Patrice Bacchin
email: patrice.bacchin@univ-tlse3.fr

Interfacially driven transport in narrow channels

Keywords: colloids, diffusio-osmosis, hydrodynamics, colloid-interface interactions

of forces by interactions and friction is rather complex. When colloids ow in a narrow channel, non-equilibrium interplay between colloid-interface interactions and hydrodynamics controls the transport ef ciencies: (i) the mass transport efciency, which governs the colloid transmission through the channels and (ii) the mixture (colloids and liquid) transport ef ciency, which determines the energy required for the liquid to ow through the channel.

A way to progress toward a better understanding of this complex interplay is to develop model and simulation tools that unravel the mechanisms taking place in such a process. The transport of colloids in narrow channels involves non equilibrium forces balance and exchanges of forces between the colloids and the solvent molecules. Such entanglement can be described at different levels with different simulation methods. At the molecular level, dynamic molecular

Introduction

The transport of colloids inside narrow channels is not only an interesting scienti c question but also a challenge in man y processes in various elds of application. Flow through pores is a common process in living bodies (kidneys, membrane cells, etc), in natural systems (aquifers) and in industrial applications ( ltration, desalting, etc) [START_REF] Shannon | Science and technology for water puri cation in the coming decades[END_REF]. Beyond these applications, the recent development of micro uidic experiments and the nano-scale engineering of interfaces have revived the question of the effect of colloid-interface interactions on transport in con ned channels and through small ori ces [START_REF] Bocquet | Flow boundary conditions from nano-to micro-scales[END_REF][START_REF] Lee | Osmotic ow through fully permeable nanochannels[END_REF].

The impact of physicochemical parameters on transport properties needs to be better understood if industrial or natural applications are to be improved but the problem of a ternary system (colloids, liquid and interface) with a strong exchange simulations [START_REF] Yoshida | Osmotic and diffusio-osmotic ow generation at high solute concentration. II. Molecular dynamics simulations[END_REF] can describe the effect of interaction forces between molecules on the dynamic transport properties. However, colloid particles are much bigger than solvent molecules and, for this reason, dynamic simulation cannot be applied to a large system involving a strong colloid concentration gradient. The coupling between resolved colloidal particles and uid ow can be modelled with a mesoscopic approach considering the solvent molecules as a continuum phase: a lattice Boltzmann (LB) method [START_REF] Cates | Simulating colloid hydrodynamics with lattice Boltzmann methods[END_REF][START_REF] Hidalgo R C | Flow of colloidal suspensions through small ori ces[END_REF], dynamical density functional theory (DDFT) [START_REF] Zimmermann | Flow of colloidal solids and uids through constrictions: dynamical density functional theory versus simulation[END_REF] or a force coupling method (FCM) [START_REF] Agbangla | Numerical investigation of channel blockage by owing microparticles[END_REF][START_REF] Agbangla | Collective dynamics of owing colloids during pore clogging[END_REF]. In the last decade, these 'dynamical forces' methods have allowed fast progress to be made in the understanding of the role played by colloidal interaction on the ow at local scale but they are still limited to several hundred particles. In contrast, the thermodynamical approach (for example, the non-equilibrium thermodynamics approach of Kedem and Katchalsky [START_REF] Kedem | Thermodynamic analysis of the permeability of biological membranes to nonelectrolytes[END_REF]) allows a large number of particles to be processed statistically; the colloid and the uid phase are both treated as continuum phases. This type of approach is then classically used to describe osmosis as driven by a gradient of water chemical potential. But osmosis is a far-from-equilibrium phenomenon, so an approach based on local equilibrium thermodynamics is not always valid. Furthermore, such an approach cannot take dissipation via local uid ow into account. Consequently, a complete theoretical description of ow driven by hydrodynamics and several thousands of multibody interfacial interactions is still lacking.

From a soft matter point of view, progress has recently been made in modelling the role played by colloid-colloid interactions but a better understanding of interfacial phenomena could be achieved by considering the mechanical role played by colloid-interface interactions. This role was qualitatively pointed out in the earlier works on Brownian diffusion at the interface [START_REF] Fermi | Thermodynamics[END_REF][START_REF] Van't Hoff | The role of osmotic pressure in the analogy between solutions and gases[END_REF][START_REF] Einstein | Investigations on the Theory of the Brownian movement ed R Fürth[END_REF]. In the 1970s, several authors [START_REF] Manning | Binary diffusion and bulk ow through a potential-energy pro le: a kinetic basis for the thermodynamic equations of ow through membranes[END_REF][START_REF] Anderson | Mechanism of osmotic ow in porous membranes[END_REF] put forward theoretical models to quantify the role of colloid-interface interactions on the transport. However, these models were not really developed and integrated in simulation code. In the past ten years, several authors have again pointed out the important effects of solute or colloid-wall interactions on transport through a narrow channel [START_REF] Cardoso | Dynamics of osmosis in a porous medium[END_REF][START_REF] Lachish | Osmosis and thermodynamics[END_REF][START_REF] Nelson | Osmosis and thermodynamics explained by solute blocking[END_REF][START_REF] Kramer | Osmosis is not driven by water dilution[END_REF]. Recently, a two-uid model has been proposed that introduces nanometric scale colloidinterface interactions in momentum and mass balances [START_REF] Bacchin | An energy map model for colloid transport[END_REF]. The two-uid model or suspension balance model [START_REF] Nott | Pressure-driven ow of suspensions: simulation and theory[END_REF][START_REF] Nott | The suspension balance model revisited[END_REF] is based on solving the eld equations written from the volume averaging of the governing equations (local momentum and mass balances) on the two phases: colloids and solvent molecules. The novelties of the approach is to solve the two-uid model on an energy landscape that represents the interfacial interactions. A 1D application of the model has been developed [START_REF] Bacchin | Colloid-interface interactions initiate osmotic ow dynamics Colloids Surf[END_REF] by considering an energy landscape that represents the overall energy barrier (induced by colloid-interface interactions) to cross the interface. It proves how the colloid-interface interaction drives the osmotic ow. For limiting cases, it underlines the compatibility of this mechanical approach with NET approaches describing the transport of colloid and solvent. Such models can help to elucidate the 'strange' transport mechanism of uids at the nano-scale [START_REF] Bocquet | Nano uidics, from bulk to interfaces[END_REF] (recently highlighted in micro uidic experiments or with nanotubes [START_REF] Farajian | Polarization, energetics, and electrorheology in carbon nanotube suspensions under an applied electric eld: an exact numerical approach[END_REF], aquaporins, etc) and therefore to help progress by designing speci c nanoscale molecule/pore interactions within arti cial nano-pores in order to optimize the transport [START_REF] Gravelle | Optimizing water permeability through the hourglass shape of aquaporins[END_REF].

