|  Microstructure of the near-wall layer of filtration-induced colloidal
assembly
---|---
|  Mohand Larbi Mokrane,a,b † Térence Desclaux,a † Jeffrey Morris,c Pierre
Josephb and Olivier Liota,b ∗
|  This paper describes an experimental study of filtration of a colloidal
suspension using microfluidic devices. A suspension of micrometer-scale
colloids flows through parallel slit-shaped pores at fixed pressure drop.
Clogs and cakes are systematically observed at pore entrance, for variable
applied pressure drop and ionic strength. Based on image analysis of the layer
of colloids close to the device wall, global and local studies are performed
to analyse in detail the near-wall layer microstructure. Whereas global
porosity of this layer does not seem to be affected by ionic strength and
applied pressure drop, a local study shows some heterogeneity: clogs are more
porous at the vicinity of the pore than far away. An analysis of medium-range
order using radial distribution function shows a slightly more organized state
at high ionic strength. This is confirmed by a local analysis using two-
dimension continuous wavelet decomposition: the typical size of crystals of
colloids is larger for low ionic strength, and it increases with distance from
the pores. We bring these results together in a phase diagram involving
colloid-colloid repulsive interactions and fluid velocity.

##

††footnotetext: a Institut de Mécanique des Fluides de Toulouse, Université de
Toulouse, CNRS, Toulouse, France.††footnotetext: b LAAS-CNRS, Université de
Toulouse, CNRS, Toulouse, France.††footnotetext: c Levich Institute and
Chemical Engineering, CUNY City College of New York, New York,
USA.††footnotetext: † Co-first: both authors contributed equally to this
work.††footnotetext: ∗ Corresponding author<EMAIL_ADDRESS>
##  1 Introduction

Filtration is a central process in many fields of application, such as water
treatment or bioprocessing. Membrane fouling and the induced hydraulic
resistance enhancement are key problems for improving filtration processes
(power consumption, membranes’ lifetime). When a suspension flows through a
membrane, many objects (colloids, cells, bacteria, aggregates) can accumulate
at the membrane surface. The efficiency of the filtration decreases as the
hydraulic resistance induced by accumulated particles rises. This has a
dramatic impact on oil recovery1, ink-jet printing2, biodetection3 or water
infiltration in soils4 and even in brain diseases5. For decades, filtration
studies have focused on membrane scale and global hydraulic response6, 7 or
membrane materials8. Since the advent of microfluidics at the beginning of the
21st century, the study of microfiltration has become a very active field of
research9. The use of microfluidic devices is indeed an efficient way to
understand the key microscopic mechanisms involved during the filtration
process. Wyss _et al._10 were precursors in this domain, followed by many
others, as detailed in a recent review11.

Different strategies and scales have been explored. At particle level, the
first steps of clogging process have been studied using confocal microscopy
12, 13. Adhesion of one particle on a pore surface involves different
parameters or mechanisms such as pore geometry14, 15 or Brownian motion16. At
pore scale, many studies proposed experimental and/or theoretical-numerical
approaches10, 17, 18, 19, 20, 21 to clogging dynamics. The upscaling of pore-
scale results towards membrane scale is a recent development using parallel
pore arrays22, 23, 21 or more complex porous-like media24, 25. This revealed
some complex mechanisms of “cross-talk” between pores during filtration
processes. Especially, previous work of our team22 revealed that pore
redistribution by Brownian diffusion accelerates the clogging process. The
proposed model is based on assumptions about clog hydraulic permeability.

In such (sub-)micrometric devices, direct measurements of flow rate under
given pressure drop are not accessible using commercial apparatus. Some
indirect measurements are possible26. But a direct analysis of the clog micro-
structure can provide insights on clog permeability. Up to now, the first
studies of pore microstructure have been performed for soft particle
microfiltration such as microgel27, 28, 29, 30. These studies focused on
microgels deformation, but they observed different structures of the
filtration cake with amorphous or crystalline behaviours depending on forcing.
To our knowledge, there is no systematic study of colloidal clog
microstructure.

Contrary to deformable microgels, colloidal particles of the form we study are
essentially rigid and the physico-chemical surface properties are crucial to
understand the way they accumulate under external forcing. At colloidal scale
($\lesssim 1\,\mu$m), typical interaction scales are similar to particle
diameter. In addition, Brownian diffusion brings an extra energy source which
can allow energy barrier crossing necessary for adhesion; or influence
particle spatial organisation31.

The Derjaguin, Landau, Verwey and Overbeek (DLVO) theory, consisting of a
superposition of van der Waals interactions and electrostatics in an aqueous
solution, is the standard approach to explain colloid-colloid or colloid-wall
interactions32, 33, 34, 35.

The repulsive colloid-colloid or surface-colloid interactions are caused by
Electrical Double Layer (EDL) interactions33, 34. The zeta potential $\zeta$
corresponds to the electric potential at the slip plane between the two sub-
layers composing the EDL. The typical thickness of the total layer, the Debye
length, corresponds to electrostatic surface charge screening. It is expressed
as:

$\kappa^{-1}=\sqrt{\dfrac{\epsilon_{r}\epsilon_{0}k_{B}T}{e^{2}\sum_{i}\rho_{\infty,i}z_{i}^{2}}},$
(1)

where $\epsilon_{r}$ is the relative dielectric constant of the fluid,
$\epsilon_{0}$ the vacuum electric permittivity, $k_{B}$ the Boltzmann
constant, $T$ the temperature, $e$ the elementary charge, $\rho_{\infty,i}$
the bulk concentration of ion $i$ and $z_{i}$ its valence. Consequently, the
Debye layer is inversely proportional to bulk concentration to the power
$1/2$: the more concentrated the solution, the thinner the EDL. The ionic
strength is defined as:

$I=\dfrac{1}{2}\sum_{i}\rho_{\infty,i}z_{i}^{2}.$ (2)

So we can write $\kappa^{-1}\propto 1/\sqrt{I}$.

