Cyril Chevarin

Xunhao Wang

Denis Bouyer
email: denis.bouyer@umontpellier.fr

Volodymyr V Tarabara
email: tarabara@msu.edu

Thierry Chartier

André Ayral

CFD-guided patterning of tubular ceramic membrane surface by stereolithograhy: Optimizing morphology at the mesoscale for improved hydrodynamic control of membrane fouling

Keywords: ceramic membrane, computational fluid dynamics, stereolithography, fouling

    CFD-guided patterning of tubular ceramic membrane surface by stereolithography: Optimizing morphology at the mesoscale for improved hydrodynamic control of membrane fouling

Introduction

Rapid advances in additive manufacturing have paved the way for the simplified fabrication of integrated systems as well as system components with complex shapes hitherto inaccessible by conventional manufacturing methods. It is particularly the case for ceramic materials [START_REF] Chartier | Rapid Prototyping of Ceramics[END_REF][START_REF] Chartier | Additive manufacturing to produce complex 3D ceramic parts[END_REF][START_REF] Chen | 3D printing of ceramics: A review[END_REF], which require high temperature sintering to confer morphological, microstructural and mechanical properties to the shaped object. These advances open up new possibilities for the design of ceramic membranes with materials. Indeed, there is a growing interest in the use of additive manufacturing methods to prepare both membranes, including ceramic filters [START_REF] Chen | Three-dimensional printing of high-flux ceramic membranes with an asymmetric structure via digital light processing[END_REF][START_REF] Dommati | A comprehensive review of recent developments in 3D printing technique for ceramic membrane fabrication for water purification[END_REF][START_REF] Wang | Optimization of UV-curable alumina suspension for digital light processing of ceramic membranes[END_REF], and membrane modules [START_REF] Balogun | 3D printing and surface imprinting technologies for water treatment: A review[END_REF][START_REF] Femmer | Print your membrane: Rapid prototyping of complex 3D-PDMS membranes via a sacrificial resist[END_REF][START_REF] Issac | Review of manufacturing three-dimensional-printed membranes for water treatment[END_REF][START_REF] Lee | The potential to enhance membrane module design with 3D printing technology[END_REF][START_REF] Low | Perspective on 3D printing of separation membranes and comparison to related unconventional fabrication techniques[END_REF].

Many of the current membrane processes are based on the principle of crossflow filtration, with key concerns regarding the minimization of concentration polarization and fouling. Fouling management often requires a regular interruption of the filtration process by cleaning sequences (e. g. backwashing) limits the lifetime of the membranes. A complementary approach to minimizing fouling and its deleterious effects effects is to promote turbulence in the feed flow. The use of turbulence promoters is possible but at a cost of an increased pressure drop in the feed flow channel. Another option is to promote turbulence closer to the membrane surface by altering its topography. This has been the subject of various studies, most of which focused on the surface patterning of polymer membranes [START_REF] Choi | Sharkskin-mimetic desalination membranes with ultralow biofouling[END_REF][START_REF] Won | Correlation of membrane fouling with topography of patterned membranes for water treatment[END_REF][START_REF] Zhao | Optimization of patterned polysulfone membranes for microalgae harvesting[END_REF] although some did employ flat plate ceramic membranes [START_REF] Lyu | 3D-printed surface-patterned ceramic membrane with enhanced performance in crossflow filtration[END_REF][START_REF] Ng | Effect of surface-patterned topographies of ceramic membranes on the filtration of activated sludge and their interaction with different particle sizes[END_REF].

Most installations use ceramic membranes in a tubular configuration, the support of which is produced by extrusion. Based on a computational fluid dynamics (CFD) approach, Yang et al. explored optimization of multi-channel ceramic membranes [START_REF] Yang | CFD-based optimization and design of multi-channel inorganic membrane tubes[END_REF]. While the number and the section of the channels are easily adjustable, the same cannot be said for their surface morphology. Twenty years ago a clever method was developed, allowing the extrusion of tubular supports with a spiral internal surface favoring the turbulence of the flow [START_REF] Broussous | New ceramic membranes designed for crossflow filtration enhancement[END_REF]. During the 2018 International Conference in Inorganic Membranes, a presentation by TAMI Industries highlighted the CFD-based prediction of the benefits of having curvilinear flow inside a tubular membrane channel, the practical implementation of such flow field using bundles of helical membranes, as well as the potential of additive processes for manufacturing such membranes [START_REF] Lescoch | Game changer in inorganic membranes[END_REF].

The present work concerns a preliminary study on the morphological design of ceramic tubular membranes. Single-channel alumina tubular membranes with an inner diameter of 7 mm and an outer diameter of 10 mm were selected, corresponding to usual specifications for current commercial membranes of this type. The inner surface the membrane was patterned with rings -nmorphological elements that cannot be introduced at the membrane surface by the conventional shaping method based on paste extrusion. The ring profile and height as well as the inter-ring distance were optimized using CFD modeling by evaluating the impact of these parameters on the turbulence near the membrane surface. CFD predictions were validated in preliminary experiments with symmetric alumina microfiltration (pore diameter of ~ 0.2 μm) membranes, with or without such rings, manufactured by an additive process (i. e. stereolithography) from alpha alumina powder. The patterned membranes and smooth membranes (controls) were comparatively evaluated in terms of fouling behavior in experiments with P. aeruginosa. suspensions.

Numerical model

Geometry

The flow hydrodynamics in the tubular membranes was simulated for both smooth and patterned membranes as shown in Figure 1. Here, "patterned" refers to a membrane with the inner (feed-facing) surface having a patterned morphology. In both configurations (smooth or patterned inner surface), the outer diameter and the membrane length were 10 mm and 50 mm, respectively. The inner diameter of both membranes was 7 mm everywhere except in locations corresponding to roughness elements ("rings") on the surface of the patterned membrane; each such "ring" was ~ 0.5 mm high, leading to a somewhat smaller inner diameter in those locations. Thanks to the axial symmetry, simulations were performed using 2D axisymmetric geometry thus notably reducing the calculation time.

Governing equations

Two feed flow rates were simulated herein, with mean velocities of 0.45 m•s -1 and 0.68 m•s -1 , respectively in agreement with the values used in filtration experiments. Given the density and viscosity of water at 23 °C, the diameter of the flow channel, and the two mean velocities, Reynolds numbers of 3100 and 4830, respectively were calculated, corresponding to the end of the transition zone between laminar and turbulent hydrodynamic regimes. Consequently, the classical Reynolds Averaged Navier-Stokes (RANS) transport equations for incompressible Newtonian fluid and turbulent regime were used in the model.

In the RANS model, the conservation of momentum equation is given by:

𝜌𝜌(𝛻𝛻. 𝑣𝑣 ⃗)𝑣𝑣 ⃗ = -𝛻𝛻𝛻𝛻 + 𝛻𝛻. �(𝜂𝜂 + 𝜂𝜂 𝑡𝑡 )[(𝛻𝛻𝑣𝑣 ⃗ + 𝛻𝛻 𝑣𝑣 ⃗ 𝑇𝑇 )]� + 𝜌𝜌𝑔𝑔 ⃗ ( 1
)
where 𝜌𝜌 is the water bulk density (kg.m -3 ), 𝑣𝑣 ⃗ is the velocity vector (m.s -1 ), 𝛻𝛻 is the static pressure (Pa), [(𝛻𝛻𝑣𝑣 ⃗ + 𝛻𝛻 𝑣𝑣 ⃗ 𝑇𝑇 )] = 𝜏𝜏 ´ is the viscous stress tensor, 𝜂𝜂 is the dynamic viscosity of the water (Pa.s), 𝜂𝜂 𝑡𝑡 is the turbulent (or eddy) viscosity (Pa.s) and 𝑔𝑔 ⃗ is the acceleration due to gravity (m.s -2 ).