The aim of this paper is to investigate how the interfacial interactions exerted by walls on colloids interplay with the colloid and solvent ows through a narrow channel constriction. The novelty is to represent the narrow channel shape by a 2D energy map that describes the landscape of colloid-wall interactions encountered by colloids for each position in the channel. This new 2D approach also relies on a strong coupling between the Stokes equation and the mass balances. The advantages of this work compared to previous simulations [START_REF] Bacchin | An energy map model for colloid transport[END_REF][START_REF] Bacchin | Colloid-interface interactions initiate osmotic ow dynamics Colloids Surf[END_REF] are to account for the effect of con nement on the interfacially driven ows that, at the end, reveal local secondary ow superimposed on the primary ow. The paper is organized as follows. Section 2 introduces the numerical model. In section 3, a limiting case, for a closed channel, is rst investigated to analyse a pure interfacially driven transport controlled by a diffusion-osmosis phenomenon [START_REF] Lachish | Osmosis and thermodynamics[END_REF]. In section 4, the ow of a colloidal dispersion through a channel is simulated to explore the strong coupling between the ow and the colloid-wall interaction and their consequences on transport ef ciencies.

Theoretical background and model development

The model is based on a two-uid model. The two-uid model (or mixture model or suspension balance model) [START_REF] Nott | The suspension balance model revisited[END_REF][START_REF] Morris | Curvilinear ows of noncolloidal suspensions: the role of normal stresses[END_REF][START_REF] Vollebregt | Suspension ow modelling in particle migration and micro ltration[END_REF] allows the velocities of the colloid phase, u c , of the uid phase, u f and of the mixture phase, u m (coming from volume averaging, φu c + (1 -φ) u f )) to be determined, together with the volume fraction of the colloid phase, φ. Solving it thus relies on the application of the momentum and mass balances written for the colloid phase, for the uid phase and, by addition, for the mixture phase. In a previous paper, such a two uid model was adapted to implement the colloid/colloid interactions (via the osmotic pressure) and the colloid/ interface interactions (via an energy map) [START_REF] Bacchin | An energy map model for colloid transport[END_REF]. The following sections establish the model (section 2.1), present the consequence of the balance between pressure and colloid-interface interactions (section 2.2) and describe the simulation conditions used in the paper (section 2.3).

Transport of colloids and uid near interfaces

When colloids ow close to an interface, multi-body interactions occur between colloidal particles and between the colloids and the interface. Classically, colloid-colloid interactions are accounted for by the osmotic pressure, Π cc , which is a pressure (an energy per unit volume) that includes the entropic contribution and multibody interactions. The gradient of the osmotic pressure leads to a thermodynamic force. This force is responsible for the Brownian diffusion (for the entropic part of the force) and for an interaction-induced diffusion (for the colloid-colloid contribution), both of which are represented by the generalized Stokes-Einstein law. On the other hand, the colloid/interface interactions can be de ned as an energy required to access a spatial position close to the interface. These interactions can be represented by an energy landscape, Π ic , that can be spatially mapped. Physically, this landscape [START_REF] Wales | Energy Landscapes[END_REF] maps the colloid-interface interaction energy per unit volume that expresses the Gibbs free energy changes caused by the interactions (similarly to the ∆G caused by a reaction), which is also the additional free energy for the introduction of a colloid inside the interfacial layer. This map represents the overall interactions between the colloids and the interface (for example, DLVO and hydration forces [START_REF] Israelachvili | Role of hydration and water structure in biological and colloidal interactions[END_REF][START_REF] Verwey | Theory of the Stability of Lyophobic Colloids[END_REF]). This colloid/interface interaction energy per unit volume, Π ic , is also a pressure and is thus a complement to the osmotic pressure accounting for the colloid/ colloid interactions, Π cc . The gradient of the energy map, ∇Π ic , gives the force experienced by the colloid close to the interface. The product of the colloid volume fraction by the force, φ∇Π ic , represents the density of the interfacial force between the colloids and the interface in newtons per unit volume and can be integrated in mechanical approaches [START_REF] Anderson | Colloid transport by interfacial forces[END_REF][START_REF] Marbach | Osmotic and diffusio-osmotic ow generation at high solute concentration[END_REF]. Summarizing, the descriptors of colloidal forces are:

• for colloid/colloid forces, the gradient of osmotic pressure, ∇Π cc , where Π cc is a colligative property (a function of the volume fraction of the dispersion, φ) • for colloid/interface forces, the density of colloid-interface forces, φ∇Π ic , where Π ic is a map of energy per unit volume (a function of the spatial positions, x, y and z).

