Marine Protat

Noémie Bodin

Frédéric Gobeaux

Florent Malloggi

Jean Daillant

Nadège Pantoustier

Patrick Guenoun

Patrick Perrin

Biocompatible Stimuli-Responsive W/O/W Multiple Emulsions Prepared by One-Step Mixing with a Single Diblock Copolymer Emulsifier

Keywords: Multiple emulsions, amphiphilic block copolymer, stimuli-responsive, phase inversion, interfaces

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INTRODUCTION

Multiple emulsions like water-in-oil-in-water emulsions (W/O/W) are an ideal platform for the encapsulation of many molecules of interest that need to be protected from external stresses before being released. In the food processing industry, the high potential of multiple emulsions to encapsulate and protect sensitive and reactive compounds such as flavors, vitamins or minerals has been demonstrated [START_REF] Muschiolik | Multiple Emulsions for Food Use[END_REF][START_REF] O'regan | Sodium Caseinate-Maltodextrin Conjugate Stabilized Double Emulsions: Encapsulation and Stability[END_REF][START_REF] Jiménez-Colmenero | Potential Applications of Multiple Emulsions in the Development of Healthy and Functional Foods[END_REF] . Multiple emulsions also show promise as delivery systems in cosmetic and pharmaceutical products, as they could enable sustained release of encapsulated compounds as well as the co-encapsulation of incompatible materials [START_REF] Gallarate | On the Stability of Ascorbic Acid in Emulsified Systems for Topical and Cosmetic Use[END_REF][START_REF] Carlotti | O/W Multiple Emulsions for Dermatological and Cosmetic Use, Obtained with Ethylene Oxide Free Emulsifiers[END_REF][START_REF] Schwarz | Optimisation of Multiple W/O/W Nanoemulsions for Dermal Delivery of Aciclovir[END_REF][START_REF] Hoppel | Natural Polymer-Stabilized Multiple Water-in-Oil-in-Water Emulsions: A Novel Dermal Drug Delivery System for 5-Fluorouracil: W/O/W for 5-Fluorouracil[END_REF] .

In most cases, the preparation of W/O/W emulsions requires a combination of hydrophilic and hydrophobic surfactants to stabilize the W/O as well as the O/W interface, and also a two-step emulsification process [START_REF] Garti | Double Emulsions -Scope, Limitations and New Achievements[END_REF] which introduces many destabilization pathways and lead to emulsions with short-term stability [START_REF] Dobrynin | Scaling Theory of Polyelectrolyte Solutions[END_REF][START_REF] Ficheux | Some Stability Criteria for Double Emulsions[END_REF][START_REF] Garti | Recent Developments in Double Emulsions for Food Applications[END_REF][START_REF] Jiao | Multiple Emulsion Stability: Pressure Balance and Interfacial Film Strength[END_REF] . Many formulation approaches have been deployed to improve stability [START_REF] Garti | Double Emulsions -Scope, Limitations and New Achievements[END_REF][START_REF] Garti | Double Emulsions: Progress and Applications[END_REF] : sophisticated combinations of emulsifiers have been selected, the proportions of the two types of emulsifiers have been optimized, co-surfactants have been added… However, the complexity of their multi-step formulation route, the lack of control of emulsions morphology and encapsulation rate, as well as the necessity of using important amounts of surfactants to obtain stable multiple emulsions, are still a limitation to large-scale applications [START_REF] Dickinson | Double Emulsions Stabilized by Food Biopolymers[END_REF] .

Recently, progress has been made in the bulk formulation process of multiple emulsions as a couple of studies have shown that they can be prepared using a one-step phase inversion mixing process and a single emulsifier. This emulsifier can be a diblock copolymer [START_REF] Hanson | Nanoscale Double Emulsions Stabilized by Single-Component Block Copolypeptides[END_REF][START_REF] Hong | One-Step Formation of W/O/W Multiple Emulsions Stabilized by Single Amphiphilic Block Copolymers[END_REF][START_REF] Besnard | Multiple Emulsions Controlled by Stimuli-Responsive Polymers[END_REF][START_REF] Zhang | Impact of Electrolytes on Double Emulsion Systems (W/O/W) Stabilized by an Amphiphilic Block Copolymer[END_REF] as well as some nanoparticles [START_REF] Binks | Types of Phase Inversion of Silica Particle Stabilized Emulsions Containing Triglyceride Oil[END_REF][START_REF] Binks | Silica Particle-Stabilized Emulsions of Silicone Oil and Water: Aspects of Emulsification[END_REF][START_REF] Binks | Influence of Propylene Glycol on Aqueous Silica Dispersions and Particle-Stabilized Emulsions[END_REF] . Hanson et al. were the first to obtain long-term stability W/O/W emulsions stabilized by a diblock polypeptide [START_REF] Hanson | Nanoscale Double Emulsions Stabilized by Single-Component Block Copolypeptides[END_REF] . Sun et al. also demonstrated the possibility to prepare stable multiple emulsions stabilized by a synthetic diblock copolymer via catastrophic phase inversion [START_REF] Sun | Influence of Asymmetric Ratio of Amphiphilic Diblock Copolymers on One-Step Formation and Stability of Multiple Emulsions[END_REF] . Modified silica particles were used by Binks et al. to prepare stable W/O/W Pickering emulsions [START_REF] Binks | Types of Phase Inversion of Silica Particle Stabilized Emulsions Containing Triglyceride Oil[END_REF] . Li et al. were the first to use a single anionic surfactant to prepare stable W/O/W emulsions via catastrophic phase inversion [START_REF] Li | Preparation of High Internal Water-Phase Double Emulsions Stabilized by a Single Anionic Surfactant for Fabricating Interconnecting Porous Polymer Microspheres[END_REF] . Recently, Tu et al. also prepared one-step pH-responsive multiple emulsions via transitional phase inversion using Janus particles as a stabilizer [START_REF] Tu | One-Step Encapsulation and Triggered Release Based on Janus Particle-Stabilized Multiple Emulsions[END_REF] .

