Nikos Kyriakou

Louis Winnubst

Martin Drobek

Sissi De Beer

Arian Nijmeijer

Alix Pizzoccaro-Zilamy

Controlled Nanoconfinement of Polyimide Networks in Mesoporous γ-Alumina Membranes for Molecular Separation of Organic Dyes

Keywords: Polyimide, nanoconfinement, mesoporous γ-alumina membranes, solvent resistant nanofiltration, organically modified membranes, surface-induced in-situ polymerization

Polyimide networks are key in the development of stable, resilient, and efficient membranes for separation applications under demanding conditions. To this aim, the controlled design of the network's nanostructure and its properties are needed. Yet, such control remains a challenge with the currently available synthesis methods. Herein, we present a simple nanofabrication approach that allows the controlled nanoconfinement, growth and covalent attachment of polyimide (PI) networks inside the mesopores of γ-alumina layers. The attachment of the PI network on the γ-alumina layer was initiated via different pre-functionalization steps who play a pivotal role in inducing the in-situ polymerization reaction at the pore entrance and/or at the inner pore surface. The nanoconfinement was found to be limited to maximal the 1.5 µm thick γ-alumina supporting layer and the resulting hybrid PI/ceramic membranes showed stable performance in a variety of solvents. These PI/ceramic membranes were found to be very efficient in the challenging separation of small organic dye molecules such as Rhodamine B (479 g mol -1 ) from toxic solvents such as dimethylformamide or dioxane. Therefore, this technique opens up possibilities for a multitude of separations. Moreover, the PI synthesis approach can be applied to other applications that rely on porosity and stability control as well, such as for advanced insulation and anticorrosion.

INTRODUCTION

Industry relies heavily on separation methods, from the purification of primary materials to the isolation of polymers, pharmaceuticals, and many other products. [START_REF] Sholl | Seven Chemical Separations to Change the World[END_REF] Conventional separation methods, such as distillation, are principally thermally driven. As a result, they can be costly, energy intensive and inefficient in separating thermally sensitive intermediates or products. [START_REF] Sholl | Seven Chemical Separations to Change the World[END_REF][START_REF] Marchetti | Molecular Separation with Organic Solvent Nanofiltration: A Critical Review[END_REF][START_REF] Vandezande | Solvent Resistant Nanofiltration: Separating on a Molecular Level[END_REF] As an alternative, nanofiltration membrane-based separation technologies are increasingly implemented in industry in either combination with distillation or extraction as hybrid processes or by totally replacing these conventional methods. The term "nanofiltration" (NF) refers to a pressure-driven membrane-based separation process in which particles and dissolved molecules smaller than 2 nm are rejected. By replacing the high energy-demanding thermally-driven separation methods (e.g. distillation) with low energydemanding pressure-driven NF membranes, one can get more energy-efficient and environmentally-friendly processes. [START_REF] Sholl | Seven Chemical Separations to Change the World[END_REF][START_REF] Marchetti | Molecular Separation with Organic Solvent Nanofiltration: A Critical Review[END_REF][START_REF] Vandezande | Solvent Resistant Nanofiltration: Separating on a Molecular Level[END_REF] Nevertheless, current membrane materials, which are often polymeric, are not always compatible with industrial streams, particularly mixtures of water and organic solvents, causing membrane failure due to degradation or dissolution of the material itself. Therefore, when designing a membrane for complex applications, one should consider not only the mixture of solutes but also the nature of the solvents involved.

Nowadays, membranes developed for such challenging applications are called solventtolerant nanofiltration membranes (STNF). [START_REF] Marchetti | Molecular Separation with Organic Solvent Nanofiltration: A Critical Review[END_REF][START_REF] Merlet | Hybrid Ceramic Membranes for Organic Solvent Nanofiltration: State-of-the-Art and Challenges[END_REF] The most common materials used to prepare STNF membranes are polymers, like polyamide-imides, [START_REF] Darvishmanesh | Mass Transport through Nanostructured Membranes: Towards a Predictive Tool[END_REF] polydimethylsiloxanes, [START_REF] Volkov | High Permeable PTMSP/PAN Composite Membranes for Solvent Nanofiltration[END_REF] etc. These polymeric membranes have generally shown high permeability and stable rejection with polar organic solvents such as alcohols or tetrahydrofuran. Yet, the performance of these membranes is unsatisfactory in presence of apolar solvents, mainly due to the degradation of the supporting layer. [START_REF] Merlet | Hybrid Ceramic Membranes for Organic Solvent Nanofiltration: State-of-the-Art and Challenges[END_REF] As a result, the permeability often becomes lower and the rejection unpredictable. Hence, such behavior hinders the implementation of this membrane technology in water/solvent streams. The combination of a chemical-inert porous ceramic support, functionalized with polyimides (PI), can overcome the instability issue of the support. PI are among the most resilient polymeric membrane-based materials used nowadays. PI are polymers characterized by a stable imide ring as a repetitive unit that exhibits good mechanical properties, chemical solvent resistance, heat resistance, and electrical properties. [START_REF] Baumgartner | Towards a General Understanding of Hydrothermal Polymerization of Polyimides[END_REF] Polyimides are prepared by the polycondensation reaction between (di)anhydrides and (di or tri) amines at temperatures between 180 and 300 °C. Due to their high thermal (> 400 °C) and chemical resistance, polyimides are extensively used as membrane materials for applications in gas separation, solvent exchange, and many others. [START_REF] Vanherck | Crosslinking Polyimides for Membrane Applications: A Review[END_REF] Commercial PI membranes such as Puramem® were found well adapted for specific STNF applications in solvents such as toluene and heptane, but only in operating conditions up to 50°C. [START_REF]Product overview -PuraMem® Membranes for organic solvent nanofiltration (OSN) -Evonik Industries[END_REF] To prevent dissolution and to increase the hydrophilicity (imide group to amic acid) of the PI membrane material in polar aprotic or chlorinated solvents, it has been reported that additional crosslinking steps are often needed. [START_REF] Vanherck | Crosslinking Polyimides for Membrane Applications: A Review[END_REF] Even though, the amic acid is more hydrophilic than imides, they nevertheless are less chemically stable. [START_REF] Cai | Characterization and Hydrolytic Stability of Poly (Amic Acid) Ammonium Salt[END_REF] Therefore, in membrane technology it is crucial to combine a high stable material, such as polyimides, which are also potentially processable to membranes. Therefore different approaches were explored to overcome the problems associated with material stability and processability. [START_REF] Morikawa | Preparation of a New Class of Polyimide-Silica Hybrid Films by Sol-Gel Process[END_REF] For example, Kuttiani Ali et al. [START_REF] Kuttiani Ali | Enhanced Removal of Aqueous Phenol with Polyimide Ultrafiltration Membranes Embedded with Deep Eutectic Solvent-Coated Nanosilica[END_REF] developed hydrophilic nanocomposite membranes for ultrafiltration by adding silica nanoparticles, pre-functionalized with deep-eutectic solvents, into a polyimide solution prior to casting. The membranes containing 2 wt.% of nanoparticles presented the best mechanical and phenol retention under a wide pH range. Moreover, recently, Wei et al. [START_REF] Wei | Combing Three-Dimensional Water Channels and Ultra-Thin Skin Layer Enable High Flux and Stability of Loose Polyimide/SiO2 Nanofiltration Membranes at Low Operating Pressure via One Step in-Situ Modification[END_REF] prepared an ultrathin polyimide/silica nanofiltration membrane by in-situ hydrolysis and condensation of tetraethoxysilane. The resulting membrane presented improved hydrophilicity, mechanical strength, and thermal stability compared to the pure PI-NF membrane. By using a similar approach, Qiang et al., [START_REF] Qiang | Bioinspired: A 3D Vertical Silicon Sponge-Inspired Construction of Organic-Inorganic Loose Mass Transfer Nanochannels for Enhancing Properties of Polyimide Nanofiltration Membranes[END_REF] formed a resistant STNF membrane. Despite the promising performances, the potential leaching of nanoparticles is not negligible and can lead to potential human and environmental exposure. [START_REF] Mueller | Nanofiltration and Nanostructured Membranes-Should They Be Considered Nanotechnology or Not?[END_REF] Thus far, many researchers have focused on introducing inorganic nanoparticles into the PI matrix. The opposite approach in which PI networks are confined in an inorganic matrix could also be employed.

Following this latter strategy, Isaacson et al. [START_REF] Isaacson | Fundamental Limits of Material Toughening in Confined Polymers[END_REF][START_REF] Isaacson | Synthesis of Polyimides in Molecular-Scale Confinement for Low-Density Hybrid Nanocomposites[END_REF] nanoconfined a PI network in a mesoporous tortuous organosilica matrix. The preparation procedure involved infiltrating polyamic acid oligomers into the porous matrix and subsequent cross-linking of the polymer units. As a result, the composite film/layer prepared showed enhanced resistance to fracture compared to the pristine mesoporous support, due to a so-called confinement-induced molecular bridging mechanism. Such confined polyimide systems could be used as a thermal barrier coating for high-temperature operations (at least up to 350 °C) and superior lightweight materials for aerospace applications. However, the possibility to use this confined PI network as a separation layer is unknown.

Studies have shown that the nanoconfinement of a crystallized polymer within nanoporous anodic aluminum oxide (AAO) templates is a suitable approach to prepare innovative systems for biosensing, optical and electrical-related applications. [START_REF] Liu | Confined Crystallization of Polymers within Nanopores[END_REF] When a polymer is confined within a micrometer thick rigid AAO template, comprising of vertically oriented large pores (10-100 nm) which are not tortuous, the crystallization behavior experiences dramatic changes as the pore size is reduced. This approach allows to modulate the polymer nanostructures for specific applications but is not suitable for practical separation applications in industry due to the limited surface area of the AAO supports when dealing with cubic meter of water and their fragility when exposed to harsh conditions. [START_REF] Pizzoccaro-Zilamy | New Generation of Mesoporous Silica Membranes Prepared by a Stöber-Solution Pore-Growth Approach[END_REF] In contrast to previous studies done on AAO support or mesoporous organosilica layer, we looked at defined and rigid mesoporous ceramic membranes with relatively low tortuosity. Alumina (γ, or α phase) membranes are commercially predominant in the market and are available in the shape of discs and tubes. [START_REF]Home -inopor -the cutting edge of nano-filtration[END_REF][START_REF]Pervaporation Membranes by Pervatech[END_REF] Compared to AAO templates, mesoporous γ-Al2O3 membranes are supported on millimeter thick α-alumina supports, and commercialized in the form of modules, making them suitable for real separations under demanding conditions (high pressure and temperature). [START_REF] Abedini | Strength and CO2 Separation Characteristics of α-Alumina Supported γ-Alumina Thin Film Membrane[END_REF][START_REF] Uhlhorn | Synthesis of Ceramic Membranes[END_REF] However, due to their relatively large pores (~5 nm), they are not selective in the nanofiltration range. Nevertheless, the γ-alumina surface is rich in free hydroxyl groups that can be used to modify the pore entrance and inner pore surface by covalently attaching molecules, oligomers, and polymer brushes to prepare hybrid membranes. [START_REF] Merlet | Hybrid Ceramic Membranes for Organic Solvent Nanofiltration: State-of-the-Art and Challenges[END_REF][START_REF] Kyriakou | New Method toward a Robust Covalently Attached Cross-Linked Nanofiltration Membrane[END_REF][START_REF] Kyriakou | Hydrolytic Stability of PEG-Grafted γ-Alumina Membranes: Alkoxysilane vs Phosphonic Acid Linking Groups[END_REF][START_REF] Tanardi | PDMS Grafting of Mesoporous γ-Alumina Membranes for Nanofiltration of Organic Solvents[END_REF][START_REF] Merlet | Comparing the Performance of Organic Solvent Nanofiltration Membranes in Non-Polar Solvents[END_REF] In fact, it has been shown that grafting-to and grafting-from reactions can be applied to modify dense substrates [START_REF] Sun | Tribological Properties of Chemically Bonded Polyimide Films on Silicon with Polyglycidyl Methacrylate Brush as Adhesive Layer[END_REF] but also the pore entrance and inner pore surface of γ-alumina mesoporous layers. [START_REF] Merlet | Hybrid Ceramic Membranes for Organic Solvent Nanofiltration: State-of-the-Art and Challenges[END_REF][START_REF] Kyriakou | New Method toward a Robust Covalently Attached Cross-Linked Nanofiltration Membrane[END_REF][START_REF] Kyriakou | Hydrolytic Stability of PEG-Grafted γ-Alumina Membranes: Alkoxysilane vs Phosphonic Acid Linking Groups[END_REF][START_REF] Tanardi | PDMS Grafting of Mesoporous γ-Alumina Membranes for Nanofiltration of Organic Solvents[END_REF][START_REF] Merlet | Comparing the Performance of Organic Solvent Nanofiltration Membranes in Non-Polar Solvents[END_REF] Sun et al. [START_REF] Sun | Tribological Properties of Chemically Bonded Polyimide Films on Silicon with Polyglycidyl Methacrylate Brush as Adhesive Layer[END_REF] used the grafting-from method to grow polyglycidyl methacrylate brushes from the surface (-induced) of silicon wafers. These brushes were utilized as an adhesive interlayer for chemically attaching a polyimide film on silicon wafers. As a result, improved friction and wear resistance was observed, compared to the polyimide films on bare silicon wafers. However, the possibility of simultaneous pore confinement and covalent attachment of a crosslinked PI network onto the ceramic support has not been demonstrated yet. Furthermore, the growth of a crystalline polymer in one step from the surface of a ceramic support as well as the applicability of such material under membrane conditions is still not shown in literature.