The total interaction energy is the combination of electrostatic and van der
Waals interactions. When the distance between two objects is reduced, a
secondary minimum appears (which is known to have a significant impact on
filtration36). The energy profile contains a primary minimum at very low
inter-particle distance. A particle in this primary minimum is at a much lower
energy than that needed to escape, thus it is adhered through van der Waals
interactions.

The second main mechanism at play in assembling colloids is Brownian motion.
When a colloid is advected by surrounding flow, the competition between
advection and Brownian diffusion can be interpreted using the dimensionless
Péclet number:

$Pe=\dfrac{\text{advective transport}}{\text{diffusive
transport}}=\dfrac{3\pi\eta d^{2}U}{k_{B}T},$ (3)

with $\eta$ the fluid viscosity, $d$ the particle diameter and $U$ the typical
advection velocity. A Péclet number smaller than 1 means that diffusion is
more important than advection, which can easily be reached when colloids are
advected at a decreasing velocity during membrane fouling process. This
mechanism (shear, Brownian motion) was added to the DLVO theory37, 38, as well
as hydrogen bonds39, to build an extended DLVO theory (XDLVO).

This theoretical framework is essential to study microfiltration of colloids.
In this work we use an experimental approach to investigate the microstructure
of filtration-induced colloidal assembly in model microfluidic systems. We
flow a suspension which progressively clogs the pore. Spinning-disk confocal
microscopy is used to determine individual positions of accumulating particles
during clogging, for different ionic strengths and applied pressure drops. We
propose an analysis of the global clog properties before focusing on more
local tools to understand the heterogeneity in the spatial organisation of the
colloidal particles.

##  2 Experimental methods

We use a model-system approach in order to tightly control the different
parameters such as pore dimension, colloid size and properties or ions
concentration.

###  2.1 Particle suspension

Our experiments consist in flowing a suspension through microfabricated slits
to observe accumulation of colloids at their entrance. We use polystyrene
beads (diameter $d=1\pm 0.02\,\mu$m, density 1.05 g.cm-3) volume-loaded with
fluorophore, so they can be easily observed using fluorescence microscopy.
Very nearly monodisperse particles used insures that the particle diameter
distribution should not influence on clog structure. The particle surfaces are
carboxylate-modified leading to a negative surface charge, with measured zeta-
potential near $-50\,$mV ($pH\approx 6$); measurements performed by dynamic
light scattering on a _Malvern_ “Zeta-Sizer” instrument.

We use suspensions with volume fraction of $\phi=4\times 10^{-5}$. For the 1
$\mu$m-diameter particles we use, this corresponds to $7.9\times 10^{4}$
particles per $\mu$L. By adding potassium chloride, two ionic strengths are
used to compare repulsive interaction effects: $I=0.5\,$mM and $I=5\,$mM. This
corresponds to $\kappa^{-1}=13\,$nm and $\kappa^{-1}=4\,$nm respectively. Note
that for the carboxylate-modified latex particles we use, the Critical
Coagulation Concentration in presence of ions K+ is $51$ mM40. Thus we work in
a salt concentration range far from spontaneous particle coagulation. The
measured pH of the suspension is about 6.

In the ionic strength range we used, the force profile between two
carboxylate-modified latex particles versus interparticle distance reveals an
electrostatic barrier whose height is about 0.5–1 nN, and without secondary
minimum41. If we consider the glass-particle interaction, the profile is very
similar with a barrier of the same order of magnitude42.

###  2.2 Microfluidic device

A two-level microfabricated device is used to mimic membrane constrictions.
Four parallel slits of cross-section $w\times h=5\times 3.2\,\mu\text{m}^{2}$
and length $L=50\,\mu$m are formed using plasma etching in silicon. Separated
by 50 $\mu$m (center to center), they are supplied using inlet and outlet
microfluidic channels (depth $18\,\mu$m, width $300\,\mu$m). A
170$\,\mu$m-thick borosilicate glass plate covers the silicon-etched wafer,
sealed using anodic bonding. In our pH and saline conditions, the zeta
potential of microfluidic chip materials is about -30 mV for silicon43 and -55
mV for borosilicate44. In the range of ionic strength we consider, surface
charge of both silicon and borosilicate are not fully screened45, 44.
Consequently, the EDL interactions between surfaces and beads will be
repulsive. The design is presented in figure 1 (top). Due to the manufacturing
process, the slits connect the corners of the cross-section of the inlet and
outlet microchannels. Figure 1 (bottom) shows a bright-field micrograph of the
microfluidic device we used. At 40$\times$ magnification the slits can easily
be observed.

Fig. 1: Top: sketch of the microfluidic chip, slits (red) connect the
microfluidic channel (blue); slits are connected to microchannel corners. Top
view is on the left, side view is on the right. Bottom: micrograph of the
slits, 40$\times$ magnification.

The flow is imposed using a _Fluigent_ pressure controller in the range
$\Delta P=$10–100 mbar with precision under 0.1 mbar. In the absence of any
clog, this corresponds to a flow rate of $Q=$37–370 nL.min-1. Note that the
hydraulic radius $r_{h}$ of one slit and of the microchannel are related as:

${r_{h}^{microchannel}=8.7\,r_{h}^{slit}}$ (4)

Thus, as the hydraulic resistance scales as $r_{h}^{-4}$, we consider that the
applied pressure drop is entirely a result of flow through the slits.

