi et al. [2020] with a learning
rate of $1e-3$ for the model parameters. We keep the other parameters of Adam
at their standard PyTorch values. We find this setting to generalize to the
partitioned network experiments but found a higher learning rate of $3e-3$ for
the hyper-parameters to be helpful. We chose the convolutional network from
Schwöbel et al. [2021] with (learned) dropout added between layers. The
model’s dropout layers are initialized to drop $10\%$ of hidden activations.
For the baseline model we keep the dropout-rate fixed and found $10\%$ to be
more stable than $30\%$.

##### CIFAR10 & RotCIFAR10

We fix a mini-batch size of $32$, leading to multiple updates per client per
round in both, the full dataset regime as well as the $5k$ data-points
setting. Similarly to the MNIST setting, we performed an initial exploration
of hyperparameters on the baseline FedAvg task and use the same ones on
partitioned networks. We used dropout on the middle layer of each block which
was initialized to 0.1 for both the baseline and partitioned networks and
whereas partitioned networks optimized it with $\mathcal{L}_{ML}$ and the
concrete relaxation from Maddison et al. [2016], the baseline kept it fixed.
For the server side optimizer we used Adam with the default betas and a
learning rate of $1e-2$, whereas for the hyperparameters we used Adam with the
default betas and a learning rate of $1e-3$. In both cases we used an
$\epsilon=1e-7$. For the local optimizer we used SGD with a learning rate of
$10^{-0.5}$ and no momentum.

###  J.3 MNIST learning curves

In Figure 10 we show learning curves for the three considered dataset sizes on
the standard MNIST task. Each learning curve is created by computing a moving
average across $10$ evaluations, each of which is performed every $10$
communication rounds, for each seed. We then compute the average and standard-
error across sees and plot those values on the y-axis. On the x-axis we denote
the total communication costs (up- and download) to showcase the partitioned
networks reduction in communication overhead. We see that especially for the
low dataset regime, training has not converged yet and we expect performance
to improve for an increased number of iterations.

Figure 10: Learning curves for MNIST experiments on $1.25k$, $5k$ and $50k$
data-points respectively.


# Directional drying of a colloidal dispersion: quantitative description with
water potential measurements using water clusters in a poly(dimethylsiloxane)
microfluidic chip

Hrishikesh Pingulkar  Sonia Maréchal  Jean-Baptiste Salmon  CNRS, Solvay, LOF,
UMR 5258, Université de Bordeaux,178 av. Schweitzer, Pessac, 33600

###### Abstract

We have developed a poly(dimethylsiloxane) (PDMS) microfluidic chip to study
the directional drying of a colloidal dispersion confined in a channel. Our
measurements on a dispersion of silica nanoparticles once again revealed the
phenomenology commonly observed for such systems: the formation of a porous
solid with linear growth in the channel at short times, slowing down at longer
times as the evaporation rate decreases. The growth of the solid is also
accompanied by mechanical stresses that are released by the delamination of
the solid from the channel walls and the formation of cracks. In addition to
these observations, we report original measurements using hydrophilic filler
in the PDMS formulation used (Sylgard-184). When the PDMS matrix is in contact
with water, water molecules pool around these hydrophilic sites, resulting in
the formation of microscopic water clusters whose size depends on the water
potential $\psi$. In our work, we have used these water clusters to estimate
the water potential profile in the channel as the porous solid grows. Using a
transport model that also takes into account solid delamination in the
channel, we then linked these water potential measurements to the hydraulic
permeability of the porous solid. These measurements finally enabled us to
show that the slowdown in the evaporation rate is due to the invasion of the
porous solid by air/water nanomenisci at a critical capillary pressure
$\psi_{\text{cap}}$.

##  I Introduction

The drying of colloidal dispersions, a common step in many manufacturing
processes, remains a fascinating subject of study whose understanding involves
many aspects from the physico-chemistry of colloids, transport phenomena, to
the mechanics of porous solids Russel (2011); Routh (2013); Goehring _et al._
(2015); Bacchin _et al._ (2018). For a general understanding of these
phenomena, many groups have focused on model cases, such as the confined
directional drying Allain and Limat (1995); Dufresne _et al._ (2003, 2006). In
such experiments, a dilute dispersion is confined within a capillary or Hele-
Shaw cell, with cross-sectional dimensions $<100~{}\mu$m, see Fig. 1a for a
schematic view. When the air/dispersion meniscus is pinned at the outlet of
the cell, solvent evaporation induces a flow at a rate $J$ (m/s) due to mass
conservation. This flow then accumulates the colloids at the end of the cell
until a consolidated solid is formed. For rigid colloids, the porosity of the
solid does not stop solvent evaporation, even for nanoparticles as small as a
few nm Dufresne _et al._ (2003, 2006), and the solid invades the cell at a
rate $\dot{x}_{c}$. At longer times, the evaporation-induced flow through the
porous solid leads to mechanical stresses that are released by instabilities:
formation of shear bands Boulogne _et al._ (2014); Kiatkirakajorn and Goehring
(2015); Yang _et al._ (2015), delamination of the solid from the cell walls
Sarkar and Tirumkudulu (2011); Xu _et al._ (2010), and formation of cracks
Allain and Limat (1995); Dufresne _et al._ (2003); Gauthier _et al._ (2007);
Inasawa and Yamaguchi (2012); Xu _et al._ (2013).

Figure 1: (a) Sketch of confined directional drying of a colloidal dispersion,
and cross-sectional view along the dashed line of the microfluidic chip shown
in (b). Dimensions: $h=30$, $h_{a}=70$, $w_{a}=300$, $\lambda=500$, and
$H\simeq 200~{}\mu$m. The water bath on top of the PDMS layer prevents
pervaporation. Colloids are shown in red, with black dots representing the
water clusters in PDMS, see Section III.2. (b) Schematic view of the
microfluidic chip. Blue indicates water, yellow fluorinated oil. I1 and I2 are
the two liquid inlets. Air flows at a controlled relative humidity
$\mathrm{RH}$ in the gray channel (width $w_{a}$). Evaporation induces a flow
at a rate $J=-\dot{x}_{m}$ in the main channel ($w=100~{}\mu$m) as long as the
air/water meniscus remains pinned at $x=0$.