In this work, we have initiated for the first time the in-situ polymerization reaction of a PI network directly from an inorganic surface, and controlled the nanoconfinement inside rigid and well-defined, tortuous γ-alumina mesoporous γ-alumina mesoporous layers as indicated in Scheme 1. Our strategy is to use two different precursors that promote the confinement of the PI network in the ceramic support. The first precursor, bearing an amino-functional group, is located at the top surface and pore entrance, while the second one, consisting of an anhydridefunctional group, is present on both the surface and within the mesopores of the γ-alumina layer. Indeed, the above functional groups induce a surface polyimidization reaction that controls the location of the network formation either at the pore entrance or inside the γ-alumina layer.

Furthermore, we show that by increasing the reaction time from 1 to 5 days, the membrane performance was significantly improved due to increase in concentration of the polyimide network into/on the mesoporous layer. To demonstrate the successful growth and confinement of the polymer inside the mesoporous layer, a combination of different surface and pore characterization techniques were employed. In addition, the nanoconfined PI-based membranes were tested in different model mixtures where its potential as solvent-resistant nanofiltration (NF) membrane was demonstrated. The concept described in this work, illustrates how a crosslinked polyimide can grow in a nanoconfined space, such as the tortuous but defined mesopores of our alumina membranes. This can be achieved by simply controlling the grafting of the initiator for the surface-induced in-situ polymerization from the surface of the alumina support. This approach can be expanded in other fields where controlled nanoconfinement of a crosslinked crystalline polymer is desired for different applications. conducted with dianhydride (PMDA, ) molecules. Via this route, the subsequent PI network formation is also extended (favored) inside the γ-alumina mesopores.

RESULT AND DISCUSSION

Synthesis and characterization of the pre-functionalized γ-alumina layer and polyimide membranes. Our strategy to pre-functionalize supports and subsequently form the polyimide (PI)-based membranes is presented in Scheme 1. The PI nanoconfined membranes were prepared by the pre-functionalization of the mesoporous γ-alumina layer with 3aminopropyl trimethoxysilane (APTES). A reliable vapor phase grafting procedure has been developed by our research group to covalently attach APTES molecules at the pore surface of the γ-alumina layer. Indeed, carefully selecting the grafting conditions and a suitable pore filling agent can lead to a homogeneous monolayer of APTES molecules without the problem of homocondensation reactions occurring between the alkoxysilane linking group. [START_REF] Kyriakou | New Method toward a Robust Covalently Attached Cross-Linked Nanofiltration Membrane[END_REF] The primary amine group of APTES can react via a condensation reaction with the dianhydride precursor of the PI network during the subsequent polyimidization reaction between pyromellitic dianhydride (PMDA) and melamine (MA). This reaction should lead to the covalent attachment of the polyimide network exclusively on the top surface and pore entrance of the mesopores γ-alumina-layer (Scheme 1, route A). Here, we employed a pore blocking agent allowing us to graft only the top surface and pore entrance of the γ-alumina layer. Hence, we assume that the rapid PI network formation from the functionalized pore entrance will limit the diffusion of monomers as well as any PI oligomer units formed in the bulk solution into the pristine pores. In this way, a PI concentration gradient is induced along the mesoporous layer with the highest concentration close to the pore entrance. To allow for the PI network formation to occur also inside the mesopores of the support, samples were modified in solution with only PMDA before the in-situ polycondensation (Scheme 1, route B). Here, PMDA can react not only with the amino group of the grafted APTES molecules but also with free hydroxyl groups at the inner pore surface of γ-alumina layer. [START_REF] Amirilargani | Molecular Separation Using Poly (Styrene-Co-Maleic Anhydride) Grafted to γ-Alumina: Surface versus Pore Modification[END_REF] Thus, the PI network formation takes place uniformly from the whole surface, including pore entrance and pore surface, of the mesoporous γ-alumina layer.

Fourier transform infrared (FTIR) analysis was employed to demonstrate the prefunctionalization of the mesoporous γ-alumina layer. The spectra of the pristine γ-alumina layer and the layers pre-functionalized respectively with APTES (sample A) and APTES+PMDA (sample B) are shown in Figure 1a. The FTIR spectrum of the pristine mesoporous layer shows a broad band centered at 3420 cm -1 which can be attributed to the stretching vibration of adsorbed water and surface hydroxyl groups. [START_REF] Coates | Interpretation of Infrared Spectra, A Practical Approach[END_REF][START_REF] Merlet | Growing to Shrink: Nano-Tunable Polystyrene Brushes inside 5 Nm Mesopores[END_REF] Functionalization of the γ-alumina layer with APTES results in the appearance of primarily a broad band between 1180 and 970 cm -1 which is associated with the formation of the Al-O-Si bond and confirms the grafting of the APTES at the top surface and pore entrance. This finding is further confirmed by the presence of the asymmetric and symmetric stretching vibration bands at 2927 and 2886 cm -1 which can be attributed to the alkyl groups (-CH2-) of the grafted APTES molecules. In addition, the sharp vibration band at 2974 cm -1 (Figure S4) attributed to the linking function (CH3-CH2OSi) is not present in the spectrum of sample A, suggesting complete hydrolysis of the functional group during the grafting reaction and thus confirms the formation of the desired Si-O-Al bond again. [START_REF] Kyriakou | New Method toward a Robust Covalently Attached Cross-Linked Nanofiltration Membrane[END_REF] In comparison with sample A, the interpretation of the FTIR spectrum of sample B is more difficult due to the number of absorption bands detected. Instead of attributing each vibration band, only the most important ones are discussed here. The formation of the imide functional group and attachment of PMDA is confirmed first by the vibration bands at 1787 and 1727 cm -1 , ascribed to the C=O bond, and the band at 1365 cm -1 , related to C-N-C bond. In addition, the anhydride group is also apparent at 1856 cm -1 which suggests either the partial reaction of PMDA with the amino-functionalized support (A) and/or the presence of unreacted and physically adsorbed PMDA on the ceramic support. The band at 2460 cm -1 attributed to the carboxylate groups (COO -) [START_REF] Tena | Thermally Treated Copoly(Ether-Imide)s Made from Bpda and Alifatic plus Aromatic Diamines. GAS Separation Properties with Different Aromatic Diamimes[END_REF] in the sample is due to the formation of amic acid group or due to PMDA grafting at the pore surface. [START_REF] Amirilargani | Molecular Separation Using Poly (Styrene-Co-Maleic Anhydride) Grafted to γ-Alumina: Surface versus Pore Modification[END_REF][START_REF] Tena | Thermally Treated Copoly(Ether-Imide)s Made from Bpda and Alifatic plus Aromatic Diamines. GAS Separation Properties with Different Aromatic Diamimes[END_REF] Cyclohexane permporometry was used to study the effect of pre-functionalization on the pore size distribution of the support. This dynamic characterization technique allows measuring the pore diameter of active pores present in the pre-functionalized γ-alumina layer.

The stepwise analysis will enable one to follow the change in pore diameter starting from the pristine γ-alumina layer and moving towards the functionalized samples (A and B in Figure 1b). The pristine γ-alumina layer exhibits a mean pore diameter of ̴ 5.5 nm and the support pre-functionalized with APTES (sample A) shows no pore size diminution. This last result differs from published reports where a pore shrinkage of 0.5 nm was observed, where no glycerol or other pore-blocking agents were used. [START_REF] Pinheiro | Development of a PDMS-Grafted Alumina Membrane and Its Evaluation as Solvent Resistant Nanofiltration Membrane[END_REF] This means that glycerol, used as the pore blocking agent, has allowed us to control the grafting reaction and to functionalize only the top surface of the γ-alumina layer. However, subsequent functionalization with PMDA (sample B) resulted in a significant reduction of the pore diameter. Compared to pristine γalumina and sample A, the oxygen permeation curve of sample B presented in Figure 1b suggests that the pore opening occurs at low cyclohexane relative pressures during the desorption step (≈ 0.1 instead of 0.4 P/P0). Considering the data acquired via cyclohexane permporometry, we cannot determine the exact pore diameter since the Kelvin equation is not valid due to the very unclear transition point obtained for this sample. Nevertheless, knowing the limit of the measurement, which corresponds to the molecular diameter of cyclohexane ( ̴ 0.5 nm), one can assume that the pore diameter of the sample B must be lower than 1 nm.

Overall, the cyclohexane permporometry results indicate the presence of PMDA in the pores of the γ-alumina layer, which can be physically or chemically adsorbed at the pore entrance and inner pore surface. However, the pore diameter of B is below the molecular size of cyclohexane (condensable liquid). Thus, pore size distribution cannot be estimated in this way.

As explained before, the attachment, growth and molecular confinement of the PI network in the γ-alumina layer was performed via two different approaches; firstly, by the direct formation of the PI network from the pre-functionalized surface of sample A, where the functional groups were at the top surface and pore entrance of the γ-alumina layer and secondly, from sample B, where the functional groups were located at the top and pore surface of the γ-alumina layer. FTIR analysis was performed on the nanoconfined PI membranes to assess the spectroscopic characteristics of the network formed after in-situ polymerization. Figure 2a displays the spectra of the four PI nanoconfined membrane samples (A-1/5 and B-1/5). All samples exhibit similar spectroscopic characteristics with minor differences in the intensity of certain bands for the two different reaction times (1 and 5 days)

and both reaction routes. The two bands at ̴ 1780 and ̴ 1720 cm -1 are ascribed to the C=O bond of the imide and are more intense with longer reaction times for both routes. The bands at 1565 and 1453 cm -1 are attributed to the stretching vibration of the triazine ring and appear in all membranes. [START_REF] Mircescu | SERS and DFT Study on Melamine[END_REF] The band at 1660 cm -1 could indicate the presence of amic acid on the γalumina layer. However, no other absorption bands confirm the presence of amide. To get a better insight into this observation, the powders (I-1 and I-5) collected from the reaction mixture were used to indirectly gain information on the nature of the material, confined in the mesopores (Figure S13 andS14). The FTIR spectra of the powders, formed in the bulk solution during in-situ polymerization of membrane samples, clearly show the formation of an amino-terminated imide network. Hence, in-situ polymerization seems to promote imide formation for both 1-and 5-days reaction time, without any indications of amic acid presence.

Finally, the band between 1390 and 1360 cm -1 is ascribed to the absorption of the C-N-C group. [START_REF] Li | Synthesis of Three Dimensional Extended Conjugated Polyimide and Application as Sodium-Ion Battery Anode[END_REF][START_REF] Wang | Porous Polyimide Framework: A Novel Versatile Adsorbent for Highly Efficient Removals of Azo Dye and Antibiotic[END_REF][START_REF] Sadhasivam | Synthesis and Characterization of Optically Active Polyimides and Their Octa(Aminophenyl)Silsesquioxane Nanocomposites[END_REF] Thus, leading to the conclusion that the FTIR analysis strongly suggests the presence of a polyimide network in the γ-alumina layer.

