Corentin Tregouët

Thomas Salez

Cécile Monteux
email: cecile.monteux@espci.fr

Mathilde Reyssat
email: mathilde.reyssat@espci.fr

Corentin Trégouët

Thomas Salez Cd

Microfluidic probing of the complex interfacial rheology of multilayer capsules

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Introduction

Droplets with valuable content can be protected from their environment by a membrane, usually elastic, which turns them into capsules. Encapsulation of droplets allows a better control on the delivery of the droplet content. Capsules are used in a very broad range of applications such as food industry [START_REF] Klinkesorn | Encapsulation of emulsified tuna oil in two-layered interfacial membranes prepared using electrostatic layer-by-layer deposition[END_REF] , drug delivery [START_REF] Delcea | Stimuliresponsive LbL capsules and nanoshells for drug delivery[END_REF][START_REF] Correa | Highly Scalable, Closed-Loop Synthesis of Drug-Loaded, Layer-by-Layer Nanoparticles[END_REF] , material science [START_REF] Yang | Microencapsulation of isocyanates for self-healing polymers[END_REF] or cosmetics [START_REF] Tirilly | Interplay of Hydrogen Bonding and Hydrophobic Interactions to Control the Mechanical Properties of Polymer Multilayers at the Oil/Water Interface[END_REF] . Various types of processes have been developed in the last decades to produce capsules, especially thanks to the development of microfluidic technics [START_REF] Gao | Stability and Mechanical Properties of Polyelectrolyte Capsules Obtained by Stepwise Assembly of Poly(styrenesulfonate sodium salt) and Poly(diallyldimethyl ammonium) Chloride onto Melamine Resin Particles[END_REF][START_REF] Antipov | Sustained release properties of polyelectrolyte multilayer capsules[END_REF][START_REF] Priest | Microfluidic polymer multilayer adsorption on liquid crystal droplets for microcapsule synthesis[END_REF][START_REF] Kantak | A 'microfluidic pinball' for on-chip generation of Layer-by-Layer polyelectrolyte microcapsules[END_REF][START_REF] Erni | Deformation modes of complex fluid interfaces[END_REF][START_REF] Polenz | Polyurea microcapsules in microfluidics: Surfactant control of soft membranes[END_REF][START_REF] Bjornmalm | Flow-Based Assembly of Layer-by-Layer Capsules through Tangential Flow Filtration[END_REF][START_REF] Dupré De Baubigny | One-Step Fabrication of pH-Responsive Membranes and Microcapsules through Interfacial H-Bond Polymer Complexation[END_REF] .

To rationalize the formulation of capsules it is necessary to understand the link between the composition of the protecting membrane, its mechanical properties and the performances of the capsules, which requires model polymer membranes as well as experimental techniques to determine capsule mechanical properties. New experimental techniques have been developed to characterize elastic polymer membranes at liquid interfaces [START_REF] Vandebril | A double wall-ring geometry for interfacial shear rheometry[END_REF][START_REF] Sarrazin | AFM Investigation of Liquid-Filled Polymer Microcapsules Elasticity[END_REF][START_REF] Xie | Interfacial rheological properties of selfassembling biopolymer microcapsules[END_REF][START_REF] Knoche | Elastometry of deflated capsules: Elastic moduli from shape and wrinkle analysis[END_REF][START_REF] Oshri | Wrinkles and folds in a fluid-supported sheet of finite size[END_REF] . The rheology of such membranes has been studied in model geometries [START_REF] Sarrazin | AFM Investigation of Liquid-Filled Polymer Microcapsules Elasticity[END_REF][START_REF] Knoche | Elastometry of deflated capsules: Elastic moduli from shape and wrinkle analysis[END_REF][START_REF] Gordon | Self-assembled polymer membrane capsules inflated by osmotic pressure[END_REF][START_REF] Carvajal | Mechanics of pendant drops and axisymmetric membranes[END_REF][START_REF] Erni | Interfacial viscoelasticity controls buckling, wrinkling and arrest in emulsion drops undergoing mass transfer[END_REF][START_REF] Ferri | Elastic nanomembrane metrology at fluid-fluid interfaces using axisymmetric drop shape analysis with anisotropic surface tensions: deviations from Young-Laplace equation[END_REF][START_REF] Kleinberger | Systematic study of alginate-based microcapsules by micropipette aspiration and confocal fluorescence microscopy[END_REF] and also on real capsules in viscous flows to mimic real conditions of fabrication and use, through various experiments [START_REF] Xie | Interfacial rheological properties of selfassembling biopolymer microcapsules[END_REF][START_REF] Kleinberger | Systematic study of alginate-based microcapsules by micropipette aspiration and confocal fluorescence microscopy[END_REF][START_REF] Walter | Shear induced deformation of microcapsules: shape oscillations and membrane folding[END_REF][START_REF] Prevot | Design of a Microfluidic System to Investigate the Mechanical Properties of Layer-by-Layer Fabricated Capsules[END_REF][START_REF] Hu | Characterizing the membrane properties of capsules flowing in a square-section microfluidic channel: Effects of the membrane constitutive law[END_REF] , theories [START_REF] Barthès-Biesel | The time-dependent deformation of a capsule freely suspended in a linear shear flow[END_REF][START_REF] Finken | Micro-capsules in shear flow[END_REF][START_REF] Higley | Semi-analytical solutions for two-dimensional elastic capsules in Stokes flow[END_REF] and simulations [START_REF] Pozrikidis | Finite deformation of liquid capsules enclosed by elastic membranes in simple shear flow[END_REF][START_REF] Park | Transient dynamics of an elastic capsule in a microfluidic constriction[END_REF][START_REF] Koolivand | Deformation of an elastic capsule in a microfluidic T-junction: settling shape and moduli determination[END_REF] . From this perspective, microfluidics is an ideal tool which allows to integrate measurement units directly in situ in the devices and to perform automated measurements on a large number of capsules.

The use of extensional flows to measure the surface properties of droplets has been first developed by Taylor [START_REF] Taylor | The Formation of Emulsions in Definable Fields of Flow[END_REF] with the so-called four-roller apparatus. Since then, this method has been adapted in microfluidics by imposing an extensional flow field to droplets at a transition between a narrow channel and a larger chamber, and has been firstly employed to measure interfacial tensions [START_REF] Cabral | Microfluidic approach for rapid multicomponent interfacial tensiometry[END_REF][START_REF] Trégouët | Transient deformation of a droplet near a microfluidic constriction : a quantitative analysis[END_REF] . In our previous work [START_REF] Trégouët | Transient deformation of a droplet near a microfluidic constriction : a quantitative analysis[END_REF] , by simulating the flow field in the large chamber we were able to determine the shear forces applying on the capsules. Assuming a competition between the viscous forces deforming the capsules and the interfacial tension force restoring its shape, we were then able to predict the shape of the droplet as a function of its position in the channel for a given interfacial tension. Inversely, the fit of our experimental data with the model enabled to deduce the interfacial tension of the droplets. Recently, similar design has been used to measure the elastic modulus of capsules [START_REF] Leclerc | Transient flow of microcapsules through convergent-divergent microchannels[END_REF][START_REF] Polenz | Monitoring reactive microencapsulation dynamics using microfluidics[END_REF][START_REF] Gires | Transient behavior and relaxation of microcapsules with a cross-linked human serum albumin membrane[END_REF] . However, for intermediate situations where several interfacial components can interfere (for example in the case of visco-elastic capsules for which the interfacial tension force is not negligible), there is no model which can rationalize the evolution of the capsules deformation over time by accounting for the relative importances of membrane viscoelastic properties and interfacial tension.

In previous studies published in our group [START_REF] Tirilly | Interplay of Hydrogen Bonding and Hydrophobic Interactions to Control the Mechanical Properties of Polymer Multilayers at the Oil/Water Interface[END_REF][START_REF] Tirilly | Interfacial Rheology of Hydrogen-Bonded Polymer Multilayers Assembled at Liquid Interfaces: Influence of Anchoring Energy and Hydrophobic Interactions[END_REF][START_REF] Tregouët | Polymer multilayers at liquid interfaces : assembly , interfacial rheology and microfluidic probing[END_REF] , we have designed model polymer membranes with controlled interfacial properties. These membranes are obtained by assembling polymer multilayers at oil water interfaces through hydrogen bonds. We have shown that the hydrophobicity of the polymers enables to control the interaction energy of the assembled layers by an interplay of hydrogen bonds and hydrophobic interactions, which tunes the shear moduli of the membrane. Moreover, controlling the anchoring energy of the chains by adding hydrophobic anchors on the chains enables to control the compression moduli of the adsorbed layers.

In the present article, our goal is to study the relation between the composition of the polymer membranes protecting the droplets and the deformation of capsules in a controlled flow field. We produce, using microfluidics, monodisperse populations of oil droplets covered by polymer multilayers of various compositions described above and we study their deformation in microfluidic extensional flows, obtained by the combination of a constriction followed by a larger chamber. We find that the dynamics of the capsules under this extensional flow strongly depends on the composition of the multilayers. We propose a coarse-grained model to extract the different characteristic properties of each polymer membrane, and evaluate the relative importances of the various factors ruling the capsule's deformation.

Materials and methods

Oil droplets in aqueous solutions are covered with polymers in a microfluidic chip, and directed through an extensional flow to measure their deformation and relaxation.

Materials

Liquids and polymers

The droplets consist of mineral oil (Sigma Aldrich, Mineral Oil Rotational Viscosity Standard of viscosity 29.04 mPa.s at 25.00 • C) flowing in different polymer solutions, prepared at 1 %w and adjusted at pH 3 with molar solutions of HCl or NaOH. In recent articles [START_REF] Tirilly | Interplay of Hydrogen Bonding and Hydrophobic Interactions to Control the Mechanical Properties of Polymer Multilayers at the Oil/Water Interface[END_REF][START_REF] Tirilly | Interfacial Rheology of Hydrogen-Bonded Polymer Multilayers Assembled at Liquid Interfaces: Influence of Anchoring Energy and Hydrophobic Interactions[END_REF] we have studied the interfacial rheological properties of polymer multi layers assembled at oil water interfaces by hydrogen bonds. The polymers used to obtain the multilayers are:

• poly(methacrylic acid), (PMAA) provided by Polysciences, M w = 50000 g/mol,

• poly(vinyl pyrrolidone), (PVP) provided by Fluka, M w = 40000 g/mol,

• covalently randomly hydrophobically modified poly(acrylic acid) PAA-0.7-C 12 and PAA-0.8-C 8 : respectively 0.7% and 0.8% of the monomers are grafted to an alkyl chain of 12 and 8 carbons respectively, the backbone being provided by Polysciences Inc., M w = 100000 g/mol. We prepared these polymers by following the protocol of Wang et al. [START_REF] Wang | Viscometric behaviour of hydrophobically modified poly(sodium acrylate)[END_REF] . The grafting density has been measured by proton NMR.