A few years ago, our team proved that appropriate polymer architectures can easily generate one-step W/O/W emulsions via transitional phase inversion [START_REF] Besnard | Multiple Emulsions Controlled by Stimuli-Responsive Polymers[END_REF][START_REF] Besnard | Breaking of the Bancroft Rule for Multiple Emulsions Stabilized by a Single Stimulable Polymer[END_REF] . Water-toluene emulsions were stabilized using PS-b-P(S-st-DMAEMA) copolymers constituted of a hydrophobic block of polystyrene (PS) and a hydrophilic block of randomly distributed units of styrene (30 mol %) copolymerized with 2-(dimethylamino)ethyl methacrylate (DMAEMA) (70 mol %).

Emulsions stabilized with these copolymers were stable for several months but were also stimuli-responsive with a capacity of encapsulation tuned by pH or temperature [START_REF] Besnard | Multiple Emulsions Controlled by Stimuli-Responsive Polymers[END_REF][START_REF] Besnard | Breaking of the Bancroft Rule for Multiple Emulsions Stabilized by a Single Stimulable Polymer[END_REF] . The latter study showed that the architecture of the polymer is important for the stabilization of multiple emulsions. A study of polymer conformations based on Cryo-TEM imaging and SANS measurements also evidenced the link existing between interfacial curvature and emulsion type [START_REF] Besnard | Breaking of the Bancroft Rule for Multiple Emulsions Stabilized by a Single Stimulable Polymer[END_REF] .

In this work, we describe another step forward by extending such advances to biocompatible systems. We used Atom Transfer Radical Polymerization to synthesize an amphiphilic PDMS-b-PDMAEMA diblock copolymer composed of a hydrophobic block of polydimethylsiloxane (PDMS) and a hydrophilic block of pure DMAEMA. PDMS and PDMAEMA polymers are materials which have already been demonstrated biocompatible properties and were used as micelles for anticancer drug delivery [START_REF] Car | pH-Responsive PDMS-B-PDMAEMA Micelles for Intracellular Anticancer Drug Delivery[END_REF][START_REF] Dinu | Engineered Non-Toxic Cationic Nanocarriers with Photo-Triggered Slow-Release Properties[END_REF] . We used this pH, ionic strength and temperature-sensitive polymer to stabilize water -Miglyol® 812 as well as water -isopropyl myristate emulsions. Miglyol® 812 and isopropyl myristate are biocompatible oils widely used in pharmaceutics and cosmetics. With both systems, we show that stable W/O/W emulsions can be generated in a one-step formulation process via transitional phase inversion with pH and / or ionic strength. Finally, we investigate the formation of W/O/W emulsions through surface tension measurements and Cryo-TEM imaging to correlate multiple emulsion formation to tension and curvature of the interface.

EXPERIMENTAL SECTION

Materials. Monohydroxyl terminated poly(dimethylsiloxane) (number average molar mass 3000 g.mol -1 ) (PDMS-OH), bromo-2-methylpropionyl bromide (BrMPBr, 98%), triethylamine (Et3N, 99.5%), anhydrous tetrahydrofuran (THF, 99.9%), 1,1,4,7,10,10hexamethyltriethylenetetramine (HMTETA, 97%), copper (I) bromide (CuBr, 99.999%), sodium hydrogenocarbonate (NaHCO3, 99%), magnesium sulfate, sodium chloride (NaCl), sodium nitrate (NaNO3), isopropyl myristate and Nile Red were purchased from Sigma-Aldrich. 2-(dimethylamino)ethyl methacrylate (DMAEMA, 99%) and dichloromethane (99%) were obtained from Acros Organics. Miglyol® 812 was generously provided by IMCD. HCl and NaOH solutions were supplied by Merck. Milli-Q water with a resistivity of 18.2 MΩ.cm was used to prepare aqueous solutions for emulsions.

All reagents were used without further purification except the DMAEMA and the THF, which were filtered through a column of basic alumina (Acros Organics) to remove inhibitors and stabilizers and stored at low temperature, at -10°C.

Synthesis of PDMS-b-PDMAEMA Block

Copolymers. The block copolymer was synthesized by Atom Transfer Radical Polymerization (ATRP) of DMAEMA using preformed functional-PDMS. The scheme of the synthesis for PDMS-b-PDMAEMA is described in Figure 1. The PDMS-macroinitiator was synthesized from monohydroxylterminated-PDMS (PDMS-OH) according to a previously described procedure [START_REF] Duquesne | Synthesis of Silicone-Methacrylate Copolymers by ATRP Using a Nickel-Based Supported Catalyst[END_REF][START_REF] Semsarzadeh | Atom Transfer Radical Polymerization of Styrene and Methyl (Meth)acrylates Initiated with Poly(dimethylsiloxane) Macroinitiator: Synthesis and Characterization of Triblock Copolymers[END_REF] with slight changes as reported below. Briefly, the PDMS macroinitiator was prepared from hydroxylterminated PDMS by using bromo-2-methylpropionyl bromide dissolved in anhydrous THF ([PDMS-OH]0 = 0.16M) and triethylamine ([Et3N]0 = 0.66M). Bromo-2-methylpropionyl bromide ([BrMPBr]0 = 0.44M) was added and the reaction was stirred at room temperature for three days. Temperature was then increased and maintained at 60°C for one hour. The polymer solution was filtered to remove salts and the solvent was removed under vacuum.

The resulting oil was dissolved in dichloromethane and washed with saturated NaHCO3 aqueous solution and water. The organic phase was finally dried over magnesium sulfate, filtered and the solvent was removed under vacuum. Yield: 73%. [START_REF] Muschiolik | Multiple Emulsions for Food Use[END_REF] Typically, Atom Transfer Radical Polymerizations (ATRP) were carried out under nitrogen atmosphere using syringe techniques. The catalyst (copper (I) bromide, 0.453 g, 0.32 mmol) was introduced in a dry round flask equipped with a three-way stopcock and a magnetic stirrer. Three nitrogen/vacuum cycles were performed. Monomer (DMAEMA, 2.44 g, 15.5 mmol), PDMS-macroinitiator (1.55 g, 0.52 mmol), soluble ligand (HMTETA, 0.158 g, 0.68 mmol) and solvent (THF, 50% of total volume) were bubbled with nitrogen and then added in a separated flask containing CuBr using a previously flame-dried stainless cannula. The polymerization of DMAEMA was then conducted under nitrogen at 60°C under magnetic stirring for three hours. Ratios between PDMS-macroinitiator, DMAEMA, CuBr and HMTETA were 1/30/1.3/2.6 respectively. The polymerization reaction was stopped by cooling down the flask under cold running water and the polymer solution was purified by passing through a column of basic alumina to remove copper. The excess of solvent was removed under vacuum and the polymer solution was precipitated in basic water to obtain a solid, which was then dried under vacuum.