The suspension balance model consists of momentum and mass balances written for the uid, the colloids and the mixture phases (equations ( 1)-( 6)). The delicate part of the model is the writing of the momentum exchanges between the phases that govern the rheological response of the system and the coupling between colloidal and hydrodynamic forces. A physics-grounded expression thus considers that [START_REF] Bacchin | An energy map model for colloid transport[END_REF]:

• for colloid/colloid forces, a reversible exchange of momentum, ∇Π cc , operates between the uid (equation (1)) and the colloid phase (equation ( 2)) to be consistent with the uctuation-dissipation theorem for Brownian objects. The colloid/colloid forces thus do not appear in the mixture balance (equation ( 3)). • for colloid/interface forces, by the action/reaction principle, the momentum due to the force, φ∇Π ic , in the colloid phase (equation ( 1)) is counterbalanced by a force applied to the interface. In this case, Newton's third law is broken for the colloid/ uid mixture. There is thus no counteracting force on the uid (equation ( 2)). For these reasons, the colloid/interface force appears in the mixture balance (equation ( 3)).

For these reasons, only the colloid/interface interactions appear in equation (3) as an external force induced by the presence of the wall (the wall can be considered as an external third phase for the two-uid model in a con ned environment). With these considerations, the set of Eulerian equations is given below for a colloidal dispersion with a volume fraction, φ, that corresponds to a particle number density, n. The set of equations below is close to the one already discussed in the 1D framework [START_REF] Bacchin | Colloid-interface interactions initiate osmotic ow dynamics Colloids Surf[END_REF]: the only change is that the global permeability representing the viscous dissipation in one direction is here replaced by a local viscous dissipation, η m ∇ 2 u m , as in the Stokes ow equation.

Momentum balance

On the dispersed phase +nF drag -∇Π cc -φ∇Π ic = 0.

(1)

On the uid

-nF drag -∇p + η m ∇ 2 u m + ∇Π cc = 0. (2)
On the mixture

-∇p + η m ∇ 2 u m -φ∇Π ic = 0. (3)

Mass balance

On the dispersed phase

∂φ ∂t = -∇ • (φu c ) . (4)
On the uid

∂ (1 -φ) ∂t = -∇ • ((1 -φ) u f ) . (5)
On the mixture

0 = ∇ • u m . (6)
For the drag force representing the forces due to the friction induced by the relative velocity between the phases and the colloid mobility, m:

F drag = u m -u c m (φ) . (7)
By combining equations ( 1)-( 7), a nal set of three equations can be obtained:

∇ • u m = 0 (8) -η m ∇ 2 u m + ∇p + φ∇Π ic = 0 (9) ∂φ ∂t = -∇ • (φu m + m (φ) V p (-∇Π cc -φ∇Π ic )) (10)
where equation ( 8) expresses the conservation of the volume for an incompressible mixture consisting of uid and colloid; equation ( 9) describes the viscous dissipation of the ow, -η m ∇ 2 u m , driven by the release of uid pressure, ∇p, and of colloid-interface interactions, φ∇Π ic; and equation [START_REF] Kedem | Thermodynamic analysis of the permeability of biological membranes to nonelectrolytes[END_REF] relates the colloid ow driven by the mixture ow, the colloid-colloid interaction and the colloid-interface interactions.

According to the original formulation of the model, the pressure p used in these equations refers to the product (1 -φ) p where p is the static pressure. For simplicity and because the pressure is not a parameter modifying the simulations, the term p will be used along the paper to represent (1 -φ) p.

The coupling of the mass ux terms (equation ( 10)) helps to describe the main mechanisms that occur during transport of colloids through a narrow channel:

the coupling of convective, φu m , and diffusive uxes, -mV p ∇Π cc , describes the accumulation of concentration [START_REF] Zydney | A concentration polarization model for the ltrate ux in cross-ow micro ltration of particulate suspensions[END_REF][START_REF] Porter | Concentration polarization with membrane ultra ltration[END_REF] that takes place when colloids accumulate at the channel bottleneck the coupling of convective ux and migration induced by colloid-interface interactions, -mV p φ∇Π ic , describes the heterogeneous critical ux phenomena [START_REF] Bacchin | Model for colloidal fouling of membranes[END_REF][START_REF] Bacchin | Colloidal surface interactions and membrane fouling: investigations at pore scale[END_REF]: a critical convective drag force needed to overcome the colloidinterface repulsion and then to lead to a deposit at the interface (heterogeneous liquid-solid transition) the coupling of diffusive ux and migration induced by colloid-interface interaction describes the Boltzmann exclusion at equilibrium that is due to the colloid-interface interaction [START_REF] Bacchin | Colloid-interface interactions initiate osmotic ow dynamics Colloids Surf[END_REF] However, a strong difference with conventional approaches can be seen in the momentum balance (equation ( 9)). The viscous dissipation term is expressed here as the combination of the applied static pressure with the density of colloid-interface forces, φ∇Π ic , as previously discussed by Anderson [32]. This coupling allows consistency to be maintained for the description of the equilibrium: in the absence of drag force (i.e. at equilibrium, u m = u c in equation ( 7)), equation ( 1) leads to φ∇Π ic = -∇Π cc and the momentum equation (equation ( 9)) thus matches the description of the equilibrium between the static pressure and the osmotic pres sure, ∇p = -φ∇Π ic = ∇Π cc . In non-equilibrium conditions, the density of colloid interface forces plays the role of a forcing term (or a local resistance) on the momentum equations of the uid ow, similarly to the ones appearing in the FCM used to describe multi-phase ows [START_REF] Agbangla | Numerical investigation of channel blockage by owing microparticles[END_REF][START_REF] Agbangla | Collective dynamics of owing colloids during pore clogging[END_REF][START_REF] Lomholt | Force-coupling method for particulate two-phase ow: Stokes ow[END_REF]. It will be shown in the next sections that this coupling allows the interfacially driven transport to be described as diffusio-osmosis and Marangoni ows.