Confined directional drying has been used in many studies to better understand
the drying of colloidal dispersions, from the mechanical instabilities
mentioned above to transport phenomena Sarkar and Tirumkudulu (2009); Goehring
_et al._ (2017); Inoue and Inasawa (2020); Hooiveld _et al._ (2023); Noirjean
_et al._ (2017); Inasawa _et al._ (2016). This is mainly due to the control of
evaporation conditions in this one-dimensional drying geometry and the
monitoring possibilities offered by this technique. In addition to these
studies, confined directional drying is also a model configuration for
studying the drying of porous media saturated with a colloidal dispersion in
the capillary regime Keita _et al._ (2013); Ghiringhelli _et al._ (2023).
Despite all these studies, the evaporation kinetics, $J$ vs $t$, and the
growth of the porous solid, $x_{c}$ vs $t$, remain open questions to this day
Pingulkar and Salmon (2023). In the case of nanoparticles with typical radii
$a\leq 20$ nm, experiments have shown that the solid layer first grows
linearly, $x_{c}\sim t$, and then follows a square root growth $x_{c}\sim
t^{0.5}$ indicating a decrease of the evaporation-induced flow rate $J$
Dufresne _et al._ (2003); Sarkar and Tirumkudulu (2011); Inasawa and Yamaguchi
(2012); Lidon and Salmon (2014). Most theoretical works suggest that this
slowdown is actually due to the minute recession of the evaporation interface
within the porous solid, adding a high resistance to evaporation linked to the
diffusion of water vapour in the dry porous layer, the so-called capillary-
limited regime Dufresne _et al._ (2006); Lidon and Salmon (2014); Wallenstein
and Russel (2011); Pingulkar and Salmon (2023). For very small colloids
($a<10~{}$nm), the water pore pressure at the evaporation interface can
decrease down to values that affect the partial pressure of water in the the
gas phase, the so-called Kelvin effect Thiery _et al._ (2017). In this flow-
limited regime, there is no resistance to evaporation in the gas phase, and
the slowdown is explained by the increasing resistance due to the flow through
the growing porous solid Pingulkar and Salmon (2023). To date, no work has
been done to differentiate experimentally between these two regimes (except
ref. Vincent _et al._ (2016), but for drying-induced permeation in nanoporous
media), nor has quantitative modelling been carried out, due to the lack of
data describing the porous solid, in particular its hydraulic permeability
$\kappa$ (m2). Furthermore, all the theoretical studies are based on one-
dimensional modelling of transport phenomena, and do not take into account the
possible role played by cracks and solid delamination on the solvent
evaporation.

Our aim in the present work is to bring new insights on the confined
directional drying of a dispersion of silica nanoparticles (mean radius
$a=11~{}$nm), using experiments in a poly(dimethylsiloxane) (PDMS)
microfluidic chip. Our experiments first enable the directional drying of the
colloidal dispersion in a highly confined geometry (cross-section $h\times
w=30\times 100~{}\mu$m2, see Fig. 1), while providing accurate measurements of
both the evaporation kinetics $J$ vs $t$, and the growth rate of the solid,
$x_{c}$ vs $t$. We also exploit the presence of hydrophilic filler in the PDMS
formulation used, to estimate the water potential $\psi$ (Pa) along the porous
solid as it grows. These original data, combined with a model taking into
account the presence of an air film linked to delamination, enable us to
estimate the permeability $\kappa$ of the porous solid and show that the
slowdown observed in our configuration corresponds to the capillary-limited
regime.

The present paper is organized as follows. In Section II, we present the
microfluidic tools used and the experimental methods employed. In Section III,
we present the main results of our work, in particular concerning the confined
directional drying of a nanoparticle dispersion and its quantitative
description. In Section IV, we conclude our work and suggest various
perspectives.

##  II Materials and Methods

###  II.1 A PDMS chip for confined directional drying

Fig. 1 shows schematically the chip we designed to study the confined
directional drying of colloidal dispersions. We used standard soft lithography
techniques to fabricate this chip with Sylgard-184 PDMS (mass ratio curing
agent/polymeric base $=1/10$). The channel network is composed of two inlets
(I1 and I2) connected to a serpentine-like channel of length $L=43$ mm, and
height$\times$width $h\times w=30\times 100~{}\mu$m2. The latter leads to a
higher channel (height $h_{a}=70~{}\mu$m) in which air at an imposed relative
humidity ($\mathrm{RH}$) flows. The height step in the channel ($h\to h_{a}$)
helps to trap an air/water meniscus when an aqueous colloidal dispersion is
injected using either inlet I1 or I2 Choi _et al._ (2012, 2015). Water
evaporation from the air/water meniscus then drives a flow at a rate $J$ as
long as the meniscus remains trapped, accumulating the colloids up to the
formation of a porous solid that finally invades the channel. Many similar
microfluidic experiments have been carried out, particularly in the context of
passive pumping Goedecke _et al._ (2002); Namasivayam _et al._ (2003) or for
evaporative assembly of colloidal materials Choi _et al._ (2012, 2015), see
ref. Bacchin _et al._ (2022) for a review.

In the absence of colloids, the evaporation rate under isothermal conditions
can be estimated byGoedecke _et al._ (2002); Namasivayam _et al._ (2003);
Bacchin _et al._ (2022):

$J=\frac{kc_{\text{sat}}}{\rho_{w}}(1-\mathrm{RH})\,,$ (1)