The high crystallinity of aromatic polyimides has been demonstrated in the literature by powder X-ray diffraction (XRD) analysis. [START_REF] Baumgartner | Towards a General Understanding of Hydrothermal Polymerization of Polyimides[END_REF] In our work, no diffraction peaks corresponding to crystalline aromatic polyimides could be detected in the membrane samples (Figure S15). The XRD pattern obtained revealed the presence of the highly intense diffraction peaks of the αalumina macroporous support which can be explained by X-rays' penetration depth (being more than 3 µm in the XRD configuration used). The absence of diffraction peaks correlated with the γ-alumina phase where the polyimide network is confined can be explained by the nanosized nature of this layer. Kim et al. [START_REF] Kim | High Loading of Nanostructured Ceramics in Polymer Composite Thick Films by Aerosol Deposition[END_REF] described a decrease in the diffraction peaks of the α-alumina phase when different polyimide/α-alumina film composites were prepared using an amorphous polymer. Interestingly, the diffraction peaks of the α-alumina phase were identical with our pristine support, thus suggesting either small amounts or even absence of the PI network in the α-alumina pores. To shed more light into the confined network's nature, XRD analysis was conducted on the powder extracted from the bulk solution at the end of the synthesis of the membranes. The powder XRD patterns between 5 -50° 2θ are provided in to a relatively amorphous material (two broad peaks between 10 and 50°), whereas at 200 °C or above [START_REF] Duan | Semiconducting Crystalline Two-Dimensional Polyimide Nanosheets with Superior Sodium Storage Properties[END_REF] a semi-crystalline structure was observed. Thus, the powder XRD results provided here confirm that the organization of the material depends both on the temperature of the polycondensation reaction and the reaction times. However, when the PI powder was treated at 300 °C for several hours, a new diffraction peak at 44° was observed, which is possibly related to degradation by-products. Thermal treatment of the PI powders at 400 °C, resulted in almost complete loss of crystallinity and increase in intensity of the diffraction peak at 44°

(Figure S17). The observed thermal degradation evidenced by powder XRD, is also corroborated by the thermogravimetric analysis (TGA) provided in Figure S18. A small weight loss for both I-1 and I-5, 2.2% and 1.4%, respectively, occurred upon heating from room temperature to 300°C. However, above 300 °C a significant weight loss occurs, particularly for sample I-1. Thus, we can assume that the PI network remains stable at temperatures below 300 °C, which is ideal for membrane applications.

It must be noted that the confinement of a polyimide network in the γ-alumina layer should lead to an enhancement of the physicochemical stability of the polyimide network as shown by Isaacson et al. [START_REF] Isaacson | Synthesis of Polyimides in Molecular-Scale Confinement for Low-Density Hybrid Nanocomposites[END_REF] Based on the powders' crystallinity and morphology, one can assume that the PI membrane samples exhibit similar structural characteristics as the polyimide powders.

If our assumptions are confirmed, it would mean that by simply varying the reaction time, we can engineer the membrane's micropores and thus enhance the membrane separation performance. Therefore, it is crucial to look closer at the PI network inside the support and well describe the nanoconfinement effect.

PI network nanoconfinement characteristics.

A series of analytical techniques were employed to investigate the influence of the supports' pre-functionalization on the extend of the PI nanoconfinement. First, water contact angle analysis was done on the PI nanoconfined samples and the pristine support to evaluate (indirectly) the polymerization effect on the support surface properties. The results are given in Figure 2b. Compared to the pristine layer, which presents a water contact angle of 14° (disappears in 6 -7 seconds) characterizing porous hydrophilic surface, the PI nanoconfined samples show an increased water contact angle (40 -65°). This observation suggests that the membrane surface is still hydrophilic (water contact angles < 90°) but certainly less than the pristine support. Interestingly, different water contact angles were obtained for the samples prepared via route A and B, respectively 63° (± 16°) and 54° (± 5°) for the same reaction time (1 day). Nevertheless, with the increase of the reaction time, a rise of 11° was measured for the samples prepared via route B, while a decrease of 19° was measured for the samples made via route A. As for porous hydrophilic surfaces, accurate estimation of a descending water contact angle is challenging. Still, the difference observed between the two routes can be inherent to the pre-functionalization step.

Further investigation and comparison of the PI-nanoconfined membrane surface morphology with that of the γ-alumina layer was conducted by atomic force microscopy (AFM) analysis, and the results are shown in Figure 2c. From the average roughness profiles, it is apparent that the surface morphology of the PI nanoconfined membranes changes with increasing reaction time. Compared with the pristine γ-alumina layer, polymerization for 1day, for both routes, does not seem to affect the surface roughness ( ̴ 3 nm). Samples reacted for 5 days, show only slight increase in surface roughness ( ̴ 4 nm) with the A-5 samples exhibiting larger differences in height on the surface than for B-5 ( ̴ 3 nm) as shown in Figure 2c. The empirical information gained from water contact angle and AFM analyses postulate a fundamental difference between the two routes, which becomes more prevalent after longer reaction times. The influence of reaction conditions on the pore diameter of the PI nanoconfined membranes was investigated by means of cyclohexane permporometry. The results are provided in Figure 2d. During the analysis, no oxygen permeation was measured at low cyclohexane partial pressures (< 0.55), which indicates the presence of micropores (pore diameter < 1.5 nm) or even of a dense sample. This suggests a pore diameter shrinkage of more than 4 nm. Indeed, the permporometry analysis demonstrates that the presence of the PI network affects the pore size of the mesoporous γ-alumina layer.

All the results described above showed that the polymer network changes the morphology of the surface only in a subtle manner. However, the pore sizes are significantly affected, indicating either an ultrathin top layer or a confined polymer in the mesoporous layer. To better understand this finding and to observe the PI network on the γ-alumina layer directly, we proceeded with HR-SEM analyses of the studied membrane samples. The top surface and cross-section High-Resolution SEM pictures of the PI nanoconfined membranes are given in Figure 3a, together with photographs of the membrane samples.

Compared to the pristine γ-alumina layer, which is naturally white (Figure S21), the PI nanoconfined membrane samples appear to be substantially covered by the polymeric network as denoted by the brownish coloration, typical for a polyimide material. Comparison of the top-surface micrographs of membrane samples prepared by the two fabrication routes (A and B) does not show any significant differences at first glance. One day of reaction leads to the formation of small particles with a sheet-like structure visible on both A-1 and B-1 samples.

Increasing the reaction time to 5 days leads to the disappearance of the sheet structure, suggesting the formation of a thin homogeneous layer. From the SEM analysis of PI powders (Figure S20) we observed that the 5-day long reaction yields a clear platelet-like structure, whereas the 1-day reaction results in a mixture of aggregates and crystallites. From the HR-SEM analysis of the top surface of both A-1 and B-1 samples, one can observe similar sheetlike structures. This indicates that the PI networks that are growing on the surface of the γalumina layer and those growing in the bulk solution exhibit similar characteristics. Hence, we expect that the PI network growth, induced from the ceramic surface, will have a similar morphology with smaller particles, mainly when infiltrated in the pores. Finally, it is expected that such platelet-like particles should ensure good coverage of the support surface, as seen on the micrographs of both A-5 and B-5 samples.

In comparison, the HR-SEM cross-sectional analysis of the membranes show a clear difference between the two synthesis routes after 1 day of reaction. For the A-1 sample, the PI network seems to be located at the γ-alumina layer top surface. In contrast, for the B-1 sample, infiltration of the PI network in the γ-alumina layer could be observed. The samples after 5 days of reaction time, respectively A-5 and B-5, also present extended infiltration of the PI network in the γ-alumina layer. This finding thus strongly indicates that longer reaction times of polymerization on pre-functionalized γ-alumina layers promoted the nanoconfinement of the PI network in the 5 nm pores of the γ-alumina layer. Evidently, the difference between the membranes A-1 and B-1 suggests that the choice of prefunctionalization can affect the extent of the PI nanoconfinement in the γ-alumina layer.

In complement, energy-dispersive X-ray spectroscopy (EDS) analysis can offer qualitative elemental information over a membrane's cross-section (Figure 3b). By measuring the ratio of carbon over aluminum along the cross-section of the different membrane samples, one can define the influence of the preparation route, and indirectly the pre-functionalization steps on the nanoconfinement of the PI network. Overall, the EDS analysis reveals that organic (polymeric) material resides in the γ-alumina layer. In route A, the network accumulates near the pore entrance, whereas in route B (samples B-1 and B-5), a spread distribution of the PI network inside the γ-alumina layer was observed. These results thus clearly indicate a link between the pre-functionalization step and the nanoconfinement of PI network inside the γalumina layer. On the hand, the reaction time merely affects the concentration of the polymeric network. In conclusion, by combining the knowledge gained from HR-SEM and EDS, we can assure that the functionalization of the support promotes the growth of the PI network during polymerization. Based on these results, a membrane formation mechanism was drawn (Scheme 1). The membranes prepared via route A, thanks to the presence of the amino group (APTES) at the top surface of the support, exhibit a higher concentration of PI near the pore entrance. Alternatively, with route B, a homogeneous polymer distribution is observed throughout the γ-alumina layer, which is attributed to the functionalization of the inner pore surface with anhydride functional groups (PMDA). These differences in membrane architecture can significantly affect their membrane performance due to an increase in the thickness of the separating layer, as schematically shown in Scheme 1. Additionally, longer reaction times lead to higher concentrations of polymer inside the γ-alumina layer, which could potentially promote formation of smaller pore diameters, and thus better separation performance. To confirm our interpretation from the HR-SEM and EDS analysis, a series of membrane separation tests have been performed with model water solutions described hereafter.

PI nanoconfined membranes performance. The PI nanoconfined membranes were tested first in aqueous solutions of Brilliant Yellow (BY, 625 g mol -1 ) or Rhodamine B (RB, 479 g mol -1 ) and compared with the pristine γ-alumina layer. The retention and water permeability results are summarized in Figure 4. All four PI nanoconfined membranes show retentions above 90% of BY (627 g mol -1 ) in water, which is a significant increase compared to the pristine γ-alumina layer (76%). However, with Rhodamine B (RB, 479 g mol -1 ), the retentions for the A-1 and B-1 samples were between 70-80%. With increasing the polyimidization reaction time, the membranes (A-5 and B-5) show RB retentions well over 90%. This can be attributed to increasing PI concentration in the pores of the γ-alumina layer.

Compared with the pristine γ-alumina layer (14% retention for RB), the separation performance of the PI nanoconfined membranes thus displays a significant improvement.

These results also suggest that, despite this increase in concentration of the polymeric network in the tortuous mesopores, there are still many open pores that are either smaller than the Rhodamine B molecule (sieving effect) or are small and charged (Donnan effect) and thus have a direct influence on the separation performance of the membranes. The water permeability results, on the other hand, suggest a clear difference between route A and B, particularly for samples treated for 5 days. It is evidenced from the increase in RB retention for sample A-5, compared to A-1 that for route A the polymer amount in γ-alumina pores is increasing and hence the pore is shrunk significantly. However, since the water permeability for A-5 remains comparable to that of A-1, this can be regarded as an indication that the polymer concentration is only increasing at the pore entrance, leading to thin selective barriers. For sample B-5, the RB retention is also increasing, compared to B-1, however here decreasing permeability is suggesting a thicker selective barrier. These preliminary results

indicate that the polymer concentration is increasing with increasing reaction time (A-5 and B-5), but the location where the polymer concentration is increased depends on the prefunctionalization of the support. This means that for route A the polymer grows only at the top surface and pore entrance, positively influencing the retention of the membrane but leaving, evidently, unaffected the water permeability. On the other hand, for route B the polymer grows in the whole or part of the γ-alumina layer, as also indicated by EDS analysis.

Hence, from this series of water permeation tests accompanied with permporometry, HR-SEM and EDS analyses we have clearly evidenced that in the synthesis of PI nanoconfined membranes, the pre-functionalization step controls the extent of polymerization inside the mesoporous support.

Sample A-5 was also tested for 5 days in RB/water solution to assess the stability of the PI nanoconfined ceramic membranes. The results are provided in Figure S27. As shown, the RB retention remains stable at approximately 98-99%. Furthermore, the water flux increases slightly after the first day (from 8 to 10 L m -2 h -1 ), but remains relatively stable in the following 4 days. Therefore, this preliminary result shows that the method used to prepare the PI nanoconfined membranes results in relatively stable membranes.