The droplets are characterized in the polymer solution leading to the last adsorbed layer.

Microfluidic chips

The droplets are produced by microfluidics using a standard flowfocusing device made of poly(dimethyl siloxane) (PDMS) in the production chip [START_REF] Trégouët | Transient deformation of a droplet near a microfluidic constriction : a quantitative analysis[END_REF] . Such a setup provides a very monodisperse collection of droplets, tunable from 50 to 70 µm in diameter depending on the experimental conditions. As described in the following, we design specific connections to add a second polymer layer. The production chip is further connected to a characterization chip realized with a photosensitive adhesive provided by Norland (NOA 81). This microfabrication technique is a way to build rigid channels confined between two glass surfaces, and able to sustain large flow rates without being deformed [START_REF] Bartolo | Microfluidic stickers[END_REF] . The inlet of the chip is connected to the droplet-production chip by a silicon tubing (Tygon) of inner diameter 800 µm, coated with a solution of Bovine Serum Albumine (BSA) (Sigma Aldrich) to prevent adhesion of the droplets inside the tubing (incubation at 1 %w solution during 12 h at room temperature), whereas its outlet is connected to a Peek tubing of inner diameter 125 µm.

Methods

Fabrication of multilayers

The set-up is sketched in Figure 1. Our first objective is to build capsules with a multilayer membrane. Once the droplets are produced [START_REF] Trégouët | Transient deformation of a droplet near a microfluidic constriction : a quantitative analysis[END_REF] , they are forced to circulate into a long serpentine to let the first polymer adsorb at their surfaces [START_REF] Priest | Microfluidic polymer multilayer adsorption on liquid crystal droplets for microcapsule synthesis[END_REF] . The addition of a second polymer layer is done in two steps. First, the droplets are transferred to a solution of HCl (pH 3) through a pinched-flow fractionation as described by Yamada et al. [START_REF] Yamada | Pinched flow fractionation: Continuous size separation of particles utilizing a laminar flow profile in a pinched microchannel[END_REF] and illustrated in Figure 1 b. Secondly, using the same method, the droplets are transferred from the HCl solution to a second polymer solution. In each case, the two solutions are miscible but have different optical indices, which makes the frontier visible using digital interferometric contrast. On this picture, the monodisperse droplets coming from the bottom left cross the confined interface and reach the top right channel. A barrier formed by a row of small posts prevents the droplets from going to the bottom right channel.

Capsule deformation

The principles of capsule deformation and observation were described previously [START_REF] Trégouët | Transient deformation of a droplet near a microfluidic constriction : a quantitative analysis[END_REF] . Briefly, the characterization chip (Figure 2)

consists in a sharp transition between a narrow channel (of width W = 40 µm) and a wide chamber (of width 3W ), similar to the chip used by Polenz et al. [START_REF] Polenz | Monitoring reactive microencapsulation dynamics using microfluidics[END_REF] . The height of the chamber (in the zdirection) is 100 µm: the droplets of diameter comprised between 50 and 70 µm are therefore not confined vertically in the chamber.

The divergent flow at the entrance of the wide channel imposes a high viscous stress on the capsules that elongate perpendicularly to the main direction of the flow. The entrance is therefore the region where we measure the deformation and the relaxation of the capsules. The flow is controlled by a pressure controller Fluigent. The droplet deformation is defined as a dimensionless number calculated from the x-axis width a and the y-axis height b of the droplet, through:

D = b -a b + a . (1)
The deformation and speed are measured for a large number of identical capsules at different positions, which correspond to different times. Such a process allows for a precise measurement thanks to statistical analysis. Indeed, for each set, we measure about 8 points per capsule, for about 25 capsules.

The experimental evolution of the deformation as a function of time is then interpreted through simulations and a theoretical model which enable the extraction of the interfacial rheological properties. These properties are finally compared with values obtained from independent techniques: the pendant drop method and the interfacial shear rheometer.

Simulations

The details of the simulations used to calculate the shear stress applied by the surrounding fluid on the capsule are provided in a previous article [START_REF] Trégouët | Transient deformation of a droplet near a microfluidic constriction : a quantitative analysis[END_REF] . Briefly, using COMSOL we simulate the laminar flow in the chamber for several, fixed, positions of the capsule. The boundary condition at the interface is chosen such that the fluid velocity matches the experimentally-measured capsule velocity. The viscous stress around the capsule is then calculated and its main components are averaged on all the points in the close vicinity of the interface.

Pendant drop

To determine independently the surface tension of the droplets, we use the pendant-drop method (Tracker apparatus, Teclis, France). Details concerning the principle of this technique can be found in an article from Rotenberg et al. [START_REF] Rotenberg | Determination of surface tension and contact angle from the shapes of axisymmetric fluid interfaces[END_REF] . Briefly, a fresh millimetric oil drop is formed in the polymer solution and the surface tension is then obtained through the analysis of its shape.

Results and discussion

Experimental results

We observe that all the capsules are initially negatively deformed (D < 0) as they exit the constriction due to the lateral confinement of the narrow channel. As they progress in the large chamber, their behavior depends on the nature of the interface. We stress that, in the following, the positions along the transport axis (x) and time (t) are equivalent, since the capsules' trajectories are recorded.

The case of the PMAA monolayers (Figure 3a) was already presented and analyzed theoretically in a previous article [START_REF] Trégouët | Transient deformation of a droplet near a microfluidic constriction : a quantitative analysis[END_REF] . In this case, the deformation quickly becomes positive at the constriction exit because of the elongational flow and then relaxes toward zero when the elongational stress decreases.

In the case of the monolayers of PAA-0.7-C 12 , the deformation initially negative becomes positive, reaches a maximum, before becoming negative again (Figure 3a), unlike the PMAA monolayers.

We observe more pronounced qualitative differences for the two other types of membrane, as shown in Figure 3b: the deformation remains negative all along the trajectory and shows no maximum for the bilayers PMAA/PVP, PAA-0.7-C 12 /PVP and PAA-0.8-C 8 /PVP. In addition, we observe that PAA-0.7-C 12 /PVP capsules relax significantly more slowly than all the other systems.

Theoretical Analysis

Capillary model

In a given flow field, a droplet adopts a steady deformation, D steady which was shown by Barthès-Biesel et al. [START_REF] Barthès-Biesel | Deformation and burst of a liquid droplet freely suspended in a linear shear field[END_REF] to depend on the viscosity ratio between the droplet and the surrounding phase λ = η droplet η bulk , the radius of the droplet r, and the interfacial tension γ between the two phases as follows [START_REF] Barthès-Biesel | Deformation and burst of a liquid droplet freely suspended in a linear shear field[END_REF] :

D steady = 19λ + 16 16λ + 16 • η bulk r γ • (e max -e min ) , (2)
where e max and e min are the eigenvalues of the deformation-rate tensor. The same authors also described the transient regime leading to this steady deformation as a first-order relaxation in time [START_REF] Barthès-Biesel | Deformation and burst of a liquid droplet freely suspended in a linear shear field[END_REF] . The derivative Ḋ of the deformation D with respect to time t thus depends on the steady deformation D steady and the actual deformation, through:

Ḋ = 1 τ ca • D steady -D , (3)
with a relaxation time τ ca defined as follows [START_REF] Barthès-Biesel | Deformation and burst of a liquid droplet freely suspended in a linear shear field[END_REF] :

τ ca = 2 5 • (2λ + 3) • 19λ + 16 16λ + 16 • η bulk r γ . ( 4
)
We showed in our previous article that this model fits well the data obtained for the PMAA monolayer droplets [START_REF] Trégouët | Transient deformation of a droplet near a microfluidic constriction : a quantitative analysis[END_REF] . However, as shown in Figure 4, this simple capillary model can not fit x (µm) (a) Deformation as a function of position of the center of mass, for capsules made of PMAA and PAA-0.7-C 12 , as indicated. The origin of x corresponds to the entrance of the wide chamber. The PMAA data was previously studied [START_REF] Trégouët | Transient deformation of a droplet near a microfluidic constriction : a quantitative analysis[END_REF] and serves here as a reference. the data obtained for the PAA-0.7-C 12 capsules. More precisely, this model predicts that the deformation reaches positive values before returning to zero deformation. However the PAA-0.7-C 12 returns to negative deformations at longer times. For the same reasons, the bilayer cases presented in Figure 3b can not be properly fitted (not shown) by the capillary model as they do not reach positive values of deformation. This means that, except for PMAA monolayers, our capsules cannot be considered as simple droplets with a homogeneous and constant interfacial tension.

However, we have several evidences that a capillary model is probably not sufficient to describe such membranes. Indeed, measuring the interfacial tension of PAA-0.7-C 12 layers using the pendant-drop method and varying the area of the drops (by deflating them), we found [START_REF] Tregouët | Polymer multilayers at liquid interfaces : assembly , interfacial rheology and microfluidic probing[END_REF] that the surface tension decreases as the area decreases, similarly to the case of interfaces with insoluble surfactants [START_REF] Lucassen | Dynamic measurements of dilational properties of a liquid interface[END_REF] . It is likely that the surface tension is not constant and homogeneous at the droplet interface as it is deformed. Such a membrane can be described by a surface dilational elastic modulus: E γ 47 .

In the case of the PMAA/PVP, we found that these bilayers exhibit both shear and dilational moduli on the order of the surface tension value according to pendant-drop and shear-rheometer experiments [START_REF] Tirilly | Interfacial Rheology of Hydrogen-Bonded Polymer Multilayers Assembled at Liquid Interfaces: Influence of Anchoring Energy and Hydrophobic Interactions[END_REF] .