###  2.3 Clog observation

The clogging dynamics and clog structure are observed using confocal
fluorescence microscopy (63$\times$ magnification). Unfortunately, because of
non-transparency of the beads, we cannot access the 3D structure of the clogs.
Confocal microscopy enables removal of background light originating from out-
of-plane particles. Thus, the particles at the wall layer are well resolved
(see figure 2), permitting accurate particle detection.

Clog formation is recorded with a frequency of 2 frames per minute. Figure 2
shows a sequence of images at increasing time under flow of the clog
formation, recorded by confocal microscopy. Clogging is stopped after 2400 s
for each experiment, so that a similar amount of suspension flows through the
device during the experiment. The clog growth dynamics is beyond the scope of
this article; some insights into this topic can be found in our previous
work22. The “final” size of a clog will depend on the (stochastic) first steps
of the clogging process. Consequently, when the experiment is stopped, some
clogs have reached their maximal size whereas other are still growing. We use
custom _Python_ scripts to analyze the images and allow detection of both clog
contour and particle position with sub-pixel precision. Figure 3 shows an
example of particle detection within a clog.

Due to the inability to unclog microfluidic devices in a satisfactory way,
each experiment was made in a new device, adding complexity to the
experimental process. A total of 11 experiments are usable, at different
pressure drops and two ionic strengths. Three to four clogs are visible on
each experiment, leading to about 40 different clogs. Each point and curve
presented below is an average of three or four clogs extracted from the same
experiment.

Fig. 2: Confocal micrography timelapse of the clog formation. Red lines
represent the slits limits. $\Delta P=20\,$mbar and $I=5\,$mM. The small clog
visible at the right-hand pore entrance ($t=0\,$s) is due to particles flowing
in the device during experiment installation. It will not affect the clog
microstructure, only the clogging trigger. Fig. 3: Detection of the clog
particles, with zoom on a portion of the clog. Red points correspond to
detected particles’ position.

##  3 Results

Since we can only access the wall layer – the one in contact with the
borosilicate glass plate – a bias on spatial organisation could be induced by
this wall. However the similar zeta potential of colloids and wall should
limit this bias, if it exists, to geometric and confinement factors. Moreover,
the trends shown here are generic to filtration processes, because they are
related to particle/particle interactions, of the same nature as the ones with
the walls of our devices. Observations detailed below remain related to
physical and physico-chemical phenomena at play in filtration. Two main
parameters are discussed: applied pressure drop and ionic strength.

###  3.1 Global wall layer properties

We start our analysis with global properties of the wall layer of the clog
obtained by microfiltration. Two attributes can be computed: apparent porosity
and radial distribution function.

We define the clog – at least the wall layer – apparent porosity as:

$\epsilon=1-\dfrac{V_{particles}}{V_{clog}}.$ (5)

$V_{particles}$ is the total volume of particles included in the wall layer of
the clog. If we note $N$ the number of particles, we get $V_{particles}=N\pi
d^{3}/6$. The wall layer clog volume $V_{clog}$ is estimated as the volume
defined by the clog area $A_{clog}$ on a thickness $d$: $V_{clog}=dA_{clog}$.
Finally we have:

$\epsilon=1-N\dfrac{\pi d^{2}}{6\,A_{clog}}.$ (6)

Both $N$ and $A_{clog}$ are computed directly from the clog pictures (figures
2 and 3). Figure 4 shows the apparent porosity for different applied pressure
drops and two ionic strengths (0.5 mM and 5 mM). We do not observe a visible
trend when changing $I$ and applied pressure drop which means that the mean
porosity of the wall layer is not affected by these two parameters and remains
in the range 0.42–0.52. It is a bit higher than the expected porosity for a 2D
random packing of monodisperse spheres. Intuitively, applying relatively weak
pressure drop allows particles to find a better location, so it could lead to
a less porous clog, which is not the case for these results. Actually, when we
study the local porosity (see next section), some spatial heterogeneities
appear.

Fig. 4: Apparent porosity of the wall layer of the clog, as a function of the
applied pressure drop and for two different ionic strengths. The two dashed
lines represent the random close packing and minimal porosity for a 2D
assembly of monodisperse spheres.

To go further than the ratio between void and solids in the clog, we analyse
the spatial organisation of the particles in the wall layer of the clog. We
consider the Radial Distribution Function (RDF) which gives insights about
average order at a distance $r$ from any particle. It can be expressed as (see
e.g. Saw _et al._46):

$g(r)=\sum_{i=1}^{N}\dfrac{\psi_{i}(r)/N}{(N-1)\left(\dfrac{dS_{r}}{A_{clog}}\right)},$
(7)

where $\psi_{i}(r)$ is the number of particle centers included in a shell
between $r-\delta r$ and $r+\delta r$. $N$ is the the total number of
particles in the clog of area $A_{clog}$. $dS_{r}$ represents the shell
surface. Edge effects represent the main limitation of this formula. Our clogs
have indeed specific and non-regular shapes. To bypass this problem, we
adapted a method proposed by Larsen & Shaw47.

Fig. 5: Radial distribution function for two applied pressure drops at
$I=0.5\,$mM. Inset: RDF conditioned by direction angle (see text for details).
Fig. 6: Radial distribution function for two applied pressure drops at
$I=5\,$mM. The blip observed at $r=0.5\,\mu$m is probably related to aliasing.