The two membranes showing the best rejection, the B-5 and A-5 samples, were subsequently tested in different solvents containing RB. The results for the solvent permeability and RB retention in solvents are given in Figure 4c and b, accordingly. Three different organic solvents were selected based on their polarity; ethanol, DMF and 1,4dioxane with polarity values of 0.654, 0.386, and 0.164 respectively. Indeed, by testing the membranes in different liquid media, we gain more insight into the membrane layer's properties. As shown in Figure 4c, the membranes perform well in all solvents. Only in ethanol, the retention performance of the PI nanoconfined membranes is lower than 90% (79% for B-5 and 83% for A-5). The results can be attributed to the nature and solubility of RB in different solvents. According to Hinckley et al., [START_REF] Hinckley | Solvatochromism and Thermochromism of Rhodamine Solutions[END_REF] RB in a solution can be present in two forms, the lactone (L) and the zwitterionic (Z) form, which are in equilibrium, and the most dominant form depends on the solvent. Hinckley et al. [START_REF] Hinckley | Solvatochromism and Thermochromism of Rhodamine Solutions[END_REF] showed that the ratio zwitterionic:lactone (charged:neutral) in water (Z:L = 4.4) and formamide (Z:L = 7.67) is higher than in ethanol (Z:L = 2.4). Since ethanol favors the neutral form of RB, compared to water and DMF, we expect that the zwitterion rejection might be related to ionic repulsions between the membrane and the solute. Thus, leading to slightly lower rejections in ethanol.

In 1,4-dioxane, probably a different mechanism is at play. The solubility of RB in 1,4dioxane is significantly lower than in ethanol, DMF or water. In this regard, the solvent is expected to have a much higher preference to the polymer (PI) than to the solute, leading to high retentions observed experimentally for the studied systems. These tests suggest that the PI network confined in the γ-alumina layer could be slightly charged since RB retentions are the best in water and DMF, where the zwitterionic form is dominant. The charge could originate either form the presence of primary terminal amines, which is also suggested from the FTIR analysis, and the monomer ratio used during membrane preparation. As such, we propose that by adjusting in-situ polymerization conditions, such as monomer ratio, one can tune the final membrane properties.

To conclude, a comparison between different membranes from literature with the A-5 membrane samples is shown in Figure 5. Evidently, A-5 is a potentially interesting membrane with permeabilities in different solvents between 1 and 1.6 L m -2 h -1 bar -1 and retentions in the NF range (469 g mol -1 ). However, since the water permeability of the pristine γ-alumina supported on α-alumina (4 -5 L m -2 h -1 bar -1 ) is relatively low, we expect that utilizing supports with thinner intermediate layer, for example of nanometer thickness, as well as an αalumina support with larger pore diameters and higher porosities can potentially improve the membrane performance even further.

CONCLUSIONS

In this work, PI networks were confined in mesoporous inorganic layers by top or innerpore surface-induced polyimidization. By pre-functionalizing of the top surface of the support, the polymeric network was confined at the top and entrance of the pores. Inner pore surface functionalization led to a homogeneous polymer distribution throughout the functionalized ceramic layer. The two monomers employed, MA and PMDA, allowed for forming a cross-linked and thus chemically resistant PI network inside the top layer of the ceramic support. By tuning the reaction time, we showed that the nanoconfinement of the polymer could also be effectively tuned. All these membranes were scrutinized through a series of characterization techniques, including SEM, FTIR, and pore diameter measurements, to demonstrate the influence of the applied methodology on their structure and final physicochemical properties. The as-prepared PI nanoconfined membranes showed attractive separation performance with good retentions of Rhodamine B (479 g mol -1 ) in water and different organic solvents. At this moment, we do not foresee how to measure the resulting molecular weight that we expect to be very small, but this will be the object of further research using Molecular Dynamic simulation.

The principal asset of the work presented here relies on a demonstration of a method, allowing to control the polymerization of a cross-linked polymer inside the confined space of the 5.5 nm pores of the γ-alumina layer for the preparation of hybrid NF membranes. This work is a forerunner for confining polymers in nanoporous substrates and regulating the location of the polymer growth. We assume that this method can be used to grow polymers with even higher chemical resistance, such as polybenzimidazoles by using a similar preparation method as for polyimides. Furthermore, this method can be advantageously used as a tool in other fields to confine cross-linked polymers with low processability inside rigid supports to form for example low density, high strength, and thermal conductive nanocomposites for microelectronic insulation [START_REF] Zhou | Creation of a Multilayer Aluminum Coating Structure Nanoparticle Polyimide Filler for Electronic Applications[END_REF][START_REF] Zhou | Excellent Thermal Conductivity and Dielectric Properties of Polyimide Composites Filled with Silica Coated Self-Passivated Aluminum Fibers and Nanoparticles[END_REF][START_REF] Li | Polyimide Aerogel Fibers with Superior Flame Resistance, Strength, Hydrophobicity, and Flexibility Made via a Universal Sol-Gel Confined Transition Strategy[END_REF] or for anticorrosion coatings. [START_REF] Sezer Hicyilmaz | Applications of Polyimide Coatings: A Review[END_REF] (anhydrous) and transferred into a reaction vessel with the glycerol filled γ-alumina layer suspended above the solution. The solution was heated to 105 °C for 3h in a sealed vessel.

After grafting, the functionalized porous support was washed with 20 mL anisole for 1h and 20 ml of water for 20 min under sonication and dried overnight at 50°C under vacuum.

Amino-functionalized supports, obtained at this stage, were denoted as A. where cp and cf are the permeate and feed solute concentrations, respectively. Retention samples were obtained at recoveries between 35 and 50%. The dye adsorption during retention measurements was calculated with equation:

Pre

𝑀 𝐴𝑑𝑠 = 𝑀 𝑓 -𝑀 𝑅 + 𝑀 𝑝 (2)
where, MAds is the amount of dye adsorbed on each membrane, Mf is the total amount of dye used at the beginning of each separation test (feed solution), MR is the amount of dye in the retentate and Mp is the amount of dye in the permeate. In all cases, the dye concentration of BY and RB was increased in the retentate to account for limited adsorption of 2 -3% for the PI nanoconfined ceramic membranes. Solute concentrations of BY and RB were calculated from Perkin-Elmer λ12 UV-Vis spectrophotometer results at the characteristic

Scheme 1 .

 1 Scheme 1. Schematic illustration of the two fabrication routes, A and B, used for the

Figure 1 :

 1 Figure 1: (a) FTIR spectra of the pre-functionalized samples (A (red) and B (orange)) and

Figure

  FigureS16for the powder samples obtained after 1 or 5 days of reaction times (denoted as I-1

Figure 2 :

 2 Figure 2: (a) FTIR analysis of the PI nanoconfined samples. The complete spectra between 4000 and 400 cm -1 are provided in SI. (b) The water contact angle of the PI nanoconfined

Figure 3 :

 3 Figure 3: (a) Images of the PI nanoconfined membranes (left) accompanied with HR-SEM

Figure 4 :

 4 Figure 4: (a) Water permeability of the PI nanoconfined membranes and the pristine layer (γ-

Figure 5 :

 5 Figure 5: Comparison of the best performing membrane from this work (A-5) with other

  -functionalization of the γ-alumina layer's inner pore surface. Under inert atmosphere in a 50 mL reaction vessel charged with 40 mg (0.18 mmol) of PMDA, 20 mL of mesitylene were added stirred for 1-2 min. A sample was then immersed in the solution and the mixture was heated to 160 °C over-night. The mixture was cooled to room temperature and the sample was washed with n-methyl-2-pyrrolidone (NMP) and acetone in a sonicated bath for 30 min. Finally, the sample (denoted as B) was dried in a vacuum oven at 50 °C over-night.Pore surface-induced polyimidization reaction. In a 50 mL reaction vessel 530 mg (2.43 mmol) of PMDA and 260 mg (2.06 mmol) MA were added. Under inert atmosphere 9 mL of anhydrous NMP, 9 mL of mesitylene and 0.9 mL of isoquinoline were added in the reaction vessel and stirred for few minutes. Afterwards, the pre-functionalized sample (A or B) was added in the mixture and was heated to 200 °C for either 1 or 5 days. After the reaction was completed, the dark brown mixture was cooled to room temperature, and the membrane was removed from the solution and washed with 20 mL of NMP in a sonicated bath for 1h. Then, the membrane was immersed in 20 mL of fresh NMP and left for 3 days at room temperature to remove unreacted monomers or ungrafted polymer. Finally, the membrane was sonicated in 20 mL of acetone and dried in a vacuum oven at 50 °C overnight. Membrane samples prepared via route A (Scheme 1) are denoted as A-1 and A-5 (or A-1/5) respectively for 1 and 5 days of reaction. Samples prepared via route B (Scheme 1) are denoted as B-1 and B-5 (or B-1/5). intermittent-contact mode in air with AFM instrument Bruker Dimension ICON. The average roughness profile of the samples was determined by imaging 1 µm 2 of each sample.Membrane performance. Permeability and retention data were collected with a custommade, dead-end filtration setup, connected via a pressure regulator valve to a nitrogen tank for pressurizing the solutions. Permeability (L m -2 h -1 bar -1 ) is expressed as the flux (L h -1 ) of water or a solvent across a membrane per unit of driving force, per square meter of exposed membrane area (2.4 cm 2 ). Flux data were collected by weighing the mass of permeate at fourtime intervals, while permeability was determined from flux data at three applied transmembrane pressures between 8 and 20 bar, by taking the slope of a linear fit of the collected flux data. All slopes were found to be linear unless otherwise noted. Retentions (R) of Brilliant Yellow (BY, Mw = 624.55 g mol -1 , 50 ppm), Rhodamine B (RB, Mw = 479.02 g mol -1 , 50 ppm) were calculated with the equation: R = 1 -cp/cf(1)

  Sissi de Beer -Sustainable Polymer Chemistry Group, Department of Molecules &Materials, MESA+ Institute for Nanotechnology, University of Twente, 7500 AE Enschede, The Netherlands; orcid.org/0000-0002-7208-6814 Email: s.j.a.debeer@utwente.nl Arian Nijmeijer -Inorganic Membranes, MESA+ Institute for Nanotechnology, University of Twente, 7500 AE Enschede, The Netherlands; Email: arian.nijmeijer@shell.com

ACKNOWLEDGMENTS

Özlem Kap from the Physic of Complex Fluids is acknowledged for the AFM analysis.

Bernard Fraisse from the Institut Charles Gerhardt, Montpellier, France and Karin Nieuwenhuijzen from the Inorganic Material Science group at the University of Twente are acknowledged for XRD and capillary measurements, respectively. Our special thanks go also to Didier Cot from Institut Européen des Membranes in Montpellier, France for the SEM observations of the membrane samples.

Funding Sources

This work is part of the research program entitled 'Solvent Tolerant Nanofiltration and reverse osmosis membranes for the purification of industrial aqueous streams' (STNF), which is taking place within the framework of the Institute for Sustainable Process Technology (ISPT, project no BL-20-12).

METHODS

Materials. Solvents ethanol (technical grade > 95%), anisole (>99%, Merck, NL), Nmethyl-2-pyrrolidone (NMP) (>99%, anhydrous, Merck, NL), mesitylene (>99%, Acros Organics, NL), isoquinoline (95%, TCI, Europe), acetone (technical grade, > 95%), 1,4dioxane (anhydrous, Sigma Aldrich, NL), dimethylformamide (> 99%, Sigma Aldrich, NL) and ethanol (analytical grade, Merck, NL) were used as received. Water was purified through a Milli-Q™ Reference Water Purification System. Glycerol (anhydrous, Merck, NL), 3aminopropyl triethoxysilane (>98%, Sigma Aldrich, NL), pyromellitic dianhydride (PMDA) (97%, Sigma Aldrich, NL), melamine (MA) (99%, Sigma Aldrich, NL), Brilliant Yellow (70%, Sigma Aldrich, NL), Rhodamine B (> 99%, Merck, NL) were used as received. The chemical structures and abbreviations can be found in Figure S1 of the Supporting Information (SI).

Support fabrication.

The α-alumina (α-Al2O3 > 99 %) flat-sheet substrates (disc: diameter 21 mm, thickness 2 mm, 80 nm pore diameter) with one polished side were purchased from Pervatech B.V., the Netherlands. The polished side was dip-coated with a boehmite sol (prepared in-house) and subsequently calcined at 650 °C for 3 h to form a γ-alumina layer of 1.5 μm in total thickness and 5.5 nm mean pore diameter. Further details for the fabrication of the γ-alumina layer can be found elsewhere. [START_REF] Uhlhorn | Synthesis of Ceramic Membranes[END_REF][START_REF] Cuperus | Permporometry: The Determination of the Size Distribution of Active Pores in UF Membranes[END_REF] The calcined supports were washed by immersion in a 2:1 v/v water/ethanol solution for at least 8h, at room temperature and then dried over-night, in a vacuum oven at 50 °C.