The standard k-ε model [START_REF] Launder | The numerical computation of turbulent flows[END_REF] (see Supplementary Information (SI), section S1) was used to simulate the turbulence, meaning that the turbulent viscosity 𝜂𝜂 𝑡𝑡 was modeled using eq. ( 2) [START_REF] Jones | The prediction of laminarization with a two-equation model of turbulence[END_REF]:

𝜂𝜂 𝑡𝑡 = 𝜌𝜌𝐶𝐶 𝜂𝜂 𝑘𝑘 2 𝜀𝜀 , (2)
where 𝐶𝐶 𝜂𝜂 is a constant, 𝑘𝑘 is the turbulent kinetic energy (m².s -2 ) and 𝜀𝜀 is the turbulent energy dissipation rate (m².s -3 ).

Boundary conditions

Fully developed flow with a mean velocity (in m.s -1 ) was considered in the model for boundary conditions in the inlet side. Two mean velocities were simulated, i. e. 0.45 m.s -1 and 0.68 m.s -1 . At the outlet side, a null static pressure boundary condition P 0 was fixed. A no-slip boundary condition was applied at the wall.

Numerical simulations

The hydrodynamic model was simulated using the commercial COMSOL Multiphysics® software, working with finite element method. Using a 2D axis-symmetric model, meshes size was ranged between 0.08 mm near the wall and 0.2 mm in the bulk of the flow. Boundaries layers and finer meshes were built near the wall surfaces to improve the numerical accuracy in the boundary layer. The number of tetrahedral meshes was close to 100 000 for the simulations without rings and 140 000 for the simulations with rings.

Experimental

Membrane manufacturing and microstructural characterization

The membranes were manufactured by the company 3DCERAM SINTO (Limoges, France) according to our specifications. The green bodies were built using the stereolithography additive process (CERAMAKER 900 machine) based on the selective polymerization of a reactive mixture system under the effect of a UV irradiation. The reactive system was a suspension of alumina particles (1.75 µm mean particle size) in a mixture of curable monomer and oligomer, with the addition of a photoinitiator. After debinding and sintering, the pieces were externally machined to achieve the requested dimensional specifications (outer diameter = 10.0 ± 0.2 mm; inner diameter = 7.0 ± 0.2 mm; length = 49.2 ± 0.2 mm).

For the rough membranes, the targeted characteristics for rings were as follows: height of 500 µm; a symmetrical profile with the first and second edges (considering the flow direction) forming an angle of 10° and 45°, respectively, the vertical of the surface of the rings; distance between two successive rings of 3 mm. The geometry of these rough membranes was confirmed by making a cutout in one of these tubes using a diamond saw, observing this sample by Keyence VHX 6000 digital microscope (VHX) and then analyzing the Keyence images with suitable software (VHX-6000_950F).

The microstructure of the membranes was investigated using a Field Emission Scanning Electron Microscope (FE-SEM; Hitachi S4800). The porosity of these macroporous samples was determined by mercury porosimetry (AutoPore IV 9500 Micromeritics).

Fouling experiments

Experimental setup and membrane permeability measurements

The membranes were tested using a crossflow filtration system (see SI, Figure S1), which included a peristaltic pump (910-0025, Thermo Scientific), a custom-made membrane unit and a back-pressure valve (SS-43S6, Swagelok). The membrane was mounted using two stainless steel push-to-connect fittings (KQG2H10-00, Grainger). To seal the membrane area under the fittings and ensure it is not available for permeation, the outside surface of the membrane under the fittings was coated with a thin layer of epoxy (LOCTITE, Henkel). The pure water flux tests for the 3D-printed membranes were performed using DI water and five different transmembrane pressure values: 1.03, 1.38, 1.72, 2.07, and 2.41 bar [START_REF] Lyu | 3D-printed surface-patterned ceramic membrane with enhanced performance in crossflow filtration[END_REF][START_REF] Launder | The numerical computation of turbulent flows[END_REF][START_REF] Brinker | Sol-Gel Science: The Physics and Chemistry of Sol-Gel Processing[END_REF]30, and 35 psi, respectively). The DI water (pH 5.8) was used as the feed solution in these tests. The crossflow velocity was maintained constant at either 0.45 m•s -1 or 0.68 m•s -1 . The permeability of each membrane was calculated based on the slope of the flux vs pressure dependence.

Bacterial suspensions: Preparation and characterization

An aqueous suspension of P. aeruginosa (Schroeter) Migula (PA01-LAC, ATCC) was used as the feed in membrane fouling tests. PA01 was cultured in Luria-Bertani (LB) broth to the exponential growth phase. The standard curve (colony-forming units per milliliter (CFU/ml) vs absorbance at 600 nm) was established.

The preparation of the bacteria stock is detailed in the SI file, section S2. The bacteria were enumerated by a colony count method. Briefly, the vortexed solutions were serially diluted, mixed with soft agar, and poured onto LB-agar plates, which were incubated at 37 °C overnight. The formed colonies were counted, and then the areal density of colony-forming units (CFU/ml) was calculated. The absorbance of the bacterial suspensions was measured by a spectrophotometer (Spectronic 21D). The hydrodynamic diameter of bacteria was measured by dynamic light scattering (DLS). In these tests, a small sample (0.5 mL) of the bacterial suspension was diluted in 1 mM KCL solution, prefiltered with 0.22 µm syringe filter, to the total volume of ~ 2 mL.

Membrane fouling tests

Prior to each crossflow filtration test, the feed suspension was prepared by diluting the bacteria stock tenfold with DI water. The volume fraction of PA01 in the feed was 3.9 10 -5 . Based on this very low value and the Einstein relation between relative viscosity and volume fraction of dispersed phase [START_REF] Einstein | A New Determination of Molecular Dimensions[END_REF], the viscosity of this suspension was considered to be equal to that of pure water. The ionic strength of the feed was 15 mM. During fouling tests, the crossflow velocity (averaged across the membrane channel cross-section) was 0.45 m•s -1 or 0.68 m•s -1 , which corresponded to the maximum crossflow rate (1570 mL•min -1 ) that the peristaltic pump could provide. Prior to challenging the membrane with the bacterial suspension, the membrane was equilibrated with 15 mM NaCl at pH 6. The permeate was collected on a mass balance with a 1 min interval. All fouling tests were done at the ambient temperature of 23 ℃ and the relative humidity of 47%. After each fouling test, the membrane was cleaned by circulating solutions in the feed channel in the following sequence: (1) DI water for 30 min, (2) 5 mM EDTA at pH 11 for 30min, (3) DI water for 30 min, (4) 2 mM SDS at pH 11 for 30 min, (5) DI water for 30 min [START_REF] Herzberg | Biofouling of reverse osmosis membranes: Role of biofilm-enhanced osmotic pressure[END_REF].

Three smooth and three rough membranes were tested under the same experimental conditions to estimate the reproducibility of the results. In order to investigate the contribution of the different fouling mechanisms to the overall decline in permeate flux, additional tests were carried out in the dead-end filtration regime following a procedure described earlier [START_REF] Wang | Pore blocking mechanisms during early stages of membrane fouling by colloids[END_REF].