Emulsions Preparation. The polymer was dissolved at the concentration of 5 g.L -1 in either Miglyol® 812 or isopropyl myristate. NaCl, HCl and NaOH solutions were used to prepare aqueous solutions at different pH and ionic strengths using MilliQ-water. A fixed amount of 3 mL of polymer-containing oil and 3 mL of aqueous solution were placed into contact at room temperature for 24 hours before emulsification. All the pH values reported in this work were measured after 24 hours of contact between both phases. Emulsions were then obtained by mixing both phases using an Ultra Turrax T10 homogenizer (8 mm head) operating at 24000 rpm for 40 seconds.

Stability Measurements.

Emulsion stability was assessed one day, one week, one month and two months after emulsification. Macroscopic stability was assessed by visual analysis of the vials containing emulsions. Emulsions are considered to be macroscopically stable as long as the dispersed phases do not form a subnatant or supernatant phase separated from the emulsion. Microscopic stability was assessed by confocal microscopy. Emulsion pictures were taken at different times after emulsification and the mean diameter of emulsion drops was calculated for each picture. After emulsification, emulsion samples were stored in the dark at room temperature.

Characterization. Size Exclusion Chromatography (SEC, Viscotek GPC max VE2001; TDA 302 triple detector array) in THF containing 2% triethylamine and 1 H NMR spectroscopy (Bruker 400MHz spectrometer) were used to determine molar masses, polydispersity index (PDI) and precise compositions of block copolymer.

Residual copper in block copolymer samples was determined using i Cap Q-inductively coupled plasma-mass spectrophotometer from Thermo Elemental. Samples were prepared by solubilization of 20 mg of polymer in 30mL HNO3 0.2M.

Confocal Laser Scanning Microscopy (CLSM) was used to determine emulsion type and stability. Confocal imaging was done in a quartz cell (0.5 mm light path) through an Olympus Fluoview FV1000 inverted confocal microscope. 150 µL emulsion samples were placed in the cell and studied at room temperature. Nile Red was used as a hydrophobic fluorophore, and the oil phase appear in blue in the pictures. We used the following criterium to discriminate between O/W and W/O/W emulsions (O/W and W/O/W emulsions are identified by blue squares and green circles respectively in the emulsion type diagram). W/O/W emulsions are defined as emulsions for which the surface of the oil drops containing water droplets covers more than 50% of the surface of the emulsion, the surfaces being measured from a series of four confocal microscopy pictures of a given sample. If the criterium is not fulfiled, the emulsions are called O/W. Diameters of emulsions drops were calculated from the images obtained by confocal microscopy. For each W/O/W emulsion, we measured the mean drop size and standard deviation (SD) of a panel of 500 oil globules using ImageJ (NIH software) with a systematic error of 10 %, mostly because some drops are visible albeit their largest diameter is not perfectly in the z plane of focus.

Interfacial tensions were measured using a spinning drop tensiometer (SVT 20N, Dataphysics). Measures were made on pre-equilibrated samples for 24 hours in presence of polymer, for different pH and ionic strengths. A capillary was filled with the equilibrated aqueous phase, avoiding formation of air bubbles and a small drop of the oil phase was formed into the capillary using a syringe. The elongation of the oil droplet stops when the centrifugal forces are balanced by the interfacial tension forces. By measuring the cylinder radius, the interfacial tension was determined from the Vonnegut's equation [START_REF] Vonnegut | Rotating Bubble Method for the Determination of Surface and Interfacial Tensions[END_REF] . For each pH and ionic strength condition, the measure was repeated at least two times.

Morphology of structures formed by the copolymer in aqueous phases after partitioning was determined from Cryo-TEM images. Drops of aqueous phase were deposited on EM grids covered with a holly carbon film (Quantifoil R2/2) previously treated by plasma glow discharge. The excess of liquid on the grids was blotted out with a filter paper and the grids were quickly immerged in liquid ethane to form a thin vitreous ice film. The whole process was performed using a Vitrobot apparatus (PEI Company). Observations were conducted at low temperature (-180°C) on a JEOL 2010 FEG microscope operated at 200 kV. The contrast arises from the differences in the electron densities of the sample molecules. The defocus was set to 7 000 -10 000 nm to obtain enough phase contrast under low-dose conditions. Images were recorded with a Gatan camera.

RESULTS AND DISCUSSION

Synthesis of PDMS-b-PDMAEMA Block

Copolymer. The PDMS-b-PDMAEMA diblock copolymer was synthesized by ATRP from a preformed PDMS-macroinitiator. We used commercially available monohydroxyl terminated PDMS as a starting material to obtain a monofunctional bromoalkyl-terminated PDMS-macroinitiator suitable for the polymerization of DMAEMA by ATRP (Figure 1).

The PDMS-macroinitiator is obtained by simple condensation reaction of well-defined hydroxyl terminated PDMS with BrMPBr [START_REF] Duquesne | Synthesis of Silicone-Methacrylate Copolymers by ATRP Using a Nickel-Based Supported Catalyst[END_REF][START_REF] Semsarzadeh | Atom Transfer Radical Polymerization of Styrene and Methyl (Meth)acrylates Initiated with Poly(dimethylsiloxane) Macroinitiator: Synthesis and Characterization of Triblock Copolymers[END_REF] , which has been used in a subsequent step for the polymerization of DMAEMA by ATRP (Figure 1). The copper-mediated living radical polymerization of DMAEMA from PDMS-macroinitiator was carried out in THF (DMAEMA/THF, 1/1 vol.) at 60°C. The target molar mass of PDMAEMA block is 5 500 g.mol -1 and [DMAEMA]/[PDMS-macroinitiator]/[CuBr]/[HMTETA] were 30/1/1.3/2.6.