Interfacial pressure and interfacially driven transport

The key point discussed in this paper is the coupling of the near-wall colloid and solvent transport phenomena. The coupling between mass and momentum balances is classically taken into account by the viscosity function that encompasses the uid/colloid interactions. Here, such coupling also comes from the colloid-interface interactions that appear in the momentum (equation ( 9)) and mass balance (equation ( 10)) with the mathematical contribution, φ∇Π ic (the interfacial force density in N m -3 ):

in the mass balance, these interactions induce an additional mass transport due to the forces between the colloids and the wall in the momentum, the interaction with the interface changes the momentum of the colloids and thus contributes to a change in the mixture momentum

The mixture momentum (equation ( 9)) is a balance between a viscous dissipation term and two elastic terms: the pressure gradient and the term coming from the colloid-interface interaction, φ∇Π ic . For greater clarity, the balance between the elastic contributions, ∇p = -φ∇Π ic , can be considered rst. A change in colloid-interface interactions can thus be associated with a uid pressure change. If the colloid-interface interactions are repulsive, (Π ic increases when approaching the interface), these interactions lead to a decrease in the uid pressure close to the interface. Physically, this means that the uid loses the mechanical reaction force (resulting from the action force of the colloid on the interface), which is dissipated in the interface (in absence of elastocapillary effect) [START_REF] Bacchin | Colloid-interface interactions initiate osmotic ow dynamics Colloids Surf[END_REF]. This decrease in uid pressure at the interface allows the total pres sure applied on the interface to be kept constant: the loss in uid pressure is equated to the pressure due to the normal force exerted by colloids on the interface. This contrib ution of the colloid-interface interaction to the uid pressure can be computed as the integration of the force density from the bulk to a position A in the system, p int = -´A ∞ φdΠ ic , the so called 'interfacial pressure'. Repulsion (attraction) with the interface leads to a negative (positive) interfacial pressure that can be associated with a local decrease (increase) in uid pressure close to the interface.

Local changes in the interfacial pressure can appear because of variation in the colloid volume fraction, φ, or variation in the colloid-interface interaction, Π ic . If such variations occur along an interface, a pressure gradient develops along the surface which, in turn, induces a ow called interfacially driven transport. Such phenomena are sketched for the cases of attractive and repulsive colloid-interface interactions in gures 1(a) and (b), respectively. In the case of attractive interaction ( gure 1(a)), the presence of colloids interacting with the wall induces a liquid overpressure. In this case, the increase in colloid concentration induces an increase of uid overpressure. A ow is then induced from the high pressure (high concentration) zone to the low concentration zone. This ow corresponds to a Marangoni effect (also called capillary driven ow) that drives the ow to low surface tension (where there is a large number of colloids having an af nity with the interface i.e. acting as a surfactant) to high surface tension (where there are few surfactant-colloids). In the case of repulsive interactions, gure 1(b), the interfacial interactions induce a decrease in the interfacial pressure in the region with a high concentration of colloids. This pressure gradient drives a ow called diffusion-osmosis ow in this case.

The interfacial force density, φ∇Π ic , drives the ow of both particles (particle exclusion in interfacial layer) and liquid (through interfacially driven ows). These interfacial forces between the colloids and the interface strongly couple the near-wall colloid and solvent dynamics. These transport phenomena thus have a hydrodynamic character that cannot be explained by thermodynamic considerations. The twouid model that accounts for colloid-interface interaction in momentum and mass balances depicts the transient initiation of the osmotic and Marangoni ows and thus generalizes and uni es the existing approaches. This description also has further consequences when the interface is no longer a solid interface but a uid one. From a thermodynamical point of view, the energy map represents the interfacial Gibbs free energy and, thus, the increase or decrease in pressure is closely related to an increase or a decrease in water activity, p int = kT Vw ln(a w ). Considering water activity instead of interfacial pressure, it can be noted in gure 1 that the osmotic or the Marangoni ows always develop from high to low water activity zones. Such local variation of water activity at the interface can have many consequences when evaporation kinetics is considered, the evaporation rate being proportional to the difference of water activity between the liquid interface and the ambient air. The local variation of water activity given by the model can thus help to consider the contribution of colloid-colloid and colloid-interface interactions on water evaporation. The interaction of colloids with the interface also leads to an anisotropy in pressure close to the interface. This stress anisotropy can be normal and/or tangential to the interface [START_REF] Squires | Micro uidics: uid physics at the nanoliter scale[END_REF]:

when there is a local pressure gradient normal to the free uid surface, it leads to an interfacial (capillary) stress that leads to a surface tension. According to Kirkwood and Buff [START_REF] Kirkwood | The statistical mechanical theory of surface tension[END_REF], the surface tension for a at interface is linked to the integral of the interfacial pressure across the interface, γ = ´∞ -∞ p int dx.

when there is a pressure gradient tangential to the surface (because of a variation of colloid volume fraction or a variation in interactions), interfacially driven ows (solvent ows with a component tangential to the surface) are initiated.