In order to take into account these considerations, we alternatively consider these membranes as purely elastic ones.

Elastic model

In the case of purely elastic membranes, i.e. in the absence of capillary effects and membrane viscosity, Barthes-Biesel and Rallison [START_REF] Barthès-Biesel | The time-dependent deformation of a capsule freely suspended in a linear shear flow[END_REF] proposed a model to describe the relaxation of a slightly deformed capsule toward its spherical shape. In such a case, the exponentional relaxation occurs in a steady liquid and is driven by the elasticity of the membrane, while the bulk viscosity of the droplet tends to slow down the dynamics. In a more general case, the deformation D relaxes toward a reference deformation D ref following the equation:

Ḋ = 1 τ el (D ref -D) , (5)
where the introduced time scale can take the two (note the possible signs) following values:

τ el± = 3(19λ + 16)(2λ + 3) 5(19λ + 24) ± √ 5377λ 2 + 14256λ + 9792 • η bulk r 3G , (6)
with G the 2D shear elastic modulus (homogeneous to an energy per unit surface) of the membrane. In the simplest case, D ref =0 and the capsules relax toward a spherical shape. As pointed out by Leclerc et al. [START_REF] Leclerc | Transient flow of microcapsules through convergent-divergent microchannels[END_REF] , the largest relaxation time rules the relaxation of the membrane which is then given by: τ el-. Note that, using the same characteristic time, a variant of Equation ( 5) has been recently proposed by Gires et al. [START_REF] Gires | Transient behavior and relaxation of microcapsules with a cross-linked human serum albumin membrane[END_REF] for viscoelastic membranes, which also does not take into account the extensional flow (but only the shear flow).

Such models, by neglecting the extensional flow, can only predict a relaxation of the capsules from a negative deformation to a spherical shape and cannot account for a positive deformation as the one observed for PAA-0.7-C 12 .

Suggested model

To account for all our experimental results, we suggest that both the interfacial tension and the elasticity of the membrane should be taken into account. The deformation should result from two contributions. The first one comes from the competition between the shear rate in the continuous phase which deforms the capsule and the surface tension which resists the deformation. This competition can be described dynamically by Equation (3). The second contribution corresponds to an elastic relaxation toward a reference shape D ref , which is not necessarily zero. Such a dynamic is described by Equation ( 5).

As previously mentioned, we point out that our membranes can be considered as elastic. Moreover, variations of the surface tension due to surface-excess variations are usually taken into the elastic effects together with the actual elasticity of the membrane. Both components contribute to the restoring forces and are defined by a global 2D effective modulus E (homogeneous to an energy per unit surface).

Furthermore, we know thanks to interfacial rheology experiments that the polymer membranes do relax: the surface tension rises after compression and decreases after dilatation (not shown), and the shape of droplets covered by such membranes evolves over time after compression of the interface, as shown in Figure 5. This means that the polymer chains in the membrane relax. At short time scales, the shape of the capsule should thus be a compromise between the initial shape (minimizing the energy for the elastic membrane that matured in the narrow channel) and the spherical shape (minimizing the area and hence the energy due to the homogeneous and isotropic surface tension). At long time scales, the membrane being viscoelastic, the polymer chains and thus the elastic stresses should relax and the droplet should retrieve its spherical shape. To include this potential membrane relaxation in our model, we allow for D ref (t) to be a function of time t that relaxes towards zero over a viscoelastic polymeric time τ pol . Fig. 5 Wrinkling and relaxation of a PMAA/PVP membrane at the dodecane-water interface, in a pendant-drop apparatus: the membrane is compressed at time t = 0, and then the inner volume is kept constant, while we observe the membrane relaxing over time. Scale: the outer diameter of the nozzle at the bottom is 1.2 mm.

The initial reference deformation D ref (0) is thus comprised between 0 (i.e. the spherical shape of the droplet before the nar-row channel) and the deformation of the capsule in the narrow channel, depending on the length of the latter and thus the maturation of the confined membrane. Figure 6 illustrates the case where the narrow channel is long enough to have the restoring stresses totally relaxed before the capsule enters the wide chamber. In such a case, the reference deformation at the entrance of the wide chamber matches the actual deformation at that point. Fig. 6 Schematic deformation of the capsule along its trajectory before the deformation chamber, with the corresponding reference deformation D ref (red dashed line). In the confining channel, when the elastic stresses relax, the reference deformation becomes the present deformation of the capsule. In the chamber, the reference deformation finally relaxes.

In our experiments, the capsules are confined in the narrow channel during a time τ conf 10 -2 s. The long-channel approximation is thus valid if τ pol τ conf , which has to be checked a posteriori.

We would like here to insist on the distinction between the time τ pol and the two other times τ ca and τ el . The two latter times describe the relaxation of the capsule's shape according to a competition between a driving force (interfacial tension or elasticity) and the bulk viscosity. On the contrary, τ pol describes the relaxation of the stresses within the membrane, and is thus related to the relaxation of the constitutive polymer chains which is the core of the membrane's viscoelasticity. The typical length scales associated to these two types of processes are thus totally different: τ ca and τ el describe the capsule as an object of a few tens of micrometers, while τ pol describe the dynamics of the polymer chains, whose size is on the order of a few tens of nanometers. Accordingly, we expect τ pol to be independent of the capsule's radius r, while we know from Equations ( 4) and ( 6) that τ ca and τ el scale linearly with r.

To summarize, inspired by the pure cases above, we propose a more general model that simply assumes a superimposition of all the different contributions, and that couples them through the following equations (valid in the large chamber only):

               Ḋ(t) = 1 τ ca (D steady (t) -D(t)) + 1 τ el (D ref (t) -D(t)) , ( 7a
)
Ḋref (t) = 1 τ pol (0 -D ref (t)) . ( 7b
)
The first term of the right-hand side of Equation (7a) comes from Equation (3). It corresponds to a capillary relaxation towards an instantaneous steady state, and it is characterized by the time scale τ ca . The second term corresponds to an elastic relaxation towards an instantaneous reference state, and it is characterized by the time scale τ el which includes the effects of different moduli: the "liquid modulus" E γ describing the variations of the surface tension due to surface-excess variations, the 2D elastic compression modulus K , and the 2D elastic shear modulus G .

We note E the 2D effective modulus which is a combination of these three moduli, and that is expected to be close to the highest of these three moduli. We can thus write that:

τ el ∝ η bulk r E . (8)
Equation (7b) accounts for the relaxation of the stresses in the membrane, and the viscoleastic return to a spherical shape. We would like to point out that a long relaxation time τ pol of the polymers usually implies a high modulus in simple systems: for instance in the Lucassen model [START_REF] Lucassen | Dynamic measurements of dilational properties of a liquid interface[END_REF] , a long relaxation time τ pol is due to a low diffusivity. This implies a high value of the liquid modulus E γ . Such systems would elastically relax over a short time τ el . In more complex systems, the link between τ pol and τ el can be less straightforward. This is in particular what we will see in the following.

Solving the above coupled equations allows to predict a theoretical deformation of the capsules, with an interfacial tension and viscoelastic effects, in a viscous extensional flow. They will be confronted to the experimental data presented above, which will provide three fitting parameters, τ ca , τ el and τ pol , and thus insights on the various mechanisms at play. Finally, as a remark, we can recast Equation (7a) into an effective single-forcing process:

Ḋ(t) = 1 τ forcing (D forcing (t) -D(t)) , (9)
where:

τ forcing = τ ca -1 + τ el -1 -1 , (10)
and:

D forcing (t) = τ forcing • τ ca -1 • D steady (t) + τ el -1 • D ref (t) . (11)
This compact form highlights the universality of the model, that could be extended to other types of responses: more complex viscoelasticity, charge effects, poroelasticity...

PAA-0.7-C 12 monolayer: quickly-relaxing low modulus

We use the general model presented in Equation ( 7) to fit the experimental data for the PAA-0. Here, we find an interfacial tension that is higher than in pendantdrop experiments, but we have a fair order-of-magnitude agreement. Moreover, we observe that τ el τ ca , which we interpret as follows: the interfacial-tension variations (i.e the liquid modulus) are low compared to the absolute value of the interfacial tension but not completely negligible. Finally, we find that τ pol τ conf , which validates the long-channel approximation discussed above: at the end of the narrow channel, just before the capsules enter the wide chamber, all the elastic stresses have relaxed. Thus, the inital reference deformation D ref (0) corresponds to the actual shape of the capsules as they enter the chamber.

PAA-0.7-C 12 /PVP bilayer: slowly-relaxing high modulus

We apply the same analysis to the capsules covered by a PAA-0.7-C 12 /PVP bilayer, as presented in Figure 8. The best fit gives τ ca = 3.25 • 10 -5 s, and thus (we recall here that τ ca also depends on the capsule's radius, which slightly varies between the different experiments) the same interfacial tension (γ = 35 mN/m) as for the PAA-0.7-C 12 monolayer. Moreover, we find the same polymeric relaxation time (τ pol = 1.2 • 10 -3 s) as in the case of the PAA-0.7-C 12 monolayer, but a significantly lower elastic time τ el = 3.0 • 10 -6 s. According to Equation (8), this implies that the effective elastic modulus E is significantly higher for the bilayer than for the monolayer, which is consistent with the added interactions between PAA-0.7-C 12 and PVP chains. The value obtained for the polymeric relaxation time (τ pol 1 ms) is also coherent with the pendant-drop experiments reported previously [START_REF] Tirilly | Interfacial Rheology of Hydrogen-Bonded Polymer Multilayers Assembled at Liquid Interfaces: Influence of Anchoring Energy and Hydrophobic Interactions[END_REF] . Therein, there was no evidence of any surface-tension inhomogeneity, or anisotropy, during compression, neither for the PAA-0.7-C 12 /PVP bilayers nor for the PAA-0.7-C 12 monolayers, which indicates that the inner relaxation times are shorter than the typical time scale of such experiments (on the order of ∼ 1 s).