with $c_{\text{sat}}$ (kg/m3) the saturation vapor concentration in air,
$\rho_{w}$ (kg/m3) the mass density of liquid water, and $k$ (m/s) a mass
transport coefficient. For diffusion-dominated transport for the water vapour
over the length scale $\lambda$ (see Fig. 1),
$k\simeq(h_{a}/h)\,D_{w}^{\mathrm{air}}/\lambda$ with $D_{w}^{\mathrm{air}}$
(m2 s-1), the diffusion coefficient of the water vapour in air Massman (1998).
Eqn (1) leads to evaporation rates ranging from $J\simeq 0.5$ to $2.3~{}\mu$m
s-1 for relative humidity ranging from $\mathrm{RH}=0.8$ to $0$. However, the
PDMS matrix is not totally impermeable to water, as water molecules can
solubilize in the elastomer, diffuse, and then evaporate into the ambient air,
a mechanism known as pervaporation. Water pervaporation in our PDMS chip
inevitably leads to a flow directed towards the fixed air/water meniscus
Randall and Doyle (2005); Verneuil _et al._ (2004), and thus superimposed on
the flow induced by evaporation. This pervaporation-induced flow can in turn
also concentrate the colloids as in the many applications reviewed in Ref.
Bacchin _et al._ (2022). Dollet et al. calculated the flow due to
pervaporation for a single linear channel of rectangular cross-section Dollet
_et al._ (2019). This relation leads to an estimated pervaporation-induced
flow $J_{p}\simeq 6~{}\mu$m s-1 for the filled serpentine channel shown in
Fig. 1, $H=200~{}\mu$m, and an external relative humidity $\mathrm{RH}=0.5$.
This flow, significantly larger than the evaporation-induced flow predicted by
eqn (1), can be reduced by increasing $H$, but not completely Bacchin _et al._
(2022). Indeed, for the serpentine channel, the assumption of a single linear
channel no longer holds for large $H$, but theoretical estimates in this case
Noblin _et al._ (2008) still lead to $J_{p}\simeq 0.1$–$1~{}\mu$m s-1 for
thicknesses $H$ of a few mm. Hence, in order to fully neglect pervaporation in
our work, the whole chip is immersed in water before and during the
experiments, as demonstrated in ref. Randall and Doyle (2005). With this
simple technique, the pervaporation-induced flows are fully eliminated after a
time scale of the order of $\tau_{p}\simeq H^{2}/D_{w}^{p}$, $D_{w}^{p}\simeq
6$–$8\times 10^{-10}$ m2 s-1 being the diffusion coefficient of water in PDMS
Bacchin _et al._ (2022). This led us to choose a thickness $H\simeq
200~{}\mu$m for our experiments, a compromise between the time required to
stop pervaporation, $\tau_{p}$ of the order of a few minutes in this case, and
the limits of microfabrication of thin PDMS chips.

###  II.2 Confined directional drying experiments

We studied a charge-stabilized dispersion of silica nanoparticles in water
commercialized under the name Ludox AS40 (monodisperse anionic grade, Sigma
Aldrich). The volume fraction of the commercial dispersion estimated using dry
extract measurements is $\varphi_{0}\simeq 0.24$ and the mean radius of the
nanoparticles is $a=11$ nmSobac _et al._ (2020). All experiments were carried
out at room temperature, $T\simeq 22^{\circ}$C.

In a typical experiment, the microfluidic chip is first immersed in a water
bath for about $1$ h, and an air stream of controlled $\mathrm{RH}$ is imposed
in the air channel (typical rate $0.4$ mL min-1, HumiSys LF $\mathrm{RH}$
generator, InstruQuest Inc.). Then, the dispersion of silica nanoparticles is
injected in the chip through inlet I1 using a small amount of excess pressure
(typically 50 mbar, MFCS, Fluigent). The dispersion invades the main channel
and the geometric step traps the air/dispersion meniscus ($x=0$). Then, a
fluorinated oil (Fluorinert FC40, 3M) is gently injected from inlet I1 towards
I2, leading to the formation of an oil/dispersion meniscus at $x_{m}=L$, the
inlet of the serpentine channel. This is only possible because our device has
two inlets, I1 and I2, connected to the same main channel. Finally, we used
bright field microscopy (Olympus IX73) at 2X magnification (spatial resolution
$3.25~{}\mu$m/pixel) using a sCMOS camera (Orca 4, Hamamatsu) to track both
the drying process along the channel and estimate the evaporation-induced flow
rate from the displacement of the oil/dispersion meniscus by :

$J=-\frac{\text{d}x_{m}}{\text{d}t}\,.$ (2)

###  II.3 Raman microspectroscopy

The confocal Raman measurements shown in Section III.2 were obtained with a
custom-made Raman microspectrometer setup coupled to an inverted microscope
(Olympus IX71). A laser beam (wavelength $532$ nm, output power $\simeq 30$
mW) is focused with a water-immersion 60X objective (NA of $1.2$) on the water
clusters observed in the PDMS film and discussed in Section III.2. The
scattered light is collected by the same objective, filtered, and directed to
the spectrometer (Andor Shamrock, grating $600$ lines mm-1, input slit width
$100~{}\mu$m). A confocal pinhole ($100~{}\mu$m) reduces the out-of-focus
contributions. Typical acquisition times were of the order of $2$ s.

##  III Results

###  III.1 Confined directional drying, global views

Fig. 2 shows a snapshot at $t=100$ min of the confined directional drying of
the dispersion of silica nanoparticles for an experiment performed with an air
stream of relative humidity $\mathrm{RH}=0.2$, see the movies M1.avi and
M2.avi †. For the sake of clarity, this snapshot zooms on the first turn in
the serpentine channel, but the full field of view makes it possible to
visualize the oil/dispersion meniscus and thus measure the evaporation rate
$J$ using eqn (2).

Figure 2: Snapshot at $t=100~{}$min zooming on the first turn in the
serpentine channel, see also movie M2.avi †. $x_{c}$ indicates the compaction
front, $x_{d}$ the delamination front, and $x_{f}$ the position of the single
crack at that instant. The scale bar is equal to $500~{}\mu$m. The colored
circles surround the positions of the water clusters in PDMS that are used to
estimate the water potential, see Section III.2 and Fig. 7 (they have almost
all disappeared at $t=100$ min). Their colors, from red to yellow, indicate
their increasing positions $x_{i}$ from the end of the channel, see the axis.

Our observations once again reveal the phenomenology commonly observed during
the confined directional drying of such dispersions Dufresne _et al._ (2003,
2006); Lidon and Salmon (2014). At initial times, the accumulation of the
nanoparticles at $x=0$, leads to the formation of a porous solid which then
invades the main channel, as it is constantly fed by an evaporation-induced
flux of colloids. A compaction front separates the dilute dispersion from this
colloidal material. Its position $x_{c}$ is easily identified by the
disappearance of the channel walls, due to the matching of the optical indices
of the PDMS and the dispersion at the compaction front. This transition from a
liquid dispersion to a solid is accompanied by the onset of mechanical
stresses Wallenstein and Russel (2011); Style and Peppin (2011). In our case,
these stresses deform the microfluidic channel, until the colloidal solid
detaches from it, as often reported for similar experiments Xu _et al._
(2010); Sarkar and Tirumkudulu (2011). The delamination front, denoted
$x_{d}$, broadly follows the compaction front but intermittently, and is
easily detected thanks to the strong optical contrast associated with the
formation of an air film. In the experiment, see Fig. 2, the solid initially
detaches from the vertical walls of the channel, then later across its entire
width. Subsequently, these mechanical stresses cause cracks to form in the
colloidal solid, as also observed many times for similar experiments Allain
and Limat (1995); Dufresne _et al._ (2003); Inasawa and Yamaguchi (2012);
Gauthier _et al._ (2007), but with specific characteristics linked to the high
confinement of the channel. The number of cracks is indeed relatively limited
(3 in the experiment presented at $t=500~{}$min), and they are transverse to
the porous solid, literally cutting it into several distinct pieces. Even at
2X magnification, our observations also revealed the presence of shear bands
in the growing solid, as often described in similar experiments Boulogne _et
al._ (2014); Yang _et al._ (2015); Kiatkirakajorn and Goehring (2015).