The number average molar mass of PDMS-b-PDMAEMA and the polymer composition were determined by 1 H NMR (Figure 2). First, the number average molar mass of the PDMSmacroinitiator was estimated by 1 H NMR to be around 3000 g.mol -1 corresponding to a polymerization degree of 38 ( 1 H NMR spectrum not shown). The methacrylate content in the PDMS-b-PDMAEMA copolymer was determined from the relative intensities of protons characteristic of the PDMAEMA (singlets assigned to both of CH2 groups and the N(CH3)2 group from PDMAEMA at 2.6 and 4.1 as well as 2.3 ppm, respectively) and PDMS (singlet assigned to both of CH3 groups at 0 ppm) blocks. As a result, the PDMS-b-PDMAEMA block copolymer was designed by PDMS38-b-PDMAEMA25. and (G)). The fact that no multiple emulsions are formed with this system is consistent with the architecture of the polymer used in this study since previous results on other systems [START_REF] Besnard | Breaking of the Bancroft Rule for Multiple Emulsions Stabilized by a Single Stimulable Polymer[END_REF] have shown that no multiple emulsions are formed when the hydrophilic block of the copolymer is purely constituted of DMAEMA. Here, it is also found out that no pH-driven phase inversion takes place for this system at I = 0 mol.L -1 . This is probably amplified by the surface-active effect of the oil which acts like a hydrophilic surfactant in this pH and I conditions and thus promotes the stabilization of O/W emulsions only. At pH > 10 and I > 1.5 mol.L -1 , the formation of W/O emulsions is observed (Figure 5.a, (I)).

At this basic pH, the polymer is fully neutralized and the high ionic strength contributes to increasing the hydrophobicity of the polymer [START_REF] Marchal | Emulsion Stabilization and Inversion Using a pH-and Temperature-Sensitive Amphiphilic Copolymer[END_REF] . The extension of the hydrophilic block of the copolymer is then significantly reduced, and could become smaller than the extension of the hydrophobic block. Such conditions are likely to promote an interface curvature towards water and thus the stabilization of W/O emulsions. emulsion (E) (Table 1). Recently, Bae et al. [START_REF] Bae | Osmotically Driven Formation of Double Emulsions Stabilized by Amphiphilic Block Copolymers[END_REF] described a mechanism to explain the formation of W/O/W emulsions in a two-step emulsification process. The first one is the making of a simple O/W emulsion by mixing and the second one consists in the spontaneous formation of the W/O/W emulsion by inner water drops formation. In the second step, water transfers to the oil under the osmotic pressure gradient caused by the presence of salt residues in oil, which are however more soluble in water than in oil. Consequently, in Bae's mechanism, a growing kinetics of the inner water drops leading to the formation of monodisperse water drops with final size controlled by the amount of salt both in oil and water is observed. In our case, the inner water droplets of our W/O/W emulsions are immediately visible after preparation and no growth kinetics is observed. Moreover our water drops size polydispersity is rather large in sharp contrast with Bae's observations. Although quite interesting, we then believe that Bae's mechanism is not sufficient to explain the formation of our W/O/W. It is thus reasonable to admit that there might be some other underlying mechanism(s) that takes place to explain the W/O/W emulsion formation in our systems. Also, the polymers used here (PDMS-b-PDMAEMA) was synthesized by ATRP and contain copper residues at a concentration of 3.4 x 10 -4 copper ions per chain. The copper concentration is thus ten times lower than the concentration of Bae's purest copolymer where multiple emulsions do not form. Consequently, the formation of multiple emulsions is unlikely due to this mechanism only in our case.

Two months after emulsification all emulsions are still macroscopically stable, to the exception of the W/O one (Figure 5,(I' present excellent macroscopic and microscopic stability, especially emulsions (E') and (F'), which show comparable water encapsulation and similar sizes of the oil globules after emulsification and two months later (Table 1).

One-Step Formation of Water -Isopropyl Myristate Emulsions Stabilized by PDMS38-b-

PDMAEMA25. Similar emulsion type diagrams were obtained with isopropyl myristate as the oil phase just after (Figure 6.a) and two months after emulsification (Figure 6.b). Table 2 presents mean drop diameters and standard deviations for W/O/W emulsions at both times. For water -isopropyl myristate emulsions stabilized by PDMS38-b-PDMAEMA25, only O/W emulsions were obtained without addition of external salt on the whole pH range (Figure 6 (J), (M) and (P)), as it was the case for the water -Miglyol® 812 ones (Figure 5, (A), (D) and (G)). As we remarked previously, this is consistent with the architecture of the polymer and with the surface-active effect of the oil (Figure 4).

For this system, all emulsions formed at pH = 1 are O/W ones as soon as the emulsification has taken place (Figure 6, (J), (K) and (L)). These emulsions present a limited microscopic stability (Table 2). At high pH and I, unstable W/O emulsions are formed (Figure 6, (R)).

At intermediate values of pH and I, a large domain of stable W/O/W emulsions is obtained (Figure 6, (N), (O) and (Q)). These emulsions display excellent macroscopic and microscopic stability (Table 2), but it can be seen on the confocal microscopy pictures that the internal water droplets have flocculated inside oil globules, showing that the water -myristate -PDMS38-b-PDMAEMA25 system is less efficient than the water -Miglyol® 812 -PDMS38-b- 3).

Table 3. Values of interfacial tension between Miglyol® 812 and water in presence of PDMS38-b-PDMAEMA25 for different conditions of pH and ionic strengths.

For both oils, the variation in the interfacial tension values from one sample to another (Table 3) appears to be too small to make the interfacial tension a decisive parameter to determine the emulsion type or even the final size of the emulsion drops. Interfacial tensions for W/O/W emulsions are slightly smaller for isopropyl myristate ( region where both are present [START_REF] Pays | Double Emulsions: A Tool for Probing Thin-Film Metastability[END_REF][START_REF] Pays | Understanding the Stability and Lifetime of Emulsions[END_REF] . This transition is linked to a decrease of film curvature, in accordance with the change of emulsion type between the two samples. However, it appears that a non-zero positive curvature is compatible with the formation of multiple emulsions.