Furthermore, we interpret the concordance of the polymer relaxation time τ pol between the monolayer and the bilayer as a consequence of the predominance of the alkyle graft anchoring on the dynamics of the interface, as shown in a previous work [START_REF] Tirilly | Interfacial Rheology of Hydrogen-Bonded Polymer Multilayers Assembled at Liquid Interfaces: Influence of Anchoring Energy and Hydrophobic Interactions[END_REF] . A second evidence for the importance of graft anchoring is the comparison with an analogous experiment performed on PAA-0.8-C 8 /PVP bilayers, also presented in Figure 8. We observe that the relaxation of such bilayers is much faster than the relaxation of PAA-0.7-C 12 /PVP bilayers. In this simple picture, shorter alkyle grafts, and thus lower anchoring energy, leads to faster relaxation. In such a case, if the relaxation of the bilayers was instead controlled by the moduli only (and thus τ el ), we would expect a lower elastic modulus for the PAA-0.8-C 8 compared to the PAA-0.7-C 12 as these chains are less strongly anchored to the interface [START_REF] Barentin | Polymer Brushes Formed by End-Capped Poly(ethylene oxide) (PEO) at the Air-Water Interface[END_REF] . In such case, according to Equation ( 8), we would observe a faster relaxation for PAA-0.7-C 12 /PVP than for PAA-0.8-C 8 /PVP -which is opposite to what we measured. This comparison between PAA-0.8-C 8 /PVP and PAA-0.7-C 12 /PVP thus indicates that the relaxation time which rules the overall relaxation of those capsules is the one for polymer relaxation τ pol , and that graft anchoring is a key factor of the underlying polymer dynamics.

PMAA/PVP bilayer: barely-relaxing high modulus

Unlike the PAA-0.7-C 12 /PVP bilayers, the PMAA/PVP bilayers show wrinkles during several seconds after compression in pendant-drop experiments, as presented in Figure 5. Accordingly, we expect to have in this case τ pol > 1 s, which has two implications. First, this time scale is much longer than the time during which the capsules are confined in the narrow channel τ conf . As a consequence, we assume that such capsules do not have enough time to undergo any viscoelastic maturation process within the narrow channel, and that the reference deformation D ref (0) at the entrance of the wide chamber is close to zero (because of the memory of the non-deformed state before confinement). Secondly, the expected value of τ pol is so high that it can be considered as infinite when compared to the typical time scales of our microfluidic measurements, which is about 1 ms. Therefore, we fix the value of the reference deformation D ref to a (almost zero) constant at all times. The result of the fit is presented in Figure 9. We find the same interfacial tension γ = 40 mN/m as for the PMAA monolayer [START_REF] Trégouët | Transient deformation of a droplet near a microfluidic constriction : a quantitative analysis[END_REF] , and an elastic time τ el = 3 • 10 -6 s similar to the elastic time of PAA-0.7-C 12 /PVP.

Discussion

We summarize in Table 1 expect from macroscopic measurements, when available. In the case of PAA-0.7-C 12 , where the modulus is a liquid dilational modulus E γ , if there is no desorption, we can show that E γ = Π, where Π is the surface pressure appearing in the state equation of Leclerc et al. [START_REF] Leclerc | Asymmetric Multiblock Copolymers at the Gas -Liquid Inter-face : Phase Diagram and Surface Pressure[END_REF] (cf Supplementary Information). The surface pressure of PAA-0.7-C 12 at the mineral oil-water interface is Π 10 mN/m, which leads to E γ 10 mN/m, in good agreement with what we obtained in our experiments. For the PAA-0.7-C 12 /PVP bilayer, no measurement of the elastic modulus has been performed at time scales smaller than τ pol 1 ms, and consequently we have no reference for comparison. In contrast, the value extracted from the experiment with PMAA/PVP is close to the value measured in the interfacial rheometer [START_REF] Tirilly | Interplay of Hydrogen Bonding and Hydrophobic Interactions to Control the Mechanical Properties of Polymer Multilayers at the Oil/Water Interface[END_REF][START_REF] Tirilly | Interfacial Rheology of Hydrogen-Bonded Polymer Multilayers Assembled at Liquid Interfaces: Influence of Anchoring Energy and Hydrophobic Interactions[END_REF] . In summary, the general model described in Equation ( 7) offers both a good understanding of the relaxation dynamics of various capsules in a microfluidic device, and a satisfactory agreement with calibrated experiments concerning the extracted physical parameters. The strong influence of the ratio τ pol /τ conf between the polymer and confinement time scales (which e.g. controls the differences observed between the PAA-0.7-C 12 /PVP and PMAA/PVP systems) further suggests that it might be interesting to perform new sets of experiments by varying the length of the narrow channel. This would be a way to extract more precisely τ pol , and to characterize further the underlying molecular mechanisms for relaxation -a key ingredient for understanding and tuning finely the membrane properties.

Conclusion

Through the use of a sharp microfluidic-channel extension we measured the shape relaxation of capsules with different kinds of polymer membranes. We observed various behaviors depending of the viscoelastic parameters of these membranes. We proposed a general model to understand this rich set of behaviors, arising from the relative importances of the outer fluid viscosity, and the interfacial tension, elastic modulus, and/or viscous modulus of the membranes. The comparison of the experiments with the model predictions allows us to rationalize all the experimental

Fig. 1 a

 1 Fig. 1 a. Design of the capsule-production microfluidic chip. The blue part corresponds to the flow focusing. The serpentines at the entrance of the fluids increase the resistance of the channels and give a better control of the stability of the droplets. The red part corresponds to an incubation channel, whereas the yellow one is made of two successive phase changes. b: zoom on a phase change set-up made of a pinched flow fractionation device used twice, to transfer the droplet from the first polymer solution to the rinsing solution, and then from the rising solution to a second polymer solution. In each case, the two solutions are miscible but have different optical indices, which makes the frontier visible using digital interferometric contrast. On this picture, the monodisperse droplets coming from the bottom left cross the confined interface and reach the top right channel. A barrier formed by a row of small posts prevents the droplets from going to the bottom right channel.

Fig. 2

 2 Fig.2Schematic of the characterization chip. The deformation, initially negative when the capsules are confined in the channel, becomes positive when they enter the chamber, and finally relaxes towards zero (spherical shape) in the middle of the chamber. The dashed rectangle represents the observation frame.

D

  exp (PMAA) D exp (PAA-0.7-C 12 )

D

  exp (PAA-0.7-C 12 / PVP) D exp (PAA-0.8-C 8 / PVP) D exp (PMAA / PVP) Deformation of multilayer capsules versus position in the chamber, for various membrane compositions as indicated.

Fig. 3

 3 Fig. 3 Deformation of capsules made of different kinds of membrane.

Fig. 4

 4 Fig.4 Deformation as a function of position for PAA-0.7-C 12 monolayer capsules (see Figure3a). The dashed and solid lines represent the predictions for the steady and theoretical deformations in the capillary model (Equations (2) and (3)), with two test surface tensions as indicated.

  7-C 12 capsules. The fit shown in Figure 7 provides: τ ca = 3.03 • 10 -5 s (i.e. an interfacial tension of 35 mN/m), τ el = 1.7 • 10 -4 s, and τ pol = 1.2 • 10 -3 s. Pendant-drop experiments for an oil-water interface (not shown) indicated that the interfacial tension between mineral oil and PAA-0.7-C 12 is about 25 mN/m and decreases under compression.

Fig. 7

 7 Fig. 7 Deformation as a function of position for PAA-0.7-C 12 monolayer capsules. The dashed lines represent the theoretical evolution of the two components : the steady deformation of the capillary model and the reference deformation of the elastic one. The solid line corresponds to the theoretical prediction of the global deformation given by Equation 7. The fit of the data provides: τ ca = 3.03 • 10 -5 s (γ = 35 mN/m), τ el = 1.7 • 10 -4 s and τ pol = 1.2 • 10 -3 s.

Fig. 8

 8 Fig. 8 Deformation as a function of position for PAA-0.7-C 12 /PVP and PAA-0.8-C 8 /PVP bilayer capsules. The dashed and solid lines represent the predictions for the forcing and theoretical deformations in the general model. For the PAA-0.7-C 12 /PVP bilayer, the best fit provides: τ ca = 3.25 • 10 -5 s (γ = 35 mN/m), τ el = 3 • 10 -6 s, and τ pol = 1.2 • 10 -3 s. For the PAA-0.8-C 8 /PVP bilayer, the best fit provides: τ ca = 3.67 • 10 -5 s (γ = 35 mN/m), τ el = 3 • 10 -6 s, and τ pol = 2.5 • 10 -4 s.

  all the results obtained by fitting the experiments to the general model. The effective 2D moduli E are calculated from τ el according to Equation (6). The orders of magnitude of the obtained moduli are in fair agreement with what we

Fig. 9

 9 Fig. 9 Deformation as a function of position for PMAA/PVP bilayer capsules. The dashed and solid lines represent the predictions for the forcing and theoretical deformations in the general model. The best fit provides: τ ca = 2.8 • 10 -5 s (γ = 40 mN/m) and τ el = 3 • 10 -6 s. We assumed τ pol 10 -3 s.

Table 1

 1 Best-fit parameters extracted from the comparison between all the experimental data and the general model. results in a unified framework. Based on this work, by screening different confinement times, we envision the possibility of probing finely, and in situ, the rheological and molecular behaviours of capsules and cells within microfluidic plateforms.