Results and discussion

Membrane design from numerical simulations

As already said in the introduction, in the frame of this pioneering study, it was decided to pattern the inner surface of the tubular membranes with simple and continuous rings. The current limitations of the existing additive manufacturing in terms of dimensional tolerances were considered for sizing these rings. With the implemented technique, i.e. stereolithography, the dimensional resolution is about 100 µm [START_REF] Chartier | Additive manufacturing to produce complex 3D ceramic parts[END_REF]. It is the reason why rings with a height and a width of 500 µm were chosen. Whatever the configuration, 𝜀𝜀 is much higher close to the top of the rings compared to the region between the rings, with a ratio close to 23 at 0.45 m•s -1 and more than 42 at 0.68 m•s -1 . A 50% increase in the inlet velocity (from 0.45 to 0.68 m•s -1 ), leads to a five-fold increase in 𝜀𝜀 at the top of trgeh rings and a three-fold increase in 𝜀𝜀 between the rings. These results confirm the intuitive predictions, highlighting the interest to disturb the flow near the membrane surface to create more turbulence (more dissipation of turbulent kinetic energy). With this geometrical configuration, the risk lies in the formation of dead zone between the rings that is the zone with local re-circulation and characterized by a lower level of turbulence. Further, when analyzing the influence the ring shape (symmetric vs asymmetric), the numerical simulations clearly show that the low ε region, localized just downstream of each ring, is smaller behind rings with the asymmetrical profile. For both inlet velocities evaluated (0.45 or 0.68 m•s -1 ), asymmetry appears to be favorable for preventing fouling phenomena.

Based on these numerical results, the asymmetric profile was selected. The following study of the impact of the inter-ring distance (see SI, Figure S2) showed that it had a relatively minor effect on the spatial distribution of ε; the inter-ring distance of 3 mm was selected as allowing the most reduction in the spatial extent of low ε zones.

Table 1 summarizes the values of the energy dissipation rate 𝜀𝜀 and the pressure loss along the membrane. These values give an indication of the cost of integrating the rings as turbulence promoters within the tubular membrane. Whatever the inlet velocity, the pressure loss increases by a factor 15 when using rough membranes rather than smooth ones. This result is not surprising because it is directly linked to the total dissipation rate of turbulent kinetic energy, and it should be related to the energy cost of backwashing frequency that should be done between two filtration cycles.

Table 1. Specific pressure loss, ∆𝑃𝑃 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 , in the feed flow along the membrane and mean value of turbulent dissipation rate, 𝜀𝜀 𝑚𝑚𝑓𝑓𝑚𝑚𝑖𝑖 , at the membrane surface for two values of the average crossflow velocity, 𝑣𝑣 𝑖𝑖𝑖𝑖 . Turbulent dissipation rate is for the membrane surface between the rings and is deduced from the simulations graphically represented in Figure S2. 1 shows that a significant advantage can be expected with the rough membranes in terms of fouling limitation, especially for the largest value of average inlet crossflow velocity. The reported values of specific pressure loss in the feed flow qualitatively account for the additional energy cost associated with the use of rough membranes. The determined values are more than one order of magnitude larger for the rough membranes but rather low in absolute value.

Smooth membrane Rough membrane

𝑣𝑣 𝑖𝑖𝑖𝑖 = 0.45 m•s -1 𝑣𝑣 𝑖𝑖𝑖𝑖 = 0.68 m•s -1 𝑣𝑣 𝑖𝑖𝑖𝑖 = 0.45 m•s -1 𝑣𝑣 𝑖𝑖𝑖𝑖 = 0.68 m•s -1 𝜀𝜀 𝑚𝑚𝑓𝑓𝑚𝑚𝑖𝑖 (m

Microstructural characterization of the manufactured membranes

Figure 4 shows the internal morphology of the rough membranes with the axisymmetric structure of the roughness elements ("rings") clearly shown (Figure 4a). The ring profile (Figure 4b) and the inter-ring spacing (Figure 4c) correspond rather well to the targeted characteristics considering the 100 µm spatial resolution of the implemented additive method. SEM imaging of the planar surface (Figure 5a) and the cross-section (Figure 5b) of the membranes showed a close packing arrangement of grains with a bimodal size distribution. The continuous matrix with submicron porosity is embedded with grains several microns in size.

Mercury porosimetry measurements yielded the porosity of ~ 33 % and the specific surface area of ~ 2.5 m 2 g -1 . These values are consistent with what can be expected to result from the limited sintering of a random close-packing arrangement of powder particles. The pore size distribution (Figure 6) is centered around an average pore size of ~ 0.2 µm. The prepared membranes can thus be classified as microfilters.

Fouling experiments

Based on the DLS measurements, the average size of bacteria in the feed suspension was ~ 0.8 µm, which is significantly larger than the measured average pore size (~ 0.2 µm) of the membrane (Figure 6). The DLS measurements estimated the diffusion coefficient, which was then converted to particle size assuming that the scatterers are spherical.

Application of the Carman-Kozeny relation to mercury porosimetry data [START_REF] Brinker | Sol-Gel Science: The Physics and Chemistry of Sol-Gel Processing[END_REF] estimates the intrinsic permeability of the membrane to be ~ 1.6 10 -16 m 2 . The expected water permeability at 23 °C is thus equal to 42 ± 16 L•m -2 •h -1 •bar -1 , which is within the experimental error from the valued measured experimentally for a smooth membrane (69 ± 15 L•m -2 •h -1 •bar -1 ).

The results obtained in crossflow filtration tests are summarized in Figure 7. Based on the application of Hermia blocking law analysis to dead end filtration data (see SI, Figures S3 andS4), two fouling regimes could be discerne: pore blockage and cake filtration. The presence of integrated rings had no effect on pore blockage but led to a significant mitigation of fouling in the cake filtration regime at the higher value of crossflow velocity (Figure 7b). This is in good agreement with the numerical simulations and a mean value of turbulent dissipation rate at the surface of the membrane between the rings being much larger for the patterned membranes.

Considerations of scale

The fouling tests revealed a significant difference in roughness-induced mitigation of fouling between the two studied crossflow velocities. For a crossflow velocity of 0.45 m•s -1 , the smooth and rough membranes seem to be affected by fouling phenomena in a similar way. For both membrane types, the permeate flux decreased by ~ 22 % after 70 mL of permeate was collected. By contrast, significant differences were observed in tests with the crossflow velocity of 0.68 m•s -1 ; in this case, whereas the smooth membranes exhibit the same 22 % flux decrease, the flux decline was only 10 % in experiments with the patterned filter. This result was very encouraging and provided the proof of concept of the patterned membranes.

Near the membrane surface, between the rings, the dissipation rate of the turbulent kinetic energy (ε) is multiplied by a factor 3.5 when the velocity is increase from 0.45 and 0.68 m•s -1 . Turbulent energy dissipation rate 𝜀𝜀 is close to 3 m 2 •s -3 for 𝑣𝑣 𝑖𝑖𝑖𝑖 = 0.45 m•s -1 and 10 m 2 •s -3 for 𝑣𝑣 𝑖𝑖𝑖𝑖 = 0.68 m•s -1 , respectively (Table 2). Near the upper part of the rings, where the turbulence is the highest, ε is multiplied by a factor 5 when increasing the velocity (60 vs 300 m 2 •s -3 ). The values of the Kolmogorov microscale, 𝜅𝜅, are calculated using eq. ( 3) [START_REF] Kolmogorov | The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers[END_REF][START_REF] Kolmogorov | The local structure of turbulence in incompressible viscous liquid[END_REF][START_REF] Kolmogorov | Dissipation of energy in locally isotropic turbulence[END_REF] and reported in Table 2.

𝜅𝜅 = � 𝜂𝜂 3 𝜀𝜀 � 1 4 � (3)
Whatever the location (between the rings corresponding to the lowest values of ε, or near the upper part of the rings corresponding to the highest values of ε), 𝜅𝜅 increases 150 % with a decrease in the mean velocity from 0.68 to 0.45 m•s -1 . Kolmogorov microscale corresponds to the lowest scale of turbulence, where viscosity dominates and the turbulent kinetic energy is dissipated. So, 𝜅𝜅 is often a relevant hydrodynamic parameter, which could be linked to fouling mechanisms. Notably, the values of 𝜅𝜅 ~ two orders of magnitude smaller than characteristic dimensions of roughness features -the size of the rings (~ 1.5 mm) or the inter-ring distance (~ 3 mm). Thus, based on purely scale-based considerations, the improvement in permeate flux (Figure 7b) is tentatively attributed to the disruption of the filter cake growth at the scale on several microns (i.e. ~ 𝜅𝜅 µm. Even if the fouling mechanisms are complex and their mitigation could be linked to several phenomena, the significant increase of the dissipation rate of the turbulent kinetic energy and the associated decrease of the Kolmogorov microscale are in good agreement with the fouling results reported in Figure 7.