Pre-functionalization of the top surface and pore entrance. The γ-alumina layer was first filled with 1-2 mL of glycerol by rubbing the viscous liquid onto the surface and letting it soak for > 10 min. The top surface of the substrate was dabbed clean with a fiber-less tissue.

Then, 21 μl of 3-aminopropyl trimethoxysilane (APTES) was dissolved in anisole After the preparation of each membrane, the remaining reaction solution was collected and filtered under vacuum to yield a dark brown powder. These powders were then washed with 50 mL of NMP and 50 mL of acetone. Finally, the powders were dried in a vacuum oven at 50 °C over-night. Powder samples collected from the solution are denoted as I-1 and I-5.

Detailed information on the spectroscopic characterization of the PI powders can be found in the SI.

Material Characterization. Fourier Transform Infrared spectroscopy (FTIR) measurements on both membrane and powder samples were done using a Perkin Elmer UATR Spectrum Two. Wavenumbers between 4000 and 550 cm -1 were scanned in reflectance mode at a resolution of 4 cm -1 for a minimum of 16 scans. Powder X-ray diffraction (XRD) patterns were recorded using a PANanalytical X'Pert PRO diffractometer at the wavelength of Cu Kα (λ = 1.5405 Å) (X-ray power: 40 kV, 40 mA) in Bragg-Brentano scanning mode. The program scanned angles (2θ) from 5 to 55 ° with a 0.026° step and a step time of 158s.

Scanning electron microscopy (SEM) images of powder and membrane samples and energydispersive X-ray spectroscopy (EDS) were obtained using a JEOL JSM-6010LA scanning electron microscope using an accelerating voltage of 5 kV. SEM samples were sputtered with 5 nm of palladium/platinum layer to avoid sample charging. High resolution scanning electron microscopy (HR-SEM) micrographs of membrane samples were obtained with a Hitachi S-4800 field-emission scanning electron microscope (Japan) using an accelerating voltage of 2 kV. Samples were metallized with platinum to favor charge release. The change in pore diameter of the membrane samples was determined by permporometry using cyclohexane as condensable vapor. The experimental procedure is described in detail elsewhere. [START_REF] Zhou | Excellent Thermal Conductivity and Dielectric Properties of Polyimide Composites Filled with Silica Coated Self-Passivated Aluminum Fibers and Nanoparticles[END_REF] Water contact angles were measured using the sessile drop method, with 2 μL drops of Milli-Q water. Atomic force microscopy (AFM) imaging was carried out in wavelength of 401.5 (BY), 543 (RB/water), 554 (RB/water and RB/1,4-dioxane) and 560 (RB/DMF) nm.

ASSOCIATED CONTENT

Supporting Information

Chemical structures of compounds used; Complete FTIR spectra of the starting compounds and synthesized samples; X-ray diffraction analysis data; Cyclohexane pore radius measurement of grafted samples; High resolution SEM images of pristine support; EDS raw data of synthesized membrane samples.

AUTHOR INFORMATION

Corresponding


Mohamed Fnadi
email: mohamed.fnadi@univ-littoral.fr

Andreas Rauh
email: andreas.rauh@uni-oldenburg.de

Validated Model Predictive Control based on Exponential Enclosures

Keywords: validated simulation, exponential enclosures, model predictive control

Introduction

Guaranteed numerical integration is a fundamental tool to solve initial value problems of ordinary differential equations (IVP-ODEs) with uncertain initial conditions and parameters in a reliable and validated way. Providing guaranteed solution enclosures to these IVP-ODEs is essential for designing and verifying linear and nonlinear feedback controllers, mainly for predictive control approaches. In the literature, many solvers have been developed, such as the DynIbex library, allowing for the computation of enclosures which are guaranteed to contain all possible system states. The DynIbex library is based on Runge-Kutta schemes to obtain tight state enclosures [START_REF] Dit Sandretto | Validated Explicit and Implicit Runge-Kutta methods[END_REF]. Nevertheless, it has been shown that -due to the computational complexity of Runge-Kutta methods [START_REF] Fnadi | Experimental Validation of a Guaranteed Nonlinear Model Predictive Control[END_REF] -fast convergence and high accuracy of the computed enclosures are not always guaranteed for finitely long integration time spans, possibly leading to an excessive duration to get the IVP-ODEs' solutions. To overcome these issues, exponential enclosure techniques for IVP-ODE problems seem to be attractive to remarkably reduce the computing time of validated methods and to approach real-time capability [START_REF] Rauh | Exponential Enclosure Techniques for Initial Value Problems with Multiple Conjugate Complex Eigenvalues[END_REF][START_REF] Rauh | Verified Simulation of Control Systems with Interval Parameters Using an Exponential State Enclosure Technique[END_REF].

The time aspect is especially crucial, because at each sampling instant, a validated nonlinear model predictive controller (NMPC) needs to compute optimal and guaranteed system inputs along a receding horizon that minimize some interval cost function and ensure compatibility constraints (such as actuator saturations or safety constraints on the state trajectories) [START_REF] Fnadi | Experimental Validation of a Guaranteed Nonlinear Model Predictive Control[END_REF]. Our motivation is to interface exponential enclosure techniques with the validated NMPC to remarkably speed up the solution.

Guaranteed Nonlinear Model Predictive Control

Consider a dynamic system defined by the following IVP-ODEs :

       ẋt = f (t, x t , u, p) x 0 ∈ [x 0 ] ⊆ IR n u ∈ [u] ⊆ IR m p ∈ [p] ⊆ IR p , (1)
where the state vector is denoted by x t , the vector of parameters by p, and the control vector by u. 1. Filtering and branching: The first step provides a sequence of guaranteed input interval boxes at each time-step k over the prediction horizon N p , denoted as

[U] k = [u] k ×[u] k+1 ×. . .×[u] k+N p -1 .
Branching and filtering procedures allow the computation of safe input intervals along the receding time horizon that satisfy the state constraints (i.e., ∀j, [x j ] ⊆ [x min,j , x max,j ], where x min,j and x max,j are the bounds for the admissible domain for each state variable) and ensure convergence to the reference interval (i.e., [

x k ] → [x r,k ]).
2. Interval optimization: Since safe inputs are computed over a finite time horizon, the optimization algorithm is launched to com-pute the optimal inputs [U] ⋆ k by minimizing as much as as possible a newly formulated interval objective function to reduce the error between predicted and reference outputs as well as the norm of the input intervals.

Exponential Enclosure Technique

Guaranteed numerical integration methods aim at computing the state enclosure sequences (t j , [x j ]) j∈N , assuming that the input and parameter boxes [u] and [p], respectively, are piecewise constant and known for each validated simulation. Here, the exponential enclosure technique will be applied to approximate the IVP-ODEs' solutions, given in [START_REF] Dit Sandretto | Validated Explicit and Implicit Runge-Kutta methods[END_REF]. It has been shown that this method improves the accuracy of the computed state enclosures and reduces the required computation time for asymptotically stable systems [START_REF] Rauh | Exponential Enclosure Techniques for Initial Value Problems with Multiple Conjugate Complex Eigenvalues[END_REF]. The dynamic model ( 1) can be reformulated by considering that the dynamic parameters are represented by constant intervals, and the input variables are assumed to be included in an augmented state vector, i.e., x T t u T (x t ) T , denoted for brevity again as x t with ẋt = f (x t ).

(

) 2
To ensure the (local) asymptotic stability of the system model in the neighborhood of a desired terminal state, we assume -as a prerequisite for the exponential enclosure approach -that a feedback controller is included in a cascaded manner in the control law u(x t ) so that the NMPC effectively computes a kind of feedforward control sequence.

To prevent the growth of the diameters of the intervals (t j , [x j ]) j∈N for asymptotically stable systems with a minimum computational capacity, the exact solution x ⋆ t can be bracketed into the following exponential state enclosures

x ⋆ t ∈ [x e ](t) = exp [Λ] • t • [x e ](0) , [x e ](0) = [x 0 ], (3)
where Λ represents a yet unknown dynamics matrix. By choosing [Λ] = diag{[λ i ]}, i = 1, . . . , n, as a diagonal matrix, its elements λ i need to have negative real parts to describe contracting state enclosures.

Using the exponential state enclosures (3) and a Picard iteration with the iteration index κ, we obtain

x ⋆ t ∈ [x e ] (κ+1) = exp [Λ] (κ+1) • t • [x e ](0) = [x e ](0) + t 0 f exp [Λ] (κ) • s • [x e ](0) ds. ( 4
)
The differentiation of ( 4) with respect to time, belonging to the integration interval t ∈ [t], leads to

ẋ⋆ [t] ∈ [Λ] (κ+1) • exp [Λ] (κ+1) • [t] • [x e ](0) = f exp [Λ] (κ) • [t] • e ](0) . (5)
Assuming a converging iteration with [Λ] (κ+1) ⊆ [Λ] (κ) and, thus, [λ i ] (κ+1) ⊆ [λ i ] (κ) , the iteration formula for [λ i ] (κ+1) can be expressed as

[λ i ] (κ+1) = f i exp [Λ] (κ) • [t] • [x e,i ](0) exp [Λ] (κ) • [t] .[x e,i ](0) , i = 1, . . . , n. (6)
The guaranteed state enclosure at the time instant t = T = sup([t]) is given by

x ⋆ t ∈ [x e ](t) = exp [Λ] • T • [x e ](0), (7)
where [Λ] is the final result of the iteration (6).

Preliminary Results using DynIbex

The NMPC strategy is applied to stabilize a nonlinear inverted pendulum with two serial joints, actuated by a DC motor whose angular speed is the input variable. To evaluate the dynamic model of the inverted pendulum, we can solve the IVP-ODEs in a validated way using the DynIbex library. Figs. 1a and 1b show the measured pendulum angle (black lines) with the computed enclosures by DynIbex using point-valued parameters p (red enclosures) and interval parameters [p] (blue enclosures). We can notice that the simulated tubes of the pendulum angle are close to the real measured signal. Moreover, we have calculated the coverage ratios between the measurements and the simulated tubes as recapped in Tab. 1. The coverage ratios confirm that the model is identified with high precision when the dynamic parameters are considered as intervals that account for different uncertainties related to the measurements and dynamic modeling. However, the accuracy of the validated simulation should be enhanced because the widths of the computed Rung-Kutta enclosures enlarge with time, and the coverage ratios are not quite satisfactory.

(a) Initial conditions:

[u 0 ] = [0 Nm, 0 Nm] and [x 0 ] = [0 • , 0 • ] [x 10 ] × [0, 0] [x 20 ] × [-98 • , -100 • ] [x 30 ] × [0, 0]
] = [0 • , 0 • ] [x 10 ] × [0, 0] [x 20 ] × [46 • , 50 • ] [x 30 ] × [0, 0] [x 40 ]
. Figs. 2a and 2b display the simulation results of the validated NMPC approach. As it can be seen in Fig. 2a, the pendulum arm starts from the downward position, and it is stabilized via the vali- dated NMPC in its vertical upright position interval [x r ] with a small settling-time (around t r5% ≈ 0.18 s). Despite its proven effectiveness in making the system output converge to the desired reference interval, it still has some drawbacks. The main ones are related globally to the computation time, which depends mainly on a large number of bisections of the initial input domain [u k ] preventing the validation of this approach in real-time. This issue can be reduced by using exponential enclosure techniques in combination with an underlying feedback controller.

  The sets [x 0 ] = [x 10 ] . . . [x n0 ] T , [u] = [u 1 ] . . . [u m ] T , and [p] = [p 1 ] . . . [p p ] T , expressed as interval boxes, are respectively the initial condition of the state vector, the interval-bounded input, and the set of feasible dynamic parameters. The proposed guaranteed NMPC encompasses two stages [2]:

  Initial conditions: [u 0 ] = [0.15 Nm, 0.18 Nm] and [x 0

Figure 1 :

 1 Figure 1: The validation of the dynamic model of a nonlinear inverted pendulum using the DynIbex library. Comparison between the actual and simulated pendulum angles at different initial conditions with point-valued and interval parameters.

  With interval-valued dynamic parameters [p] 51% 61% With point-valued dynamic parameters p 38% 34%

  (a) Tube for the pendulum angle. (b) Input Intervals.

Figure 2 :

 2 Figure 2: Validated NMPC results starting from the downward position.

Table 1 :

 1 Coverage rates between the model and physical reality.


Nasko Karamanov

Hopkins ' On

Picard

O N Hopkins' Picard Group P Ic

AT THE PRIME 3

.

