# Colloid aggregation simulation in a Poiseuille flow using SRD-MD simulations

Hanady Semaan, Manuella Cerbelaud, Julien Gerhards, Benoit Crespin, Riccardo Ferrando, Arnaud Videcoq

► To cite this version:

Hanady Semaan, Manuella Cerbelaud, Julien Gerhards, Benoit Crespin, Riccardo Ferrando, et al. Colloid aggregation simulation in a Poiseuille flow using SRD-MD simulations. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 2025, 707, pp.135896. 10.1016/j.colsurfa.2024.135896. hal-05004876

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# Colloid aggregation simulation in a Poiseuille flow using SRD-MD simulations

Hanady Semaan<sup>a,b</sup>, Manuella Cerbelaud<sup>a ID,*</sup>, Julien Gerhards<sup>a</sup>, Benoit Crespin<sup>b ID</sup>, Riccardo Ferrando<sup>c</sup>, Arnaud Videcoq<sup>a</sup>

## GRAPHICAL ABSTRACT

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

**Keywords:**

Colloidal suspensions
SRD-MD
Poiseuille flow
Colloidal interaction
Aggregation

## ABSTRACT

The use of flowing colloidal suspensions in confined environments is of great relevance in many scientific fields, from microfluidics to ceramic manufacturing processes. Despite efforts to understand the structuring of colloids under flows, structuring of attractive colloids in a non-uniform flow remains less studied. In this article, Stochastic Rotation Dynamics - Molecular Dynamics (SRD-MD) simulations are used to study the aggregation of particles subjected to Poiseuille flow between two plates. Simulations show that the Poiseuille flow strongly influences colloid aggregation. They reveal that, depending on the potential wells depth used, the same flow can either lead to the fragmentation of aggregates or to the formation of compact and ordered aggregates. The aggregation and reorganization dynamics of these aggregates are explained by the velocity gradient and hydrodynamic effects.

## 1. Introduction

Understanding the aggregation of colloidal suspensions is crucial across diverse sectors, ranging from ceramic processing to coatings, paints, inks, drug delivery, and food processing. In the field of ceramics, for example, suspensions are often used during the shaping process [1]. In these various processes, suspensions can be subjected to different shear conditions, and it is important to understand their behavior under flow. Nowadays, the development of additive processes such as inkjet

printing or microextrusion makes it increasingly important to take into account not only flow effects, but also the confinement of suspensions. The flow of suspensions in confined environments has also been an important topic in the field of microfluidics for several years [2].

Given the large number of parameters that influence the behavior of suspensions, numerical simulations can help identify the key parameters. Various simulations have already been proposed to understand the aggregation of colloids subjected to flow. These must include hydrodynamic effects. Indeed, during the aggregation of colloids under flow,

<footnote><sup>a</sup> Univ. Limoges, CNRS, IRCER, UMR 7315, F-87000 Limoges, France</footnote>

<footnote><sup>b</sup> Univ. Limoges, CNRS, XLIM, UMR 7252, F-87000 Limoges, France</footnote>

<footnote><sup>c</sup> Physics Department, University of Genoa, Via Dodecaneso 33, 16146, Genoa, Italy</footnote>

<footnote>* Corresponding author.
E-mail address: manuella.cerbelaud@unilim.fr (M. Cerbelaud).</footnote>
the combined effects of electrostatic and hydrodynamic forces play a role in the stability of suspensions and in the colloid arrangement [3].

The effect of shear on aggregate formation or disaggregation, for example, has been analyzed by Stokesian dynamics [4–7]. Aggregate rotation, breakage and alignment effects have been used to explain the structure of aggregates under flow. More recently, Lorenzo et al. have also used Brownian dynamics including hydrodynamics to understand the coupled effect of particle interactions and shear on the structure of suspensions [8]. However, these simulations do not take into account the presence of walls, which leads to non uniform shear and which can influence agglomeration. Aggregate growth on walls can indeed be detrimental in certain processes, for example by reducing the cross-sectional area or even by clogging the channel. The formation of deposits on walls, for example, was analyzed in the 90s by square lattice models [9], or more recently by models coupling the Discrete Element Method (DEM) with Computational Fluid Dynamics (CFD) [10–12].

Techniques capable of simulating the behavior of colloidal suspensions in environments confined between walls include SRD-MD (Stochastic Rotation Dynamics - Molecular Dynamics), also known as multiparticle collision dynamics (MPCD) [13–15]. This technique, introduced by Malevanets and Kapral, enables the simulations of non uniform suspension flows by taking hydrodynamic effects into account. Various studies have shown, for example, that it is capable of reproducing a Poiseuille flow [16,17]. In addition, this method has proved ideally suited to investigate inertial focusing in microfluidic channels [18,19]. In these studies, colloid interactions are generally repulsive.