CONCLUSION

In this paper, we synthesized a biocompatible copolymer that can be used as a single emulsifier to stabilize W/O/W emulsions using two different biocompatible oils, isopropyl myristate and Miglyol® 812. These emulsions are formulated in a one-step emulsification process induced by transitional phase inversion using pH and ionic strength fine adjustments.

The formulation of our system is indeed well-controlled and highly reproducible. It leads to the making of W/O/W emulsions with long-term stability in a large domain of pH and ionic strength conditions. The interfacial tension is not a key parameter to predict emulsion type even if the formation of multiple emulsions often requires low interfacial tensions. However, the tuning of the polymer films interfacial curvature and hence microemulsions morphology is likely to be a determinant parameter to promote emulsion type in general and in particular the formation of W/O/W emulsions.

Figure 1 .

 1 Figure 1. Synthesis of a PDMS-b-PDMAEMA block copolymer from a monohydroxyl-

Figure 2 . 1 H

 21 Figure 2. 1 H NMR spectra of PDMS38-b-PDMAEMA25 in CDCl3

Figure 3 .Figure 4 .

 34 Figure 3. SEC traces of PDMS38-b-PDMAEMA25 and its PDMS-macroinitiator

Figure 5 .Table 1 .

 51 Figure 5. Emulsion type diagrams and fluorescence confocal microscopy images for water -

Finally, a remarkably

  large domain is evidenced in which stable W/O/W emulsions are formulated (Figure 5.a, (B), (C), (E), (F) and (H)). These W/O/W emulsions are readily formed in one emulsification step using a single polymeric emulsifier at pH and ionic strength conditions intermediate to those that lead to the formation of O/W and W/O emulsions. A control over a wide range of oil droplet sizes is reached by adjusting the conditions of pH and ionic strength of the water phase at constant mixing power. Indeed the mean diameter for the oil drops of the W/O/W emulsions decreases from 200 µm for emulsion (B) to 19 µm for

  )), which separates into oil and aqueous phases. The O/W domain has also enlarged at the expense of the W/O/W emulsion one. More precisely, the multiple emulsions prepared at pH = 1 were turned into O/W ones, irrespective of ionic strength (Figures 5.b, (B') and (C')). In particular, for all emulsions prepared at pH = 1, oil droplets have coarsened with a diameter increase of about 30% (Table1), showing a lack of microscopic stability. However, all other W/O/W emulsions (Figure5.b (E'), (F') and (G'))

Figure 6 .

 6 Figure 6. Emulsion type diagrams and fluorescence confocal microscopy images for water -

  Study of Interfacial Tensions between Oil and Water inPresence of Polymer. To get insight on emulsion type diagrams, interfacial tensions between both oils and water in the presence of PDMS38-b-PDMAEMA25 were measured for different values of pH and ionic strength (Table

Figure 7 .

 7 Figure 7. Cryo-TEM pictures of water -isopropyl myristate microemulsions formed in water

Table 1

 1

presents mean drop diameters and standard deviations for W/O/W emulsions at both times.

Table 2 .

 2 Comparison of mean oil drop diameter and standard deviation (SD) just after and two months after emulsification for water -isopropyl myristate W/O/W emulsions stabilized by PDMS38-b-PDMAEMA25. The error in the determination of mean oil drop diameters can be estimated to be around 10 %. Only oil globules were taken into account for the determination of mean drop diameter and SD.

  PDMAEMA25 at stabilizing the W/O interface of W/O/W emulsions. This can be due to the increased density difference between isopropyl myristate and water compared to Miglyol® 812, which causes the internal droplets to sediment inside the oil globules and promote flocculation.

	Sample (t = 0)	N	O	Q
	Mean diameter / SD (µm)	14/6	15/6	23/13
	Sample (t = 2 months)	N'	O'	Q'
	Mean diameter / SD (µm)	13/5	14/5	25/12

Table 3 ,

 3 (N) and (O)) than forMiglyol® 812 (Table3, (E) and (F)), which may explain the smallest drop size for W/O/W emulsions stabilized by isopropyl myristate (Table1 and 2).

	Miglyol® 812	(A)	(B)	(C)	(D)	(E)	(F)	(G)	(H)	(I)
	γ (mN.m -1 )	2.0 0.4	2.0 0,4	4.7 1.0	1.1 0.2	1.5 0.3	0.9 0.2	2.1 0.4	1.4 0.3	0.8 0.2
	Isopropyl myristate	(J)	(K)	(L)	(M)	(N)	(O)	(P)	(Q)	(R)
	γ (mN.m -1 )	0.4 0.1	1.6 0.3	3.6 0.8	3.8 0.8	0.3 0.1	0.4 0.1	2.0 0.4	1.4 0.3	0.5 0.1

H NMR (CDCl 3 , 400 MHz): PDMS 38 -b-PDMAEMA

Probing interfacial curvature with Cryo-TEM measurements. As it was previously suggested that interfacial curvature likely imposes the type of emulsion for W/O and O/W types [START_REF] Besnard | Breaking of the Bancroft Rule for Multiple Emulsions Stabilized by a Single Stimulable Polymer[END_REF] , a reasonable assumption would be that the curvature of the water -oil interface in presence of polymer is close to zero in the domain where W/O/W are formed. Indeed, this could lead to the stabilization of both types of interfaces.

The nature of microemulsions formed in the aqueous phase after partitioning was determined using Cryo-TEM measurement (Figure 7). Cryo-TEM imaging was only carried out on water -isopropyl myristate microemulsions because no images could be obtained for water -Miglyol® 812 ones, which is probably related to an incompatibility between the nature of the oil and the freezing process.

b). The microemulsion morphology changes between the two

samples can be attributed to modifications in the interfacial polymer layer. Indeed, microemulsions are usually well described as an assembly of flexible films dominated by the curvature energy as introduced by Helfrich [START_REF] Helfrich | Elastic Properties of Lipid Bilayers: Theory and Possible Experiments[END_REF] . It was indeed shown that for a certain range of the curvature parameters, a transition occurs from spherical to cylindrical micelles through a

AUTHOR INFORMATION

Corresponding Author. * E-mail: patrick.perrin@espci.fr ; patrick.guenoun@cea.fr ACKOWLEDGEMENT We thank the RTRA Triangle de la Physique for its funding through the Imagemul project.