Potential of stereolithography for ceramic membrane design

Through creating desired surface morphology at the nanoscale, additive manufacturing methods help control flow to mitigate deposition of colloids and larger particle and, thereby, mitigate membrane fouling. However, the methods impose limitations on the material properties of the membrane that supports such surface features. First, small alumina particle size helps improving both the spatial resolution of the stereolithography method and the mechanical properties of membranes post-sintering; however, the smaller size of the primary particles leads to a low final pore size. The pore diameter of ~ 0.2 µm reported in this work is more than an order of magnitude smaller than the pore size achievable by extrusion of pastes formulated from coarser and monodispersed powders. Addition of pore-forming agents in the stereolithography formulation is thus recommended to increase the pore size and achieve a better balance between resolution and permeability. Second, symmetrical patterned microfiltration membranes could be used as a porous support for the production of patterned ultrafiltration or nanofiltration membranes by deposition of successive thin layers by slip casting.

Conclusions

CFD simulations were successfully used for guiding the design of 3D-printed tubular ceramic membranes and optimizing the surface morphology at the mesoscale in order improve the hydrodynamic control of membrane fouling. CFD predictions were corroborated by the experimental data on membrane fouling. The mesoscale design of microfiltration membranes by additive manufacturing can be combined with preparing the lower porosity separation layer by conventional methods of separation layer. Such optimal combination of membrane fabrication methods may be practically implemented through a multistep/multi-method morphological design as an alternative to changing the casting mixture composition for pore size control.

Figure 1 .

 1 Figure 1. Geometry of smooth and patterned tubular membranes.

  Rough membrane -symmetrical profile (c) Rough membrane -asymmetrical profile

Figure 2 .Figure 3 .

 23 Figure 2. Dissipation rate of turbulent kinetic energy, ε, in the vicinity of the rings on the surface of the tubular membrane for 𝑣𝑣 𝑖𝑖𝑖𝑖 = 0.45 m•s -1 : (a) smooth membrane, (b) symmetrical profile, (c) asymmetrical profile.

Figure 4 .Figure 5 .

 45 Figure 4. Geometry of the rough membrane: (a) image of the inner surface of the membrane cutout. (b) and (c): Characteristic sizes of roughness elements.

Figure 6 .

 6 Figure 6. Pore size distribution of ceramic membranes and the size distribution of bacteria in the feed suspension.

Figure 7 .

 7 Figure 7. Normalized permeate flux recorded in crossflow filtration tests with smooth and rough membranes. (a) a crossflow velocity of 0.45 m•s -1 ; (b) and a crossflow velocity of 0.68 m•s -1 . Empty symbols correspond to the filtration stage where fouling is due to pore blocking while filled symbols correspond to the cake filtration regime.

Table 2 .

 2 Kolmogorov microscale 𝜅𝜅 in low 𝜀𝜀 and high 𝜀𝜀 zones for two values of the average crossflow velocities

	Location along the	Mean crossflow	Turbulent energy	Kolmogorov
	membrane surface	velocity, 𝑣𝑣 𝑖𝑖𝑖𝑖 (m•s -1 )	dissipation rate, 𝜀𝜀 (m 2 •s -3 )	microscale, 𝜅𝜅 (µm)
	Between the rings	0.45	3.2	23.6
		0.68	10.5	17.6
	Above the ring	0.45	60	11.4
		0.68	300	7.6

Acknowledgments

We are grateful to Mr. Didier Cot and Dr. Martin Drobek (both at Institut Européen des Membranes) for performing SEM and Hg porosimetry analyses, respectively. We also thank 3DCERAM SINTO (Limoges, France) for the manufacturing the prototype membranes and providing them to our team. This material is 14 based upon work supported in part by the U.S. National Science Foundation under grant No. OISE-1952438, which supported the cooperation between Institut Européen des Membranes and Michigan State University.

Author statement

Cyril Chevarin: Methodology, Investigation, Analysis, Writing -Reviewing and Editing

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Valentin Vaillant-Coindard

Florian Chotard

Benjamin Théron

Cédric Balan

Jérôme Bayardon

Raluca Malacea-Kabbara

Ewen Bodio

Yoann Rousselin

Paul Fleurat-Lessard

Pierre Le Gendre

Bis(salicylamidine) Ligands (FAlen): A Variant of Salen with "à la Carte" Denticity

come

INTRODUCTION

Tetradentate N2O2 Schiff-bases, derived from salicylaldehydes and 1,2-diamines, also known under the name of salen, are ubiquitous ligands able to give complexes with most metal ions of the periodic table (Figure 1). 1 Metal-salen complexes find application in many fields ranging from catalysis, 2 chemical sensors, [START_REF] Strianese | Salen-type aluminum and zinc complexes as two-faced Janus compounds: contribution to molecular sensing and polymerization catalysis[END_REF] molecular magnets, [START_REF] Miyasaka | Magnetic assemblies based on Mn(III) salen analogues[END_REF] energy storage [START_REF] Freire | Metallo(salen) complexes as versatile building blocks for the fabrication of molecular materials and devices with tuned properties[END_REF] to biological applications. [START_REF] Erxleben | Transition metal salen complexes in bioinorganic and medicinal chemistry[END_REF] In catalysis, salen complexes are considered as benchmark catalysts for a wide array of reactions, like the coupling of CO2 and epoxides into cyclic or polymeric carbonates, 7 the stereocontrolled polymerization of rac-lactide (LA) [START_REF] Hormnirun | Study of ligand substituent effects on the rate and stereoselectivity of lactide polymerization using aluminum salen-type initiators[END_REF] or the stereoselective epoxidation of unfunctionalized alkenes. [START_REF] Jacobsen | Highly enantioselective epoxidation catalysts derived from 1,2-diaminocyclohexane[END_REF] One of the keys to the success of salen-type ligands is their ease of access and modularity.

Their steric and electronic characteristics can indeed be easily tuned by choosing appropriate diimine bond and phenyl ring substituents. More drastic modifications of the ligand were also made by replacing either one or both imines functions with an amine 10 or with an iminophosphorane. [START_REF] Marín | Salen Ligands: What Does the Phosphorus Change?[END_REF] These systems were found to be very efficient, sometimes even outperforming the parent salen ligands. 12 In this context, we recently developed phenoxy-amidines (FA), a new variant of phenoxy-imines (FI), aka hemi-salen, in which the imine function is replaced by a N,N,N'-trisubstituted amidine (Figure 1). [START_REF] Vaillant-Coindard | Phenoxy-Amidine Ligands: Toward Lactic Acid-Tolerant Catalysts for Lactide Ring-Opening Polymerization[END_REF] Amidines differ from imines in that the nitrogen lone pair electrons of the amidine NR2 group is involved in resonance, thus making them much more basic and more stable towards nucleophiles than imines. We envisioned that the strong σ and π-donor character of the amidine function will provide additional stability to the metal ion resulting in robust complexes. The first proof of concept was obtained with phenoxy-amidine-Zn complexes that can promote the ROP of rac-lactide contaminated with up to 4 mol% lactic acid (a protic impurity often present in technical grade lactide), while phenoxy-imine-Zn analog catalysts are inactive under the same conditions. Based on the same strategy, we sought to develop an amidine-variant of salen ligands. Herein, we report the synthesis of these ligands, which we named FAlen (Figure 1), their coordination chemistry with Al as well as preliminary results in ROP of rac-lactide.