Introduction

Let C be a symmetric monoidal category with product ∧ and unit I. We say that an object X in C is invertible if there exists an object Y in C such that X ∧ Y ∼ = I. If the collection of equivalence classes of invertible objects is a set, then the product defines a group structure on it. We denote this group by P ic(C), the Picard group of C .

For example, by [START_REF] Michael | Constructions of elements in Picard groups, Topology and representation theory[END_REF] the homomorphism Z → P ic(S) : n → S n defines an isomorphism between the integers and the Picard group of S, the stable homotopy category.

Let K n be the category of K(n)-local spectra, where K(n) is the n-th Morava K-theory at the prime p. The unit in K n is given by L K(n) S 0 and the product of two K(n)-local spectra by X ∧ Y := L K(n) (X ∧ Y ) (as the ordinary smash product of two K(n)-local spectra need not be K(n)-local). Hopkins' Picard group is the group P ic(K n ) which we denote by P ic n . The first account of it appears in [START_REF] Strickland | On the p-adic interpolation of stable homotopy groups[END_REF] and the case n = 1 is treated in details in [START_REF] Michael | Constructions of elements in Picard groups, Topology and representation theory[END_REF] where also some examples of elements of P ic 2 at the prime p = 2 are given.

In this paper we are interested in P ic 2 at the prime p = 3. One way to study K n is through the functor E n * X := π * L K(n) (E n ∧ X) where E n is the Lubin-Tate (commutative ring) spectrum with coefficients ring E n * ∼ = WF p n [[u 1 , . . . , u n-1 ]][u, u -1 ], where the power series ring is over the Witt vectors WF p n . Recall that E n is acted on by the (Big) Morava Stabilizer group G n = S n ⋊ Gal(F p n / F p ) by E ∞ -maps (Goerss-Hopkins, Hopkins-Miller). Let EG n be the category of profinite E n * [[G n ]]-modules, i.e. E n * -modules with an G n -action compatible with the action of G n on E n * . The tensor product (over (E n ) * ) gives a monoidal structure on EG n . Proposition 1.1. [START_REF] Michael | Constructions of elements in Picard groups, Topology and representation theory[END_REF] Let X ∈ K n . Then the following conditions are equivalent :

a) X is invertible in K n ; b) E n * X is free E n * -module of rank 1; c) E n * X is invertible in EG n .
1.1. Let P ic alg n := P ic(EG n ). By Proposition 1.1 there is a homomorphism :

ǫ n : P ic n → P ic alg n X → E n * X .
1

Let P ic alg,0 n be the subgroup of P ic alg n of index 2 of modules concentrated in even degrees. Let M ∈ P ic alg,0 n and ι M be a generator of M in degree 0, as an (E n ) *module . Then for all g ∈ G n there exists a unique element

u g ∈ (E n ) × 0 such that g * (ι M ) = u g ι M .
The map θ M : g → u g is a crossed homomorphism and is a well defined element in H 1 (G n ; (E n ) × 0 ) that does not depend on ι M . Thus we have a homomorphism P ic alg,0 n

→ H 1 (G n ; (E n ) × 0 ). Proposition 1.2. [5] P ic alg,0 n ∼ = H 1 (G n ; (E n ) × 0
) . Not much is known for the kernel κ n of ǫ n (cf. [START_REF] Strickland | On the p-adic interpolation of stable homotopy groups[END_REF]). When n 2 ≤ 2(p -1) and n > 1 or when n = 1 and p > 2 it is known to be trivial. It is conjectured (Hopkins, [START_REF] Strickland | On the p-adic interpolation of stable homotopy groups[END_REF]) that k n is a finite p-group.

The next theorem is an unpublished result of Goerss, Henn, Mahowald and Rezk.

Theorem 1.3. At the prime p = 3, κ 2 ∼ = Z/3 × Z/3 .
The next two theorems describe some known results for P ic n .

Theorem 1.4.

[6]

P ic 1 ∼ = Z 2 × Z/2 × Z/4 p = 2 P ic 1 ∼ = Z p × Z/2(p -1) p > 2 .
The spectrum S 1 is a generator of P ic 1 in the case p > 2. In an unpublished result and using Shimomura's calculations of π * L 2 S at primes p > 3 Hopkins shows Theorem 1.5. For primes p > 3

P ic 2 ∼ = Z 2 p × Z/2(p 2 -1)
. The main result of this paper is the following theorem.

Theorem 1.6. At the prime 3

P ic alg 2 ∼ = Z 2 3 × Z/16 generated by (E 2 ) * S 1 and (E 2 ) * S 0 [det],
where det is a suitable character of G 2 .

Theorem 1.3 and Theorem 1.6 imply the following theorem.

Theorem 1.7. At the prime 3

P ic 2 ∼ = Z 2 3 × Z/3 × Z/3 × Z/16 . 1.
2. This paper is organized as follows. In section 2 we recall the basic properties of the Morava stabilizer group G n and describe some important subgroups in the case n = 2 and p = 3. We also recall the GHMR resolution of [START_REF] Goerss | A resolution of the K(2)-local sphere at the prime 3[END_REF] and the spectral sequence of [START_REF] Henn | The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited[END_REF] and [START_REF] Karamanov | À propos de la cohomologie du deuxième groupe stabilisateur de Morava; application aux calculs de π * L K(2) V (0) et du groupe pic 2 de Hopkins[END_REF] that we use for the most difficult part of our calculation. In section 3 we define two elements of P ic alg n that turn out to be generators in the case of P ic alg 2 . In section 4 we present three short exact sequences that we use to simplify the calculations. In Section 5 we describe the part of the first page of the spectral sequence that is needed for the calculations. The final calculations for P ic alg,0 2 are done in section 6, and P ic alg 2 is treated in the last section. The author would like to thank Hans-Werner Henn for many usefull discussions and for sharing his knowledge on the subject.

On the Morava stabilizer group and the GHMR resolution

In this section we recall some basic properties of the Morava stabilizer group and some important finite subgroups in the case n = 2 and p = 3. We also describe the main tool of this work, that is the algebraic GHMR resolution of the K(2)-local sphere constructed in [START_REF] Goerss | A resolution of the K(2)-local sphere at the prime 3[END_REF]. This resolution is used in [START_REF] Henn | The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited[END_REF] and [START_REF] Karamanov | À propos de la cohomologie du deuxième groupe stabilisateur de Morava; application aux calculs de π * L K(2) V (0) et du groupe pic 2 de Hopkins[END_REF] to determine the homotopy of the mod-3 Moore spectrum localized at K(2). We will use some of the calculations of [START_REF] Henn | The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited[END_REF] and [START_REF] Karamanov | À propos de la cohomologie du deuxième groupe stabilisateur de Morava; application aux calculs de π * L K(2) V (0) et du groupe pic 2 de Hopkins[END_REF] and the spectral sequence used there. For more details the reader is referred to the corresponding papers.

2.1. Recall that S n is the group of automorphisms of the Honda formal group law Γ n with p-series [p] Γn (x) = x p n , that is, the group of units in the endomorphism ring End(Γ n ). Let O n be the non-commutative ring extension of WF p n (the Witt vectors over F p n , that we denote by W from now on) generated by an element S satisfying S n = p and Sw = w σ S where w ∈ W and σ is the lift of the Frobenius automorphism of F p n . Then End(Γ n ) can be identified with O n . For example, in the case n = 2 and p = 3 each element g of S 2 can be written as g = g 1 + g 2 S with g 1 ∈ W × and g 2 ∈ W.

Right multiplication of S n on End(Γ n ) defines a homomorphism S n → GL(W).

Composition with the determinant can be extended to G n to obtain a homomorphism G n → W × ⋊ Gal(F p n / F p ) and it is easy to check that this lands in

Z × p ⋊ Gal(F p n / F p ).
The quotient of Z × p with its torsion subgroup, isomorphic to Z/(p -1), can be identified with Z p and we get a homomorphism called reduced determinant or reduced norm:

G n → Z p . The kernel of this homomorphism is denoted by G 1 n and in the case when p -1 divides n we have n by S 1 n . 2.4. Let n = 2 and p = 3 from now on. Let ω be a primitive eighth root of unity, φ ∈ Gal(F 9 /F 3 ) the generator, t := ω 2 , ψ := ωφ and a := 1 2 (1 + ωS). It is easy to verify that a is an element of order 3 . These elements satisfy ψa = aψ, tψ = ψt 3 , ta = a 2 t and ψ 2 = t 2 . Then a, ψ and t generate a subgroup of order 24, denoted G 24 , ω and φ a subgroup isomorphic to the semi-dihedral group of order 16, denoted SD 16 . The elements ω 2 and φ generate a subgroup of SD 16 isomorphic to the quaternion group of order 8, denoted Q 8 and we have ( 1)

G n ∼ = G 1 n × Z p . 2.
SD 16 ∼ = Q 8 ⋊ Gal(F 9 /F 3 ) .

The action of the element

a on F 9 [[u 1 ]][u, u -1 ] is described in [5, Cor. 4.7].
For our purposes we only need the following formulae :

a * u ≡ (1 + (1 + ω 2 )u 1 )u mod (u 3 1 ) a * u 1 ≡ u 1 -(1 + ω 2 )u 2 1 mod (u 3 1
) . The (integral) action of ω is given by [START_REF] Devinatz | The action of the Morava stabilizer group on the Lubin-Tate moduli space of lifts[END_REF] ω * u 1 = ω 2 u 1 and ω * u = ωu and the Frobenius φ acts Z 3 -linearly by extending the action of the Frobenius on W via

(3) φ * u 1 = u 1 and φ * u = u .

2.6. The GHMR resolution. In [START_REF] Goerss | A resolution of the K(2)-local sphere at the prime 3[END_REF] a resolution of the trivial G 1 2 -module Z 3 is constructed that has the following form

0 -→ C 3 ∂3 -→ C 2 ∂2 -→ C 1 ∂1 -→ C 0 ∂0 -→ Z 3 -→ 0 where C 0 = C 3 = Z 3 [[G 1 2 ]] ⊗ Z3[G24] Z 3 and C 1 = C 2 = Z 3 [[G 1 2 ]] ⊗ Z3[SD16] χ
(4) E s,t 1 = Ext t Z3[[G 1 2 ]] (C s ; M ) =⇒ H s+t (G 1 2 ; M )
in which E s,t 1 = 0 for 0 < s < 3 and t > 0, and for s ≥ 0 and t > 3, and also

E 0,t 1 ∼ = E 3,t 1 ∼ = H t (G 24 ; M ) and E 1,0 1 ∼ = E 2,0 1 ∼ = Hom SD16 (χ; M ) . Note that Hom SD16 (χ; M ) ∼ = {m ∈ M | ω * m = φ * m = -m}.
2.7. Let N 0 be the kernel of ∂ 0 and j : N 0 → C 0 the inclusion. As explained in the remark after [5, Lemma 6.1] the differentials in the spectral sequence can be evaluated if we know projective resolutions Q • of N 0 and P • of C 0 as well as a chain map φ : Q • → P • covering j. These data can be assembled in a double complex

T •• with T •0 = P • , T •1 = Q • , vertical differentials δ P and δ Q and horizontal differentials (-1) n φ n : Q n → P n .
The filtration of the spectral sequence of this double complex agrees (up to reindexing) with that of the spectral sequence of the lemma. Hence extension problems in the spectral sequence (4) can be studied by using the double complex. As in [START_REF] Henn | The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited[END_REF] we obtain a resolution

P • := Z 3 [[G 1 2 ]] ⊗ Z3[G24] P ′ • induced from an explicit resolution of the trivial G 24 -module Z 3 .
Lemma 2.2. [5, Lemma 6.2] Let χ be the Z 3 [Q 8 ]-module whose underlying Z 3module is Z 3 and on which t acts by multiplication by -1 and ψ by the identity. Then the trivial Z 3 [G 24 ]-module Z 3 admits a projective resolution P ′

• of period 4 of the following form

a 2 -a -→ χ ↑ G24 Q8 e+a+a 2 -→ χ ↑ G24 Q8 a 2 -a -→ χ ↑ G24 Q8 e+a+a 2 -→ χ ↑ G24 Q8 a 2 -a -→ 1 ↑ G24 Q8 -→ Z 3 .
We obtain , by e i the generators e ⊗ 1 of C i and by e i the generators e ⊗ 1 of P i , then by [START_REF] Henn | The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited[END_REF]Lemma 6.3] there is a chain map φ : Q • → P • covering the homomorphism j such that φ 0 : Q 0 = C 1 → P 0 sends e 1 to (e -ω) e 0 .