Fig. 3a gathers the tracking of both the compaction and delamination fronts,
$x_{c}$ and $x_{d}$ respectively, along with the positions $x_{f}$ of the
cracks when they appear. Initially, the compaction front grows linearly along
the channel, $x_{c}\sim t$, and then slows down for $t\gtrsim 80$ min
following at long time scale $x_{c}\sim t^{\alpha}$, with $\alpha\simeq 0.4$.
This smooth growth does not seem to be affected by the intermittent dynamics
of the delamination front, nor by the sudden appearance of cracks. In
addition, other experiments carried out under the same conditions led to very
similar observations for $x_{c}$ vs $t$, but with noticeable differences for
the delamination front and the number and positions of the cracks,
consistently with the stochastic nature of these phenomena.

Figure 3: (a) Compaction front $x_{c}$ ( $\circ$), delamination front $x_{d}$
( $\diamond$) and positions of cracks $x_{f}$ as they appear ($\square$) as a
function of time $t$. Inset: $x_{c}$ vs $t$ in log-log scale. (b) Position of
the meniscus $x_{m}$ against the compaction front $x_{c}$. The continuous line
is a linear fit. (c) Evaporation rate $J$ estimated from the displacement of
the meniscus using eqn (2).

Fig. 3b shows the position of the oil/dispersion meniscus $x_{m}$, versus the
compaction front $x_{c}$. The relationship between these two quantities is
affine, and linked to the colloid conservation as discussed later. Finally,
Fig. 3c shows the evaporation rate $J$ estimated from eqn (2) as a function of
time. This curve shows that $J$ decreases continuously from $J\simeq
1.3~{}\mu$m s-1 to around $J\simeq 100~{}$nm s-1 at long times. All the
observations described above, as well as the values of the dynamics $x_{c}$ vs
$t$ and $J$ vs $t$ shown in Fig. 3 are consistent with published data for
similar experiments Dufresne _et al._ (2003, 2006); Sarkar and Tirumkudulu
(2011); Inasawa and Yamaguchi (2012), see in particular ref. Lidon and Salmon
(2014). On the other hand, it is important to point out that our data suggest
a slowdown at long times following $x_{c}\sim t^{\alpha}$, with an exponent
$\alpha\simeq 0.4$ slightly lower than the $0.5$ value often reported in the
references cited above. Although it is always tricky to estimate such exponent
from fits over small time ranges, we believe that this over-slowdown is
significant and possibly due to the permeation of water from the reservoir on
top of the PDMS chip towards the channel, as discussed in Section III.6.

###  III.2 Water clusters in PDMS

In addition to the observations described above, our experiments revealed a
new phenomenon. When the chip is immersed in the water bath, a large number of
small inclusions appear in the PDMS, absorbing or scattering light, therefore
suggesting impurities, see the black dots on Fig. 2. Local observations at
higher magnifications show that these inclusions are highly dispersed, with
sizes ranging from fractions of a micron to nearly 10 $\mu$m for the largest.
When the chip is removed from the water bath, these inclusions disappear, and
the PDMS becomes completely transparent again. As shown in Fig. 4a, the
inclusions appear again exactly at the same positions as before when the chip
is re-immersed in water, suggesting the presence of preferential sites within
the PDMS matrix. Confocal Raman spectroscopy measurements on these inclusions
revealed the OH stretching vibrations of the water molecules, suggesting that
they are in fact microscopic water clusters, see Fig. 4b and 4c.

Figure 4: (a) Images ($325\times 325$ $\mu$m2) of a PDMS layer $200~{}\mu$m
thick, which, from left to right, is first immersed in a water bath, then
removed and exposed to ambient air, before being immersed again in the water
bath. (b) and (c): Raman spectra ($\cdot$) measured at the position of a water
cluster in a PDMS film in (b) ambient air and (c) when immersed in water. Both
spectra are normalized by the Raman peak at $\nu\simeq 2900$ cm-1 due the PDMS
matrix. The red curves correspond to the Raman spectrum of PDMS measured in a
cluster-free region, and blue curves to the spectrum of water. The insets show
zooms on the OH stretching vibrations in the spectral range
$\nu=[3000$–$3800]$ cm-1.

Such observations are not new, as silicone elastomers are known to become
milky when immersed in water, due to the formation of water clusters that
scatter light. The origin of these clusters is actually linked to the presence
in the matrix of hydrophilic sites, either impurities or filler materials,
which can significantly absorb water molecules Watson and Baron (1996). In the
case of Sylgard-184, the poly(dimethylsiloxane) used in the present study,
these sites are probably silica filler with high specific surface area, added
to the commercial blend by the manufacturer to improve the mechanical
properties of the elastomer Harley _et al._ (2012). Harley et al. have
measured the role of these hydrophilic sites on the equilibrium sorption of
water vapour in cross-linked Sylgard-184 Harley _et al._ (2012). For low water
vapour activity ($\mathrm{RH}<0.1$), these sites allow the immobilization of
water molecules via Langmuir-type adsorption, whereas the sorption of water
increases drastically at higher water vapour activity,
$\mathrm{RH}>0.7$–$0.8$. These authors attributed this effect to the
clustering of the water molecules around the Langmuir sites. These results led
us to the possibility that the water clusters we observe are also sensitive to
the water activity $a_{w}$ in which the PDMS is immersed, and thus on the
water potential $\psi=RT/V_{m}\log(a_{w})$, with $V_{m}\simeq 1.805\times
10^{-5}$ m3mol-1, the liquid water molar volume at room temperature
($T=22^{\circ}$C), and $R$ the universal gas constant.