		γ (mN/m)	τ ca (s)	τ el (s)	τ pol (s)	E (mN/m)
	PMAA 35	40	2.33 • 10 -5	-	-	-
	PAA-0.7-C 12	35	3.03 • 10 -5	1.7 • 10 -4	1.2 • 10 -3	1.3 • 10 1
	PAA-0.7-C 12 /PVP	35	3.25 • 10 -5	3 • 10 -6	1.2 • 10 -3	6.9 • 10 2
	PAA-0.8-C 8 /PVP	35	3.67 • 10 -5	3 • 10 -6	2.5 • 10 -4	7.7 • 10 2
	PMAA/PVP	40	2.8 • 10 -5	3 • 10 -6	10 -3	6.9 • 10 2

[START_REF] Klinkesorn | Encapsulation of emulsified tuna oil in two-layered interfacial membranes prepared using electrostatic layer-by-layer deposition[END_REF][START_REF] Delcea | Stimuliresponsive LbL capsules and nanoshells for drug delivery[END_REF][START_REF] Correa | Highly Scalable, Closed-Loop Synthesis of Drug-Loaded, Layer-by-Layer Nanoparticles[END_REF][START_REF] Yang | Microencapsulation of isocyanates for self-healing polymers[END_REF][START_REF] Tirilly | Interplay of Hydrogen Bonding and Hydrophobic Interactions to Control the Mechanical Properties of Polymer Multilayers at the Oil/Water Interface[END_REF][START_REF] Gao | Stability and Mechanical Properties of Polyelectrolyte Capsules Obtained by Stepwise Assembly of Poly(styrenesulfonate sodium salt) and Poly(diallyldimethyl ammonium) Chloride onto Melamine Resin Particles[END_REF][START_REF] Antipov | Sustained release properties of polyelectrolyte multilayer capsules[END_REF][START_REF] Priest | Microfluidic polymer multilayer adsorption on liquid crystal droplets for microcapsule synthesis[END_REF][START_REF] Kantak | A 'microfluidic pinball' for on-chip generation of Layer-by-Layer polyelectrolyte microcapsules[END_REF][START_REF] Erni | Deformation modes of complex fluid interfaces[END_REF]

Acknowledgements

This work was financially supported by ANR JCJC INTERPOL. The authors thank Mamisoa Nomena and Samuel Poincloux for their precious help and advice in this work. They also thank Oliver Bäumchen, Ingmar Polenz, Jean-Christophe Baret, Julien Dupré de Baubigny, Nadège Pantoustier and Patrick Perrin for fruitful discussions.


Caroline Hache

Stéphane Honoré

Guillaume Hache

Guillaume Hache Involving

Involving patients in a workshop focused on communication skills: a

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Representations

• "Excellent idea to involve patients" / I appreciated " the interaction with patients" / " this simulation involving real patients" • I appreciated " the absence of patient/pharmacist barrier " / "the opportunity to practice interviews without pressure or shame" / "freedom" • "Amazing experience ! every course should be conducted like this one, thank you.

• "Gathering patients and future health professionals is extremely enriching. Please do it again." • "I enjoyed to be involved in this session, exchanging views, learning and having fun" • "I appreciated the opportunity to interact with the students" • "I loved the atmosphere" ü Scoring the learners' competencies

Figure 1 :

 1 Figure 1 : the pharmacists' patient care process, from JCPP


Laurent Legentil

Sorya Belaz

Jean-Pierre Gangneux

Florence Robert-Gangneux

Vincent Ferrières
email: vincent.ferrieres@ensc-rennes.fr

Synthesis and biological properties of galactofuranosyl-containing fluorescent dyes

Keywords:

come

Graphical Abstract

The presence of hexoses with striking five-membered ring was highlighted only in non-mammal living kingdom. As a consequence, these natural compounds, and the related biological pathways, are targets of choice for the development of some molecular tools useful to increase basic knowledge and/or to fight some related diseases. These diseases include especially tuberculosis and leishmanial infections. The first one is caused by Mycobacterium tuberculosis which infects millions of people each year throughout the world. 1 On the other hand, the leishmaniasis disease is mainly endemic in Eastern Africa, Brazil, India, Middle East and Mediterranean countries. Signs of infection by Leishmania species can vary and range from skin disorders to deadly visceral forms. 2 In both cases, WHO calls on governments, scientists, civil society and private companies to unite to stop the constant growth of these diseases.

Several teams have shown that furanosyl-containing conjugates may impact both M. tuberculosis and L. donovani lifecycle. 3 For instance, it was established that non-natural nucleotide-furanoses can be recognized and transferred by galactofuranosyl transferases on growing galactan. 4,5 Such nucleotides act at an early stage of this biological process and they significantly disturb the biosynthesis of this key galactan. Several UDP-analogues were also produced in order to inhibit this transferase and also the key UDP-galactopyranose mutase. [6][7][8][9][10][11][12] Considering recent knowledges related to the glycoconjugates part of Mycobacteria and Leishmania cell walls, some studies have focused on the biological impact of furanosyl-containing glycolipids. 2,[13][14][15][16][17][18][19] Until now, the best results were obtained using a bicatenar thiofuranoside as a potent biocide agent against M. tuberculosis. 15 Nevertheless, the corresponding mode of action was not elucidated.

Octyl galactofuranoside (GalfOct) is a very simple glycolipid, whose structure however differs from the one of standard alkyl glycosides by the presence of the carbohydrate polar head in a furanose form. Leveraging its amphiphilic properties 20 has recently led us to investigate the impact of this compound on microorganisms that are able to biosynthesize and to metabolize furanosyl-containing conjugates. 21 A more detailed study was performed on Leishmania donovani using a small library of GalfOct derivatives. 18 The furanose form clearly contributes to a significant decrease of membrane fluidity leading to profound alterations in the leishmanial promastigote membrane and organelles. It was also shown that GalfOct was able to decrease intramacrophagic amastigote multiplication. This result led us to propose herein furanosyl-containing fluorescent dyes to characterize the intracellular distribution of GalfOct and to help

A RT I C L E I N F O A BS T RA C T
understanding of the mechanism of interaction of furanosides with the intracellular parasites.

Considering the respective size of the furanolipid and the most common fluorescent dyes, significant alterations of physicochemical and biological properties cannot be ruled out. Nevertheless, we were encouraged by recent studies in which a lipid chain was partly substitute by a fluorescent tag. [22][23][24] Therefore, we focused our attention on the conjugation of our furanoside with xanthene derivatives like (i) fluorescein, known for its low cellular permeability in assays involving living cells, or (ii) the red-shifted fluorescent rhodamine 6G because its positive charge strengthens interactions with negatively charged membranes. 25 Interestingly, these two related structures show a maximum of fluorescence in different pH range and would thus allow to monitor the trafficking of GalfOct in presence of infected macrophages, whatever the nature of the microenvironment. Moreover, the latter probe overcomes in vivo auto-fluorescent issues and ensures good diffusion through organic tissues.

The synthesis of the targeted fluorophores was based on a common intermediate 4 bearing a masked terminal amino function on the lipid part. Its synthesis started from GalfOct that was obtained in one step according to a modified Fischer glycosylation of octanol. 20 Subsequent benzoylation, acetolysis and bromination under standard conditions afforded the furanosyl donor 3 in a good 85% overall yield. The desired building block 4 was then obtained with 95 % yield by glycosylation of the commercially available N-Fmoc-protected 6-amino-1-hexanol using silver(I) triflate as the promoter. Further full removal of basic-sensitive esters and carbamate groups was efficiently performed under Zemplen conditions. Finally selective thiocarbamoylation in a dimethylformamide (DMF) solution in the presence of diisopropylethylamine (DIPEA) using fluorescein isothiocyanate (FITC) afforded the desired fluorescein conjugate 6 in 77% yield. Its structure was corroborated by 1D and 2D NMR spectroscopy: signal at δ 4.86 ppm correlated with a 3 J H1,H2 1.9 Hz coupling constant, and signal at δ 109.4 ppm for C-1 highlighted the five-membered ring and the desired 1,2-trans configuration. Maximum excitation and emission wavelengths were measured in phosphate buffered saline (PBS) solution (pH 7) at 468 nm and 520 nm, respectively. The rhodamine conjugate was prepared slightly differently, following a procedure described by Adamczyk and Grote. 26 Transacylation of the starting ethyl ester rhodamine 6G in DMF was thus performed in the presence of piperidine. This base allowed selective deprotection of the terminal amino function that could subsequently add to the carbonyl atom ethyl ester of rhodamine 6G to give, after elimination of ethanol, the desired benzoylated derivative 7. The reaction was performed under microwave activation (100 W) at 90 °C and allowed to access the target conjugate 7 in only 30 minutes in 68% yield. The presence of a 13 C NMR signal at δ 65 ppm characteristic of the sp 3 hybridized carbon attach to a nitrogen indicated the major formation of the closed spirolactam form. Final debenzoylation was carried out as described above. The resulting water soluble rhodamine 2'-amide 8 was obtained as the open cationic specie following acidification with hydrochloric acid. This was corroborated by the disappearance of the 13 C NMR signal at δ 65 ppm. The measurement of fluorescent properties afforded  ex =528 nm and  em =552 nm in PBS.

The synthetic furanosyl-containing fluorescent probes were then investigated using macrophages infected by leishmanial parasites. Two effects were monitored: the trafficking of GalfOct labelled with fluorescein or rhodamine, and the possible alteration of the biological properties following the labelling. On one hand, L. donovani promastigotes grown at stationary phase Scheme 1. Synthesis of the galactofuranosyl-containing fluorophores 6 and 8.

(>5 days of culture) were used to infect human macrophages overnight (doi 10:1). After 3 washes, infected macrophages were treated for 6 or 18 hours with 80 µM of conjugate 6 or 8, or with 80 µM of fluorescein, rhodamine 6G or of GalfOct used as controls (Figures 1A-1D). After three washes and fixation with 4% formalin, nuclei were labeled with Hoechst 33342 at 0.001 µM for 30 minutes. Then pictures of each condition were taken using epifluorescence microscopy (Nie, Nikon®) with a 50x oil immersion lens. All conditions were done in triplicate and repeated three times. Conjugate 8 was also tested once (in triplicate) using confocal microscopy on uninfected cells (Figure 1E).

On another hand, amastigotes growth inhibition rate was calculated on the infected cells incubated for 48 hours with 80 µM of either compounds 6 or 8, fluorescein, rhodamine 6G or GalfOct as treated control or medium alone as untreated control. Then slides were washed three times and colored with May Grünwald Giemsa stain. The enumeration of infected cells and intraparasitic multiplication was determined via microscopic examination using a 100x oil immersion lens. A minimum of 200 macrophages were counted on each well for every condition. Amastigotes multiplication rate was computed taken into account infected cells harboring two or more amastigotes in their cytoplasm. All conditions were done in quadruplicate and repeated twice. Statistical analysis was performed using Mann-Whitney test and GraphPad Prism 5® software. Results were considered significant for p < 0.05.