Cryo-TEM observations were done thanks to "Investissements d'Avenir" LabEx PALM (ANR-10-LABX-0039-PALM). We gratefully acknowledge Mohamed Hanafi for the SEC analysis. We thank IMCD for having generously provided the Miglyol® 812 we used in our experiments.


Ngoc-Khuyen Le
email: ngoc-khuyen.le@telecom-paristech.fr

Anaïs Vergne
email: anais.vergne@telecom-paristech.fr

Philippe Martins
email: philippe.martins@telecom-paristech.fr

Laurent Decreusefond Ltci

Distributed Computation of the Čech Complex and Applications in Wireless Networks

In this paper, we introduce a distributed algorithm to compute the Čech complex. This algorithm is aimed at solving coverage problems in self organized wireless networks. Two applications based on the distributed computation of the Čech complex are proposed. The first application detects coverage holes while the later one optimizes coverage of wireless networks.

I. INTRODUCTION

Coverage is a key factor that determines the quality of service in wireless networks. In recent applications to solve coverage problems in wireless networks, simplicial complex is often utilized to represent the network topology. Unlike graph, which represents only the neighborhood of cells, simplicial complex presents the relationship of cells with a higher dimension.

The Čech complex is a simplicial complex that represents exactly the topology of the network [START_REF] Vin De Silva | Blind swarms for coverage in 2-D[END_REF]. The Čech complex is the right tool to describe and optimize the coverage for wireless networks. However, the computation of the Čech complex is complicated. The Čech complex represents a collection of cells by a simplex if they have a non-empty intersection. In [START_REF] Dantchev | Efficient construction of the ech complex[END_REF], an algorithm to compute the Čech complex is introduced, but this algorithm is designed to be utilized in graphics science. Therefore, this algorithm only works with a collection of same sized cells. Although an algorithm to compute the Čech complex for a collection of differently sized cells is proposed in [START_REF] Le | Construction of the generalized cech complex[END_REF], this algorithm is still centralized.

In [START_REF] Le | Simplicial homology based energy saving algorithms for wireless networks[END_REF], authors introduce an application based on the Čech complex to optimize the coverage for wireless networks. However, as the computation of the Čech complex is still available in centralized way, this application is also centralized. The future wireless networks will be self organized and therefore require distributed computations. In [START_REF] Silva | Coordinate-free coverage in sensor networks with controlled boundaries via homology[END_REF], [START_REF] Ghrist | Coverage and hole-detection in sensor networks via homology[END_REF], [START_REF] Feng Yan | Connectivity-based distributed coverage hole detection in wireless sensor networks[END_REF], some algorithms to detect coverage holes in wireless networks are developed. Although these algorithms are available in both centralized and distributed way, they use the Rips complex instead of the Čech complex to represent the topology of the network. The Rips complex is an approximation of the Čech complex. The Rips complex represents a collection of cells by a simplex if every pair of cells in this collection are intersected. Therefore, the Rips complex may not capture exactly the topology of the network. So, some coverage holes may be undiscovered [START_REF] Yan | Accuracy of homology based approaches for coverage hole detection in wireless sensor networks[END_REF].

In this paper, we introduce the distributed computation of the Čech complex for a given collection of differently sized cells. This algorithm is aimed at solving coverage problems in wireless networks. We also propose two applications that are based on the distributed computation of the Čech complex. Given a wireless network, the first application detects coverage holes while the second one optimizes its coverage. The optimized coverage reduces as much as possible the intersection among cells. As a result, the waste power due to interference within intersected regions is avoided efficiently. Both of these applications are distributed.

This paper is organized as follows. Section II introduces a short background of simplicial homology and its applications in wireless networks. Section III describes all details about the distributed computation of the Čech complex. Section IV proposes a distributed coverage hole detection and a distributed coverage optimization for wireless networks. The last section concludes our paper.

II. SIMPLICIAL HOMOLOGY AND APPLICATIONS

In this section, we introduce some notions of simplicial homology and its applications. For further details about the simplicial homology, see documents [START_REF] Munkres | Elements of algebraic topology[END_REF] and [START_REF] Hatcher | Algebraic topology[END_REF].

Given a set of vertices V , a k-simplex is an unordered subset {v 0 , v 1 , . . . , v k }, where v i ∈ V and v i = v j for all i = j. The number k is its dimension. See Figure 1 for some instances. Any subset of vertices set of a simplex is a face of this simplex. An oriented simplex is an ordered type of

v 0 0-simplex 1-simplex v 0 v 1 2-simplex v 0 v 1 v 2 3-simplex v 0 v 1 v 2 v 3
Fig. 1: An example of simplices.

simplex. The orientation of a simplex changes if two vertices of this simplex swap their position. The change of orientation is represented by a negative sign as:

[v 0 , v 1 , . . . , v i , v j , . . . , v k ] = -[v 0 , v 1 , . . . , v j , v i , . . . , v k ]
An abstract simplicial complex is a collection of simplices such that: every face of a simplex is also in the simplicial complex.

Let X be a simplicial complex. For each k ≥ 0, we define a vector space C k (X) whose basis is a set of oriented ksimplices of X. If k is bigger than the highest dimension of X, let C k (X) = 0. We define the boundary operator to be a linear map ∂ : C k → C k-1 as follows:

∂[v 0 , v 1 , . . . , v k ] = k i=0 (-1) i [v 0 , v 1 , . . . , v i-1 , v i+1 , . . . , v k ]
This formula suggests that the boundary of a k-simplex is the collection of its (k-1)-faces, as illustrated in Figure 2. For example, the boundary of a segment is its two endpoints. A filled triangle is bounded by its three segments. A tetrahedron has its boundary comprised of its four faces which are four triangles.

v 0 v 1 ∂ -→ v 0 + v 1 -∂ -→ 0 v 0 v 1 v 2 ∂ -→ v 0 v 1 v 1 v 2 v 2 v 0 ∂ -→ 0 v 0 v 1 v 2 v 3 ∂ -→ v 0 v 1 v 2 v 3 ∂ -→ 0 Fig. 2: Boundary operator.
The composition of boundary operators gives a chain of complexes:

• • • ∂ -→ C k+1 ∂ -→ C k ∂ -→ C k-1 • • • ∂ -→ C 1 ∂ -→ C 0 ∂ -→ 0
Consider two subsets of C k (X): cycle-subset and boundarysubset, denoted as Z k (X) and B k (X) respectively. Let ker be the kernel and im be the image. By definition, we have:

Z k (X) = ker(∂ : C k → C k-1 ) B k (X) = im(∂ : C k+1 → C k )
Z k (X) includes cycles which are not boundaries while B k (X) includes only boundaries. A k-cycle u is said to be homologous with a k-cycle v if their difference is a k-boundary:

[u] ≡ [v] ⇐⇒ u -v ∈ B k (X). A simple computation shows that ∂ • ∂ = 0.
This result means that a boundary has no boundary. Thus, the k-homology of X is the quotient vector space:

H k (X) = Z k (X)\B k (X)
The dimension of H k (X) is called the k-th Betti number:

β k = dim H k = dim Z k -dim B k (1)
This number has an important meaning for coverage problems. The k-th Betti number counts the number of kdimensional holes in a simplicial complex. For example, β 0 counts the connected components while β 1 counts the coverage holes, etc.

Definition 1 ( Čech complex): Given (M, d) a metric space, ω a finite set of points in M and ǫ(ω) a sequence of real positive numbers, the Čech complex with parameter ǫ(ω) of ω, denoted Čǫ(ω) (ω) is the abstract simplicial complex whose k-simplices correspond to non-empty intersection of (k + 1) balls of radius ǫ(ω) centered at the (k + 1) distinct points of ω.

If we choose ǫ(ω) to be the cell's coverage range R, the Čech complex verifies the exact coverage of the system. In the Čech complex, each cell is represented by a vertex. A covered space between cells corresponds to a filled triangle, tetrahedron, etc. In contrast, a coverage hole between cells corresponds to an empty (or non-filled) triangle, rectangle, etc.

III. DISTRIBUTED COMPUTATION OF ČECH COMPLEX

A. System model

We consider a wireless network composed of N distinct cells. We assume that each cell uses isotropic propagation. The coverage of the i-th cell is modeled as:

c i (v i , r i ) = {x ∈ R 2 : x -v i ≤ r i },
where . is the Euclidean distance, the vertex v i represents the base station location and r i is coverage radius of the ith cell. Let U be the collection of cells, then U = {c i , i = 0, 1, . . . , (N -1)}. The Čech complex of U, Č(U), is defined as the Čech complex of the wireless network. In the Čech complex, each vertex, i.e. a 0-simplex, v i corresponds to the ith cell c i in the network. An edge, i.e. a 1-simplex, represents the connection, or the intersection, between two cells. Each k-simplex, where k ≥ 2, represents the common intersection of the coverage of together (k + 1) corresponding cells of this simplex. For example, in the Figure 3, the 2-simplex [v 2 , v 3 , v 6 ] means the overlap of coverage of cell c 2 , cell c 3 and cell c 6 . There is no coverage hole inside these cells. The higher dimensional simplex is, the higher overlap times is. The 3-simplex [v 0 , v 1 , v 2 , v 6 ] means the four corresponding cells: c 0 , c 1 , c 2 and c 6 , together, have a common intersection. In contrast, a chain of 1-simplices indicates a coverage hole inside corresponding cells of the chain. For example, the chain

[v 3 , v 4 ] + [v 4 , v 5 ] + [v 5 , v 6 ] + [v 6 , v 3 ]
shows a coverage hole inside four cells c 3 , c 4 , c 5 and c 6 . To analyze the network topology, we use characteristics of the homology of the Čech complex. As the definition of the Čech complex, each k-simplex represents the common intersection of together its (k + 1) corresponding cells. We denote S k the collection of all ksimplices of the complex. Each vertex, that is a 0-simplex, corresponds to a cell of the network. The collection of 0simplices, S 0 , is then obviously the list of corresponding vertices of cells.

c 0 c 1 c 2 c 3 c 6 c 5 c 4 v 0 v 1 v 2 v 3 v 6 v 5
S 0 = {v i | i = 0, 1, . . . , (N -1)}.
Each 1-simplex is an edge that connects two overlapping cells. In other words, it is a pair of two neighbor cells. If two vertices belong to the vertices set of a k-simplex, where k ≥ 2, then they form a 1-simplex. So, they are neighbors. Therefore, each cell needs to detect all its neighbors before computing all its simplices.

B. Neighbors detection

We assume that each cell c i can communicate with other cells over radio within a distance d i = 2r i . We assume that there are enough frequency slots for cells to communicate over radio without collision. Every cell is also connected by a backhaul network. At the initial state, each cell broadcasts a ping message with its position and its radius over radio channel. If a cell receives a ping message, it verifies if the cell that sent this ping message is a neighbor. If they are neighbors, then the cell that received the ping message sends a relationship confirmation together with its position and its radius to the cell that sent the ping message by using the backhaul network. After receiving the confirmation, the cell that sent the ping message adds the cell that sent the confirmation into its collection of neighbors. We assume that all the cells can reply the confirmation within a period t ack . After this period t ack , every cell detects its collection of neighbors. We denote the collection of neighbors of the cell c i as N i .

C. Distributed simplices computation

When the collection of neighbors is available, each cell computes its simplices by verifying the intersection among it and its neighbors. For the details about the intersection verification, see [START_REF] Le | Construction of the generalized cech complex[END_REF]. As each pair of neighbors forms a 1simplex, the collection of 1-simplices of each cell is easily found. To find all k-simplices, where k ≥ 2, each cell verifies if a group of it and its k neighbors has common intersection. If these cells have common intersection, then this group forms a k-simplex. However, neighborhood is a two-way relationship. Therefore, the verification of intersection could be duplicated by different cells that are neighbors. To avoid the redundant duplication, each cell verifies the intersection by following a right hand rule. This rule is that each cell verifies only with neighbors that are on its right hand side. If there is a neighbor which has the same horizontal coordinate, it verifies only this neighbor if it has higher vertical coordinate. If a simplex is found by a cell, this cell transmits this simplex to every cell that belongs to this simplex. As a result, every cell detects all its simplices. For example, in Figure 3, the cell c 2 verifies the intersection with only the cell c 3 and c 6 . It detects the simplex [v 2 , v 3 , v 6 ]. It receives its other simplices from the neighbor c 0 and c 1 . The Algorithm 1 reports the distributed computation of Čech complex for each cell. We denote the cell that is computing the simplices as c i . The highest dimension of simplices that are considered is dim max . The output of this algorithm, the collection of simplices of the cell c i is denoted as Či .