We computed the RDF for different configurations $(I,\Delta P)$ on the whole
cake’s wall layer formed at the pore entrance. Figures 5 and 6 show the RDF
for $(I=0.5\,\text{mM},\leavevmode\nobreak\ \Delta P=[30;80]\,\text{mbar})$
and $(I=5\,\text{mM},\leavevmode\nobreak\ \Delta P=[10;80]\,\text{mbar})$
respectively. Inset in figure 5 shows the RDF ($\Delta P=80\,$mbar,
$I=0.5\,$mM) computed for different orientations. The angular domain is
divided in 9 segments and nine RDF are computed by selecting only particles in
each angular segment. This leads to nine RDF. Results are angle-interpolated
to obtain a radial chart where color represents the $g(r)$ amplitude. This
shows an isotropy of colloids spatial organisation.

For the total RDF, we do not observe a clear difference when changing the
applied pressure drop at fixed ionic strength. The comparison of $I=0.5\,$mM
and $I=5\,$mM (not shown here) at given pressure drop does not reveal notable
difference either. The small differences in peak positions and heights,
visible in figure 5, are difficult to analyse because of RDF resolution.
Nevertheless, these RDF can give some insights about average spatial
organisation of particles in a filtration clog/cake. The oscillations observed
on the RDF of a 2D amorphous crystal can be modeled as48, 49, 50:

$g(r)-1=\dfrac{K}{r^{1/2}}\exp\left(-\dfrac{r}{\lambda}\right)\sin\left(\dfrac{2\pi
r}{D}+\phi\right),$ (8)

where $K$, $\lambda$, $D$ and $\phi$ are constant coefficients. $D$ represents
the oscillation period while $\lambda$ is a “screening length” quantifying the
decay of a local spatial organisation (not to be confused with the Debye
screening length). The higher $\lambda$, the more spatially organized the
material. This expression is an asymptotic behaviour and is a signature of the
medium-range order of the considered 2D assembly. For this reason, we fit the
experimental RDF excluding the first peak. Inset in figure 7 shows an example
of fitted RDF (actually of $r^{1/2}(g(r)-1)$).

Fig. 7: Screening length as a function of applied pressure drop for the two
different ionic strengths. Error bars correspond to fit’s 95%-confidence
bounds. Inset: normalized RDF for $\Delta P=80$ mbar and $I=0.5\,$mM. The
solid line represents the fit obtained with equation 8.

Figure 7 presents the screening length $\lambda$ versus the applied pressure
drop $\Delta P$ for $I=0.5$ mM and $I=5\,$mM. Two results emerge from this
plot. First, the screening length is higher for low ionic strength. Second,
screening length seems to decrease at high pressure drop. This could be the
signature of the competition between hydrodynamic forcing and electrostatic
repulsion. At low $I$ and $\Delta P$, particle adhesion will be prevented or
delayed allowing for colloids to self-organize, whereas at high $I$ and
$\Delta P$, adhesion is facilitated which leads to a more amorphous clog. This
interpretation will be detailed in the section 4. In fact our results averaged
on the whole clog’s wall layer are rather dispersed. This is due to
heterogeneity of the clogs: very amorphous regions coexist with perfectly
crystalline ones. This was already observed in filtration cakes during
microgel filtration27. However our situation is quite different because we
study hard particles, and physico-chemical colloid-colloid interactions differ
widely from microgel-microgel ones.

Fig. 8: Angularly averaged local porosity for (left) $I=0.5\,$mM and (right)
$I=5\,$mM as a function of the distance from the pore entrance. The
uncertainty range corresponds to standard deviation obtained with several
clogs in the same experiment (when possible). Short plateau for small $r$ is
an extrapolation of the smallest exploitable radius $r$.

###  3.2 Local analysis

In order to quantify these heterogeneities, we propose a local analysis of
clog microstructure, considering the local porosity and local colloid
organisation.

####  3.2.1 Clog local porosity

Fig. 9: Examples of 2D map of local porosity $\epsilon_{loc}$. (a)
$I=0.5\,\text{mM}$ and $\Delta P=30\,\text{mbar}$; (b) $I=5\,\text{mM}$ and
$\Delta P=10\,\text{mbar}$. Red lines represent pore edges. At the pore
entrance, uniform patch is a consequence of the local porosity processing. The
radial approach does not permit to detail the porosity at this location, and
the porosity at $r=0$ is replicated on a surface corresponding to the surface
of $60\,\mu$m2 used to compute local porosity.

The averaged porosity measurements presented above are not sufficient to
understand in detail the clog microstructure and underlying physical and
physico-chemical mechanisms at play in colloid assembly under filtration. We
propose a systematic study of the local porosity $\varepsilon_{loc}$ of each
clog. We define the local porosity at a given point of the clog (in polar
coordinates $\overrightarrow{r}$) as the porosity in a radially-oriented
curved-trapezoidal box of approximately $60\,\mu$m2 centered on
$\overrightarrow{r}$ (corresponding to a 8-colloid wide box). Figure 9 (a) and
9 (b) show two examples for $(I=0.5\,\text{mM},\Delta P=30\,\text{mbar})$ and
$(I=5\,\text{mM},\Delta P=10\,\text{mbar})$ respectively. One can observe
qualitatively that the porosity decreases with the distance from the pore
entrance, with a porosity divided by approximately 2. This can be counter-
intuitive as a compression of the clog could happen during the clogging
process. We did not observe such a compression. It can be easily attributed to
van der Waals adhesion of the particles which prevents them from moving.