Figure 5: (a) Sketch of the calibration experiment of the water clusters in
PDMS. A solution at a given water potential $\psi$ flows in a wide channel
(height $\simeq 200~{}\mu$m, width $\simeq 1$ mm) over a thin layer of PDMS
bonded on a glass slide ($\simeq 20~{}\mu$m). The drawing shows only the
bottom PDMS layer. (b) Relative normalized intensity $I_{n}$ against $\psi$
for glycerol solution ($\blacklozenge$) and NaCl solutions ($\blacksquare$).
The red curve is a fit by eqn (3) with $\psi_{c}\simeq-8~{}$MPa. Errorbars
show the standard deviations based on data from eight different water
clusters. Inset: same data against the corresponding water activity
$a_{w}=\exp[V_{m}\psi/(RT)]$.

Fig. 5a shows the experiments we did for testing this assumption. A thin PDMS
film is coated on a glass slide, and we added a wide channel ($\simeq 1~{}$mm
in width, $\simeq 200~{}\mu$m in height) on top to inject aqueous solutions of
known water potential, binary mixtures water/NaCl (at molalities up to $5$
mol/kg) Fontana (2007) and water/glycerol (at glycerol mass fractions up to
$0.6$) Ninni _et al._ (2000); Bouchaudy _et al._ (2018). For $\psi\leq-40$ MPa
($a_{w}\leq 0.75$), we do not observe any water clusters at the magnification
10X, while the clusters grow in size when the water potential increases up to
$\psi=0$ ($a_{w}=1$, pure water). For this thickness of PDMS layer, $\simeq
20~{}\mu$m, we did not observe any kinetic effect as the appearance or
disappearance of the clusters follows the imposed water potential almost
instantaneously. To quantify more precisely these observations, we averaged
the light intensity collected by the camera over a few microns around several
clusters, then subtracted the intensity measured in a cluster-free region to
eliminate the fluctuations of the illumination. This quantity is finally
normalized between $0$ and $1$ and plotted in Fig. 5b against $\psi$. Despite
the errorbars, mainly due to the dispersity of the observed inclusions, these
measurements show that the water clusters are indeed sensitive to the water
potential, as the normalized intensity $I_{n}$ can roughly be fitted by:

$\displaystyle I_{n}=\exp(-\psi/\psi_{c})\,,$ (3)

with $\psi_{c}\simeq-8~{}$MPa. This range of water potentials and of
corresponding water activity $a_{w}$ is consistent with the observations of
Harley et al. Harley _et al._ (2012) that have reported a drastic increase of
the water solubility in Sylgard-184 PDMS for vapour activity
$\mathrm{RH}>0.7$–$0.8$. As noted by Harley et al. Harley _et al._ (2012), the
exact mechanism for the water clustering around the hydrophilic sites, and its
dependence with the water potential remain unknown. This mechanism could be
similar to the osmotic pumping mechanism that has been proposed to explain the
high water uptake of vulcanized rubbers, which is also due to the clustering
of water molecules around hydrophilic impurities Fedors (1980); Thomas and
Muniandy (1987). However, the origin of an osmotic force induced by the
hydration of fillers in Sylgard-184 remains to be demonstrated, and the role
of the PDMS elastomer elasticity remains to be investigated.

###  III.3 Local estimates of the water potential

In the following, we exploit these clusters as sensors of the water potential
in the PDMS matrix. Fig. 6 shows several snapshots zooming at the end of the
evaporation zone, $x=0$, evidencing the growth of the colloidal solid at
initial times. These images also reveal that some of the initially visible
inclusions near the channel disappear over time, as the solid invades the
channel.

Figure 6: Snapshots at different times $t$ zooming in on the beginning of the
serpentine channel, see also movie M3.avi †. Arrows point to a water cluster
that disappears at $t\simeq 100~{}$min. The scale bar is equal to
$300~{}\mu$m. The change of the texture of the solid at its end for $t\gtrsim
100$ min reveals the shear bands initially present, and suggests the invasion
of air/water nanomenisci, see Section III.5.

To better quantify this effect, we have selected 14 water clusters that
disappear with time, located very close to the channel and with positions
along the channel ranging from $x_{i}=0.2$ to $3$ mm, see the colored circles
in Fig. 2. As for the calibration shown in Fig. 5, we have calculated the
normalized intensity $I_{n}$ measured by the camera for each water cluster,
see Fig. 7.

Figure 7: Relative normalized intensity $I_{n}$ against time $t$ for the water
clusters shown in Fig. 2. Only 6 curves are shown for the sake of clarity, and
the color gradient indicates the increasing positions $x_{i}$ of the clusters
along the channel (from $x_{i}\simeq 0.2$ to $3$ mm). The continuous lines are
guides for the eye. Inset: Corresponding water potential $\psi$ using the
calibration given by eqn (3) with $\psi_{c}=-8$ MPa, vs $x_{c}-x_{i}$ for the
positions $x_{i}$ shown in the main graph.

This plot clearly evidences a front of disappearance of these inclusions,
parallel to the progression of the porous solid in the channel.

In order to make a rough estimate of the water potential at the locations of
these different clusters, these curves are converted into water potential
using eqn (3) and plotted against $x_{c}-x_{i}$, see the inset of Fig. 7.
These data unambiguously show that the water potential in the vicinity of the
microfluidic channel at a position $x_{i}$ decreases over time as the
colloidal solid grows beyond this position in the channel (i.e., when
$x_{c}-x_{i}>0$). In the following, we will use these data to estimate the
water potential $\psi$ at the different positions $x_{i}$ along the channel.

###  III.4 Model

We now present a simplified model of the directional drying in our
experiments. Fig. 8a shows a schematic view along the channel of the growth of
the porous solid at a given time $t$. In this picture, we do not consider the
concentrated layer of unconsolidated colloids upstream the compaction front
($x>x_{c}$) even if the later can extend over large scales due to collective
diffusion in such charge-stabilized dispersions Goehring _et al._ (2017);
Sobac _et al._ (2020). The observed delamination implies that the solid has a
smaller cross-section than the channel cross-section $(h\times w)$. In the
following, we neglect the small shrinkage of the solid across its width (a few
microns for $w=100~{}\mu$m) and we assume that most of the shrinkage occurs
across the height as also observed in ref. Sarkar and Tirumkudulu (2011) and
Xu _et al._ (2010). This is mainly due to the relative high aspect ratio of
the channel ($h\times w=30\times 100~{}\mu$m2) and the small thickness of the
PDMS chip ($H=200~{}\mu$m).