RESULTS AND DISCUSSION

We first reacted ethylene diamine (0.5 equivalents) with the chloroiminium chloride derivative generated from N,N-diethyl-2-methoxybenzamide (Scheme 1). After deprotection of the phenol group from methyl ether, the bis-(phenol-amidinium) salt [L1H4]Br2 was obtained in 73% yield (Scheme 1). Its 1 H NMR spectrum shows low field shifted protons at 9.7 ppm and at 9.59 ppm corresponding to OH and NH protons, respectively. The NEt2 protons give two set of signals due to restricted rotation around the N-C amidine bound. A diastereotopic split is further observed for the methylene protons of both the ethyl groups and the ethylene bridge due to atropisomer axial chirality, generated by the restricted rotation of Csp 2 -Csp 2 bond of the phenol moieties. The structure of [L1H4]Br2 was confirmed by XRD analysis (Figure 2). The two amidinium moieties For the coordination chemistry study, the proligand L1H2 was generated in situ from the well characterized bisamidinium salt [L1H4]Br2 using two equivalents of KHMDS. The alkane metathesis pathway from L1H2 and AlMe3 (1 equiv.) yielded a new compound 1 whose 1 H NMR spectrum was not consistent with the expected (L1)AlMe-like complex (Scheme 2). Instead, a labile proton at 7.8 ppm and two AlMe signals at -1.08 ppm and -0.61ppm were observed. The compound was crystallized and its XRD analysis shows the FAlen L1 ligand in κ 2 O coordination mode. Two methyl groups complete the coordination sphere of aluminum giving a dimethyldiphenoxyaluminate with a tetrahedral geometry (τ4' = 0.91) [START_REF] Okuniewski | Coordination polymers and molecular structures among complexes of mercury(II) halides with selected 1-benzoylthioureas[END_REF] . One of the two amidine functions is protonated and serves as counter-ion. The expected trans-cis amidine isomerization previously observed with phenoxy-amidine ligands did not occur in this case, giving rise to a FAlen ligand coordinated in a κ 2 O fashion.

Scheme 2. Synthesis of the FAlen aluminate complex 1. We then decided to develop a new FAlen ligand L2 by introducing a phenylene rather than an ethylene linker between two N,N-dimethyl-substituted phenoxy-amidine groups. We hypothesized that this should result in a ligand with a better preorganized structure to wrap the metal ion. The proligand L2H2 was first synthesized in 36% overall yield according to the same procedure as L1H2 using 2-methoxy-N,N-dimethylbenzamide and o-phenylenediamine. The yield was further improved substantially to 67% by in-situ generating Vilsmeier-type reagent from unprotected 2hydroxy-N,N-dimethylbenzamide and thionyl chloride, and subsequent addition of o-phenylene diamine (Scheme 3).

Scheme 3. Synthesis of the FAlen proligand L2H2.

In the solid-state, the structure of L2H2 shows a zwitterionic form with one of the two phenolamidine fragments protonated at N1' and the other deprotonated at O, both being connected by an intramolecular (O -•••HN + ) hydrogen bond. Both amidine and amidinium moieties feature transconfiguration and small ΔCN values of 0.061(3) and 0.001(3) Å, respectively (Figure 4). The The reaction of L2H2 with one equivalent of AlMe3 in THF at 60°C for 2 h afforded (L2)AlMe complex 2 in 81% yield (Scheme 4). X-Ray analysis of crystals of 2 indicates that the Al atom has a pseudo-tetrahedral geometry with the FAlen L2 ligand in κ 3 O,O,N coordination mode and a methyl ligand occupying the fourth coordination site (Figure 5).

Scheme 4. Synthesis of the FAlen methyl Al complex 2.

The Al atom is located in the FAlen cavity at a distance of 2.97 Å from the trans-amidine atom N1', which is shorter than the sum of the van der Waals radii (3.39 Å). Assuming that both amidine functions in the proligand L2H2 are trans in solution, in agreement with the solid-state structure and the known greater thermodynamic stability of trans-amidine, it means that trans-cis isomerization of one of the two amidine functions occurred upon coordination. Surprisingly, the 1 H and 13 C NMR spectra of 2 in CD2Cl2 at 298 K shows a set of signals consistent with a Cs symmetric structure. We hypothesized that a dynamical process must take place in solution and can explain this inconsistency. To get some insights on the fluxional behavior of 2, we performed DFT calculation at the M06-2x/6-31+G(d,p) computational level (in DCM) (Figure 6). First of all, it shows that amidine functions easily isomerize during coordination with the metal by rotation around the C=N bond with very shallow energy barriers (ΔG ‡ ≤ 12.9 kcal.mol -1 ). These energy values are consistent with a dynamic process occurring in solution at room temperature or even below (the 1 H NMR spectra of 2 did not change significantly down to 193 K, see Fig. S27). The FAlen ligand can thus momentarily have the two amidine functions in a cis-configuration, ideally suited for κ 4coordination and the formation of stepped and umbrella-shaped complexes (B1/B2 and C). All these complexes are in fast equilibrium in solution at room temperature, but the enantiomeric κ 3coordinated complexes (A1/A2) were found more thermodynamically stable than the κ 4 -forms (B1/B2 and C), as observed experimentally. It should be noted that a much higher energy barrier (ΔG ‡ = 20.4 kcal.mol -1 ) was estimated for trans-cis amidine isomerization within a hemiFAlen proligand (2-hydroxy-N,N-dimethyl-N'-propylbenzamidine), in agreement with values previously reported in the literature (See SI, Fig. S48-S49). 15 We then hypothesized that by decreasing the electron density at the Al center, the κ 4 -FAlen coordination mode should be thermodynamically favored over the κ 3 -form. For this, we carried out the alcoholysis of 2 with Schrock's alcohol (Scheme 5). After 3 days at 45°C in CD2Cl2, a single complex 3 was formed. It's 1 H NMR spectrum was consistent with a Cs symmetric structure, showing only 6 signals for the aromatic protons and one broad signal for the NMe2 groups. The high field shifted AlMe signal of 2 located at δ = -1.12 ppm was replaced by a signal at δ = 0.67 ppm corresponding to the AlOC(CF3)2Me group. Scheme 5. Synthesis of the FAlen alkoxy Al complex 3.

Suitable crystals for X-Ray diffraction study were obtained by slow diffusion of pentane in a dichloromethane solution of 3 (Figure 7). In the solid state, the Al atom has a square-based pyramidal geometry (τ5 = 0,01) [START_REF] Addison | Synthesis, structure, and spectroscopic properties of copper(II) compounds containing nitrogen-sulphur donor ligands; the crystal and molecular structure of aqua[1,7-bis(N-methylbenzimidazol-2′-yl)-2,6dithiaheptane]copper(II) perchlorate[END_REF] with the FAlen ligand in basal position and in κ 4 -coordination mode. The alkoxy ligand occupies the apical position, in the cavity formed by the FAlen ligand in an umbrella-shaped conformation. The Al-O1/O1' and Al-N1/N1' distances are 1.816(2)-1.815(2) Å and 1.977(2)-1.987(2) Å, respectively, in the same range as those observed in analogous Salen(Al) complexes. 17 DFT calculations carried out on complex 3 revealed a dynamic behavior similar to that of 2 (see SI, Fig. S50), with however lower energy for the κ 4 -complex (C') than for the κ 3 -form (A1') (Figure 8). NBO studies were then carried out in order to evaluate the residual charge of aluminum in these different complexes and to see if a correlation exists between the electronic richness of the metal and the coordination mode of the ligand. Expectedly, NBO analysis show higher positive charge on Al in the alkoxy complex with respect to the alkyl complex. Coordination of the second amidine function decreases the charge on Al in both cases. It should be noted that the most stable complex in each series has almost the same charge on Al. It therefore seems that the FAlen ligands can adapt their mode of coordination and their geometry according to the electronic richness of the metal ion. Suitable crystals for X-Ray diffraction study were obtained by diffusion technique from the NMR sample (CD2Cl2/pentane/Py-D5). The ORTEP view presented on Figure 9 shows a dimeric structure in which the two cationic Al units are bridged by the oxygen atoms of one of the phenoxy groups of each FAlen ligand. The aluminum atom adopts a pseudo-octahedral geometry defined by the four O2N2 atoms of the FAlen ligand in the equatorial plane. The O atom of a bridging aryloxy group and the N atom of an additional pyridine ligand occupy the two axial positions.