Simulation of interfacially driven ow in a narrow channel

In this paper, the set of equations (( 8)-( 10)) is solved in a geometry representing a channel with solid (non-deformable) walls with a non-dimensional form:

∇ • Pe = 0 (11) ∇ • ( µ (φ) ∇Pe) = ∇ p + φ∇ Π ic + (1 -φ)∇ Π if (12) ∂φ ∂ t = -∇ • φPe + K (φ) -∇ Π cc -φ∇ Π ic . (13)
In table 1, the dimensionless variables are de ned and their links with the dimensional variables are quanti ed. The nondimensional terms are obtained by dividing by the diffusion force so that the advection term becomes a Péclet number in equations ( 11)- [START_REF] Einstein | Investigations on the Theory of the Brownian movement ed R Fürth[END_REF]. For the nondimensionalization, the diffusion coef cient is considered at dilute condition, D 0 = m 0 kT, where m 0 is the mobility of colloids having a volume, V p . The corresponding Reynolds number is thus the Péclet number divided by the Schmidt number, Sc. The set of data used in these simulations was de ned to characterize When the colloids are attracted by the interface (a) the colloid-interface interaction contributes to a local increase in the interfacial pressure contribution, -´φdΠ ic . This increase is greater when the colloid concentration is higher (bottom part of the gure). This tangential pressure gradient leads to ow directed toward the lower concentration: Marangoni ow. For colloids experiencing repulsion with the interface (b), the interaction leads to a local decrease of pressure close to the interface that is more pronounced for higher colloid concentration. These pressure variations induce a ow toward the higher concentration. This diffusio-osmotic phenomenon causes the osmotic ows. a dispersion of 10 nm in diameter in a system with a characteristic size (the pore size) of 1 micrometre. Such a size ratio ensures that the dispersion is treated as a continuous medium and the Eulerian approach is correctly used. Under these size conditions, the non-dimensional viscosity is equal to 5.55 × 10 -6 . The dependence of viscosity on the volume fraction is not taken into account. The osmotic pressure is de ned by a van't Hoff law for an ideal dispersion: the colloid-colloid interactions are not taken into account and the diffusion coef cient remains independent of the colloid volume fraction. In these simulations, the mechanisms induced by colloid/colloid interactions and by the coupling between volume fraction and viscosity are switched off. Such simulations thus focus on the effect of colloid/interface interactions on the uid dynamics. These simulations can represent the ow of a rather dilute colloidal dispersion close to an interacting interface. Equations ( 11)-( 13) could be solved in speci c geometries by introducing solid walls as boundary conditions with noslipping conditions. In this paper, another approach is chosen: the equations are solved for the whole domain but with a local penalization method [START_REF] Angot | A penalization method to take into account obstacles in incompressible viscous ows[END_REF] in the Stokes equation to account for the presence of solid walls. In equation ( 12), a term, Π if , is thus added to penalize the ow in the solid domain. This term physically express the uid-wall interaction that forces the ow away from the interface. This way of writing the equation has the advantage of treating the wall interactions similarly: the presence of the narrow channel in the ow is represented through the interactions that the wall interface exerts on the uid, Π if , together with the interaction it exerts on the colloids, Π ic . These interactions are a function of the distance to the wall, which is determined through a level set method. The penalization for the uid is a very stiff exponential function that applies in a very thin interfacial layer close to the interface. To be negligible, the interfacial layer for uid-interface interactions is less than one tenth of the interfacial layer for colloid-interface interactions. The interaction between the colloids and the wall are also represented by an exponentially decreasing function similar to the one that could be obtained by the DLVO theory.

In this paper, the decay length is taken to be 0.1 (one tenth of the pore diameter) and the maximum energy at the wall is xed at 100. These values were chosen to be close to those calculated for 10 nm spheres dispersed in 10 -5 M solution with zeta potential of 80 mV for both particles and walls. The resulting colloid-interface interaction map is plotted in gure 2. The set of equations was solved with the partial differential equation solver Fipy [START_REF] Guyer | FiPy: partial differential equations with python[END_REF] ( nite element volume) implemented on the Python platform Canopy (Enthought, Austin). Simulations were performed with periodic conditions on the top and bottom boundaries: this corresponds to the ow through a network of narrow channels. Such simulations with periodic conditions enable the description of the possible interferences in mass accumulation between two consecutive channels. To have a complete analysis of this effect, the size ratio between the channel bottleneck and the pillar can be modi ed by changing the geometry. The full code used for the solving is given in SI 6 in the supplementary information (available at stacks.iop.org/JPhysCM/30/294001/mmedia). Simulations are presented in the next sections (i) for no net ow conditions through the channel to illustrate the diffusion-osmosis phenomena (section 3) and (ii) for ltration conditions through the channels with a counter-osmotic pres sure (section 4).

Pure diffusio-osmosis in closed channel

In this section, the model is used to describe a diffusioosmotic ow by simulating the ow induced by a gradient of concentration. The simulations are performed in a 2D geometry representing a pore closed at one end (right boundary in gure 3). The global x velocity is thus zero. At the open side of the channel (left boundary), colloids arrive by diffusion. This diffusion induces a gradient of concentration along the pore axis that, in turn, should lead to an osmotic ow: the so called diffusio-osmosis phenomenon.

Simulations allow the momentum and the mass balance (equations ( 12) and ( 13) respectively) to be solved and the continuity equation (equation [START_REF] Fermi | Thermodynamics[END_REF]) to be satis ed. The mixture velocities along x and y, the pressure eld and the volume the generation of an osmotic ow (with negative x velocities) when the colloid concentration reaches the pore wall (diffusio-osmosis) the coupling of the osmotic ow with forced convection (with positive x velocities) to keep a zero net ow across the pore section (diffusio-osmotic/forced convection secondary ow) the return to equilibrium (zero uid velocities and zero mass ux) when the diffusion homogenizes the concentration along the pore axis.

