Guillaume Hoyez

Jolanta Rousseau

Cyril Rousseau

Sébastien Saitzek

James King

Petra Ágota Szilágyi

Christophe Volkringer

Thierry Loiseau

Frédéric Hapiot

Eric Monflier

Anne Ponchel

Cyclodextrins: a new and effective class of co-modulators for aqueous zirconium-MOF syntheses

Keywords: MOF formulas, textural properties, XRD patterns, FTIR and NMR spectra, CO2 sorption isotherms, N2 sorption isotherms, DSC, TEM, dispersion

come

Introduction

Metal-organic frameworks (MOFs) have attracted much attention during the past decade as they exhibit great potential in many applications such as gas storage, drug delivery, catalysis, separation, imaging and sensing. MOFs consists of crystalline networks resulting from coordination of polytopic organic linkers onto oxo/hydroxo clusters. 1,2 Their applicability in catalysis has now been amply demonstrated, 3,4,5,6 and their effectiveness is greatly dependent on their specific surface area (SSA) and presence of defects within the MOF network. 7,8,9,10,11 Since the seminal works by Schaate et al. in 2011, 12 it has been assumed that modulators mainly allow adjusting reaction kinetics of the nucleation and the growth of MOFs, however, they also alter the regular arrangement of their constitutive oxo/hydroxo clusters and polytopic organic linkers to create defects, e.g. missing linkers or missing nodes. 13,14 Modulators are typically non-bridging ligands, which compete with linkers by coordination to the metal clusters. 15 So far, monocarboxylic acids such as benzoic, formic, acetic and trifluoroacetic acid are amongst the most employed. The number of defects in the MOF lattices is strongly correlated with the steric hindrance, the concentration and/or acidity of the modulator. 16 Small quantities of modulators prevent the formation of the kinetically preferred disordered MOFs, the reason for this is that coordination equilibria between metal clusters, linkers and modulators regulate the rate of crystal growth and therefore the framework extension. Such exchange processes lead to the controlled growth of phase-pure crystalline frameworks that otherwise would only result in the formation of an amorphous powder. 17 Conversely, large modulator concentrations result in the formation of highly defective MOFs. 18 Vacancy concentrations can be controlled in a systematic manner, as shown by Cliffe et al. 19 Herein, we propose an innovative strategy to influence the crystal growth of Zr-MOFs using a saccharide-based modulator. Specifically, we have set out to systematically investigate the effect saccharides play in altering the Zr-MOF formation in conjunction with monocarboxylate acid modulators. One might expect that the MOF structure at the nanoscale might be affected by the coordination of saccharide hydroxyl groups to the Zr-node. To validate our hypothesis, we selected the benchmark MOF UiO-66-NH2 as the model. 20,21 UiO-66-NH2 is a zirconium-based metal-organic framework, which consists of [Zr6O4(OH)4] 12+ secondary building units (SBU) linked together by 2-aminoterephthalate carboxylates (BDC-NH2) (Figure 1).

To our knowledge, nothing has been described so far on the utilization of saccharides as co-modulators to tune the structure of UiO-66-NH2. In line with this, we attempted the hydrothermal synthesis of UiO-66-NH2 in the presence of both acetic acid and saccharides. We especially compared the ability of two acyclic saccharides -sorbitol and -D-methyl glucopyranose-and three cyclic oligosaccharides --, and -cyclodextrins (CD)-to alter the UiO-66 morphological and structural properties across a range of concentrations. Through a detailed analysis of the MOF structures obtained in each case, we demonstrate that it is possible to synthetically control the specific surface area (SSA) and porosity of the UiO-66-NH2 crystallites through a judicious choice of saccharides

Results and discussion

In this study, we deliberately chose water as the solvent because it is environmentally friendly, cheap, and suitable for scaled-up synthesis. Initially, we envisioned synthesizing the historically wellknown UiO-66 22 from ZrOCl2•8H2O and terephthalic acid in water using acetic acid as a modulator. However, all our attempts failed irrespective of the experimental conditions because of the insolubility of terephthalic acid in water/acetic acid mixtures. We then turned our efforts to the synthesis of selected UiO-66-NH2 from ZrOCl2•8H2O and H2BDC-NH2, on account of their appreciable solubility in water. Recently, some of us succeeded in synthesizing UiO-66-NH2 from the disodium salt of 2-aminoterephthalate in ambient aqueous conditions using acetic acid modulator. 23 This study allowed us to demonstrate the need for the pre-treatment of zirconium cluster in the 50-70 °C temperature range, a condition requirement we assigned to the formation of the hexanuclear zirconium complexes, identified as the necessary building block for framework assembly. The reaction conditions selected were inspired from the literature. 24,25 To further enable the solubilization/dispersion of all reagents in water, H2BDC-NH2 and saccharides were carefully ground prior synthesis. The subsequently increased particle surface resulted in a considerably better dissolution rate. 26 The grinding procedure took place using a laboratory-scale vibrating ball-mill for 10 min at 30 Hz. The ground reactants were then mixed with ZrOCl2•8H2O in a water/acetic acid mixture (10 mL/10 mL), and the solution was stirred at 90 °C for 24 h. After cooling down the solution at room temperature, the resulting solid materials were subjected to repeated centrifugation/washing steps using DMF, water and methanol consecutively.