SRD-MD has already been used to analyze aggregation of suspensions in a channel. For example, Nikoubashman studied the behavior of Janus particle suspensions capable of forming micelles [20]. A detailed analysis of the effect of flow on the aggregates obtained is presented. More recently, Han et al. have used SRD-MD to study particle aggregation in a microfluidic channel [21]. They observed that shearing of aggregating colloids leads to the formation of crystalline aggregates. However, the authors focus their study on the aggregation under a particular confinement condition where the size of the aggregates is close to that of the channel.

In this article, we propose to use SRD-MD to extend the study of aggregation under flow by examining the aggregation of colloids in a Poiseuille flow induced between two plates sufficiently spaced to obtain aggregates smaller than the wall separation distance. Particular attention will be paid to the aggregate shapes obtained by varying the flow and/or the attraction potential well depth. The aim is to understand how flow modulates the shape and structure of aggregates as a function of particle interactions. It is indeed known that the potential well depth can influence the aggregation under flow [8]. Here, we propose to use particles interacting with a generalized Lennard-Jones potential whose well depth can be modified.

First, the SRD-MD simulations employed in this study will be presented. The simulations will initially be used to analyze velocity profiles in suspensions composed of colloids that repel each other. Subsequently, they will be used to analyze aggregation under flow.

## 2. Model and methods of simulation

The suspensions are simulated using the hybrid SRD-MD method. In this method, the liquid phase is described by SRD, a coarse-grained model with point particles whose dynamics consist of a succession of two steps: free streaming and collision. In the free streaming step, the fluid particles move freely over the time step $\Delta t_{SRD}$ without interacting with each other:

$$r_i(t + \Delta t_{SRD}) = r_i(t) + v_i(t) \Delta t_{SRD}, \quad (1)$$

where $r_i(t)$ and $v_i(t)$ represent position and velocity of particle $i$, respectively.

<table><caption>**Table 1**
Parameters used in SRD-MD simulations in order to reproduce the diffusion coefficient of a colloid in water at 293 K.</caption><thead><tr><th>Parameter</th><th>Symbol</th><th>Value</th></tr></thead><tbody><tr><td>Mass of fluid particles</td><td>m<sub>f</sub></td><td>6.75 × 10<sup>-19</sup> kg</td></tr><tr><td>Mass of colloids</td><td>m<sub>c</sub></td><td>2.49 × 10<sup>-16</sup> kg</td></tr><tr><td>Radius of colloids</td><td>a<sub>c</sub></td><td>300 nm</td></tr><tr><td>SRD angle</td><td>α</td><td>130°</td></tr><tr><td>SRD cell size</td><td>a<sub>0</sub></td><td>a<sub>c</sub>/2 = 1.5 × 10<sup>-7</sup> m</td></tr><tr><td>Average number of fluid particles per cell</td><td>γ</td><td>5</td></tr><tr><td>Dimensionless mean free path</td><td>λ</td><td>0.1</td></tr><tr><td>SRD time step</td><td>Δt<sub>SRD</sub></td><td>5.18 × 10<sup>-5</sup> s</td></tr><tr><td>Temperature</td><td>T</td><td>0.0036 K</td></tr><tr><td>MD time step</td><td>Δt<sub>MD</sub></td><td>Δt<sub>SRD</sub>/8 = 6.48 × 10<sup>-6</sup> s</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Schematic representation of the simulation box.</figcaption></figure>

For the collision step, the simulation box is divided into cubic cells of size $a_0$ and the particle velocities relative to the center-of-mass velocity of the cell $v_{c.m.}$ are rotated by a fixed angle $\alpha$ around an axis chosen randomly for each collision cell:

$$v_i(t + \Delta t_{SRD}) = v_{c.m.}(t) + R_\alpha[v_i(t) - v_{c.m.}(t)], \quad (2)$$

where $R_\alpha$ is the rotation matrix.