Q • from splicing the exact complex 0 → C 3 → C 2 → C 1 → N 0 →

We denote by E the spectral sequence for

M = E 2 * /(3) ∼ = F 9 [[u 1 ]][u, u -1 ]
. The structure of the E 1 -page is well known (cf. [START_REF] Goerss | The homotopy of L 2 V (1) for the prime 3, Categorical decomposition techniques in algebraic topology (Isle of Skye[END_REF] for the group S n , the case of G n can be deduced in the same way). Recall that v 1 = u 1 u -2 is invariant modulo 3 with respect to the action of G 2 and therefore all the differentials in the spectral sequence are v 1 -linear. The element α ∈ H 1 (G 24 ; (E 2 ) 4 ) is defined as the modulo 3 reduction of δ 0 (v 1 ), where δ 0 is the Bockstein with respect to the short exact sequence

Proposition 2.3. As F 3 [β, v 1 ]-modules (β acting trivially on E s, * , * 1 if s = 1, 2) E s, * , * 1 ∼ =    F 3 [[v 6 1 ∆ -1 ]][∆ ±1 , v 1 , α, α, β]/(α 2 , α 2 , v 1 α, v 1 α, α α + v 1 β)e s s = 0, 3 ω 2 u 4 F 3 [[u 4 1 ]][v 1 , u ±8 ]e s s =
0 -→ E 2 * -→ E 2 * -→ E 2 * /(3) -→ 0 . The element α ∈ H 1 (G 24 ; (E 2 ) 12 ) is defined as δ 1 (v 2 )
, where v 2 = u -8 and δ 1 is the Bockstein with respect to the short exact sequence

0 -→ E 2 * /(3) -→ E 2 * /(3) -→ E 2 * /(3, u 1 ) -→ 0 and β ∈ H 2 (G 24 ; (E 2 ) 12 ) is the modulo 3 reduction of δ 0 δ 1 (v 2 ).
The definition of ∆ is more complicated and we have the following formula (cf. [5, Prop. 5.1]).

(5) ∆ ≡ ω 2 (1 -ω 2 u 2 1 + u 4 1 )u -12 mod (u 6 1 ) .

One of the main results in [START_REF] Karamanov | À propos de la cohomologie du deuxième groupe stabilisateur de Morava; application aux calculs de π * L K(2) V (0) et du groupe pic 2 de Hopkins[END_REF] (see also [START_REF] Henn | The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited[END_REF]Thm. 1.2]) is the following theorem.

Theorem 2.4. There are elements

∆ k ∈ E 0,0,24k 1 b 2k+1 ∈ E 1,0,8(2k+1) 1 b 2k+1 ∈ E 2,0,8(2k+1) 1 for each k ∈ Z satisfying ∆ k ≡ ∆ k e 0 b 2k+1 ≡ ω 2 u -4(2k+1) e 1 b 2k+1 ≡ ω 2 u -4(2k+1) e 2
(where the first congruence is modulo (u 2 1 ) and the last two modulo (u 4 1 )) such that

d 1 (∆ k ) = b 2(3m+1)+1 ≡ (-1) m+1 ω 2 (1 + u 4 1 )u -12k k = 2m + 1 (-1) m+1 mv 4•3 n -2 1 b 2•3 n (3m-1)+1 k = 2 • 3 n m, m ≡ 0 mod (3) d 1 (b 2k+1 ) =    (-1) n v 6•3 n +2 1 b 3 n+1 (6m+1) k = 3 n+1 (3m + 1) (-1) n v 10•3 n +2 1 b 3 n (18m+11) k = 3 n (9m + 8) 0 otherwise.
3. Two elements of P ic alg,0 n 3.1. We have two distinguished elements in P ic alg,0 n ∼ = H 1 (G n ; (E n ) × 0 ). In the case n = 2 and p = 3 these will generate the first cohomology. The first one is given by the crossed homomorphism :

η : G n → (E n ) × 0 g → g * u u .
The second one is given as the composition of the reduced norm and the canonical inclusion :

det : G n → Z × p → (E n ) × 0 . We denote the corresponding elements of H 1 (G n ; (E n ) × 0 )
again by η and det. Note that by the isomorphism of Proposition 1.2 the element (E n ) * S 2 ∈ P ic alg,0 n is sent to η (as u ∈ (E n ) -2 gives rise to a generator of (E n ) 0 S 2 ) .

The reduction W[[u

1 ]] × → W × is equivariant with respect to the inclusion W × → G n .
Taking into account the Galois group we obtain a homomorphism : Proof. Recall that when p > 2 then W × ∼ = W × F p n with the obvious Galois action. Thus

red : H 1 (G n ; (E n ) × 0 ) → H 1 (W × ⋊ Gal; W × ) → H 1 (W × ; W × ) Gal
H 1 (W × ; W × ) Gal ∼ = End(W × ) Gal ∼ = Z n p × Z/(p n -1
). The image of det is given by the composition

W × → G n det → Z × p → (E n ) × 0 → W × . If g = g 0 + g 1 S + • • • + g n-1 S n-1 with g i ∈ W and w ∈ W then gw = (g 0 + g 1 S + • • • + g n-1 S n-1 )w = g 0 w + g 1 w φ S + • • • g n-1 w φ n-1 S n-1
and the compostion above sends w to ww φ . . . w φ n-1 . The image of η is given by the composition

W × → G n η → (E n ) × 0 → W × and
this is easily verified to be the identity.

Reductions

In this short section we present three short exact sequences that we use in our calculations. The last two were also used by Hopkins in the case n = 2 and p > 3.

The first one (6)

1 → G 1 2 → G 2 → Z 3 → 1 was described in section 2. We use the Lyndon-Hochschild-Serre spectral sequence associated to [START_REF] Michael | Constructions of elements in Picard groups, Topology and representation theory[END_REF]

to calculate H 1 (G 2 ; W[[u 1 ]] × ). The main difficulty is computing H 1 (G 1 2 ; W[[u 1 ]] ×
). This is done in Theorem 6.4.

4.2. The reduction modulo 3 gives a short exact sequence

(7) 0 → W[[u 1 ]] exp(p-) -→ W[[u 1 ]] × -→ F 9 [[u 1 ]] × → 0
We will use the long exact sequence associated to [START_REF] Karamanov | À propos de la cohomologie du deuxième groupe stabilisateur de Morava; application aux calculs de π * L K(2) V (0) et du groupe pic 2 de Hopkins[END_REF]to calculate

H 1 (G 1 2 ; W[[u 1 ]]) . The difficult part is H 1 (G 1 2 ; F 9 [[u 1 ]] × ) (Corollary 6.
2). 4.3. We have another short exact sequence coming from the reduction modulo u 1

(8) 1 → U 1 → F 9 [[u 1 ]] × → F × 9 → 1 where U 1 := {h ∈ F 9 [[u 1 ]] × | h ≡ 1 mod (u 1 )}. The hard part is to calculate H 1 (G 1 2 ; U 1 )
. This is by far the hardest part of this work (cf. Theorem 5.10). Note that the group U 1 is 3-profinite.

The spectral sequence

We use the spectral sequence (4) with M = U 1 and denote it by E to distinguish it from the (additive) case M = E 2 * /(3) that we also make use of. We start with the E 1 -page. As we only need to calculate the first cohomology, it is sufficient to determine E 0,0

1 , E 0,1 1 and E 1,0 1 ∼ = E 2,0 1
and the corresponding differentials and extension problems.

The term E

0,1 1 . Proposition 5.1. H 1 (G 24 ; F 9 [[u 1 ]] × ) ∼ = Z/6.
Proof. Let F 9 ((u 1 )) × be the multiplicative group of the field of fractions of

F 9 [[u 1 ]]. Each element of F 9 ((u 1 )) × is of the form u n 1 f with f ∈ F 9 [[u 1 ]] × and n ∈ Z. The map F 9 ((u 1 )) × → Z : u n 1 f → n is a group homomorphism with kernel F 9 [[u 1 ]] × . Thus we have a short exact se- quence of G 24 -modules (9) 1 → F 9 [[u 1 ]] × → F 9 ((u 1 )) × → Z → 1
where G 24 acts trivially on Z. By Hilbert 90, the multiplicative version, we have H 1 (G 24 ; F 9 ((u 1 )) × ) = 0 and thus the long exact sequence induced by (9) yields

H 0 (G 24 ; F 9 [[u 1 ]] × ) → H 0 (G 24 ; F 9 ((u 1 )) × ) → H 0 (G 24 ; Z) ։ H 1 (G 24 ; F 9 [[u 1 ]] × ). By Proposition 2.3 we have H 0 (G 24 ; F 9 [[u 1 ]] × ) ∼ = F 3 [[v 6 1 ∆ -1 ]
] × and by a similar argument we conclude H 0 (G 24 ; F 9 ((u 1 )) × ) ∼ = F 3 ((v 6 1 ∆ -1 )) × . By ( 5) we have v 6 1 ∆ -1 ≡ u 6 1 mod (u 8 1 ) so the image of the homomorphism

H 0 (G 24 ; F 9 ((u 1 )) × ) → H 0 (G 24 ; Z) ∼ = Z
is 6Z, and the result follows.

Note that η is not defined on U 1 (as for example ω * u/u = ωu ∈ U 1 ), but 8η is well defined. Proof. The short exact sequence (8) induces a long exact sequence

→ H 1 (G 24 ; U 1 ) → H 1 (G 24 ; F 9 [[u 1 ]] × ) → H 1 (G 24 ; F × 9 ) →
The group in the middle is isomorphic to Z/6 by Proposition 5.1 and the group on the right is 2-torsion. The group U 1 is 3-profinite, so there is no 2-torsion in H * (G 24 ; U 1 ). As H 0 (G 24 ; F × 9 ) is 2-torsion, the first morphism above is injective. For the second part of the proposition we use the resolution P ′

• of G 24 constucted in Lemma 2.2, to show that the cocycle of 8η can not be a cobord. Recall that η was defined as a crossed homomorphism and thus it can be easily described using the standard (bar) resolution (cf. [START_REF] Brown | Cohomology of groups[END_REF]III.3]). A cocycle representing the image of 8η in the standard resolution B • of G 24 is given by B 1 → U 1 : [g] → g * u 8 /u 8 . By comparing these resolutions we find a representing cocycle in P ′

• . A homomorphism φ • : P ′ • → B • over the identity of Z 3 is given by φ 0 :

P ′ 0 → B 0 φ 1 : P ′ 1 → B 1 e ′ 0 → 1 8 g∈Q8 g e ′ 1 → 1 8 ( g∈Q8 χ(g -1 )g)a[a]
where e ′ i are the generators e ⊗ 1 of P ′ i and {[g]} g∈G24 is a G 24 -basis of B 1 (cf. [1, I.5]). Thus the composition

P ′ 1 → B 1 → U 1 : e ′ 1 → a 2 * u 8 a * u 8
is the desired cocycle. Using the formula from 2.5 we obtain

a 2 * u 8 a * u 8 ≡ 1 -(1 + ω 2 )u 1 mod (u 2 1
) . Now we will show that this cocycle can not be a cobord. A morphism from P ′ 0 → U 1 sends e ′ 0 to a Q 8 -invariant element h of U 1 . Thus by Proposition 5.3 we have h ≡ 1 mod (u 2 1 ) and thus the composition P ′ 1 → P ′ 0 → U 1 sends e ′ 1 to an element congruent to 1 modulo u 2 1 which is not the case for 8η. 5.1. The 0-th line. In the following proposition we give the structure of the 0-th line of the first page of our spectral sequence. We end up with a nice description of the corresponding groups as products of copies of the 3-adics.

Recall that v 1 = u 1 u -2 is in degree 4 and ∆ is in degree 24. Thus v 6 1 ∆ -1 is in degree 0. [START_REF] Henn | The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited[END_REF] we will use the image of d 1 to define h k . We allow the generators h k to be congruent to 1 -ω 2 u 4k+2 1 modulo higher powers of u 1 . Note that 12m + 2 = 4 • 3m + 2 thus for m ≥ 1 and m ≡ 0 mod (3) we can define

Proposition 5.3. a) E 0,0 1 ∼ = {g ∈ ((E 2 ) 0 /3) G24 ∼ = F 3 [[v 6 1 ∆ -1 ]] × | g ≡ 1 mod (u 1 )} . b) Let g k ∈ E 0,0 1 be such that g k ≡ 1 + v 6k 1 ∆ -k mod (u 6k+2 1 ). Then E 0,0 1 ∼ = k≥1 Z 3 {g k } k ≡0 mod (3) c) E 1,0 1 = {h ∈ U 1 | ∃k ∈ (E 2 /3) Q8 0 ∼ = F 3 [[ω 2 u 2 1 ]], h = ω * k/k} d) Let h k ∈ E 1,0 1 be such that h k ≡ 1 + ω 2 u 4k+2 1 mod (u 4k+4 1 ). Then E 1,0 1 ∼ = E 2,0 1 ∼ = k≥0 Z 3 {h k } k ≡1 mod (3)

As in

h 3m := d 1 (g 2m )
and then by definition these elements are in the kernel of d 1 .