Figure 8: (a) Schematic view of the transverse section along the serpentine
channel showing both the compaction and delamination fronts at a given time
$t$ (not to the scale). The dotted line shows the cluster-free region in PDMS
(black dots), where the water potential $\psi\ll\psi_{c}=-8$ MPa. (b)
Corresponding profile $\psi$ vs $x$ at time $t$. $x_{i}$ shows the position of
a given cluster and its water potential $\psi_{i}$.
$\psi_{\text{ext}}=RT/V_{m}\log(\mathrm{RH})$ is the water potential in the
air stream of imposed relative humidity $\mathrm{RH}$, and
$\psi^{\star}=\psi(x=0)$.

In the configuration sketched in Fig. 8, the colloid mass balance is:

$(h_{d}\varphi_{d}-h\varphi_{0})\frac{\mathrm{d}x_{c}}{\mathrm{d}t}\simeq
h\varphi_{0}\,J\,,$ (4)

$\varphi_{d}$ being the volume fraction of the porous solid, and $h_{d}$ its
thickness, both assumed constant. This relation can be derived strictly from
convection-diffusion models Style and Peppin (2011) (see also ref. Pingulkar
and Salmon (2023) discussing its validity for a non constant evaporation
rate), and includes the variation of height $h\to h_{d}$ at the compaction
front. This relation is consistent with the linear relationship observed
between the compaction front $x_{c}$ and the position of the meniscus $x_{m}$,
see Fig. 3b and eqn (2). As shown by Lesaine et al. using precise dry extract
measurements Piroird _et al._ (2016); Lesaine _et al._ (2021), the volume
fraction of dried layers of similar Ludox dispersions is $\varphi_{d}\simeq
0.64$ for evaporation rates larger than $\dot{E}\geq 10$ nm/s. Values close to
$\varphi_{d}\simeq 0.64$ have also been measured using in situ small-angle
X-ray scattering on similar nanosilica dispersions drying in similar
configurations Boulogne _et al._ (2014); Yang _et al._ (2018). In the
following, we thus assume the same value, and the linear fit shown in Fig. 3b
along with eqn (4) leads to $h_{d}/h\simeq 0.82$. This value corresponds to a
thickness of $\simeq 5~{}\mu$m for the air gap shown in Fig. 8a, and is
consistent with observations at high magnifications of the dried solid across
its height (data not shown).

Estimating the rate of evaporation in the configuration shown in Fig. 8 a
priori requires a three-dimensional model taking into account both the
transport of the water vapour in the air gap downstream of the delamination
front ($x<x_{d}$) and the transport of liquid water through the porous solid
($x<x_{c}$). For this model, we follow an approach similar to that used in
ref. Vincent _et al._ (2016) and Pingulkar and Salmon (2023) based on the use
of the water potential $\psi$ as it allows to describe water independently of
its state: liquid in the dispersion and in the porous solid, to vapor in the
gas phase. We also assume local thermodynamic equilibrium and quasi-static
conditions, i.e., continuity of the water potential. We also consider a porous
solid fully saturated with water, and we do not initially consider the role
played by cracks, see below for a discussion.

The mass flux of liquid water $J^{m}_{l}$ (kg m-2 s-1) through the colloidal
solid is driven by gradients of water potential and described by the Darcy’s
law:

$\displaystyle J^{m}_{l}=-\rho_{w}\frac{\kappa}{\eta_{w}}\nabla\psi\,,$ (5)

with $\kappa$ the permeability of the solid, and $\eta_{w}$ (Pa s) the
viscosity of water. In the gas phase, the mass flux of water $J^{m}_{g}$ (kg
m-2 s-1) is driven by gradients of water vapour concentration $c_{w}$ (kg
m-3), and given by:

$\displaystyle J^{m}_{g}=-D_{w}^{\text{air}}\nabla c_{w}\,.$ (6)

This relationship can be rewritten using the definition of the water potential
in the gas phase:

$\displaystyle\psi=\frac{RT}{V_{m}}\log\left(\frac{c_{w}}{c_{\text{sat}}}\right)\,,$
(7)

leading to:

$\displaystyle
J^{m}_{g}=-D_{w}^{\text{air}}c_{\text{sat}}\frac{V_{m}}{RT}\exp\left(\frac{V_{m}\psi}{RT}\right)\nabla\psi\,.$
(8)

Continuity for the fluid flow (assuming constant density $\rho_{w}$) and
quasi-static conditions in the gas phase imposes $\text{div}(J^{m}_{l})=0$ and
$\text{div}(J^{m}_{g})=0$ respectively, and thus:

$\displaystyle\nabla\cdot\left[-\rho_{w}\frac{\kappa}{\eta_{w}}\nabla\psi\right]=0\,,$
(9)
$\displaystyle\nabla\cdot\left[-D_{w}^{\text{air}}c_{\text{sat}}\frac{V_{m}}{RT}\exp\left(\frac{V_{m}\psi}{RT}\right)\nabla\psi\right]=0\,.$
(10)

Eqn (9) and (10) can be seen as two steady state diffusion equations for the
water potential $\psi$ with the effective diffusion coefficients,
$\rho_{w}\kappa/\eta_{w}$ for the liquid phase and
$D_{w}^{\text{air}}c_{\text{sat}}\frac{V_{m}}{RT}\exp\left(\frac{V_{m}\psi}{RT}\right)$
for the vapour phase, the latter being a function of $\psi$. Boundary
conditions for these equations are given by $\psi(x_{c},t)=0$ (the liquid
dispersion, neglecting the osmotic contribution of the unconsolidated colloids
upstream the compaction front),
$\psi(-\lambda,t)=\psi_{\mathrm{ext}}=RT/V_{m}\log(\mathrm{RH})$ (imposed
relative humidity in the air stream), and impermeability on the glass slide
$\nabla\psi\cdot n=0$. The boundary condition on the PDMS walls is less
obvious as the PDMS matrix is itself permeable to water. We will assume for
simplicity that the transport in the PDMS matrix is negligible, and that the
PDMS walls are almost impermeable (see Section III.6 for a discussion of this
approximation). Eqn (9) and (10) along with the boundary conditions given
above can be solved numerically for a given geometric configuration, such as
the one shown schematically in Fig. 8a. Such a numerical resolution should
then lead to the mass fluxes given by eqn (5) and (8), and thus to the global
evaporation rate $J$. Nevertheless, we show below using simple theoretical
arguments that the vapour transport through the air gap is negligible, and
that the evaporation rate $J$ can be described by a simple one-dimensional
equation.