On the other hand, the dynamics of colloids is obtained by molecular dynamics (MD). Here, to couple both dynamics (SRD and MD), a repulsive force between fluid particles and colloids is introduced in the MD steps. Details on this kind of simulations can be found elsewhere [22,23]. In this study, colloids are spherical particles with a radius of $a_c = 300$ nm. The interactions between fluid particles and colloids are modeled by an inverse power (IP) potential:

$$V_{cf}^{IP}(r) = \begin{cases} \epsilon_{cf} \left(\frac{\sigma_{cf}}{r}\right)^{12}, & r \le r_c = 2.5\sigma_{cf} \\ 0, & r > r_c \end{cases} \quad (3)$$

where $\epsilon_{cf} = 2.5k_B T$, with $k_B$ the Boltzmann constant, $T$ the temperature, and $\sigma_{cf} = 0.8a_c$. The value of $\sigma_{cf}$ allows a slight penetration of the fluid particles at the surface of the colloids, which avoids a non-realistic depletion attraction [24]. The SRD-MD parameters are chosen as explained in Ref. [22]. The length and mass scales are chosen to reproduce the colloidal system under study. The time scale is chosen to reproduce the diffusion coefficient of the colloids studied in water at 293 K. Once these parameters have been defined, the simulation temperature is fixed. The complete list of parameters for our simulations is given in Table 1.

Simulations are performed between two infinite walls separated by a distance of $50a_0$ (Fig. 1). The size ratio between the walls spacing and the particle diameter is then equal to 12.5. This distance allows the introduction of a significant number of colloids between the two walls, and under these conditions, the initial clusters formed are smaller than the channel size, which is close to conditions often observed in
microfluidics. Periodic conditions are applied in the X and Y directions. Walls are introduced in the Z direction. To model the walls, the bounce back conditions are applied for the fluid particles as reported in previous works [17,24–26]. When fluid particles penetrate a colloid after bouncing, they are projected onto the surface of the colloid. To avoid slip conditions on the walls, virtual particles are used as proposed in the literature [17,27]. For this purpose, a SRD cell is added in the Z direction and walls are considered at coordinates $z_b = 0.5a_0$ and $z_t = 50.5a_0$. During the SRD collision steps, cells containing the walls that have a number of fluid particles lower than the average number of fluid particles per cell $\gamma$ are filled with virtual particles to obtain in total $\gamma$ particles in the cell. The simulation box is thus fixed at (200a₀, 50a₀, 51a₀). The collisions between the colloids and the walls are modeled by introducing repulsive interactions, modeled by an IP potential:

$$U_{\text{rep}}(z) = k_{\text{B}}T \left(\frac{a_c}{z}\right)^{36} \quad (4)$$

with $z$ the distance between the wall and the colloid.

Flow is induced in the X direction by introducing a constant acceleration $g$ for the fluid particles as proposed in the literature [26,28]. Flow is implemented in the streaming phase as follows:

$$r_i(t + \Delta t_{\text{SRD}}) = r_i(t) + v_i(t) \Delta t_{\text{SRD}} + \frac{1}{2} g \Delta t_{\text{SRD}}^2. \quad (5)$$

Three different accelerations will be used: $g_1 = 0.5 \times 10^{-3} \text{ m s}^{-2}$, $g_2 = 1 \times 10^{-3} \text{ m s}^{-2}$ and $g_3 = 2 \times 10^{-3} \text{ m s}^{-2}$. In order to maintain the temperature, a local thermostat called “Local Simple Scaling” is implemented [28,29]. This thermostat consists in rescaling the relative velocities ($v_i - v_{c.m.}$) in each collision cell by a factor linked to the kinetic energy of a collision cell. Collision step is thus performed as follows:

$$v_i(t + \Delta t_{\text{SRD}}) = v_{c.m.}(t) + R_{\alpha} \left[ \sqrt{\frac{3(n_f - 1)k_B T}{\sum_{i=1}^{n_f} m_f (v_i(t) - v_{c.m.}(t))^2}} (v_i(t) - v_{c.m.}(t)) \right]. \quad (6)$$

where $n_f$ is the number of fluid particles in the collision cell containing $i$.

Initially, colloids and fluid particles are randomly distributed in the simulation box in such a way that there is no interpenetration between colloids and that the penetration of fluid particles in a colloid is limited to $0.8a_c$. The interactions between colloids are either attractive or repulsive. When they are repulsive, interactions are modeled by an IP potential:

$$V_{cc}^{IP}(r) = \begin{cases} \epsilon_{cc} \left(\frac{\sigma_{cc}}{r}\right)^{12}, & \text{if } r \le r_c = 2.5\sigma_{cc} \\ 0, & \text{if } r > r_c \end{cases} \quad (7)$$

where $r$ is the distance between two colloids, $\epsilon_{cc} = 2.5k_B T$; $\sigma_{cc} = 2a_c$.