For the other case, note that 1 + ω 2 (u 12m+6 1

+ u 12m+10 1 ) ≡ (1 + ω 2 u 12m+6 1 )(1 + ω 2 u 12m+10 1 ) ≡ (1 -ω 2 u 4m+2 1 ) 3 (1 + ω 2 u 12m+10 1 ) mod (u 12m+12 1
). Thus if h m is already definied we could use d 1 (g 1+2m ) to define h 3m+2 by

h 3m+2 := d 1 (g 1+2m )h 3 m . As h m ≡ 1 ± ω 2 u 4m+2 1 ≡ 1 ± v 4m+2 1 b -1-2m mod (u 4m+6 1
) we can use Lemma 5.5 to calculate d 1 . In order to do that we will use the partition of N introduced in Theorem 2.4. If -1 -2m = 1 + 2(3l + 1) then 1 + 2m is divisible by 3. The other cases are of relevance : if -1 -2m = 1 + 2 • 3 n (3l -1) then m = 3 n (-3l + 1) -1 with l ≡ 0 mod (3), if -1 -2m = 1 + 2 • 3 n (9l + 8) then m = 3 n (-9l -8) -1 with l ≤ -1 and the last one, if -1 -2m = 1 + 2 • 3 n+1 (3l + 1) then m = 3 n+1 (-3l -1) -1 with l ≤ -1 . In each of these cases , if h m is defined then h 3m+2 belongs to the same family with n augmented by one. This allows to define recursevely, for l, n ≥ 0 : h 3 n+1 (3l+1)-1 := h 3 3 n (3l+1)-1 d 1 (g 1+2•3 n (3l+1) ) l ≡ 0 mod (3) h 3 n+1 (9l+1)-1 := h 3 3 n (9l+1)-1 d 1 (g 1+2•3 n (9l+1) ) h 3 n+2 (3l+2)-1 := h 3 3 n+1 (3l+2)-1 d 1 (g 1+2•3 n+1 (3l+2) ) . For this to work we need to define the corresponding element of each family for n = 0. In the first case h 3l is already defined. In the other cases we define: A more complete description of d 1 is given with the following proposition.

3 .

 3 The element S generates a two sided maximal ideal m in O n with quotient O n /m ∼ = F p n . The strict Morava stabilizer group S n is the kernel of the homomorphism O × n → F × p n induced by reduction modulo m. We denote its intersection with G 1

  0 with the projective resolution P • of C 3 = C 0 (as C 1 and C 2 are projective). If we denote by e the unit of G 1 2

1 , 2 0 s > 3 where

 13 |e s | = (s, 0, 0) |v 1 | = (0, 0, 4) |∆| = (0, 0, 24) |β| = (0, 2, 12) |α| = (0, 1, 4) | α| = (0, 1, 12) .

  and the last homomorphism is induced by the short exact sequence 1 → W × → W × ⋊ Gal → Gal → 1 and the corresponding spectral sequence. Proposition 3.1. Let n = 2 and p > 2. Then the image of the homomorphism red is (topologically) generated by the images of η and det.

∼

  = H 1 (G 24 ; U 1 ) ∼ = Z/3 generated by the restriction of 8η.

v 36l+2 1 b 1 b 1 b 1 . 1 + ω 2 u 36l+14 1 mod (u 36l+18 1 ) 1 mod (u 36l+34 1 )

 111111111 -1-18l ) l > 0 h 9l+5 := ω * (1 + v 36l+22 -11-18l )/(1 + v 36l+22 -11-18l ) l ≥ 0 .Note that b -1-18l and b -11-18l belong to the two families of Theorem 2.4 that have non trivial image under d 1 . In both of these cases we can apply Lemma 5.4. This would not have been the case if we would have defined h 0 as 1 + v 2 1 b -1 as then Lemma 5.5 does not give the enough precision. The following proposition and the recursive definition of the generators describe d 1 : E The proof uses Theorem 2.4 and Lemma 5.5. Proposition 5.7.z 1,l := d 1 (h 9l ) ≡ z 2,l := d 1 (h 9l+5 ) ≡ 1 + ω 2 u 36l+30 .

Proposition 5 . 8 .Z 3 Z 3 Z 3 n≥0Z 3 Z 3

 5833333 The following complexes are exact :n≥0 Z 3 {g 1+2•3 n (3l+1) } → n≥0 {h 3 n (3l+1)-1 } → 1 for l ∈ 3N n≥0 {g 1+2•3 n (9l+1) } → n≥0 {h 3 n (9l+1)-1 } → Z 3 {z 1,l } for l > 0 {g 1+2•3 n+1 (3l+2) } → n≥0 {h 3 n+1 (3l+2)-1 } → Z 3 {z 2,l } for l ≥ 0 .

  and χ is the non trivial character of SD 16 defined over Z 3 , on which ω and φ act by multiplication by -1. The complete ring Z p [[G]] is by definition lim U,n Z p /p n [G/U ] where U runs through the open subgroups of G. Then we have the following lemma (cf. [5, Lemma 6.1]).

	Lemma 2.1. Let M be a left G 1 2 -module. Then there is a first quadrant cohomo-logical spectral sequence E * , *

r , r ≥ 1 with

Proof. Proposition 2.3 implies (a). By (a) each element g ∈ E 0,0 1 can be written as a product k≥1 g λ k k with λ k ∈ {-1, 0, 1}. As g 3k ≡ g 3 k mod (u 18k+2

) we obtain the result. To get the action of Q 8 on (E 2 /3) 0 we use formulae [START_REF] Devinatz | The action of the Morava stabilizer group on the Lubin-Tate moduli space of lifts[END_REF] and (3) and then (c) follows. As ω 2 = -ω 6 we have h -1 3k+1 ≡ 1 + ω 6 u 4(3k+1)+2 1

≡ (1 + ω 2 u 4k+2 1

) and we obtain (d).

5.2. The goal of what follows is to construct families of generators {g k } k≥1,k ≡0 mod [START_REF] Goerss | A resolution of the K(2)-local sphere at the prime 3[END_REF] and {h k } k≥0,k ≡1 mod [START_REF] Goerss | A resolution of the K(2)-local sphere at the prime 3[END_REF] as in the previous proposition on which d 1 is easy to describe.

We start with a particular element m ∈ E 1,0 1 that is related to 8η and plays the same role as the element b 1 in [START_REF] Henn | The homotopy of the K(2)-local Moore spectrum at the prime 3 revisited[END_REF].

Proposition 5.4. There exists

Proof. We imitate the proof of [5, Prop. 5.5] and use paragraph 2.7. By definition 8η is a permanent cycle so there are cochains c

and such that c represents the restriction of 8η in H 1 (G 24 ; U 1 ). From the proof of the Proposition 5.2 we have an explicit cocycle c 1 for 8η in the resolution P • , so there exists h c ∈ Hom Z3[[G 1 2 ]] (P 0 ; U 1 ) such that c = c 1 + δ P (h c ). As 24η = 1 in H 1 (G 24 ; U 1 ) (Proposition 5.2) there exists h ∈ Hom Z3[[G 1 2 ]] (P 0 ; U 1 ) such that δ P (h) = 3c 1 . In the double complex the cochain 3c + 3d is cohomologous to 3d -δ Q (h + 3h c ). One can easily check that h = u 24 /(u 8 (a * u 8 )(a 2 * u 8 )). Then 3d -(e -ω) * (h + 3h c ) is a cocycle concentrated in Hom Z3[[G 1 2 ]] (T 1,0 , U 1 ) representing 24η. The formula for m can be deduced using the formulae for the action of a and ω given in paragraph 2.5.

The next lemma is elementary but crutial as it relates the differentials of E and E, and thus suggests that we could use the generators from Theorem 2.4 to construct convenient generators g k and h k .

Proof. By Lemma 5.5 we have

1 ) where we have used the v 1 -linearity of d 1 . Then the result follows from the formulae from Theorem 2.4.

The first homology of the following complex

is isomorphic to Z 3 {h 0 }. In each complex the infinite matrix of the first homomorphism has the following form

and when non trivial the infinite matrix of the second morphism has the following form

Proof. The proof is a consequence of Proposition 5.3, Proposition 5.4, the definitions of the generators in paragraph 5.3 and Proposition 5.8.

Corollary 5.9.

Proof. This is consequence of the previous proposition. Indeed, the relevant part of the 0-th line of the first page of the spectral sequence is the product over l of the four complexes of the previous proposition together with the exact complex

Theorem 5.10. H 1 (G 1 2 ; U 1 ) ∼ = Z 3 is generated by 8η. Proof. We only need to resolve the extension problem

In this section we prove the main theorem by using the short exact sequences from Section 4. The element η again plays an important role in the proof. Proposition 6.1. H 1 (G 1 2 ; F × 9 ) ∼ = F × 9 . Proof. There is a short exact sequence 1 → S 1 2 → G 1 2 → SD 16 → 1 that gives a spectral sequence and as S 1 2 acts trivially on F × 9 we have H * (G 1 2 ; F × 9 ) ∼ = H * (SD 16 ; F × 9 ) . There is another short exact sequence 1 →< ω >→ SD 16 → Gal → 1 (where Gal := Gal(F 9 /F 3 )) and thus a spectral sequence

. By using the standard resolution it is easily seen that the group H * (< ω >; F × 9 ) is isomorphic to F × 9 in each degree as ω acts trivially on F × 9 . The group H 1 (< ω >; F × 9 ) is generated by the identity which is Galois invariant thus H 0 (Gal; H 1 (< ω >;

and by Hilbert 90

)) has to be trivial.

9 generated by η. Proof. The short exact sequence (8) induces a long exact sequence

. By Theorem 5.10 we have H 1 (G 1 2 ; U 1 ) ∼ = Z 3 and by Proposition 6.1 we have

As U 1 is a 3-profinite group, the first and the last homomorphisms are trivial.

] is 3-profinite and there is a resolution of finite type (Lazard) of the trivial G 1 2 -module Z 3 and (b) follows. Multiplication by 3 induces a short exact sequence

which induces a long exact sequence

) -→ From (a) and the long exact sequence above it follows that the homomorphism

As G is 3-profinite, it is the limit of finite 3-groups G/I n . Thus the homomorphism G/I n → G/I n induced by the multiplication by 3 is surjective and therefore G is trivial.

9 generated by η. Proof. We use the long exact sequence in H 1 (G 1 2 ; -) induced from the short exact sequence [START_REF] Karamanov | À propos de la cohomologie du deuxième groupe stabilisateur de Morava; application aux calculs de π * L K(2) V (0) et du groupe pic 2 de Hopkins[END_REF]. The homomorphism

) is injective by Proposition 6.3 (c) and also surjective as the image of η ∈ H 1 (G 1 2 ; W[[u 1 ]] × ) is a generator (by Corollary 6.2).

Finally we get to the proof of 1.6(a).

3 × F × 9 generated by η and det. Proof. We use the short exact sequence [START_REF] Michael | Constructions of elements in Picard groups, Topology and representation theory[END_REF]. We have

The theorem follows from

Note the det is the image of the identity and the image of η is a generator.

P ic alg

In this short section we prove of Theorem 1.6 i.e. we calculate P ic alg 2 . We are left with the short exact sequence 0 → P ic alg,0 2 → P ic alg 2 → Z/2 → 0 that comes from the definition of P ic alg,0 2 (cf. paragraph 1.1). Note that the isomorphism of Proposition 1.2 sends (E 2 ) * S 2 to η (cf. paragraph 3.1). Thus (E 2 ) * S 2 is an element of P ic alg,0 1 that generates Z 3 × Z/8 in P ic alg 2 . But (E 2 ) * S 1 is not an element of P ic alg,0 2 , therefore its image in the above sequence is a generator of Z/2. Thus (E 2 ) * S 1 itself must generate Z 3 × Z/16.