Fig. 3 :

 3 Fig. 3: Cells and their Čech representation.

Algorithm 1 Distributed computation of Čech complex

Require: c i the cell that is computing; Ensure: Či the collection of simplices of the cell c i ; broadcast {ping, v i , r i } over radio; S 0 = v i ; while count time < t ack do if {ping, v j , r j } is received from c j then if d(v i , v j ) < r i + r j then add v j to N i ; if x i < x j then add v j to S 0 ; add [v i , v j ] to S 1 ; end if if x i == x j and y i < y j then add v j to S 0 ; add [v i , v j ] to S 1 ; end if end if end if end while for k = 2 to dim max do for l = 2 to C k |S0| do ŝ = l-th combination of k vertices in S 0 ; ŝ = ŝ ∪ {v i }; ĉ = corresponding cells of ŝ; if intersection of all cells in ĉ is not empty then add ŝ to S k ; end if end for end for Či = {S 0 , S 1 , . . . , S k }; send Či to every corresponding cell of vertices in S 0 ;

The global Čech complex that represents the topology of the whole network is sometimes needed. There should be a master cell that controls the topology of the network. This global Čech complex can be easily built by integrating the simplices computed from every cell. Each cell sends its computed simplices that contain only the vertices satisfied the right hand rule. One more time, this rule is useful, it avoids sending the duplicated simplices.

IV. APPLICATIONS

A. Coverage hole detection

Coverage hole detection is one fundamental application in solving coverage problem. We introduce a distributed algorithm to detect all the edges that are on the cover (the boundary) of coverage holes. We assume that each cell has computed its simplices as in the Algorithm 1. If an edge (a 1-simplex) is not a cover edge of a coverage hole, then it musts connect with at least two neighbors that are on two different sides of this edge by filled triangles (a 2-simplex). Otherwise, the edge is a cover edge of a coverage hole. So, each cell cooperates with its neighbors to find all its cover edges. If a pair of neighbors can not find at least two cells that are their common neighbors and are on different sides of the edge formed by this pair, then the edge formed by this pair is a cover edge.

Algorithm 2 Distributed coverage hole detection

Require: c i (v i , r i ) the cell that is on a cover edge; Ensure: boundary cycle that contains v i ; The cells that are on cover edge can cooperate to find the cycle that is boundary of a coverage hole. If a cell c i is on a cover edge, it sends a boundary detection message {boundary detection, v i } to the other cell of this cover edge. If a cell, a receiver, received a boundary detection message from a sender, and it does not correspond to any vertex in the message, it adds its vertex to the end of this message, then it forwards the message to its neighbors that are on cover edges and are not the sender. If this receiver detects that its vertex is on the list of vertices in the message, then it remove all vertices that are above its vertex in the list. It announces the remained vertices as a boundary cycle that covers a coverage hole. We assume that all the message are sent and received in a period t 0 . After this period, each cell detects all its boundary cycles. The Algorithm 2 describes the details of the coverage hole detection.

B. Coverage optimization

Consider a wireless network, we maximize its coverage and minimize its total transmission power at the same time. Firstly, we ensure a maximal coverage for the network. Each cell is turned on and is set to work with the highest transmission power. At this initial state, the network has the largest coverage. However, many cells are hardly overlapped. The overlap region between cells causes the waste of transmission power due to interference. We can optimize the transmission power by minimizing the overlap region. However, the global coverage of the network should be conserved. In other words, the two Betti numbers β 0 and β 1 of the Čech complex of the network should be unmodified. The optimization problem can be written as:

where β * 0 and β * 1 are the Betti numbers of the Čech complex of the network at the initial state where every cell is working with the maximal transmission power.

In this section, we introduce a distributed algorithm to optimize the coverage as well as to save energy for the network. This algorithm is applied for each cell in the network. We assume that all the fenced cells and boundary cells are already known. Only the cells that are not fenced or boundary cells can try to reduce the coverage radius.

At the first step, each cell needs to search for its neighbors as well as its simplices by following the Algorithm 1. Once the neighbors set is established, each cell now starts its reduction process. On each cell, there is a timer which counts down to zero. The timer is set to a uniform random value from 1 to t max , where t max is the maximal value of the timer. When the timer of a cell is expired, this cell tries to do a reduction. If two cells that are neighbors try to reduce their radius at the same time, the coverage hole may not be detected due to the outdated information about the radius of each other. Therefore, before trying to reduce the radius, each cell sends a "pause" message to its neighbors. Then, the cell reduces its coverage radius and verifies the coverage. The radius reduction of one cell only makes topology change in the local region that is comprised of this cell and its neighbors. If there is a new coverage hole, it must be inside this local region. This means that if there is no new coverage hole after the radius reduction, the Betti numbers β 0 and β 1 of the Čech complex of this local region are unchanged. The verification of the network coverage can be reduced to the coverage verification in only this local region as in the Algorithm 3. If no hole appears, the cell confirms the reduction and sends the new value of coverage radius to its neighbors. It also sends the "continue" message to its neighbors to tell them that they can continue. If a cell received a "pause" message, it pauses its process and waits until the message "continue" is received. Then, it continues its process normally. There is a special case where two neighbor cells whose timers expire at the same time send the "pause" message to each other simultaneously. One of these two cells receives a "pause" from another before it sends "continue" message to other cells. This cell cancels the current reduction step and sets a new value for its timer and waits to retry.

If a cell tries to reduce its coverage radius and makes a coverage hole, it reverses its coverage radius to the previous value and stops the reduction process. This cell is set to irreducible.

The distributed energy saving algorithm applied for each cell is described in the Algorithm 4.

V. CONCLUSION

This paper introduces the distributed computation of the Čech complex aimed for wireless networks. Two distributed applications that are based on this computation of the Čech complex are also proposed. The first application detects the coverage hole and the second one optimizes the coverage for the network.