We perform an angular average at fixed $r$ of local porosity to obtain the
porosity as a function of the distance $r$ to the pore entrance. Note that
only locations whose absolute value of the $x$-projection of
$\overrightarrow{r}$ is lower than half of inter-pore distance (25 $\mu$m) are
selected. This prevents taking into account parts of the cake in the
“influence region” of a neighbour clog. We made this first-order choice
because we do not have information of the influence of clog interaction on
their microstructure. Interactions between clogs (overlapping for instance)
could locally affect the velocity field and so the local hydrodynamic forcing.
This could lead to more dispersion of the results. Figure 8 shows this
quantity denoted $\langle\varepsilon_{loc}\rangle_{\theta}$ as a function of
$r$ for $I=0.5\,$mM and $I=5\,$mM, and various $\Delta P$. When possible, this
quantity is averaged over several clogs in the same experiment. One observes
systematically a decrease of clog local porosity in the range
$r\in[0,12]\,\mu$m. Furthermore, figure 8 (right) does not show significant
difference when changing the applied pressure drop, especially at low $r$. For
$I=0.5\,$mM, there is more variability of the local porosity for
$r\in[0,12]\,\mu$m. It reveals the stochastic facet of colloid-surface
adhesion and the initial steps of the clogging process, as discussed in
previous works15, 12, 16. The way and position first particles stick on the
surface will affect the first portions of the clog and change locally the
porosity (and so the global clog permeability). In summary, porosity is shown
to be higher close to the pores, but without any strong systematic influence
of ionic strength or applied pressure drop.

####  3.2.2 Colloid spatial organisation

Fig. 10: Examples of 2D Fast Fourier Transform applied to different regions of
a clog ($I=0.5\,$mM, $\Delta P=30$ mbar). Predominately hexagonal, amorphous
and square organization is seen, from left to right.

A first way to study local organisation of clogs consists in computing the 2D
Fourier transform of different portions of the clogs’ wall layer. Figure 10
shows three examples of power spectra computed by 2D Fast Fourier Transform
(FFT) on three different regions of a clog. One can differentiate three
different local organisations: hexagonal, square and random. For each, the 2D
FFT reveals a typical signature. Around the central spot, secondary spots
separated by an angle specific to the lattice (60o for hexagonal arrangement
and 90o for square arrangement) appear for the organized regions. On the
contrary the more amorphous region reveals a kind of annular 2D FFT without
spatial regularity.

The power spectrum obtained by 2D FFT will depend on the window dimension
chosen for computation. To generalize this reciprocal-space analysis, we use a
wavelet decomposition which is better suited for the local analysis aimed
here. While Fourier decomposition uses a base with infinite-space support
(sine and cosine), wavelet decomposition is a projection on a base composed of
finite-space support elementary functions. Since we have a heterogeneous
spatial organisation of the clogs, a Fourier transform does not allow a more
quantitative local analysis of the local microstructure than figure 10.
Continuous Wavelet Transform (CWT) overcomes this difficulty. Each point of
the image and its surrounding environment can be decomposed in functions which
generate a basis: the wavelets***They correspond to the sine and cosine
functions in Fourier decomposition.. They are localized both in space and
frequency51. Wavelet transformation is a widely used technique in many fields
such as image processing52 or glass structure analysis53, 54. Each wavelet is
built from a single “mother” function $\psi(\overrightarrow{X})$. A specific
rotation, translation and expansion generates each wavelet for a given
position $\overrightarrow{X}=(x,y)$ :

$\Psi_{s,\Theta}(\overrightarrow{X})=\dfrac{1}{s}\psi\left(R_{\Theta}^{-1}\left(\dfrac{\overrightarrow{X}}{s}\right)\right),$
(9)

with

$R_{\Theta}=\begin{pmatrix}\cos\Theta&-\sin\Theta\\\
\sin\Theta&\cos\Theta\end{pmatrix}.$ (10)

$s$ is the expansion factor (or period) of the wavelet and $\Theta$ the
rotation factor. We have chosen the classical 2D Morlet wavelet as the
“mother” wavelet. It is a complex function which conserves phase information
with good angular selectivity55. So it is a good candidate to study
crystalline arrangements. A detailed explanation of the use of 2D-CWT is
available in Chen & Chu (2017)56. Figure 11 shows an example of 2D Morlet
“mother” wavelet.

Fig. 11: Example of 2D Morlet wavelet used for wavelet analysis; $\Theta=0$
and $s=1\,\mu$m.

The result of applying a CWT on an image is a 2D map with angle $\Theta$ in
abscissa and period $s$ in ordinate. Some examples are shown in figure 12.
Whereas for a Fourier decomposition, the power spectrum obtained from the
transform can be plotted on a classic graph, coefficients derived from CWT are
two-variable functions. Consequently, color on insets of figure 12 represents
the power that is carried by each elementary function
$\Psi_{s,\Theta}(\overrightarrow{X})$ for a given $(\Theta,s)$. The higher the
power, the more important the corresponding elementary function contributes to
the picture (2D signal). One can observe that for a point selected in an
apparently hexagonal-lattice region, yellow spots appear with a periodicity of
about 60o. The period corresponds to the typical inter-particle distance,
which is a bit under 1$\,\mu$m (one particle diameter). The same observation
can be made for an apparently square-lattice region (periodicity of 90o). For
an amorphous region, no regular pattern appears on the CWT. Continuous Wavelet
Transform thus enables extraction of quantitative information on local typical
period and angular distributions.

Fig. 12: Examples of Continuous Wavelet Transform applied to different
locations of a clog ($I=0.5\,$mM, $\Delta P=30$ mbar). The colorbar represents
the power and is common to the three plots. $\langle\cdot\rangle_{s}$ means
the average of the power spectrum amplitude on the period $s$.