On one hand, no trace of fluorescein was found either in the membrane or inside the macrophage (Figure 1D). On the other hand, compounds 6, 8 and rhodamine 6G were shown to enter in all macrophages after 6 hour incubation (Figures 1A to 1C) and to stay in cytoplasm for at least 18 hours (data not shown). Intracytoplasmic distribution was uniform throughout the cytoplasm and no co-labelling with Hoechst was observed in nuclei (Figure 1). No accumulation dyes was observed around amastigote nuclei as well. Regarding antiparasitic effect, compound 6 and fluorescein alone presented no significant effect, whereas compound 8 significantly reduced macrophage infection (p < 0.05) and amastigote multiplication (p < 0.05) after 48 hours treatment, even without accumulation around amastigote nuclei (Figure 2). Surprisingly, there was a significantly higher efficacy of the compound 8 compared to GalfOct (p < 0.05) (Figure 2). In addition, we observed phenotypic surface alterations on MGGstained slides after treatment with rhodamine 6G alone (data not shown).

Thus the effect of GalfOct-rhodamine 6G 8 could be favored by a cytotoxic effect on macrophages of rhodamine 6G itself even if there was no significant effect of rhodamine 6G on macrophagic infection (Figure 2). In addition modification of the global charge or the hydrophilic properties induced by the presence of covalently linked rhodamine 6G could explain the better antiparasitc effect of compound 8 compared to GalfOct. These properties were described elsewhere as responsible for the cytotoxicity of rhodamine 6G compared to other rhodamine on other cell lines. 27,28 It could have also participated to the stiffening of the membrane induced first by the alkyl furanoside. 18 Taken together, these observations suggest a synergistic effect of rhodamine 6G and GalfOct. Remarkably, and despite the same site of anchorage, the fluorescein ring inhibited the antiparasitic effect of GalfOct. The lack of effect of compound 6 was not due to an inhibition of intracellular diffusion of compound 6, as fluorescence was observed inside cells (Figure 1), but most likely to a modification of GalfOct properties due to its linkage to fluorescein. As fluorescein itself has low cellular permeability, its detection inside cells suggests that the labeled compound gained entry through the GalfOct moiety.

In conclusion, this paper reports for the first time the synthesis of two fluorescent labeled galactofuranosyl-containing conjugates, one green fluorescein-based probe and one red rhodamine 6G derivative. In addition to the determination of their physicochemical properties, we also evaluated their impact in an in vitro model. This study highlighted the difficulty to foresee the impact of a grafted tag onto a very small molecule on the resulting biological properties. Herein, the Galf derivative bearing a fluorescein dye showed low toxicity on macrophages while the presence of rhodamine 6G at the end of the lipid chain of GalfOct significantly improved antileishmanial activity. Further investigations on the action mechanism and in particular the impact on the membrane of the parasite are under way. Therefore, one should be careful when drawing conclusions on cellular interactions using labeled compounds, or before their use in in vivo models.

Figure 1 .

 1 Figure 1. Cell imaging after labeling with fluorescent compounds. Epifluorescence pictures (A, B, C and D) of Leishmania infected macrophages (arrow) treated for 6 hours with compound 6 (A), compound 8 (B), rhodamine 6G (C) or fluoresceine (D) (50 x oil immersion lens). Confocal picture (E) of uninfected macrophages treated with compound 8 for 6 hours (20 x lens)

Figure 2 .

 2 Figure 2. Effect of labeled or native GalfOct on cell invasion and Leishmania parasite growth. Rate of infected macrophages (A) or intracytoplasmic amastigote multiplication (B) after 48 h treatment with compound 6 or 8, compared to untreated macrophages (C) or macrophages treated with the reference furanose (GalfOct) or with rhodamine 6G. All compounds were used at a concentration of 80 µM.

Acknowledgments

The authors are grateful to the GlycoOuest network, supported by the Région Bretagne and the Région Pays de la Loire.

Supplementary Material

Experimental details, NMR data and description of biological tests are available in Supplementary material.


Carlos Lange-Bassani

Herri Ana Cameirão
email: cameirao@emse.fr

Jean Michel

Amadeu K Sum

Rigoberto E M Morales
email: rmorales@utfpr.edu.br

Encit

Brazilian

Carlos L Bassani

Jean-Michel Herri

Revisited model for inward and outward growth of gas hydrate particles in water-in-oil emulsions

Keywords: flow assurance, gas hydrates, particle growth, mass transfer

come

Introduction

Gas clathrate hydrates are crystals formed by the imprisonment of gas molecules inside cages of water-bonded molecules [START_REF] Sloan | Clathrate hydrates of natural gases[END_REF]. Hydrate formation is usually related to high pressure and low temperature conditions, which is often the case of offshore oil and gas production operations. The presence of natural gas and seawater in the pipelines may cause nucleation of those crystals, which may grow and agglomerate causing partial or complete pipeline obstruction, also known as pipe plugging. Plugging due to hydrate formation is considered one of the most challenging flow assurance issues in the oil and gas industry in the last decade [START_REF] Cardoso | Design options for avoiding hydrates in deep offshore production[END_REF].

The correct prediction of the hydrate formation rate -with related gas and water consumption rates -is important to assure flow along the pipeline without any partial or complete blockages. Several growth kinetic models have been proposed in literature to estimate the gas hydrate formation rate [see Ribeiro Jr and Lage [START_REF] Ribeiro | Modelling of hydrate formation kinetics: State-of-the-art and future directions[END_REF] and Yin et al. [START_REF] Yin | Review of gas hydrate dissociation kinetic models for energy recovery[END_REF] for a compilation of kinetic models] whereas fewer of them are predictive models and capable of estimate the particle size evolution in time [START_REF] Englezos | Kinetics of formation of methane and ethane gas hydrates[END_REF][START_REF] Herri | Methane hydrate crystallization mechanism from in-situ particle sizing[END_REF][START_REF] Turner | Methane hydrate formation and an inward growing shell model in water-in-oil dispersions[END_REF][START_REF] Shi | An inward and outward natural gas hydrates growth shell model considering intrinsic kinetics, mass and heat transfer[END_REF][START_REF] Gong | Natural gas hydrate shell model in gas-slurry pipeline flow[END_REF][START_REF] Balakin | Modelling agglomeration and deposition of gas hydrates in industrial pipelines with combined CFD-PBM technique[END_REF][START_REF] Sampaio | Non-isothermal population balance model of the formation and dissociation of gas hydrates[END_REF]. For gas-oil-water systems, the models are scarce and reduced to growth phenomenon only [START_REF] Turner | Methane hydrate formation and an inward growing shell model in water-in-oil dispersions[END_REF][START_REF] Shi | An inward and outward natural gas hydrates growth shell model considering intrinsic kinetics, mass and heat transfer[END_REF]. The kinetic models are said to be apparatus-dependent due to the lack of closure parameters, which are usually micro or meso scale variables of difficult measurement, and therefore all models have a consequent necessity of some sort of curve fitting.

In the present study, the models of Turner et al. [START_REF] Turner | Methane hydrate formation and an inward growing shell model in water-in-oil dispersions[END_REF] and Shi et al. [START_REF] Shi | An inward and outward natural gas hydrates growth shell model considering intrinsic kinetics, mass and heat transfer[END_REF] are revisited and extended: (i) the gas solubility in the hydrate shell is considered [START_REF] Bird | Transport Phenomena[END_REF][START_REF] Barrer | Diffusion in and through solids[END_REF]; (ii) the gas concentration distribution along the shell is coupled with the gas consumption in the outer surface, (iii) gas absorption by the bulk is coupled with gas consumption due to hydrate formation [START_REF] Englezos | Kinetics of formation of methane and ethane gas hydrates[END_REF][START_REF] Herri | Methane hydrate crystallization mechanism from in-situ particle sizing[END_REF]; (iv) the entire particle population is considered for the hydrate formation rate; (v) and some minor corrections of the literature models are made (use of hydrate properties for growth rates instead of water properties in Gong et al. [START_REF] Gong | Natural gas hydrate shell model in gas-slurry pipeline flow[END_REF] and Shi et al. [START_REF] Shi | An inward and outward natural gas hydrates growth shell model considering intrinsic kinetics, mass and heat transfer[END_REF] and correction of mixed molar and mass basis in Turner et al. [START_REF] Turner | Methane hydrate formation and an inward growing shell model in water-in-oil dispersions[END_REF]).

Mathematical model

Water is considered flowing as an emulsion, i.e., small and spherical droplets homogeneously distributed in oil continuous phase [START_REF] Brauner | The prediction of dispersed flows boundaries in liquid-liquid and gas-liquid systems[END_REF]. In this kind of system and whenever pressure and temperature conditions are favorable, gas hydrates form as a shell around the water droplets [START_REF] Sloan | Natural gas hydrates in flow assurance[END_REF], as depicted in Fig. 1(a). The main phenomena associated to growth kinetics of this kind of particle is presented in Fig. 1(b-g). Inner growth is related to gas diffusion through the hydrate shell, which also competes with mass transfer between the particle and the bulk and crystal integration process in the inner surface. The core shrinkage makes water from the core to be expulsed through the hydrate porous shell, thus turning possible crystal integration in the outer surface to happen, with consequent outer growth.

The major assumptions of the model are: (i) the inner and outer surfaces are considered to be at the same temperature as the bulk, although hydrate formation is an exothermic mass change process; (ii) hydrate formation takes place only at the inner and outer surface of the particle, not in the capillaries of the hydrate porous shell; (iii) the hydrate shell is non-deformable; (iv) the particle is spherical; (v) the water expulsion from the water core is due only to core shrinkage, and no capillary-induced flow or head loss in the capillaries are taken into account.

Hydrate is consider to form at a rate proportional to a fugacity (f) driving force relatively to the equilibrium condition (eq.). The local fugacity is transformed to the local concentration by the use of Henry's constant of the gas in water, defined as

H C f  [mol/(m 3 Pa)],
where C is the concentration in [mol/m 3 ]. Therefore, the gas consumption rate due to hydrate formation in [mol/s] comes:

  g i i eq i i eq w hyd dn C k A f f k A f dt H            (1)
where H w is related to Henry's constant of the gas inside water, A i is the interfacial surface of the growing surface and k i is the constant of proportionality of the crystal integration process (related to the time it takes for the gas and water to form the hydrogen bonds of the hydrate crystalline structure; or yet the time a growth unit takes to be integrated to the already existent crystal structure; [START_REF] Mersmann | Crystallization Technology Handbook -Second Edition Revised and Expanded[END_REF]. The local concentration C comes from mass transfer modeling. Gas diffusion along the hydrate shell is modeled by Fick's Law in spherical coordinates, Eq. ( 2). The boundary conditions are shown in Eq. (3) and are interpreted as following: (i) the gas diffused through the hydrate shell is consumed by the crystal integration process in the inner surface; and (ii) the gas transferred between the outer surface of the particle and the bulk will be consumed by both inner and outer growth.