When interactions between colloids are attractive, they are modeled by a generalized Lennard-Jones (GLJ) potential:

$$V_{cc}^{GLJ}(r) = \begin{cases} 4\epsilon \left[ \left( \frac{\sigma_{cc}}{r} \right)^{36} - \left( \frac{\sigma_{cc}}{r} \right)^{18} \right] & \text{if } r \le r_c \\ 0 & \text{if } r > r_c \end{cases} \quad (8)$$

where $\epsilon$ represents the potential well depth and $r_c = 1.4\sigma_{cc}$. This potential has already been used in some of our previous simulations to study colloidal aggregation [30,31]. It was chosen because it simplifies calculations compared with the more conventional DLVO potential developed by Derjaguin, Landau, Verwey and Overbeek. The chosen power enables to obtain interaction range values similar to those of the DLVO potential, i.e. the particles will interact over similar approach distances.

In the following, the study will focus on dilute suspensions of smooth spherical particles, and in a first approximation, the friction forces will be neglected.

<figure><img src="image_5.png" /><figcaption>Fig. 2. Time evolution of the velocity profile for fluid particles only, for the acceleration $g_2 = 1 \times 10^{-3} \text{ m s}^{-2}$. The analytical Poiseuille flow is also reported (solid line).</figcaption></figure>

# 3. Results and discussion

## 3.1. Flow of a stable suspension (repulsive interactions between colloids)

In this section, we study the behavior of the fluid without colloids and of the fluid with a stable suspension, made of colloids interacting only by repulsive forces. Our aim is to check the velocity profiles of the fluid alone and of non-aggregated colloids immersed in the fluid. Simulations are performed with colloids that repel each other and thus interact through the $V_{cc}^{IP}$ potential. The interactions between the colloids and the walls are repulsive. Flow is induced by the acceleration $g_2 = 1 \times 10^{-3} \text{ m s}^{-2}$. The numbers of particles included in the simulations are reported in Table 2.

Simulations are first realized without colloids. The velocity profile of the fluid particles in the Z direction (between the two plates) is calculated and reported in Fig. 2 at different times of the transient regime. This profile is compared to the analytical one determined for the fluid as [15]:

$$V_x(z) = \frac{\rho g}{2\eta} (L_z - z) \times z, \quad (9)$$

with $\rho$ the fluid density, $\eta$ the dynamic viscosity and $L_z$ the distance between the plates ($L_z = 50a_0$). Fig. 2 shows that the velocity profile of the fluid particles reaches the analytical Poiseuille profile after 1 second, confirming that the parameters used in these simulations are capable of reproducing Poiseuille flow.

Simulations are then performed by adding colloids with different volume fractions reported in Table 2. All simulations lead to parabolic velocity profiles, shown in Fig. 3a, which are similar for fluid particles and colloids. The coupling between the dynamics of the colloids and that of the fluid particles allows the colloids to be conveyed by the fluid. Results also show that an increase of the colloid volume fraction leads to a decrease of the velocity. This phenomenon is related to an increase of the suspension viscosity. It is indeed known that the viscosity of hard sphere suspensions increases with colloidal concentration [32,33], which decreases the flow velocity in accordance with Eq. (9).

Fig. 3b shows the velocity profiles obtained for different values of acceleration $g$ and a colloid volume fraction of 5%. The flow velocity decreases with $g$, as described by Eq. (9). The profile is similar to the analytical one, proving that the SRD-MD simulations used in this study are able to reproduce the behavior under a Poiseuille flow of a colloidal suspension composed of repulsive colloids. These simulations will thus be used to study the aggregation mechanisms under flow.
<table><caption>**Table 2**
Numbers of fluid particles and colloids for different volume fractions of colloidal particles $\phi$.</caption><thead><tr><th>Colloid volume fraction $\phi$</th><th>0%</th><th>1%</th><th>5%</th><th>10%</th><th>15%</th><th>20%</th></tr></thead><tbody><tr><td>Number of fluid particles</td><td>2500000</td><td>2487217</td><td>2436003</td><td>2372006</td><td>2308009</td><td>2244013</td></tr><tr><td>Number of colloids</td><td>-</td><td>149</td><td>746</td><td>1492</td><td>2238</td><td>2984</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** (a) Velocity profile obtained at $t = 1.5$ s for fluid particles and colloids for different colloid volume fractions with acceleration $g_2 = 1 \times 10^{-3}$ m s$^{-2}$. (b) Velocity profile for fluid particles and colloids for different accelerations: $g_1 = 0.5 \times 10^{-3}$ m s$^{-2}$, $g_2 = 1 \times 10^{-3}$ m s$^{-2}$, $g_3 = 2 \times 10^{-3}$ m s$^{-2}$ for a colloid volume fraction of 5%. The analytical Poiseuille flow is also reported (solid lines).</figcaption></figure>