Thus, the ethylene-bridged FAlen ligand L1 was shown to be able to adopt a κ 4 -coordination modes with an umbrella-shaped conformation like L2 in complex 3. The bond lengths Al-O, Al-O', and Al-N are broadly comparable to those reported for similar octahedral aluminum cations with Salen-type ligands. Similarly, the reaction of B(C6F5)3 with complex 2 afforded the cationic complex 5 which was then characterized by NMR in CD2Cl2/Pyridine-D5 and crystallized as bis(pyridine) adduct of a dicationic dinuclear Al complex (Figure 9).

Salen aluminum complexes have been widely used as initiator for the Ring Opening

Polymerization (ROP) of lactide. Spassky firstly showed that simple (salen)AlOMe can initiate the ROP of rac-lactide with moderate activity but with slight isotactic bias. [START_REF] Le Borgne | Ring-opening oligomerization reactions using aluminium complexes of schiff's bases as initiators[END_REF] These results triggered intensive research in the field and a great variety of salen-type ligands with different diimine linkers and substituents on the phenolate rings have been synthesized and tested allowing the identification of highly isoselective catalysts, albeit moderately active. [START_REF] Hormnirun | Study of ligand substituent effects on the rate and stereoselectivity of lactide polymerization using aluminum salen-type initiators[END_REF] We therefore choose this benchmark reaction to test the efficiency of the FAlen ligands. Polymerization runs were first conducted at 90°C in toluene in the presence of 1 equiv. of i PrOH as the alcohol source (Table 1).

The FAlen dimethyl aluminate complex 1 and the (FAlen)AlMe complex 2 led to comparable results affording PLAs in 94-95% conversion after 24 h (entries 1 and 5). The cationic system 4 and 5 were much less active in the same conditions, affording only 48% and 18% of conversion, respectively (entries 8-9). At lower temperatures, complex 1 was clearly the most effective achieving complete conversion of the monomer after 7 h at 50°C while complex 2 afforded only 68% conversion after 15 h at 70°C (entries 3 and 6). At room temperature, in dichloromethane, complex 1 is the only one still active with monomer conversion reaching 74% after 9 h (entry 4).

The GPC analysis of the different polymer samples showed a good agreement between experimental and theoretical Mn values and relatively narrow molecular weight distributions (Đ = 1.11-1.44). NMR analysis showed PLA chains with slight isotactic bias (Pm = 0.53-0.62). The kinetic of the reaction with complex 1, at 25 °C in CH2Cl2 (Fig. 10), showed a linear relationship between ln([LA]0/[LA]t) and time with a rate constant kapp of 0.0027 ± 0.0003 min -1 comparable to the one previously obtained with the parent (salen)AlMe complex, at 70 °C and higher catalyst loading (kapp = 0.0035 min -1 , [LA]0/[initiator]/[PhCH2OH] = 50/1/1, toluene, 70 °C). [START_REF] Hormnirun | Study of ligand substituent effects on the rate and stereoselectivity of lactide polymerization using aluminum salen-type initiators[END_REF] The evolution of the Mn and PDI values with conversion confirmed the well-controlled character of the polymerization, when promoted by complex 1 and isopropanol as co-initiator. Noteworthy, the MALDI-ToF spectrum obtained in these conditions showed only one series of PLA chains terminated with isopropoxy ester with a peak spacing of 144 Da and no intercalated peak due to secondary transesterification reactions (See SI, Fig. S47). Measured by GPC in THF (30 ˚C) using PS standards and corrected by applying the appropriate correcting factor (0.58). e Calculated using Mn,theo = [rac-LA]0/[catalyst]0 × MLA × conversion. f Determined from the methine region of the HD 1 H NMR spectrum. g GPC chromatogram presents bimodal profile.

Figure 10. Left: First-order logarithmic plot for the polymerization of rac-LA at 25°C in DCM using 1 with i PrOH as initiator (100:1:1). Right: Mn and PDI against conversion.

CONCLUSION

We have described a new variant of the ubiquitous salen by incorporating two amidine moieties in the ligand backbone. These new ligands, named FAlen, provide access to a series of aluminum complexes and showed quite unique coordination chemistry compared to salen. In the solid state,

Mn (g/mol)

k app = 0.0027 ± 0.0003 aluminate complex 1 was the most efficient complex, being able to initiate the ROP of LA at room temperature making it much more active than its parent salen complex. Efforts are currently underway to study further the flexible coordination mode of FAlen with other metals and to develop new FAlen ligands with different diamidine linkers and substituents on the phenolate platforms.

EXPERIMENTAL SECTION

General considerations

All reactions, except when indicated, were carried out under an atmosphere of argon using Assignment of 1 H and [START_REF] Vaillant-Coindard | Phenoxy-Amidine Ligands: Toward Lactic Acid-Tolerant Catalysts for Lactide Ring-Opening Polymerization[END_REF] C signals (when possible) was done through the use of 2D experiences (COSY, HSQC and HMBC). Nomenclature: the positions of carbon and hydrogen atoms in the aromatic rings of FAlen ligands were labelled according to Figure 10.

Synthesis of N,N-diethyl-2-methoxy-chlorobenziminium chloride. N,N-diethyl-2-

methoxybenzamide (1 equiv., 25 mmol, 5.18 g) was dissolved in toluene (25 mL) and oxalyl chloride (5 equiv., 125 mmol, 10.9 mL) was added. The mixture was stirred 16 h at 60 °C. The volatiles were evaporated, and the residue was washed with diethyl ether (5 x 50 mL), and dried to give the pure product as a beige solid (

Synthesis of [L1H4][Br2]

. N,N-diethyl-2-methoxy-chlorobenziminium chloride (2 equiv., 4 mmol, 1.05 g) was solubilized in DCM (40 mL) and cooled to 0°C. Triethylamine (10 equiv., 20 mmol, 2.70 mL) was added, and then slowly ethylenediamine (1 equiv., 1 M in DCM, 2.00 mmol, 2.00 mL). The mixture was stirred 16 h while warming to rt. The mixture was hydrolysed with NaOH solution (1 M, 30 mL), diluted with water (30 mL), and extracted with DCM (3 x 40 mL).

The gathered organic layers were dried with MgSO4, and concentrated to give the crude methoxy-protected FAlen ligand as an oil. The oil was solubilized in DCM (40 mL) and cooled to 0°C.