       (2)
, , molar diffusion rate through gas consumption due to crystal the hydrateshell integration in theinner surface

in in h r r h h in i i in i eq w r r C dC D A kA f dr H                 ; , , ,
gas consumption molar diffusion rate through gas mass transfer between due to outer gr the hydrate shell bulk and particle

out out h r r g i h h out i m out i b h r r out C dn dC D A h A C dr S dt                 owth  (3)
where D h is the gas diffusivity in the hydrate shell, C b is the gas concentration in the bulk phase, S h is the gas solubility inside hydrate in [mol of gas/mol of hydrate for a given pressure], indexes in and out refer to inner and outer surface of the particle and index i refers to one particle only. The gas concentration distribution along the hydrate shell then comes: Figure 1. (a) Gas hydrate shell formed around a water droplet in oil continuous phase, which is in contact with the gas free phase. (b) Gas absorption by the bulk competes with gas consumption due to hydrate formation. (c) Gas consumption due to hydrate formation creates a concentration boundary layer around the particle. (d) Gas diffuses through the hydrate shell. (e) Once the gas reaches the inner surface, crystal integration process takes place, causing inner growth. (f) Core-shrinkage makes water to be expulsed through the porous hydrate shell. (g) Once water reaches the outer surface, crystal integration process happens with consequent outer growth.

Gas

  2 , 2 2 2 1 1 1 1 4 1 1 1 1 w h g i in i in h h e q h b w e q m out h h w h out in out m i out in H D dn r r k r S C C S C H f h d t r S D H D r r h k r r                                        (4)
The gas molar flux through the hydrate shell equals the amount of gas consumed to form hydrates in the inner growth and comes from derivation the concentration distribution, as showed in Eq. ( 5) (left). Yet knowing that hydrate formation follows the (pseudo)-stoichiometry of

  2 1 1 G H O Hyd     
, where  is the hydration number and is related mainly to the hydrate crystalline structure, then the hydrate formation rate can be related to the gas consumption rate, Eq. ( 5) (right). The word pseudo-stoichiometry is used because  may vary during hydrate growth depending on the occupancy factor of the gas molecules in the water cages, but for simplification purposes  is considered constant in this study. Furthermore, the linear growth rate of the inner surface of the particle in dr dt  is related to the molar hydrate formation rate, Eq. ( 6).

, 2

4 in g i h in h r r in dn dC r D dt dr    ;   1 g h in in dn dn dt dt             (5)
, 3 core volume [START_REF] Yin | Review of gas hydrate dissociation kinetic models for energy recovery[END_REF] 3

h i h in h in dn d r dt M dt             , 2 1 4 h i in h h in in dn dr M dt dt r     ( 6
)
where  is the density, M is the molar mass and index h refers to the hydrate phase. It is important to notice that Eq. ( 6) is corrected from Gong et al. [START_REF] Gong | Natural gas hydrate shell model in gas-slurry pipeline flow[END_REF] by using the hydrate properties and not the water properties (water properties are mistakenly used when considering that the water core shrinks; while actually it is the hydrate shell that grows inwardly). From Eqs. ( 5) in ( 6), the inner growth rate comes:

    , 2 2 2 2 1 4 1 1 1 g i h h b w eq m out out in h h in w h in h i n o u t i m o u t dn S S C H f h d t r dr M dt r H S r D r r k h r                                 (7)
This expression is the corrected and extended form of Eq. ( 3) of Shi et al. [START_REF] Shi | An inward and outward natural gas hydrates growth shell model considering intrinsic kinetics, mass and heat transfer[END_REF] considering hydrate properties instead of water, the outer consumption of gas, the mass transfer between the particle and the bulk and the gas solubility inside the hydrate shell. When considering that gas diffusion through the hydrate shell is the limiting factor, that no outer growth happens, that the equilibrium concentration is near zero, Eq. ( 7) converges to Eq. ( 12) of Turner et al. [START_REF] Turner | Methane hydrate formation and an inward growing shell model in water-in-oil dispersions[END_REF], where the hydrate molar mass M h should be used for not mixing molar and mass basis.

Outward growth is limited to water availability in the outer surface, which is in contact with the oil. Shi et al. [START_REF] Shi | An inward and outward natural gas hydrates growth shell model considering intrinsic kinetics, mass and heat transfer[END_REF] considered a capillary-induced flow through the hydrate porous shell that drains out water from the core (hydrophilic nature of gas hydrate). However, the consideration of a rigid shell as did by them would imply in not respecting the water mass balance of the water inside the particle core, since the capillary-induced flow is independent on the inward growth. Considering no head loss in the capillaries, the water volumetric flowrate through the hydrate shell is the difference between the core volume shrinkage and the amount of water consumed to form hydrates, which are related to the inner growth rate as:

core w in d d Q dt dt                      2 4 1 1 w h in in h w M dr Q r M dt                        (8)
This expression is only valid if a rigid shell is considered, where capillary-induced flow would quickly stops due to a depressurization of the core and consequent counter-acting force from outside to inside of the core. However, capillary-induced flow may deform the shell and, if a fully deformable shell is considered, then the capillary-induced flow relates to the core volume deformation so as to maintain pressure of inner and outer surfaces. The capillaryinduced flow is however dependent on capillary parameters such as radius, tortuosity and density of capillaries, which evolve in time (porous will fill-up with time). Probably the hydrate shell is not fully deformable neither fully rigid and both mechanisms will happen together. To avoid any capillary parameters, the assumption of rigid shell is herein considered.

Once the permeation rate of water through the hydrate shell is known, the gas consumption due to outer growth comes: , , 2 min ; 4

g i out o w out i eq w o out dn C Q r k f dt M H                   (9)
where at the maximum, all the permeated water will be converted into hydrates instantly (assumption of Shi et al., 2010); but if enough water arrives at the particle outer surface, then crystal integration process may limit outer growth. The gas concentration in the outer surface in the oil phase is

dn k k Q r k C H f k H S H f dt M H S H                                ( 10
) 1 2 2 2 1 1 1 1 h in i mo u t out m w in w h i n o u t i S r k h r k h H r H D r r k                           (11)
and by considering Eqs. ( 5) and ( 6) for the outer radius, the outer growth rate comes:

      2 2 1 min ; 4 1 4 out h w i i out out b w eq out w o eq h w o o out dr M k k Q r k C H f k H H f dt M H H r                                   (12)
Knowing the inner and outer growth rate, Eqs. ( 7) and ( 12) (valid for one single particle), the hydrate formation rate of the entire particle population is estimated by Eq. ( 13). The related gas and water consumption rates comes from the (pseudo)-stoichiometry of hydrate formation and are calculated as shown in Eq. ( 14).

                      (13)
 

1 1 g h hyd dn dn dt dt    ;   1 w h hyd dn dn dt dt     (14)
Finally, the gas concentration in the bulk comes from a mass balance between the gas absorbed by the bulk and the gas consumed due to hydrate formation [START_REF] Englezos | Kinetics of formation of methane and ethane gas hydrates[END_REF][START_REF] Herri | Methane hydrate crystallization mechanism from in-situ particle sizing[END_REF]:

    / , , 2 2 /
, , gas absorption by the bulk gas consumption due to hydrate formation 1 4 1

g o g o out i in i b h g o b out i in i i o o h k A dr dr dC C C r r dt M dt dt                    (15)
where g o k is the absorption coefficient of gas by the oil bulk, /

g o A is the gas-oil interfacial surface, o  is the volume of the oil bulk, / g o o g C H f 
is the gas concentration in the gas-oil interface, H o is Henry's constant of the gas inside the oil and f g is the gas fugacity in the gas free phase.

Solution method

The model is composed of three ODEs for the inner growth rate (Eq. ( 7)), the outer growth rate (Eq. ( 12)) and the bulk concentration (Eq. ( 15)). This system of ODEs is solved using Runge-Kutta method of 4 th order. As initial conditions, the bulk is considered saturated and the inner and outer radius are considered equal to the droplet radius:

0 , b s a t o o g t t C C H f    ; 0 0 in in droplet t t t t r r r     (16)
where 0 t t  is the time step where hydrates nucleated and a hydrate shell of infinitesimal width already contours the water droplet. The water permeation rate, Eq. ( 8), is evaluated explicitly and is considered zero for the initial value, 0 0 t t Q   . All the particles are considered to be equal and not to agglomerate. Therefore, the sum term of Eq. ( 15)

becomes a multiplication by the number of particles i in the system. The main difficulty of the presented model is to estimate the closure parameters, which often come from meso or micro scale properties. The interfacial surface between oil and gas / g o A can be estimated through gas-liquid flow models (e.g., [START_REF] Taitel | A consistent approach for calculating pressure drop in inclined slug flow[END_REF]). The droplet radius droplet r can be estimated through liquid-liquid models [START_REF] Brauner | The prediction of dispersed flows boundaries in liquid-liquid and gas-liquid systems[END_REF][START_REF] Turner | Direct conversion of water droplets to methane hydrate in crude oil[END_REF].

The equilibrium fugacity of hydrate formation equals the gas fugacity in the equilibrium pressure for the given system temperature, , eq eq g T P f f  . The gas fugacity is estimated through an EoS (e.g., Peng-Robinson). The equilibrium pressure for the given pressure is estimated through hydrate equilibrium models [START_REF] Sloan | Clathrate hydrates of natural gases[END_REF]. Hydrate properties of common former gases (e.g., methane, ethane) may be found in literature [START_REF] Sloan | Natural gas hydrates in flow assurance[END_REF][START_REF] Jung | Properties and phenomena relevant to CH4-CO2 replacement in hydrate-bearing sediments[END_REF], but sometimes only an order of magnitude of the parameter exists. Also, hydrate properties may evolve with time due to cage occupancy and hydrate porosity evolution. Gas and water properties are well known in literature. Oil properties may be experimentally measured or estimated through compositional models (e.g., Multiflash TM ).