## 3.2. Size, morphology and structure of aggregates produced under flow conditions

In the following, we study suspensions with colloids interacting by attractive force which may induce aggregation. If not otherwise specified, simulations are carried out with a colloid volume fraction of 5%. Initially, particles are randomly distributed in the box and their velocities obey to the Poiseuille flow. To obtain these initial configurations, simulations are first performed with repulsive particles during 1.5 s. Time is then set to $t = 0$ s, and simulations are carried out during 30 s with attractive interactions between colloids ($V_{cc}^{GLJ}$). Snapshots of simulations obtained for potential well depths of $4k_B T$ and $10k_B T$ can be observed in Fig. 4. These values were chosen because previous simulations have shown that, at rest, this type of values gives rise to different aggregate structures: one is more compact and organized, while the other is more branched [34,35].

In both cases, the flow modifies the aggregate structures but in a different way. When colloids interact with a well depth of $4k_B T$, in the absence of flow, rather compact and organized aggregates are observed. This is in agreement with previous studies which show that for this small well depth, colloid reorganization can lead to crystallization [34]. As the flow velocity increases, i.e. when $g$ increases, the aggregates become smaller and the number of isolated particles increases. This is confirmed by Fig. 5, which shows the average number of particles per aggregate as a function of time. In the absence of flow, at 30 s, the aggregates are composed of 200 particles on average. As $g$ increases, the average number of particles per aggregate decreases. It is divided by a factor of about 10 for $g_3$. As already known experimentally, shape and size of aggregates obtained under flow is governed by the competition between the aggregation due to attraction and break up due to the hydrodynamic forces [3,37]. In case of a well depth of $4k_B T$, the aggregation between particles is not strong enough to

dominate and the hydrodynamic forces induced by the flow are thus sufficient to favor more breakage in this system. All these observations are in agreement with experimental observations performed in weakly aggregated suspensions [37].

The case of particles interacting with a well depth of $10k_B T$ is different. Fig. 4b shows that in the absence of flow, the aggregates are not compact, they are branched, and they are not organized. When the flow velocity increases, as in the case of $4k_B T$, the aggregates become smaller, however breakage is much less pronounced. At the end of the simulations, large aggregates are still observed and some isolated particles are only visible for $g_3$. Since aggregation is stronger with a well depth of $10k_B T$ than with a well depth of $4k_B T$, and the hydrodynamic forces are not dominant here, so aggregates can be formed under flow and remain, even when $g$ is increased. However, it is observed that for higher values of $g$ ($g_2$ and $g_3$), hydrodynamic effects begin to break up the aggregates. Fig. 5b shows this clearly and highlights the difference in behavior compared to $4k_B T$. This figure also reveals that the kinetics of aggregate growth at short times (0 - 5 s) is faster with flow. At this point, the aggregates grow by coalescence. The velocity gradient in the Z direction causes them to move at different velocities and increases the probability of encounter, whereas in the absence of flow the aggregates encounter each other solely by diffusion. Fig. 4b also shows that flow makes the aggregates much more compact and also more organized.

To quantify more precisely the compactness of aggregates as a function of flow velocity, their effective volume fraction $\phi_{eff}$ was calculated, as proposed in Ref. [38]. Rather than using a radius of gyration to estimate the aggregate volume, this method considers the ellipsoid comprising the aggregate. This is more precise for aggregates elongated by flow. For each aggregate, the principal moments of inertia ($I_1$, $I_2$ and $I_3$) were obtained by diagonalization of the moment of inertia
<figure><img src="image_7.png" /><figcaption>Fig. 4. Snapshots of simulations at t = 30 s without flow or with $g_1 = 0.5 \times 10^{-3}$ m s$^{-2}$, $g_2 = 1 \times 10^{-3}$ m s$^{-2}$ and $g_3 = 2 \times 10^{-3}$ m s$^{-2}$. Colloids interact with (a) $\epsilon = 4k_B T$ and (b) $\epsilon = 10k_B T$ and are colored according to the aggregates to which they belong. Snapshots are obtained by Ovito [36].</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 5. Average number of particles per aggregate as a function of time without flow and for the different accelerations ($g_1$, $g_2$, $g_3$). The potential well depth is: (a) $\epsilon = 4k_B T$ and (b) $\epsilon = 10k_B T$. Results are averaged over three independent simulations.</figcaption></figure>

tensor. Considering $I_1 \le I_2 \le I_3$, the lengths of the semi-principal axes of the equivalent ellipsoid ($a_{ell}$, $b_{ell}$ and $c_{ell}$) are obtained by:

$$a_{ell} = \sqrt{\frac{I_2 + I_3 - I_1}{N}}, \quad b_{ell} = \sqrt{\frac{I_3 + I_1 - I_2}{N}}, \qquad (10)$$

$$c_{ell} = \sqrt{\frac{I_1 + I_2 - I_3}{N}},$$

with N the number of particles in the aggregate. The effective volume fraction is then defined as:

$$\phi_{eff} = \frac{N}{a_{ell}b_{ell}c_{ell}} \qquad (11)$$

Fig. 6 shows the evolution of $\phi_{eff}$ as a function of time. In the system with $10k_B T$ (Fig. 6b), in the absence of flow, the effective density decreases with time to 10% after 20 s. This reflects the evolution of the morphology of the aggregates during the coalescence process and their difficult reorganization due to the strong bonds between the particles.

As the coalescence events progress, the aggregates become increasingly elongated and branched. However, the flow completely modifies the behavior. The effective density remains essentially at the same level over time for $g_1$, just below 30%, or increases slightly from 30% to 40% for $g_2$. This means that the flow is very efficient in reorganizing the aggregates after coalescence events, making them more compact. With $g_1$, the effective density value shows that reorganization is as efficient as having a potential well depth of $4k_B T$ without flow (Fig. 6a). With $g_2$, the flow is even more efficient. In the case of $4k_B T$, the smallest flow ($g_1$) also improves the compactness of the aggregates compared to the case without flow, their effective density reaching 40%.

These results can be compared with experimental studies on gel structuring under the effect of flow. Conrad et al. have shown that a moderate flow rate destructures a gel and leads to the formation of dense aggregates, while a higher flow rate induces the destruction of aggregates [39]. The present simulations show similar behaviors. Even
<figure><img src="image_9.png" /><figcaption>**Fig. 6.** Effective volume fraction of aggregates ($\phi_{eff}$) as a function of time in the absence of flow and for different accelerations ($g_1, g_2, g_3$). (a) Colloids interact with a potential well depth $\epsilon = 4k_B T$ and (b) with $\epsilon = 10k_B T$. Results are averaged over three independent simulations.</figcaption></figure>

if the initial conditions and colloid volume fractions are different, we can assume that the structure obtained experimentally and numerically is due to the same phenomena.

Han et al. have reported that flow can promote the ordering of aggregates in their simulations [21]. Similarly, in this study, reorganization modifies not only the morphology of the aggregates, making them more compact, but also the order within the aggregates. This can be assessed by an order parameter, P12, which is the percentage of particles with 12 nearest neighbors, i.e. with the highest possible coordination in an ordered aggregate. Fig. 7 shows this order parameter as a function of time. With a potential well depth of $4k_B T$ (Fig. 7a), the aggregates are ordered without flow, with P12 = 12%. As with compactness, the smallest flow ($g_1$) clearly improves the ordering, as evidenced by a P12 value twice as large at 30 s. In contrast, stronger flows ($g_2$ and $g_3$) completely destroy the ordering (P12 close to zero). Conversely, with a potential well depth of $10k_B T$ (Fig. 7b), the aggregates show virtually no order in the absence of flow, and P12 increases progressively with the value of $g$.

Let us now analyze why flow is so effective at reorganizing aggregates, as long as it does not destroy them. In the literature, rotation and breakage of aggregates have been mentioned to explain the reorganization of fractal aggregates in Stokesian Dynamics simulations [6,7]. We examine the simulations here to see the reorganization mechanisms at work in this case, where the aggregates do not have a small fractal dimension. It is clear that the flow generates a velocity gradient along the Z axis. As a consequence, particles in an aggregate are carried along at different velocities depending on their Z coordinate. When the aggregates are small and compact, the velocity gradient forces them to rotate. Fig. 8 illustrates this behavior. When the aggregates are larger and less compact, the velocity gradient favors the breaking of bonds and the rolling up of the aggregates on themselves, forcing them to reorganize in a more compact and ordered way, as illustrated in Fig. 9.

All previous simulations were performed with a colloid volume fraction of 5%. Fig. 10 shows simulation snapshots for colloid volume fractions of 1%, 10%, 15% and 20%. For these simulations, the behavior is similar to that observed at 5%, except that the aggregate size increases with the colloid volume fraction. The images clearly show that the flow is very efficient in producing compact and ordered aggregates. Notably at 15 and 20%, with $g_2$ and $g_3$, large patches of colloidal crystals are obtained.