Boron tribromide (6 equiv., 1 M solution in DCM, 12.0 mmol, 12.0 mL) was slowly added, and the mixture stirred 16 h while warming to rt. The mixture was cooled to 0°C and slowly quenched with water (50 mL), and extracted with DCM (3 x 50 mL), and further extracted with chloroform (114 mg, 0.2 mmol) in 10 mL of dried THF. After stirring 4 h at r.t., a 2M solution of AlMe3 in hexanes (100 µL, 0.2 mmol, 1 equiv.) in 2 mL of THF was added and the reaction mixture was stirred for another 2 h at r.t. The supernatant was cannulated and after removing the THF, the solid was washed twice with 5 mL of pentane to give aluminum complex 1 as a beige solid (58 mg, 62% Synthesis of L2H2. 2.0 g (12.1 mmol) of 2-hydroxy-N,N-dimethylbenzamide were solubilized into 10 mL of SOCl2. The mixture was stirred overnight at r.t. and the solvent was then evaporated under vacuum. 15 mL of dried CH2Cl2 was added and further evaporated. A solution of the crude product and 8.4 mL (60.5 mmol, 5 equiv.) of triethylamine in 50 mL of dried CH2Cl2 was then cooled to 0 °C and 598.6 mg (5.5 mmol, 0.45 equiv.) of o-phenylenediamine were added. The mixture was allowed to reach r.t. and stirred for 2 h 30. After hydrolysis with 50 mL of distilled water, the product was extracted with CH2Cl2 (4 x 50 mL). The organic phases were combined and dried over MgSO4 before being evaporated. The crude product was taken back in 50 mL of diethyl ether and a 1M solution of HCl in diethyl ether was added up to pH = 1. The brownish product was filtrated under vacuum and washed with 3x40 mL of diethyl ether. The whitish solid was suspended in 10 mL of distilled water and a 5M aqueous solution of NaOH was added up to pH = 9-10. The product was extracted with 50 mL of CH2Cl2. The organic phase was washed with 3x10 mL of distilled water before being dried over MgSO4 and evaporated under vacuum. The crude product was purified by column chromatography on silica gel using MeOH as eluent to afford 1.48

g

Synthesis of 5.

In glovebox, 88.5 mg (0.2 mmol) of complex 2 and 102.4 mg (0.2 mmol, 1 equiv.) of BCF were introduced in a Schlenk tube and solubilized in 4 mL of dry CH2Cl2. The mixture was stirred at r.t. for 1 h and the volatiles were evaporated under vacuum in glovebox. The crude product was then washed with pentane and dried under vacuum to afford 157 mg of complex Polymer synthesis procedure: Under argon, 288.3 mg (2.00 mmol) of rac-Lactide and 0.01 equiv. of catalyst are weighed in a vial then mixed in the chosen solvent. Afterwards, 0.01 equiv.

of co-initiator is added with a 5µL Hamilton then the vial is crimped and let stirred at the desired temperature.

Gel Permeation analysis:

The number-average, weight-average masses (Mn and Mw respectively) and molar distribution (Mw/Mn) of the polyester samples were determined by size exclusion chromatography at 30°C with the Agilent 1260 Infinity system equipped with Varian 390-LC refractometer detector. Tetrahydrofuran (THF) was used as the eluent and the flow rate was set up at 1.0 mL/min. A 5 µm pre-column and two aligned Polypore 5 µm (7.5 × 300 mm) were used. Calibration were performed using polystyrene standards (900-1 000 000 g/mol) and raw values of Mn were thus obtained.

MALDI-ToF analysis:

Mass spectra were acquired on a time-of-flight mass spectrometer (MALDI-ToF-Microflex LRF, Brucker Daltonics). An external quadratic multi-point calibration was carried out before each measurement using polyethylene glycol (PEG) mixed in methanol

Figure 1 .

 1 Figure 1. Phenoxy-imine (FI), salen, phenoxy-amidine (FA) and FAlen ligands.

  Scheme 1. Synthesis of the FAlen proligand L1H2.

Figure 2 .

 2 Figure 2. ORTEP views of [L1H4]Br2 (top) and [L1H2]2•H2O (bottom). Thermal ellipsoids are

Figure 3 :

 3 Figure 3: ORTEP views of 1. Thermal ellipsoids are drawn at 30% probability plot. Hydrogen

  1 H NMR spectrum of L2H2 in CD2Cl2 at 298 K is consistent with a C2-symmetric compound. It indeed shows only six signals in the aromatic proton region and two broad signals at δ = 3.21 and 2.77 ppm for the NMe2 groups due to the restricted rotation about the amidine bounds. The apparent inconsistency between the solid-state and solution structures of L2H2 and the absence of phenol proton signals could be explained by rapid exchanges of protons in solution between the phenol and amidine functions.

Figure 4 :

 4 Figure 4: ORTEP views of L2H2•MeOH. Thermal ellipsoids are drawn at 30% probability plot. H atoms not involved in H-bonds are omitted for clarity. Selected distances (Å): L2H2 C1'=N1'H +

Figure 5 :

 5 Figure 5: ORTEP views of complex 2. Thermal ellipsoids are drawn at 50% probability plot.

Figure 6 :

 6 Figure 6: Dynamic behavior of complex 2. Gibbs Free energy profiles computed at the M06-2X/6-

Figure 7 .

 7 Figure 7. ORTEP views of 3. Thermal ellipsoids are drawn at 30% probability plot. Hydrogen

Figure 8 .Scheme 6 .

 86 Figure 8. NBO charges on Al and free Gibbs energies of the κ 3 and κ 4 -coordination modes in

Figure 9 .

 9 Figure 9. ORTEP views of the bis(pyridine) adduct of the dimeric form of complexes 4 and 5.

(

  FAlen)Al(III) complexes were found either in κ 2 O,O', κ 3 N,O,O' or κ 4 N,N',O,O'coordination mode. DFT calculations have shown that amidines easily isomerize within the complexes which confers fluxional properties and versatile denticity to FAlen. Finally, (FAlen)Al complexes were tested for the ROP of rac-Lactide in the presence of i PrOH as co-initiator. The (FAlen-κ 2 )AlMe2

Figure 10 .

 10 Figure 10. Numbering scheme for the assignment of the NMR spectra

  CDCl3): δ (ppm) = 8.09 (dd, J = 7.7 Hz, 1.7 Hz, 1H, H7), 7.61 (ddd, J = 8.6, 7.5 Hz, 1.7 Hz, 1H, H5), 7.20 (pseudo td, J = 7.6 Hz, 0.9 Hz, 1H, H6), 7.06 (d, J = 8.5 Hz, 1H, H4), 4.82 (m, 1H, N(CH2CH3)2), 4.45 (m, 1H, N(CH2CH3)2), 4.29 (m, 2H, N(CH2CH3)2), 3.94 (s, 3H, OMe), 1.66 (t, J = 7.4 Hz, 3H, N(CH2CH3)2), 1.46 (t, J = 7.3 Hz, 3H, N(CH2CH3)2). 13 C{ 1 H} NMR (75 MHz, CDCl3): δ (ppm) = 171.7 (C1), 154.6 (C2), 135.4 (C5), 129.6 (C7), 122.3 (C6), 120.4 (C2), 111.7 (C4), 56.5 (CH3 + CH2 overlapping, OMe, N(CH2CH3)2), 54.7 (N(CH2CH3)2), 13.1 (N(CH2CH3)2), 12.6 (N(CH2CH3)2).

( 3 x

 3 40 mL). The gathered organic layers were dried with MgSO4, and concentrated to give the bis(salicylamidinium) [L1H4][Br2] as a beige solid (840 mg, 73 %). HR-MS (ESI-pos): calcd for [C24H35N4O2] + [M+H] + : 411.27545. Found: 411.27406 (-3.4 ppm). 1 H NMR (500 MHz, CD2Cl2) : δ (ppm) = 9.70 (s, 2H, OH), 9.59 (broad signal, 2H, NH), 7.60 (d, J = 8.3 Hz, 2H, H4), 7.39 (ddd, J = 8.5, 5.5, 3.5 Hz, 2H, H5), 6.98-6.92 (m, 4H overlapping, H6 + H7), 4.13 (m, 2H, N(CH2CH3)2), 3.87 (m, 2H, N(CH2CH3)2), 3.38-3.22 (m, 6H, H8 + N(CH2CH3)2), 3.22-3.15 (m, 2H, H8), 1.44 (t, J = 7.2 Hz, 6H, N(CH2CH3)2), 1.12 (t, J = 7.2 Hz, 6H, N(CH2CH3)2). { 1 H} 13 C NMR (125 MHz, CD2Cl2): δ (ppm) = 163.5 (C1), 154.4 (C3), 133.7 (C5), 126.9 (C7), 120.7 (C6), 118. 5 (C4), 114.6 (C2), 48.2 (N(CH2CH3)2), 44.9 (N(CH2CH3)2), 44.7 (C8), 13.7 (N(CH2CH3)2), 12.3 (N(CH2CH3)2). Synthesis of 1. In glovebox, KHMDS (79.8 mg, 0.4 mmol, 2 equiv.) were added to [L1H4][Br2]

5. 15 .