The ratio of the two effective diffusion coefficients in eqn (9) and (10) is
given by:

$\displaystyle\epsilon=\left.\left(\frac{D_{w}^{\text{air}}c_{\text{sat}}V_{m}}{RT}\right)\exp\left(\frac{V_{m}\psi}{RT}\right)\middle/\left(\frac{\rho_{w}\kappa}{\eta_{w}}\right)\,\right.\,.$
(11)

This ratio compares the mass flux of water through the gas phase with that
through the porous solid for a given water potential gradient. Assuming that
the permeability $\kappa$ of the solid is correctly described by the Carman-
Kozeny relation Rogers and Wirth (2013):

$\displaystyle\kappa_{CK}=\frac{(1-\varphi_{d})^{3}}{45\varphi_{d}^{2}}a^{2}\,,$
(12)

$\kappa_{CK}=3.1\times 10^{-19}$ m2, $\epsilon\ll 1$ even for high water
potential ($\psi=0$). Furthermore, the aspect ratio $h/x_{d}$ in the geometry
shown in Fig. 8a is extremely small and the water potential is therefore
approximately uniform over the channel cross-section, i.e.,
$\psi\simeq\psi(x,t)$. Because $\epsilon\ll 1$, the mass flux of liquid water
through the porous solid is much higher than the water mass flux in air, and
the global mass balance imposes $(hw)\rho_{w}J\simeq(h_{d}w)J^{m}_{l}$. Eqn
(5) then shows that the water potential decreases linearly along the channel
following the Darcy’s law:

$\displaystyle\psi(x,t)\simeq\frac{h}{h_{d}}\frac{\eta_{w}J}{\kappa}[x-x_{c}(t)]\,,$
(13)

as if the air gap due to delamination was not playing a role (negligible
transport through the gas phase). This linear decrease is shown schematically
in Fig. 8b. This assumption is also found in work on confined directional
drying Dufresne _et al._ (2003, 2006), which implicitly assumes that the gas
phase in the cracks and air films due to delamination is in equilibrium with
the liquid phase inside the porous solid. It should be noted, however, that
vapor in equilibrium in the air gap may have a humidity $\mathrm{RH}$
significantly smaller than $1$, due to the potentially low water potential
values, see Section III.5.

The simplified model presented above does not take into account the cracks
observed in the porous solid, see Fig. 2. Yet these cracks literally cut the
solid into several pieces, calling into question the description of a
continuous liquid flow as described by Darcy’s law, eqn (5). Nevertheless, it
is quite conceivable that water is transported through the vapour phase
between the different solid pieces. As the thickness $\xi$ of the cracks is
very small compared to $x_{c}$, $\xi\simeq 20$–$40~{}\mu$m , eqn (9) and (10)
can be used to show that the resistance added by the cracks is negligible
despite the fact that $\epsilon\ll 1$. This confirms the continuous
description of eqn (13) and also probably explains why the appearance of
cracks does not significantly affect the dynamics of both the compaction front
$x_{c}$ and the evaporation rate $J$ in our experiments, see Fig. 3.

###  III.5 Application to experiments

As demonstrated in Section III.3, the water clusters in PDMS allow a rough
estimate of the water potential at their location $x_{i}$ along the channel,
see the inset of Fig. 7. The clusters chosen in Fig. 2 are located very close
to the side walls of the channel, so we can assume they give good estimates of
the water potential in the channel at $x=x_{i}$. As described above, the water
potential profile decreases linearly along the channel from $\psi(x_{c},t)=0$,
see eqn (13), so that we can estimate the permeability $\kappa$ from the
measured values $\psi(x_{i},t)$ at the different clusters using:

$\displaystyle\kappa=\frac{h}{h_{d}}\frac{\eta_{w}J}{\psi(x_{i},t)}[x_{i}-x_{c}(t)]\,.$
(14)

In order to eliminate the noise associated with very small values of water
potential and $x_{i}-x_{c}$, we have only considered data from the inset in
Fig. 7 ranging from $\psi(x_{i},t)=-25$ to $-5$ MPa to estimate $\kappa$ using
eqn (14) and the data on evaporation rate $J$ and compaction front $x_{c}$
shown in Fig. 3. This analysis leads to the $\kappa$ values shown in Fig. 9a.

Figure 9: (a) Permeability values $\kappa$ estimated from eqn (14) as a
function of the position $x_{i}$ of the chosen cluster (see color code in Fig.
2). The errorbars are calculated from the standard deviation estimated for
each cluster. (b) Estimated water potential $\psi^{\star}$ at $x=0$ using eqn
(15) and $\kappa\simeq 1.3\times 10^{-19}$ m2 ($\circ$). The shaded area
corresponds to the same estimates but with a difference of $\pm 0.4\times
10^{-19}~{}$m2 for $\kappa$.

Despite the dispersion, no trend is observed in the permeability estimate,
neither with the position $x_{i}$ of the different clusters, nor with the time
$t$ for a given cluster (not shown). Moreover, we found the same trends and
values for another experiment carried out under the same conditions, and it
allows us to reasonably confirm that $\kappa\simeq 1.3\pm 0.4\times 10^{-19}$
m2. This permeability value is slightly smaller than the one estimated by the
Carman-Kozeny relation, $\kappa\simeq 0.4\kappa_{CK}$, possibly due to the
polydispersity of the nanoparticles. It is worth mentioning that this lower
value of permeability does not affect the hypothesis formulated above for the
ratio of the two effective diffusion coefficients given in eqn (11), as
$\epsilon\ll 1$ even for this value of $\kappa$.

Fig. 9b finally shows the extrapolated water potential at $x=0$:

$\displaystyle\psi^{\star}=-\frac{h}{h_{d}}\frac{\eta_{w}J}{\kappa}x_{c}\,,$
(15)

assuming $\kappa=1.3\times 10^{-19}$ m2. Lidon et al. have also reported
similar measurements of the product $Jx_{c}$ versus $t$ for similar
dispersions Lidon and Salmon (2014), see eqn (15), but we go here a step
further with our measurements of $\kappa$ enabling us to estimate the water
potential at $x=0$. $\psi^{\star}$ first decreases and reaches a plateau for
$t\gtrsim 80$ min around $\psi^{\star}\simeq-27$ MPa, and then slightly grows
for $t\geq 150$ min up to $\psi^{\star}\simeq-17~{}$MPa at $t\simeq
320~{}$min. The plateau observed for $t\gtrsim 80$ min coincides with the
transition between the linear growth rate for the compaction front, $x_{c}\sim
t$, and the slowdown observed on longer time scales, see Fig. 3a.