### 3.3. Spatial distribution of colloids during flow aggregation: dynamical Segré-Silberberg effect

Flow also has an effect on the spatial distribution of colloids as aggregation proceeds. Fig. 11 shows different stages of aggregation under a flow obtained with $g_2 = 10^{-3} \text{ m s}^{-2}$. The potential well depth of the colloidal attraction is fixed at $10k_B T$ and the colloid volume fraction at 5%.

At the beginning of the simulations, particles are uniformly distributed in the box. This is clearly observed in Fig. 11 which shows a similar number of particles along the z axis. Then, small aggregates form and a preferential position of the particles in the box is observed. At $t = 0.5$ s, the curve of the particle number as a function of $z$ shows indeed two distinct peaks, indicating that aggregates are preferentially located in two distinct layers between the walls. Finally, aggregates continue to grow and are observed in the middle of the simulation box.

This displacement of aggregates during their formation may be due to Segré and Silberberg effects. These effects, which appear in confined suspensions flowing in a channel, can be explained by the competition of two hydrodynamic forces: wall interaction forces and shear gradient lift forces. Repulsive interactions with the walls repel the colloids away from the channel walls. Conversely, shear gradient lift forces operate similarly, but in the opposite direction. The competition between these two opposing forces is known to allow the formation of a ring of particles in microfluidic channels, the so-called Segré and Silberberg effect.

This effect has been observed in SRD-MD simulations [18,21]. It is related to the size ratio between the entities and the channel [21,40]. Three different situations can be observed [40]. When the particles are small, they are generally not significantly affected by the hydrodynamic forces, allowing them to remain dispersed in the channel. Conversely, when particles are large, their interactions with the channel walls become dominant. This often results in a concentration of particles along the central axis of the flow. For particles of intermediate size, competition between the interaction forces with the wall and the shear gradient lift forces leads to a peak concentration of particles at an intermediate position in the channel.

In simulations, the Segré and Silberberg effect is then generally analyzed for a fixed ratio between the size of the channel and the size of the flowing entities (for example, particles or colloids aggregates in Refs. [18] and [21], respectively). In our case, the situation is
<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Order parameter P12 as a function of time in the absence of flow and for different accelerations ($g_1, g_2, g_3$). (a) Colloids interact with a potential well depth $\epsilon = 4k_B T$ and (b) with $\epsilon = 10k_B T$. Results are averaged over three independent simulations.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Snapshots of an isolated aggregate (extracted from a complete simulation) showing its rotation during 0.17 s of simulation time, as it moves with the flow from the top left corner of the observation window to the top right corner. The aggregate is close to the top wall of the simulation box and is clearly rotating counter-clockwise (See movies 1 and 2 in Electronic Supplementary Information). The snapshots are obtained by Ovito [36].</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Snapshots of an isolated aggregate (extracted from a complete simulation). This succession of images shows in particular the reorganization of the front part of the aggregate as it flows through the channel during 0.41 s of simulation time (see movies 1 and 3 in Electronic Supplementary Information). Snapshots are obtained by Ovito [36].</figcaption></figure>

different, because we observe the Segré-Silberberg effect acting in a dynamic way on the growing aggregates which change size during the evolution of the system. Initially, particles are not aggregated and are therefore small enough to not be affected by the hydrodynamic forces and remain dispersed in the simulation box. Then, particles

start to aggregate forming larger entities. After some time, aggregates size becomes sufficient to feel the interaction forces with the wall and the shear gradient lift forces, and aggregates migrate under the effect of Segré and Silberberg. They are preferentially located in two layers, where they continue to grow by coalescence. When they grow
<figure><img src="image_13.png" /><figcaption>**Fig. 10.** Snapshots of simulations at t = 30 s, with a potential well depth $\epsilon = 10k_B T$, for different colloid volume fractions: (a) 1%, (b) 10%, (c) 15% and (d) 20%, and with different accelerations: without flow and with a flow produced with $g_1 = 0.5 \times 10^{-3} \text{ m s}^{-2}$, $g_2 = 1 \times 10^{-3} \text{ m s}^{-2}$ and $g_3 = 2 \times 10^{-3} \text{ m s}^{-2}$. The colloids are colored according to the aggregates to which they belong. Snapshots are obtained by Ovito [36].</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. 11.** Snapshots of simulation obtained with a flow induced by $g_2 = 10^{-3} \text{ m s}^{-2}$ at different times. Particles interact via the potential of Eq. (8) with a well depth of $\epsilon = 10k_B T$. Below the snapshots, the averaged distribution of particles in the z direction is reported. Snapshots are obtained by Ovito [36].</figcaption></figure>

sufficiently, the effect of the lift forces decreases and they can once again spread out in the middle of the channel.