The results obtained from CWT can be averaged along “period” or “angle”
direction. The “period” averaging gives a peak whose full width at half
maximum gives some information on typical inter-particle distance
distribution. That gives information very similar (not shown here) as the
local porosity analysis (figure 8). The average on the angle provides a
succession of peaks, especially when a crystalline lattice is detected. Then
we are able to compute the mean inter-peak distance. This quantity allows
separation of three distinct regions: amorphous, hexagonal-lattice and square-
lattice. Practically, hexagonal lattice corresponds to mean inter-peak angular
distance in the range 55-65o and square lattice corresponds to mean inter-peak
angular distance in the range 85-95o. Other angular distances are considered
as related to amorphous regions. Figure 13 shows an example of a map obtained
from this analysis (for two different ionic strengths). One can observe that
hexagonal-lattice regions are largely predominant compared to square-lattice
ones. Moreover, for $I=0.5\,$mM the hexagonal-lattice regions seem larger when
the distance from the pores increases, whereas for $I=5\,$mM, they are quite
homogeneously distributed.

Fig. 13: Map of structures determined by CWT for $\Delta P=80\,$mbar, (a)
$I=0.5\,$mM and (b) $I=5\,$mM. Blue regions correspond to hexagonal lattice
and red ones to square lattice. White regions are amorphous. Black overhangs
at the bottom of the cakes face pore entrances. Note that the white strip
around the cake corresponds to a non-analyzed zone due to CWT edge effects.

In fact, maps presented in figure 13 are noisy. A lot of locations fit with
crystalline zones whereas their typical size is not compatible with such a
description. To denoise the maps, we apply an erosion-dilatation algorithm.
Each blue or red connected region is eroded from the edge over 0.5$\,\mu$m
(corresponding to one particle diameter total erosion). Remaining regions are
then dilated by the same length. This allows us to isolate the sufficiently-
extended crystalline-lattice regions.

To perform a local analysis of colloid arrangement in the filter cake, we
adopt the same method as we used for porosity. We use polar coordinates with
origin at pore entrance. We define shells of radius $r$ for $\theta\in[0,\pi]$
with thickness $\delta r\sim 6\mu$m. In this shell, we can define the
probability to have points included in a crystalline region
$P^{\theta}_{\text{crystal}}(r)$. Note that only locations whose absolute
value of the $x$-projection of $\overrightarrow{r}$ is lower than half of
inter-pore distance (25 $\mu$m) are selected. Again, the basis for this is
that it prevents taking into account parts of the cake in the “influence
region” of a neighbour clog, to count one same position for two different
clogs. Figure 14 shows this quantity as a function of $r$ for (left)
$I=0.5\,$mM and (right) $I=5\,$mM, and different $\Delta P$.

Fig. 14: Probability to encounter a crystalline region for (left) $I=0.5\,$mM
and (right) $I=5\,$mM as a function of the distance from the pore entrance.
The uncertainty range corresponds to standard deviation obtained with several
clogs in the same experiment (when possible). Legend is common to both plots.

One observes again a systematic inhomogeneity of the clog microstructure. The
probability to encounter crystalline regions increases with $r$ then reaches a
plateau. As we move away from the pore entrance, the clog microstructure is
more and more crystalline. For $I=0.5\,$mM, this plateau is between 0.3 and
0.4, and seems to decrease as $\Delta P$ rises. For $I=5\,$mM, the difference
when changing $\Delta P$ is less clear, but globally the plateaus are around
0.2, lower than for $I=0.5\,$mM.

##  4 Discussion: regimes of particle adhesion

The results presented above are split in two categories: global approach and
local analysis. The mean cake porosity is in the range 0.42–0.52 and does not
depend on applied pressure drop or ionic strength. This averaged approach
suggests that there is no influence of the forcing and physico-chemical
properties on clog microstructure. However, the local analysis gives some
decisive insights. Clogs are more porous at the vicinity of the pore entrance,
then become more and more compact before reaching a plateau at a distance
larger than $\sim 12\,\mu$m from pore entrance. The porosity for small $r$ is
clearly higher for low ionic strength than for $I=5\,$mM. This heterogeneity
is totally concealed when one makes an average on the whole cake’s wall layer.

Concerning the spatial organisation, a more refined study of radial
distribution function shows a subtle effect: the screening length –
corresponding to the medium-range order – is higher at low ionic strength and
seems to decrease at high applied pressure drop. A local wavelet decomposition
reveals crystalline regions – mainly hexagonal lattices. The crystalline-
region proportion increases with the distance from the pore and the plateau is
higher for $I=0.5\,mM$. Similar results were obtained numerically by Agbangla
_et al._18. In the absence of inter-particle repulsive forces, particles are
aggregated cake-like, without visible crystalline organisation. Addition of
repulsive forces lead to formation of more organised arches of particles at
the pore entrance.

We propose an interpretation based on competition between hydrodynamic forcing
and colloid-colloid repulsive interactions. To cross the energy barrier and
cause adhesion of two colloidal particles, an external forcing is necessary.
It can be provided by drag force due to hydrodynamic advection. Note that this
drag force should be influenced by hydrodynamic interactions57, 58 which
appear when a particle is travelling close to the wall or near another
particle. It could affect the energy provided by the advection, at a distance
up to several particle radii. Extra energy can also come from Brownian motion.
Let us consider a particle arriving in contact with the clog. We propose three
different mechanisms:

  1. 1.

ballistic regime: the drag force applied on the impacting colloid is large
enough to drive direct adhesion with colloid(s) in the clog;

  2. 2.

diffusive regime with adhesion: the drag force applied on the impacted colloid
has decreased. Brownian motion however causes the particle to explore local
configurations where the flow is still strong enough to lead to adhesion;

  3. 3.

diffusive regime without adhesion: there is not enough energy to observe
colloid-colloid adhesion, the impacting colloid is simply constrained to be
part of a “repulsive glass”.