Figure 4 represents the velocities (presented in a nondimensional form as a Péclet number here) in the x direction ( gure 4(a)) and in the y direction ( gure 4(b)) at time t = 2 (i.e. when the diffusio-osmosis phenomenon is at its maximum). The stream line representation ( gure 4(c)) illustrates the presence of the secondary ow with negative x velocities close to the wall (the osmotic ow) and a positive x velocity in the pore centre (the forced convection developing to ensure a zero net ow through the pore). These secondary ow cells have positive and negative y velocities at their ends ( gure 4(b)). The ows are characterized by stationary planes where the x velocities are zero. These stationary planes can be seen in gure 5, which presents the x velocity pro le along y at the pore entrance (x = 3). The stationary planes are located at 30% and 70% of the pore diameter. These locations correspond to 2 -0.5 and 1 -2 -0.5 that can be determined from the analytical solution for an interfacial ow combined with forced convection in Cartesian coordinates (as with electroosmosis ow).

The secondary ows become established progressively with time. Figure 6 presents the variation with time of the osmotic ow rate de ned as the integration along y of the negative x velocity presented in gure 5. There is no osmotic uid ow at the initial time when the colloids are not interacting with the pore wall. As soon as the colloids interact with the walls, the contribution of φ∇ Π ic in the momentum equation leads to the osmotic ow in the direction of higher φ value. The maximum ow rate occurs for t = 2 and therefore the ow starts to decrease mainly because the concentration gradient decreases along the pore axis. The volume fractions of colloids are given in SI 2. For longer times (i.e. when diffusion homogenizes the concentration) the osmotic ow vanishes.

To conclude, the diffusio-osmosis test case of a pore with one end closed allows us to check the ability of the model to describe (i) how osmotic solvent ow transiently initiates from colloid/interface interactions, (ii) how interfacially driven transport and forced convection act together to generate secondary ows, and (iii) how the system returns to equilibrium. The diffusio-osmosis mechanism can be described according to the following steps. The diffusion of colloids toward the interface generates a concentration gradient parallel to the pore wall. These concentration gradients lead to a local force density, φ∇ Π ic (plotted in SI 3) that initiates the diffusio-osmotic ows. In the case of a closed system, forced convection becomes established in the opposite direction to satisfy the solvent continuity and, combined with the diffusio-osmotic ow, leads to secondary ows close to the interface. Similar simulations can be run for colloid-interface attraction instead of repulsion. In this case, the secondary ow appears but in the opposite direction. The liquid ow is then directed toward the zone of low colloid concentration. This ow is a solute-capillary Marangoni ow with a ow toward the zone where colloids having an af nity with the interface are less concentrated (or zone of higher surface tension).

Osmotic ows in channel during ltration

In this section, simulations are performed with a geometry similar to that used in the previous section but with a ow rate through the channels (the narrow channel geometry is presented in gure 2(a)). In ltration conditions, the mass boundary conditions are a constant concentration on the left side (inlet of a ow with a given concentration) and a concentration gradient at zero on the right (outlet ow). The ow boundary conditions in the right and the left domain boundaries are a constant Péclet number along x ( xing the net ux through the pore) and a Péclet number along y at zero. The xed Péclet number along x, Pe (also corresponding to a uid velocity or a Reynolds number as presented in table 1) is then xed by the boundary conditions. The boundary conditions are periodic on the top and bottom meshes. This corresponds to a ltration case with a constant ow rate through the narrow channel: the pressure drop thus increases if colloids accumulate at the pore bottleneck and offer resistance to the ow. These simulations then depict the reverse osmosis situation where a counter osmotic ow acts against a forced convection. The previous case, described in section 3, can be seen as the asymptotic case for a Péclet number of zero, where diffusion-osmosis occurs in the absence of a net ow through the channel.

Simulations have been performed for different Péclet numbers between 0.1 and 10. The full data of the transient simulations are given in an animated panel presenting the evolution of the main parameters with time in supplementary materials (SI 4). To analyse the simulation results, the simulation with a Péclet number equal to 3 will be considered rst. Figure 7(a) maps the volume fraction accumulated at the pore entrance for the nal time t = 10 (a quasi-steady state is reached at this time) and gure 7(b) presents the variation of the volume fraction along the channel axis with time.

Colloids accumulate because of partial retention due to the repulsive barrier along the wall of the pore channel ( gure 2(a)): the water can ow in these regions whereas the colloids are expelled. The accumulation takes the form of a hemispherical plug at the pore bottleneck ( gure 7(a)) where the volume fraction of colloids can be ve times higher than that coming from the bulk on the left. The plug builds up progressively with time and reaches a steady state shape when the retro-diffusion in the bulk balances the convective mass ux on the left boundary. The thickness of the upstream area thus acts as a mass boundary layer that is progressively lled by the accumulation.

The ow is modi ed by the accumulation, which leads to a concentration gradient along the channel and then to the diffusio-osmosis phenomenon. This osmotic ow is opposite to the forced convection and appears as negative x velocities in gure 8(a). These osmotic ows are the direct consequence of the contribution of the colloid-wall interaction (term φ∇Π ic in the momentum balance). This interfacial contribution acts as if there was a decrease in pressure close to the wall (where ∇Π ic is large) or in the concentrated zone in the upstream channel zone (where φ is large) [START_REF] Bacchin | Colloid-interface interactions initiate osmotic ow dynamics Colloids Surf[END_REF]. The map of the contribution of φ∇Π ic is given in supplementary materials (SI 5). It should be noted that negative x velocities also appear where the colloid accumulates (around the position (1,1) in gures 7(a) and 8(a)). Such negative ows are due to the strong gradients of colloid concentration in the zone where colloids interact with the interface ( gure 8(a)), which induces osmosis.

The negative osmotic ows result in an additional pressure drop that keeps the permeate ow rate constant across the channel. In gure 9, the Péclet number ( xed as a boundary condition) is plotted versus the pressure drop at quasi-steady state (determined by simulations). This plot is a non-dimensional form of the classic plot of the permeate ux versus the transmembrane pressure used to quantify the ltration results. Evolution of the half osmotic ow rate (resulting from the integration along y of the negative x velocities at the pore entrance gure 5) with time. At the beginning, when colloids are not interacting with the wall, the osmotic ow rate is zero. The osmotic ow is maximum when the colloid concentration gradient reaches the pore wall and vanishes again when the diffusion homogenizes the concentration in the pore.