The displacement of clusters growing under flow has already been seen experimentally. Observations on thermogelling nanoemulsions in a channel flow have indeed shown a heterogeneous distribution of the gellified clusters in the channel according to their size [40].

Finally, in our simulations, the migration of aggregates away from the central area at the early stage of aggregate coalescence may also partly explain the faster aggregation under flow observed in Fig. 5b. In fact, the confinement of the small aggregates in two layers has two consequences. Firstly, this confinement tends to concentrate them

and, secondly, they are located in an area where they are subject to a higher velocity gradient. Both effects contribute to accelerating their coalescence.

## 4. Conclusion

In this paper, Stochastic Rotation Dynamics - Molecular Dynamics simulations are used to study colloidal aggregation in a Poiseuille flow between two infinite walls. We first verified that the analytical profile of Poiseuille flow is well obtained both for fluid particles alone and for colloidal particles dispersed in the fluid. Then, for weak colloidal
attraction, for which reorganization is easy and compact, ordered ag-
gregates are obtained in the absence of flow. A moderate flow may
slightly improve compactness and ordering, but the main effect of
flow in this case is the destruction of the aggregates. In contrast, for
strong colloidal attraction, which produces aggregates that are neither
compact nor ordered in the absence of flow, flow accelerates the growth
of the aggregates and is able to improve their compactness and cause
order to emerge within their internal structure, to levels comparable
to those of weak colloidal attraction. As long as the flow is not strong
enough to completely destroy the aggregates, the generated velocity
gradient causes certain bonds to break and the aggregates to roll up on
themselves, allowing them to adopt a more compact shape and a more
ordered structure.

This has already been suggested in the literature, but this study
provides a more detailed analysis of the phenomena at work in the
reorganization of aggregates. Our simulations reveal a new form of
the Segré and Sielberberg effect. Our results confirm the effect of
Sielberberg effect. This effect acts while colloidal aggregate grow,
causing the displacement of their average position in the channel due
to the interplay between the interaction forces with the wall and the
shear gradient. This has the effect of placing the aggregates where the
velocity gradient is particularly high and concentrating them, which
favors both their coalescence and their reorganization.

This study confirms the role of hydrodynamics in the aggregation
under flow and completes previous literature studies by studying in
detail the interplay of flow and colloid interactions. The suspensions
studied here remain rather simple, however, consisting of a single
type of spherical particles. Experimentally, suspensions are often more
complex, made up of several non-spherical particles of different types
and sizes. In order to get closer to these systems, it is envisaged to
study more complex suspensions in the future, for example containing
different types of particles.

The understanding of shear suspensions behavior may help to con-
trol the size, morphology and structure of colloidal aggregates under
flow in order to fully understand the behavior of suspensions experi-
mentally set in motion in confined environments. In fine, this should
help to improve industrial processes.

CRediT authorship contribution statement

Hanady Semaan: Writing – original draft, Visualization, Software, Investigation, Formal analysis. Manuella Cerbelaud: Writing – original draft, Supervision, Software, Methodology, Formal analysis, Conceptualization. Julien Gerhards: Software. Benoit Crespin: Writing – review & editing, Software. Riccardo Ferrando: Writing – review & editing, Supervision. Arnaud Videcoq: Writing – original draft, Supervision, Methodology, Formal analysis, Conceptualization.

Declaration of competing interest

The authors declare the following financial interests/personal rela-
tionships which may be considered as potential competing interests:
Manuella Cerbelaud and Arnaud Videcoq have patent #Method for pro-
ducing and depositing a composition (WO2023/194348 A1) pending
to Licensee. If there are other authors, they declare that they have
no known competing financial interests or personal relationships that
could have appeared to influence the work reported in this paper.

Acknowledgments

The authors would like to thank Roland G. Winkler for his construc-
tive discussions concerning the simulation codes. This research was
funded, in whole or in part, by l'Agence Nationale de la Recherche
(ANR), project ANR-20-CE46-0004. Authors thank also the Région
Nouvelle Aquitaine for the cofunding of the SOMA-DNS project and
the Agence Nationale de la Recherche (ANR) for an institutional grant

under the Investments for the future program with the reference ANR-
18-EURE-0017 TACTIC. This work was performed using HPC/AI re-
sources from YAGA, owned by the P&T unit (INSERM U1248), the
IRCER institute (UMR CNRS 7315) and the XLIM institute (UMR CNRS
7252), and hosted by the University of Limoges.

**Appendix A. Supplementary data**

Supplementary material related to this article can be found online
at https://doi.org/10.1016/j.jcolsurfa.2024.135896.

Data availability

Data will be made available on request.

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