The consequences on porosity and spatial organisation can be summed up as
follows for each regime:

  1. 1.

there is no time for particles to self-organize so porosity is high and
structure is amorphous;

  2. 2.

Brownian motion allows a particle to explore a larger energy landscape before
adhesion, and to find a more constrained position. This leads to less porous
and more crystalline structure;

  3. 3.

the structure is similar to regime 2, but porosity could be slightly lower.

Let us insist about the role of Brownian motion. It cannot bring extra energy
sufficient to cross DLVO energy barrier and lead to particle adhesion: the
typical Brownian motion energy is $\sim k_{b}T$ whereas the energy barrier is
few tens of $k_{b}T$59. Consequently, in regime 2, adhesion is still provoked
by the drag force. But Brownian motion acts as a repulsive force allowing
fluid to lubricate the interactions. Numerical simulations showed that
Brownian motion facilitates yield of a colloidal gel under shear stress60, 61.
This is the way a Brownian particle could access a larger energy landscape and
help the clog self-organization. This statement is rather difficult to access
experimentally by the methods we used, and it should deserve a specific study
with higher acquisition framerate, possibly higher spatial resolution and
specific designs.

Two other mechanisms could stimulate appearance and stability of the crystal
zones, especially for the regime 3 (diffusive regime without adhesion). Larsen
& Grier62 showed that long-range attractive interactions between like-charge
colloids can allow the formation of metastable colloidal crystals. Moreover,
presence of a wall lets appear attractive interactions between two colloids.
Their study was made with very similar colloids (size, surface charge) as we
used, but with an ionic strength 100 times lower than the ones we fixed (much
larger repulsive interactions). Consequently we can assess that these
mechanisms should play a role in the appearance of this third regime, when the
relative contribution of repulsive interactions becomes dramatic.

Figure 15 shows a schematic phase diagram for these three regimes. Two
parameters define the three regions of the phase diagram, corresponding to the
three regimes described below: (i) hydrodynamic forcing (flow rate) and (ii)
electric repulsive interactions (decreasing with increasing ionic strength).
For weak repulsion and high hydrodynamic forcing, ballistic regime is
dominant. For high repulsion and weak hydrodynamic forcing, diffusive regime
without adhesion is dominant. For intermediate repulsion and hydrodynamic
forcing, one can observe diffusive regime with adhesion. A change in Brownian
motion intensity (determined by particle size, temperature, fluid viscosity)
could slightly move the boundaries of the diagram.

Fig. 15: Schematic diagram phase representing the three regimes for different
ionic strength and hydrodynamic forcing. Shaded particle edges represent the
EDL. Their overlapping mimes particle adhesion. Arrows represent possible
displacements in the phase diagram during an experiment.

We transpose this interpretation to our data. If we look at figure 7, the
higher screening length (and hence the increased order) for $I=0.5\,$mM than
for $I=5\,$mM could indicate a regime difference: diffusive (with or without
adhesion) for $I=0.5\,$mM; ballistic for $I=5\,$mM. Furthermore, the plateau
of $P^{\theta}_{\text{crystal}}(r)$ (figure 14) is higher for $I=0.5\,$mM than
for $I=5\,$mM. This is consistent with a more organized microstructure, and so
with the predominance of a diffusive regime at high interparticle
electrostatic repulsion, and of the ballistic regime at high ionic strength.
For $I=0.5\,mM$, the plateau seems lower as $\Delta P$ rises. This is also
consistent with the appearance of a ballistic regime.

We could hope for sharp transitions when changing ionic strength or pressure
drop. Actually, hydrodynamic forcing conditions are time-dependent. Since we
work at fixed pressure drop, the flow rate decreases as the clog grows. Such
situation where pressure drop is fixed can be encountered in real crossflow
filtration devices. Consequently, velocity (and so drag force) of an arriving
particle will decrease with time. For a given experiment, salt concentration
is fixed, and thus the resulting double layer repulsion is also fixed. The
phase diagram is traversed along a horizontal line from right to left. Two
possible trajectories in the phase diagram are proposed in figure 15 for two
different ionic strengths. Consequently, a regime transition is expected
during the clogging process. This is consistent with local analyses. The
decrease of $\langle\varepsilon_{loc}\rangle_{\theta}$ with $r$ for both
$I=0.5\,$mM and $I=5\,$mM is compatible with such a transition (see figure 8).
Moreover, the increase of $P^{\theta}_{\text{crystal}}$ with $r$, as shown in
figure 14, reveals an increase of crystal regions proportion. This is also
consistent with a transition inside the clog, from ballistic to one of the two
diffusive regimes.

We globally observe some dispersion when changing the applied pressure drop.
We should be able to extract a typical length of the local porosity decrease
with $r$ – and of the crystal proportion increase. Such a characteristic
length should be dependent on $I$ and $\delta P$, which is not obvious on
presented plots (figures 8 and 14). This assumption would be true if the first
steps of the clogging process were reproducible among two experiments.
Unfortunately, the stochastic facet of clogging makes a more accurate analysis
difficult. The hydraulic resistance increase (and subsequent flow rate
decline) can vary significantly from one experiment to another, depending on
the local structure of the initial clog, next to the pore, which strongly
influences the clog’s hydrodynamic resistance.

Furthermore, regimes 2 and 3 are actually two sub-regimes of a more general
diffusive regime. We propose these two sub-regimes because of observations
made after pressure release. A release of hydrodynamic forcing revealed that a
fraction of the clog remains stuck to the membrane (and therefore adhesive)
whereas another part is re-suspended in the surrounding fluid, as presented in
figure 16. Particles still stuck after pressure release and, in addition to
the amorphous region, large parts of the remaining clogs/cakes’ wall layer
have a crystalline microstructure – revealing that in the crystalline zone,
particles can be adhered or not.