In this kind of plot, the dotted line represents the pressure drop simulated for the solvent water and the horizontal gap between the dotted line and the one obtained when ltering the colloid represents the additional counter osmotic pressure.

It can be noted that the results presented in gure 9 when the accumulation is rather homogeneous at the channel entrance (for small Péclet number) are comparable to the one determined when the equations are solved in one dimension [START_REF] Bacchin | Colloid-interface interactions initiate osmotic ow dynamics Colloids Surf[END_REF] and with the Non Equilibrium Thermodynamics approach [START_REF] Kedem | Thermodynamic analysis of the permeability of biological membranes to nonelectrolytes[END_REF]. For higher Péclet, it can be seen that the counter osmotic pressure does not vary monotonously when the ow rate increases. For a Péclet number around 7, a maximum in the counter osmotic pressure is observed, together with a marked increase in transmission. This change is also associated with a very different pattern for the colloid accumulation as sketched in gure 10. The pattern of the accumulation takes different forms:

• a homogeneous accumulated layer when operating at low Péclet number. The diffusion of colloids is high enough to homogenize the concentration along the y direction. • an axial plug with diffuse accumulated colloids at the channel bottleneck for intermediate Péclet number. This pattern is responsible for the highest accumulation and the highest counter osmotic pressure at a Péclet number of 7. • an accumulation on the channel pillars with only a partial axial accumulation for larger Péclet numbers. These Péclet numbers also relate to greater transmission through the channel: the ow is able to overcome the colloid-wall interactions.

The transition between an axial plug ( gure 10(b)) and pillar accumulation ( gure 10(c)) occurs sharply above the threshold Péclet number of 7. A similar critical transition has been studied experimentally and theoretically [START_REF] Bacchin | Critical and sustainable uxes: theory, experiments and applications[END_REF] by researchers studying the accumulation and transmission of colloids across the membrane. Such a transition is responsible for a sudden change in the fouling regime [START_REF] Bacchin | Model for colloidal fouling of membranes[END_REF][START_REF] Bacchin | Critical and sustainable uxes: theory, experiments and applications[END_REF] or colloidal transmission across a pore [START_REF] Kim | Effect of electrostatic, hydrodynamic, and Brownian forces on particle trajectories and sieving in normal ow ltration[END_REF]. The threshold appears when the drag force induced by the number of Péclet exceeds the repulsive force between the colloids and the wall [START_REF] Bacchin | Model for colloidal fouling of membranes[END_REF][START_REF] Bacchin | Colloidal surface interactions and membrane fouling: investigations at pore scale[END_REF]. This paper illustrates this concept with a local 2D approach and examines the consequences of this imbalance on the type of fouling (from axial to pillar accumulation) and on transmission ( gure 9(b)). The breaking of the accumulated layer is also observed for the same Péclet number but over time. Figure 11 presents the way the pillar accumulation pattern is initiated over time for a simulation performed with a Péclet number of 10. Initially, the accumulation grows in the form of a lm of homogeneous thickness. With time, a thinning of the accumulation is observed at the edge of the conical bottleneck ( gure 11(b)) before a breakage of the accumulation between an axial plug and the pillar accumulation ( gure 11(c)). For further time, the axial plug disappears, whereas the pillar accumulation becomes thicker and leads to the steady state accumulation presented in gure 10(c).

These results illustrate how the coupling between diffusioosmosis and hydrodynamics controls the ow in a channel. A complex force chain operates between the components of the ternary system: interface/colloid/solvent molecules [START_REF] Bacchin | Colloid-interface interactions initiate osmotic ow dynamics Colloids Surf[END_REF]. This force chain can be summarized as follows: (i) colloids interact with the interface (ii) the colloid/solvent mixture loses momentum due to the reaction force acting on the interface (Newton's third law applies for the ternary interface/colloid/ solvent system but is violated for the colloid/solvent pair) (iii) for colloidal particles, according to the dissipation uctuation theorem, this force anisotropy normal to the interface (nonzero transverse stress) leads to a net force density on the uid near the interface. The three components of the ternary system are then intimately connected by the force chain: at the end, because of colloid/interface interactions, a net force acts on the solvent molecules.

From mechanical balances, the simulations show how this net force modi es the Stokes ow and thus controls the nearwall non-equilibrium dynamics of colloids and solvent molecules. Ultimately, describing the interfacially driven ows (diffusio-osmosis and Marangoni effect) appearing inside the liquid phase is a key point but it is also important, when considering boundary conditions, to depict the surface tension and evaporation kinetics. This local description of the ow dynamics could also help to understand how the pore architecture, e.g. its hourglass shape, changes the transport ef ciencies. This kind of model should be useful to depict other conditions where diffusio-osmosis takes place, such as diffusio-phoresis [START_REF] Malgaretti | Selfdiffusiophoresis induced by uid interfaces[END_REF] and transport of self-propelling active colloids. The model could also be applied to conditions where a strong coupling between ow, surface interactions and electrochemical reaction can induce ion-depletion zone and allow membraneless seawater desalination [START_REF] Knust | Electrochemically mediated seawater desalination[END_REF]. The non-equilibrium conditions could also lead to Rayleigh-Taylor instabilities that could be a way to decrease narrow channel fouling [START_REF] Lohaus | Feed ow patterns of combined Rayleigh-Bénard convection and membrane permeation[END_REF].