The blocking point are the kinetic parameters. The absorption coefficient of gas by the oil g o k is dependent on the oil composition and the shear rate of the flow, and usually comes from experimentation [START_REF] Englezos | Kinetics of formation of methane and ethane gas hydrates[END_REF][START_REF] Herri | Methane hydrate crystallization mechanism from in-situ particle sizing[END_REF][START_REF] Melchuna | Experimental study and modeling of methane hydrates crystallization under flow from emulsions with variable fraction of water and anti-agglomerant[END_REF]. The constant of proportionality of the crystal integration process k i is a micro scale parameter and usually curve fitted from experiments [START_REF] Englezos | Kinetics of formation of methane and ethane gas hydrates[END_REF][START_REF] Shi | An inward and outward natural gas hydrates growth shell model considering intrinsic kinetics, mass and heat transfer[END_REF]. The mass transfer coefficient between the particle and the bulk may be estimated through proper correlations [START_REF] Bird | Transport Phenomena[END_REF], which are valid when the concentration layers around the particles do not interact among themselves, i.e., the bulk exists -a continuous phase with spatially homogeneous concentration. This is probably the case of dilute flow, and for higher particle fraction in the flow the mass transfer coefficient is expected to decrease.

Results and discussions

The model is compared with experimental results from Melchuna et al. [START_REF] Melchuna | Experimental study and modeling of methane hydrates crystallization under flow from emulsions with variable fraction of water and anti-agglomerant[END_REF][START_REF] Melchuna | Topological modeling of methane hydrate crystallization from low to high water cut emulsion systems[END_REF]. They measured the amount of gas consumption due to hydrate formation in a flowloop of 40 m length and 10.2 mm ID. The fluids used were methane, deionized water and Kerdane (a light oil with C11 to C14 composition). The case of 40% water cut and 1.36 m/s of mixture superficial velocity was chosen. The main characteristics of the experiment are shown in Table 1. The values of the closure parameters used for the model evaluation (with the range of magnitude suggested in literature) are presented in Table 2 The number of particles i in the system is estimated by the use of the droplet radius and the total volume of water in the system as:

3 3 4 w total droplet droplet WC i r          (17)
where WC is the water cut (ratio between volume of water w  and total volume of the system total  ). Factor  is introduced in to represent an efficiency of particles that are actually consuming gas. This efficiency may be related: i. To the amount of droplets that are actually converted to particles. ii. To the assumption of an average droplet radius (monodisperse) instead of a distribution of droplets with different sizes and therefore different mass transfer resistances (polydisperse). iii. To the amount of particles that are actually in contact to the bulk, that is, an efficiency due to the impossibility of mass transfer between the particles and the bulk for a dense dispersion and for long distances from the oil-gas surface to the particle. The presented model converges with the experimental result of Melchuna et al. [START_REF] Melchuna | Experimental study and modeling of methane hydrates crystallization under flow from emulsions with variable fraction of water and anti-agglomerant[END_REF] for an efficiency value of 2.4%

 

. This low value is probably related to effect iii, since in the experiments of Melchuna et al. [START_REF] Melchuna | Experimental study and modeling of methane hydrates crystallization under flow from emulsions with variable fraction of water and anti-agglomerant[END_REF] the contact surface between oil and gas is restricted to a small region in the top of the riser section (called 'separator'), whereas the flowloop presents a semi-industrial scale of 40-m length, with a capacity of 10 liters of liquid. This means that probably only the particles inside the liquid volume near the 'separator' interacts with the gas-oil interface and grow. Following this idea, only the particles inside the 240 ml of the flowloop near the 'separator' are growing, while the rest of the experiment is mostly propagating the slurry flow to capture agglomeration, deposition and plugging tendencies of the oil-water-hydrate flow.

Figure 2 presents the comparison of the proposed model against the experimental data of Melchuna et al. [START_REF] Melchuna | Experimental study and modeling of methane hydrates crystallization under flow from emulsions with variable fraction of water and anti-agglomerant[END_REF] for the evolution in time of the gas consumption due to hydrate formation. The model is capable of representing the experimental results with the order of magnitude of the closure parameters presented in literature (Table 2) and for a curve fitting of 2.4%  

. Although the model cannot be considered as fully validated (since the curve fitting may mask the omission of important phenomena for hydrate growth or even mistakes during the modeling), the trend of the amount of gas consumption is considered as validated. The trend is composed of an initial faster hydrate formation rate, followed by a decrease in the gas consumption rate and an asymptote trend. This asymptote trend is related to the growth of the hydrate shell and related increase of the diffusion resistance. 5.5 6.5 10 mol/(m sPa)

i k     [5]
(data regression without considering mass transfer resistances around the particle, therefore ki is expected to be higher) (data regressed for a gas composition with ~85% of methane considering mass transfer resistances)

                     With: 298.15 T K   . Validity:   20 T T K    [22]

  

Experimental measurements of Melchuna [START_REF] Melchuna | Experimental study and modeling of methane hydrates crystallization under flow from emulsions with variable fraction of water and anti-agglomerant[END_REF] for the analyzed case of Table 1 Properties of methane sI hydrates (assumption that diffusivity of methane in Kerdane is one order of magnitude higher than methane in water) FBRM measurements of Melchuna [START_REF] Melchuna | Experimental study and modeling of methane hydrates crystallization under flow from emulsions with variable fraction of water and anti-agglomerant[END_REF] for the analyzed case of Table 1 [START_REF] Melchuna | Experimental study and modeling of methane hydrates crystallization under flow from emulsions with variable fraction of water and anti-agglomerant[END_REF] for the amount of gas consumed over time due to hydrate formation.

Further improvements

The inward and outward growth model of hydrate particles based on gas diffusion through solid is capable of predicting the time scale of hydrate formation only if a high solubility of gas in the hydrate structure is adopted. The value adopted in Table 2 is however optimist, coming from compatible gases inside metal alloys or ceramics at high temperature (e.g., helium in Pyrex at 200 o C). This is however not probably the case of methane (a much larger molecule than helium) inside the hydrate crystalline structure at a temperature where hydrates would form (~ 4 o C). Furthermore, the use of implies in a negative sign for the second term inside the minimum function of Eq. ( 12), which is not Physically acceptable. This is an indication that the closure parameters are ill-posed. The results of the last section were generated considering the water permeation rate as the limiting step and neglecting the outer crystal integration due to this ill-posed closure. Literature often mention an effective diffusion of the gas through the solid hydrate structure and through the liquid trapped in the capillaries. However, if ever water is continuously being expulsed through the particle, then the capillaries are filled-up with water. In this case, gas consumption would happen in the capillary walls and inner growth would probably not happen unless: (i) diffusion through the capillaries is much faster than crystallization; or (ii) the shell width is very small, which happens at the beginning of the hydrate formation. If the capillaries are filled-up with oil, then inner growth may happen, but outer growth will not happen.

Conclusions

The present article revisited and extended the predictive models for hydrate growth in gas-oil-water systems in literature. The model was compared with experimental data and the trend of gas consumption over time was validated, though closure parameters are still scarce in literature for model implementation in other scenarios then the herein chosen. The model captures the fast initial growth rate of the hydrates, with an asymptotic trend as the hydrate shell imposes higher mass transfer resistances. The mass transfer resistances are associated with gas diffusion and water permeation trough the hydrate shell, which are related to the shell width and therefore increase in time as the shell becomes thicker.

The analysis did in this work further highlights the omission in the use of the gas solubility inside the hydrate solid structure, which may imply in much smaller time scales of hydrate formation than the ones experimentally observed. Furthermore, diffusion through the liquid trapped in the capillaries would imply in either a non-existing shrinking core (if the liquid trapped is water, with consequent crystallization in the capillary walls) or a non-existing outer growth (if the liquid trapped is oil, and therefore water never reaches the outer surface). Although the inward and outward growth model explains the asymptotic trend of hydrate formation due to mass transfer limitation, future work shall analyze the mass transfer and crystallization inside of the capillaries of the hydrate porous structure.

Figure 2 .

 2 Figure 2. Comparison of model results against experimental data of Melchuna et al.[START_REF] Melchuna | Experimental study and modeling of methane hydrates crystallization under flow from emulsions with variable fraction of water and anti-agglomerant[END_REF] for the amount of gas consumed over time due to hydrate formation.

Table 1 .

 1 . Characteristics of the experiments of Melchuna et al.[START_REF] Melchuna | Experimental study and modeling of methane hydrates crystallization under flow from emulsions with variable fraction of water and anti-agglomerant[END_REF].

	Fluids	Methane, deionized water, Kerdane
	Water cut	40%
	Pressure	80 bar
	Temperature	275.5 K
	Mixture superficial velocity	1.36 m/s
	Total liquid volume in the flowloop	10 L

Table 2 .

 2 Closure parameters used for model evaluation and suggested range of order of magnitude in literature.

	Parameter	Order of magnitude from literature			Used value
	Gas diffusivity in hydrate	 Davies et al. (2008, apud Jung et al., 2010)  13 2 3.4 7.6 10 m s h D     Validity: 3 15MPa P   and 263 268 K T  	h D		13 3.4 10 m s 2  
	Solubility of gas in solids	3 10 10   Valid for different gas species in different solids, but no experiments of 5 molgas molsolid atm h S P   [13] methane in methane sI hydrates.	S	h	1 molgas 10 molsolid  (at 80 bar) 
	Mass transfer coefficient around a spherical particle	 , 2 0.52Re Sc 0.52 1 3  4 3 1 3 2 Re o D     J = mixture superficial velocity  / 2 g o m out D h  r / Sc o g o D   o out o J r   Validity: With: 15μm out r  , 2 Re 10   , Sh 3.5 	 (for the range of rout  3 0.15 1.31 10     calculated by the model) m m h s
		Armenante and Kirwan (1989, apud Herri et al., 1999)
		12	2
	Constant of
	proportionality
	of crystal
	integration
	process

Acknowledgements

The authors acknowledge the financial support of Région AURA Auvergne Rhône-Alpes through the project COOPERA FluEnergy and TE/CENPES/PETROBRAS.