Nevertheless, we are not able to distinguish from the clog structure analysis
the two diffusive regimes (with and without adhesion). Further experiments,
based on the unclogging process, are necessary to discriminate these regimes.
This could be a way to refine our interpretation. In addition, the fact that
particles remain motionless after releasing the pressure drop is a clue that
they are stuck to the glass plate.

Fig. 16: Unclogging after pressure release. Clog before (top) and after
(bottom) pressure release.

An important point of our experiments concerns the absence of particle motion
inside the clogs. As shown in figure 2, the particles belonging to a clog at a
given time are at the same location at a later time. We never observed
particle motion during the different image sequences acquired for this work.
Such a motionless behaviour is also a clue of particle adhesion, or their
belonging to a “repulsive glass” where hydrodynamic forcing freezes non-
adhering particles in a given position – the one they took up when they
arrived at the clog. Moreover, this immobility is an indication that the fluid
flowing through the clog does not affect clog microstructure over time.
Particles belonging to an amorphous zone are indeed stuck to the surface and
cannot move anymore.

Note that these analyses are made using only the wall colloid layer. Although
the glass plate at contact with the colloids could affect the absolute
structure and organisation of this layer, the transitions and differences
observed are consistent with a physical interpretation in the 3D bulk of the
clog, and are reproducible. Moreover, if the glass plate was the main
ingredient at play in wall-layer colloid organisation, it should make uniform
the wall-layer microstructure. Unfortunately, the polystyrene particles used
in these experiments are opaque and do not permit a 3D analysis using confocal
microscopy. Working with optical-index-matched particles, which could enable
analysis in the depth of clogs, is not at all straighforward without changing
the interactions. As a matter of fact, van der Waals interactions, at play in
the force balance (see DLVO in first section) are influenced by electric
polarizability, related to the optical properties. Furthermore, real
filtration devices can comprise lateral walls. This work proposes some
insights about clog microstructure, and call other experimental studies to
investigate the 3D structure and understand the way particles pile up on the
wall layer and affect its microstructure – use of opaque colloids did not
allow a 3D study in our case.

##  5 Conclusion and perspectives

The filtration experiments we performed using microfluidic devices allow us to
study the wall layer microstructure of clogs for two different parameters:
applied pressure drop and ionic strength. We observed a clog development at
the entrance of the pore, followed by growth of a filtration cake. Resolution
was good enough to detect particles’ positions. An analysis of clog porosity
did not reveal strong global difference when changing pressure drop or salt
concentration. At first glance, a study of colloid spatial organisation using
radial distribution function also did not reveal clear influence of pressure
drop and ionic strength. Nevertheless, by fitting these functions, we were
able to extract a screening length which represents the medium-range order.
This screening length is higher for low ionic strength, revealing a better
spatial organisation.

A local analysis revealed spatial heterogeneity of the clogs. Porosity is
higher at the vicinity of the pore then decreases with the distance from the
pore. This observation is systematic for all pressure drops and ionic
strengths. This heterogeneity is also valid for colloid spatial organisation.
Fourier transform analysis of images of the clogs showed the presence of
amorphous, square-lattice and hexagonal-lattice regions. Using 2D continuous
wavelet transform, we were able to perform a local and systematic analysis of
spatial organisation of the clogs. It revealed a large predominance of
hexagonal-lattice regions compared to square-lattice ones. Moreover, the
proportion of crystalline regions increases with the distance from the pore
entrance before reaching a plateau. It is higher for high interparticle
repulsion, and seems to be lower as pressure drop rises.

We gathered all these observations in a new framework based on a phase
diagram. Three regimes are accessible, depending on flow and repulsive
interaction intensity. One is a ballistic regime, where addition of a colloid
to a clog is the result of a direct adhesion due to drag force. Two diffusive
regimes (with or without adhesion) are due to lower hydrodynamic forcing or
higher repulsion. Whereas the ballistic regime does not allow organisation of
the particles, the two others let allow time to an arriving colloid to “plug
the holes”. Where there is adhesion, it is always due to drag forces. Brownian
motion can only help to organize the clog. This framework is compatible with
all our experimental observations. These results could be compared with the
distinction between diffusion-limited aggregation (DLA) and reaction-limited
aggregation (RLA)63. DLA is a rapid process in which particles immediately
adhere to the aggregate, and provides loosely packed aggregates, whereas RLA
is a slower process that requires each particle to make multiple attempts,
allowing exploration of the existing aggregate and results in a denser (higher
fractal dimension) aggregates.

Several kinds of experiments could complete this study: 3D analysis using
confocal microscopy (but that would require index matching, which is not
straightforward), X-ray microtomography and different chip designs to observe
clogs from the side, in order to refine our results in the third dimension;
unclogging analysis to distinguish the two diffusive regimes; acquisitions
with higher framerate and spatial resolution to observe the subtle effect of
Brownian motion on clog self-organization. The second point could be the
opportunity to study the unclogging dynamics when hydrodynamic forcing is
released.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

We acknowledge the Fédération FERMAT and University of Toulouse (Project
NEMESIS) for funding these researches. This work was partly supported by LAAS-
CNRS micro and nanotechnologies platform member of the French RENATECH
network. We acknowledge Julien Lefort and Maëlle Ogier for technical and
microscopy support. We warmly thank Patrice Bacchin and Paul Duru for the
fruitful discussions and their manuscript revise.

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