 15 1 H NMR (500 MHz, CD2Cl2 + Pyridine-D5, 298K): δ (ppm) = 7.26-7.18 (m, 4H, H5 + H7), 6.91 (d, J = 7.8 Hz, 2H, H6), 6.58 (t, J = 7.8 Hz, 2H, H4), 3.64 (s, 4H, H8), 3.38 (q, J = 7.0 Hz, 8H, N(CH2CH3)2), 1.11 (t, J = 7.0 Hz, 12H, N(CH2CH3)2), 0.69 (s, 3H, BMe). { 1 H} 13 C NMR (125 MHz, CD2Cl2 + Pyridine-D5): δ (ppm) = 172.53 (C1), 166.64 (C3), 148.30 (m, CF), 139.31 (m, CF), 138.21 (m, CF), 137.38 (m, CF), 135.78 (C5 or C7), 132.83 (C7 or C5), 123.80 (C4), 1118.66 (C2), 117.47 (C6), 47.84 (N(CH2CH3)2), 47.03 (C9), 15.01 (N(CH2CH3)2), 11.20 (BMe). 19 F NMR (470 MHz, CD2Cl2 + Pyridine-D5): δ (ppm) = -132.81 (d, J = 19.1 Hz, 2F), -165.0 (t, J = 20.2 Hz, 1F), -167.59 (m, 2F).

Table 1 .

 1 ROP of rac-lactide mediated by FAlen-Al complexes 1

,2, 4 and 5 a

  Polymerization conditions: [rac-LA]0 = 1 M, 100 equiv. of rac-LA, 1 equiv of i PrOH and 1 equiv of metal catalyst, toluene. b Reactions performed in CH2Cl2. c Monomer conversion.d

	Entry	Init.	T (°C)	t (h)	Conv (%) c	Mn, exp d Mn, theo e	Đ	Pm f
	1	1	90	24	94	11 400	13 500	1.31	0.53
	2	1	70	15	99	10 200	14 400	1.44	0.63
	3	1	50	7	99	12 000	14 400	1.11	0.62
	4	1 b	25	9	74	7 500	10 700	1.11	0.57
	5	2	90	24	95	12 100	13 700	1.37 g	0.54
	6	2	70	15	68	8 300	9 800	1.22	0.56
	7	2 b	25	9	0	-	-	-	-
	8	4	90	24	48	4 500	6 900	1.11	-
	9	5	90	24	18	-	-	-	-

a

  Bruker 500 Avance Neo, or Bruker 600 Avance HD spectrometers. All acquisitions were performed at 298 K. Chemical shifts are quoted in parts per million (δ) relative to TMS (for 1 H and 13 C). For 1 H and[START_REF] Vaillant-Coindard | Phenoxy-Amidine Ligands: Toward Lactic Acid-Tolerant Catalysts for Lactide Ring-Opening Polymerization[END_REF] C spectra, values were determined by using solvent residual signals (e.g.

	CHCl3 in CDCl3) as internal standards. The apparent multiplicity of the 1 H signals is reported.

conventional Schlenk techniques and Ar glovebox. DCM, diethyl ether, THF, toluene, and pentane were dried using a MBRAUN SPS 800. Analyses were performed at the "Plateforme d'Analyses Chimiques de Synthèse Moléculaire de l'Université de Bourgogne". Elemental analyses were performed by Mr Marcel Soustelle and Ms Tiffanie Régnier on CHNS ThermoFisher Scientific Flash EA 1112 analyzer. Reagents were commercially available and used as received. High resolution mass spectra were recorded on a Thermo LTQ Orbitrap XL ESI-MS (ElectroSpray Ionization Mass Spectrometry). All X-Ray experimental procedure and crystal data are detailed in SI. NMR spectra ( 1 H, 13 C) were recorded on Bruker 300 Avance Neo, Bruker 400 Avance Neo,

  5 as a yellowish powder (82% yield). Elemental Analysis: calcd for (C43H27AlBF15N4O2)100(CH2Cl2)48: C, 52.47; H, 2.83; N, 5.63. Found: C, 52.54; H, 3.12; N, 5.50. J = 19.9 Hz, 2F), -165.03 (t, J = 20.7 Hz, 1F), -167.59 (m, 2F).

	1 H NMR (400 MHz, CD2Cl2 + Pyridine-D5): δ (ppm) = 7.46-7.31 (overlapping signals, 4H,
	H7/H5), 7.17-7.05 (overlapping signals, 4H, H4 + H9), 6.81 (m, 2H, H10), 6.67 (t, J = 7.8 Hz, 2H,
	H6), 3.13 (broad signal, 12H, NMe2), 0.89 (s, 3H, BMe). { 1 H} 13 C NMR (101 MHz, CD2Cl2 +
	Pyridine-D5): δ (ppm) = 168.26 (C3), 165.29 (C1), 138.87 (C8), 136.14 (C5/C7), 132.45 (C5/C7),
	126.55 (C9), 124.61 (C4), 122.23 (C10), 117.21 (C2 + C6), 44.70 (NMe2), 10.98 (CH3, BMe,
	deducted from HSQC). CF not seen.

19

F NMR (470 MHz, CD2Cl2 + Pyridine-D5): δ (ppm) = -132.77

(d,

We thank Dr Quentin Bonnin and Ms Marie-José Penouilh-Suzette (PACSMUB) for technical assistance with NMR spectroscopy and MALDI-ToF analysis. Prof. Jean-Pierre Couvercelle is warmly thanked for his assistance in GPC measurements.

with dithranol (DIT). Analysis were performed with DIT as matrix (10 mg/mL) and sodium trifluoroacetic acid (10 mg/mL) as an additive. The polymer (10 mg/mL), the matrix and the additive were mixed in a volumetric ratio of 1/1/0.5 in THF. All the analysis was performed in positive reflectron mode.

Computational details: All Quantum mechanics calculations were performed using the Gaussian 16 package. 19 Energy and forces were computed by density functional theory with the hybrid meta GGA M06-2X exchange-correlation functional. [START_REF] Zhao | The M06 Suite of Density Functionals for Main Group Thermochemistry, Thermochemical Kinetics, Noncovalent Interactions, Excited States, and Transition Elements: Two New Functionals and Systematic Testing of Four M06-Class Functionals and 12 Other Functionals[END_REF] The DCM solvent effects were modelled by means of the continuum model as implemented in Gaussian. Geometries were optimized and characterized with the 6-31+G(d,p) basis set. Gibbs free energies have been computed at 298K and 1 atm using unscaled density functional frequencies. Transition states were localized using the string theory [START_REF] Ren | String method for the study of rare events[END_REF] as implemented in Opt'n Path. 22 Molecular structures are drawn with the Chemcraft software. 23

ASSOCIATED CONTENT

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.xxxxxxx. full experimental details and analyses for compounds [L1H4]Br2, [L2H2], 1-5 ; computational details ; GPC analysis and MALDI-TOF spectra ; crystallographic tables (PDF); Crystallographic Information Files (CIF).

AUTHOR INFORMATION

Corresponding Authors

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

Funding Sources

Financial support from Ministère de l'Enseignement Supérieur et de la Recherche, the Centre National de la Recherche Scientifique (CNRS), ANR (MORFAL, ANR-22-CE07-0011), are gratefully acknowledged. Conseil Régional de Bourgogne Franche-Comté is acknowledged for financing the PhD grant of B.T. under the "Itinéraire Chercheurs Entrepreneurs" (ICE) program.

Calculations were performed using HPC resources from CCUB (Université de Bourgogne).

ACKNOWLEDGMENT