Conclusions

In summary, a two phase ow model and 2D simulations allow the dynamics of owing colloids to be explored when they interact with an interface. From mechanical balances, the colloid/interface interactions lead to a net force acting on the solvent molecules near the interface, which can be ascribed to an interfacial pressure gradient. Interfacially driven transport, like diffusion-osmosis or the Marangoni effect, originates from gradients in interfacial pressure along the interface, for example, due to a gradient in colloid volume fraction. The model explicitly shows how colloid/interface interactions initiate interfacially driven transport. The model thus gives a clear local scale interpretation of the osmosis phenomenon within a mechanical approach. The comprehensive approach generalizes and uni es the description of colloidal dispersion ow in a con ned system.

Interesting consequences emerge when the transport of colloids through a narrow channel is considered. Secondary ows develop in the narrow channel bottleneck because of the coupling between diffusio-osmosis and forced ow. The dynamics of these osmosis secondary ows has been analysed here and their consequences on the counter pressure and on the colloid transmission through the channel discussed. These local ows also control the way the colloids accumulate at the channel bottleneck. The simulations shed light on the existence of a critical Péclet number that leads to a transition between axial plug formation and accumulation on channel pillars.

Figure 1 .

 1 Figure 1. Representation of the interfacially driven transport in presence of a gradient of colloid concentration tangentially to an interface.When the colloids are attracted by the interface (a) the colloid-interface interaction contributes to a local increase in the interfacial pressure contribution, -´φdΠ ic . This increase is greater when the colloid concentration is higher (bottom part of the gure). This tangential pressure gradient leads to ow directed toward the lower concentration: Marangoni ow. For colloids experiencing repulsion with the interface (b), the interaction leads to a local decrease of pressure close to the interface that is more pronounced for higher colloid concentration. These pressure variations induce a ow toward the higher concentration. This diffusio-osmotic phenomenon causes the osmotic ows.

Table 1 .

 1 The dimensionless quantities used to de ne the dynamic osmotic problem. The correspondence with the dimensional quantities is given for the conditions of a = 10 -8 m, δ =10 -6 m, µ = 0 -3 Pa • s, and T = 298 K. Quantity Dimensionless form Correspondence Velocity Péclet Pe = umδ m0kT u (m) = 2.18 10 -6 Pe = 0.1 Re Reynolds Re = Pe/Sc Viscosity µ -1 • s) = 5.31 10 +9 K (φ)

Figure 2 .

 2 Figure 2. (a) 2D representation of the narrow channel geometry (the pillars are represented in brown) and of the colloid-wall interaction magnitude (with the colour map) (b) colloid-wall interaction energy across the narrow channel opening for x = 2.5.

Figure 3 .

 3 Figure 3. Representation of the colloid-interface energy map in the semi-closed channel (axis of the channel in the x direction). The boundary conditions are periodic at the top (y = 2) and bottom (y = 0), wall conditions on the right (zero velocities and zero mass ux at x = 4) and a mass inlet with no ow on the left (constant concentration and zero velocities at x = 0).

Figure 4 .

 4 Figure 4. Intensities of x velocities and y velocities (represented by a dimensionless péclet number) in (a) and (b) respectively, for a time t = 2. These variations correspond to two secondary ows with negative x velocities close to the pore walls and a return ow (positive x velocites) in the pore centre. The representation of stream lines in c) highlights the two recirculation regions at the channel inlet.

Figure 5 .

 5 Figure 5. x velocity pro le along y at the pore entrance (x = 3) for different times. The stationary planes are located at 30% and 70% of the pore diameter.

Figure 6 .

 6 Figure 6. Evolution of the half osmotic ow rate (resulting from the integration along y of the negative x velocities at the pore entrance gure 5) with time. At the beginning, when colloids are not interacting with the wall, the osmotic ow rate is zero. The osmotic ow is maximum when the colloid concentration gradient reaches the pore wall and vanishes again when the diffusion homogenizes the concentration in the pore.

Figure 7 .

 7 Figure 7. Spatiotemporal variation of colloid volume fraction at Pe = 3. (a) Maps the volume fraction on the domain at t = 10. (b) Presents the volume fraction pro le along the channel axis, x (for y = 1) for different simulation times with a time step of 0.25 from t = 0 to t = 10.

Figure 8 .

 8 Figure 8. Intensities of x velocities and y velocities (represented by dimensionless local péclet number) in (a) and (b) respectively for a time t = 10 and for a net overall Péclet number along x of 3. These variations correspond to two secondary ows with negative x velocities close to the pore walls and a return ow (positive x velocites) in the pore centre. The stream lines are represented in (c).

Figure 9 .

 9 Figure 9. (a) Péclet number as a function of the pressure drop across the channel (full red line). The dotted line represents the same data when only the solvent is ltered. (b) Colloid transmission as a function of the Pe number. The inserts represent the volume fraction map (details given in gure 10).

Figure 10 .

 10 Figure 10. Colloid accumulation for (a) Pe = 1, (b) Pe = 3 and (c) Pe = 10 for a time t = 10 corresponding to a quasi-steady state for all cases. The accumulation patterns are very different with (a) a homogeneous layer for low Péclet number (high diffusion that homogenizes the concentration along y) (b) an axial plug at the channel bottleneck for intermediate Péclet number and (c) an accumulation on the channel pillars with only a partial axial accumulation for high Péclet number.

Figure 11 .

 11 Figure 11. Dynamic evolution of the accumulation (t = 0.2, 0.3, 0.5) for Pe = 10. Breakage of the accumulated lm (a) to give axial plug (b) and accumulation on channel pillars (c). The steady state (t = 10) accumulation on channel pillars is presented in gure 10(c).

Acknowledgments

The author thanks Yannick Hallez, Fabien Chauvet, Léo Garcia and Micheline Abbas for fruitful discussions.

ORCID iDs

Patrice Bacchin https://orcid.org/0000-0002-7737-1315