Also, this transition almost coincides with an apparent change of the texture
of the solid material at its end, see the snapshots in Fig. 6, and movie
M3.avi †. For $t\gtrsim 100$ min, the texture of the porous solid around $x=0$
loses its homogeneity, thus increasing the contrast of the shear bands, which
were barely visible at shorter times Kiatkirakajorn and Goehring (2015); Yang
_et al._ (2015). All these observations suggest that the value
$\psi^{\star}\simeq-27$ MPa, much larger than the water potential imposed in
the air stream $\psi_{\text{ext}}\simeq-220$ MPa, actually corresponds to the
capillary pressure $\psi_{\mathrm{cap}}$ at which air/water nanomenisci invade
the porous solid, leading to a higher optical contrast. It is traditionally
assumed that this air invasion adds a resistance to the water mass transfer,
therefore slowing down the evaporation rate Dufresne _et al._ (2006);
Pingulkar and Salmon (2023). Our quantitative estimate,
$\psi_{\mathrm{cap}}\simeq-27$ MPa (with a range of confidence from $-37$ to
$-20$ MPa because of the uncertainty on $\kappa$, see Fig. 9b) made possible
thanks to our measurements of $\kappa$, is roughly consistent with standard
estimates given by $\psi_{\mathrm{cap}}=-\alpha\gamma/a$, with $\alpha$ a
geometrical prefactor which, according to studies, leads to predictions
ranging from $\psi_{\mathrm{cap}}\simeq-35$ MPa as in ref. Pingulkar and
Salmon (2023) to $\psi_{\mathrm{cap}}=-85~{}$MPa as in ref. Dufresne _et al._
(2003).

###  III.6 Discussions

As soon as air invades the porous solid, the description presented in Fig. 8
is no longer valid, as an additional resistance must be taken into account
because of the transport of the vapour by diffusion across a dry layer of
solid. In particular, the extrapolated values of $\psi^{\star}$ using eqn (15)
are no longer valid either, and our model cannot therefore interpret the rise
in $\psi^{\star}$ observed at long times in Fig. 9b ($t>150~{}$min). In this
capillary-limited regime, i.e., $\psi_{\text{cap}}>\psi_{\text{ext}}$, the
quantitative prediction of the evaporation rate remains an open question. All
theoretical works Wallenstein and Russel (2011); Lidon and Salmon (2014);
Pingulkar and Salmon (2023) consider a one-dimensional description of
directional drying. These models suggest that desaturation takes place from
the tip of the solid at $x=0$, and over a very small length due to the strong
resistance to mass transfer by vapour diffusion through the porous dry layer.
The change of the texture of the solid at long times shown in Fig. 6 suggests
that the invasion of the nanomenisci occurs over a long length scale within
the channel, possibly leading to the progressive desaturation of the porous
solid across its height. These observations challenge the theoretical work
cited above and call for new models and experimental measurements.

Finally, throughout our approach, we have assumed that the channel walls are
impermeable to water, but there may also be water transfer through the PDMS
matrix. Considering typical values for water solubility and water diffusion
coefficient in PDMS Harley _et al._ (2012), it is easy to show using a similar
approach leading to eqn (9) and (10) that the associated mass transfer
resistance is very high, supporting our simplification. However, it is
possible that mass transfer through the PDMS matrix plays a role at long
times, when the driving force of evaporation along the channel becomes weaker
and weaker ($J\to 0$). Indeed, the water bath imposes a water potential
$\psi=0$ at the top of the PDMS layer embedding the channel (thickness
$H=200~{}\mu$m, see Fig. 1). This implies that there is a minute flux of water
from the bath towards the channel, as soon as $\psi(x,t)<0$ in the channel.
This flux could possibly explain the over-slowdown observed for the compaction
front at long times ($x_{c}\sim t^{0.4}$), because it further reduces the
driving force of evaporation within the channel. All these considerations call
for advanced, three-dimensional modeling of mass transfer in this type of
experiment to confirm these assumptions.

##  IV Conclusions and outlooks

The model described in the previous sections, correlated with original
measurements of the water potential, has enabled us to estimate both the Darcy
permeability $\kappa$ and capillary pressure $\psi_{\text{cap}}$ of the porous
solid made of silica nanoparticles. We are not aware of any measurements of
these two quantities by other independent techniques for comparison, but the
measured values are consistent with the theoretical predictions. Our
experimental measurements open up a number of possibilities, in particular for
the quantitative description of drying-induced stresses, as the value of
$\kappa$ plays a fundamental role in poro-elastic modeling Style and Peppin
(2011); Chekchaki and Lazarus (2013); Bouchaudy and Salmon (2019); Hennessy
_et al._ (2022). In our experiments for instance, the knowledge of $\kappa$
enables us to estimate the water potential $\psi$, and therefore the pore
pressure within the solid, a quantity related to the tensile component of the
stresses during drying Style and Peppin (2011); Bouchaudy and Salmon (2019);
Hennessy _et al._ (2022). The data shown in the inset of Fig. 9 correlated
with the observations gathered in Fig. 3 can then be used to link the value of
the pore pressure to the various observed instabilities: shear bands,
delamination, and cracks.

The quantitative description of confined directional drying presented in this
work was made possible by original, but approximate, measurements of the water
potential using (uncontrolled) hydrophilic sites within the PDMS matrix. Such
observations have rarely been reported in the literature (in particular, the
data in Fig. 4 and 5) and open up numerous perspectives that go well beyond
the study of this work. We believe it is essential to better understand these
phenomena, and even to use calibrated inclusions for more precise and
controlled measurements of the water potential, because the significantly
large errorbars of the calibration curve shown in Fig. 5 are likely due to the
polydispersity of the water clusters evidenced in our work. This approach
shares strong similarities with the methods developed by Jain et al. based on
fluorescent nanogels in order to probe in situ water potential in plant leaves
with high precision Jain _et al._ (2021). It seems to us that these innovative
approaches could also open up many new prospects for understanding drying of
colloidal dispersions.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

We acknowledge Solvay, CNRS and the ANR program grant no. ANR-18-CE06-0021 for
financial support.

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