To assess the influence of saccharides as co-modulators in the production of UiO-66-NH2, we first considered sorbitol in combination with acetic acid. Note that the amount of sorbitol was chosen for easy comparison with -CD (which consists of 6 -Dglucopyranoside units). After several consecutive cycles of centrifugation and washing, the powder was dried and analyzed by XRD (Figure 2). Poorly defined XRD patterns were observed corresponding to amorphous products, indicative of the poor performance of sorbitol as co-modulator for the growth of MOF crystallites. Notably, changing from sorbitol to -D-methylglucopyranoside under the same experimental conditions resulted in the formation of UiO-66-NH2 material of more crystalline nature as revealed by our XRD data (Figure 2). The powder pattern showed typical reflections of the face-centered-cubic structure of UiO-66-NH2 with the Fm-3m space group (ESI, Figure S1).  -CD was then used as a co-modulator under the same experimental conditions. Mixtures of acetic acid and -CD gave similar XRD patterns, irrespective of the number of equivalents of -CD with respect to the ligand, which is indicative of isostructural compounds (ESI, Figure S2). Indeed, no significant modification of the diffraction patterns was noticed upon increasing the proportion of -CD up to 1 equivalent, suggesting that the lattice parameter and the crystallinity were not altered. We attribute the slight peak broadening observed at low modulator concentrations to the existence of nanocrystallites, according to previous reports. 27 The formation of UiO-66-NH2 through the coordination interactions between Zr6-clusters and BDC-NH2 ligands was further demonstrated through the alteration of several IR bands (ESI, Figures S3 andS4). The absorption bands were similar to those observed with -D-methyl glucopyranoside except the bands assigned to acetic acid. 22,28 The absorption bands at 1496, 1382, 1336 cm -1 were attributed to the stretching modes of the aromatic BDC-NH2 bonds, and the bands at 1568 and 1423 cm -1 were assigned to the stretching, asymmetric (as) and symmetric (s) vibrations of COO -groups, respectively (1257 cm -1 CO simple). 22,29 Additional information about the UiO-66-NH2 structures derived from -CD as co-modulator was further extracted from the N2 adsorption-desorption isotherms (Figure 3-A). For comparison, the isotherm of the control UiO-66-NH2 prepared in the same conditions but without -CD is also given. Overall, they display characteristic features of Type I adsorption isotherms, typical of microporous solids in which most pores have a size below 20 Å. The major gas consumption in the isotherms occurred at fairly low relative pressure (P/P°<0.1) before reaching a plateau. Table 1 lists the textural characteristics of the samples obtained with various amounts of the α-CD modulator, in terms of specific surface area (BET and Langmuir), micropore surface and volume, total volume and pore b) c) a)

size. With only acetic acid as modulator (no saccharide), the BET surface area was found to be 611 m 2 •g -1 (Table 1, entry 1), which is in relative agreement with what was previously measured on hydrothermally prepared UiO-66-NH2 samples in the literature (833 m 2 •g -1 and 888 m 2 •g -1 ). 14,23 The slight discrepancy could be attributed to the activation procedures, especially the preliminary grinding of H2BDC-NH2. Indeed, in our hands, the BET surface area was 826 m 2 •g - 1 when the linker was not ground. Table 1. Textural characteristics of UiO-66-NH2 synthesized in the presence of various amounts of -CD as co-modulator. Note that all BET surface areas were calculated using the consistency criteria recommended by Rouquerol et al. 30 for selecting the appropriate linear pressure range (i.e. 0.001 -0.05). A proportion of micropore (Vmicro/Vtotal) of ca. 54 % was found, this value being very consistent with the observation of Zhou et al. (55.9 %). 31 We then investigated the influence the amount of -CD on the textural properties of UiO-66-NH2. Notably, the porosity was greatly enhanced when the Zr-MOFs were produced in the presence of acetic acid/-CD mixtures. For instance, using an α-CD/linker molar ratio of as low as 0.25, the BET specific surface area increased from 611 to 1071 m 2 •g -1 and the total pore volume from 0.268 to 0.429 cm 3 •g -1 (Table 1, entries 2-4). The micropore surface area and micropore volume were also enhanced by the use of -CD, by an extent ranging from 424 to 775 m 2 •g -1 and from 0.144 to 0.268 cm 3 •g - 1 respectively. The microporous structure can be verified by the pore size distribution (PSD) derived from nonlocal density functional theory (NLDFT). The PSD was altered in the micropore region, showing a bimodal distribution with pores diameters centered at about 11.8 Å and 14.8 Å (Figure 3-B), in good agreement with the literature. 32,33 The porosity of the UiO-66-NH2 samples was further enhanced upon the addition of higher quantities of -CD, as illustrated in Figure 4 through their BET and Langmuir specific surface areas. UiO-66-NH2 prepared with 1 equivalent of -CD with respect to the H2BDC-NH2 ligand featured a BET specific surface area of 1451 m 2 •g -1 , a Langmuir surface of 1589 m 2 •g -1 (Table 1, entry 7). SSA calculated for UiO-66-NH2 synthesized using sorbitol or-D-methylglucopyranoside as saccharide sources were added for comparison.  calculated using the Langmuir method,  calculated using the BET method.

As usually observed, slightly higher Langmuir surface areas were obtained for all the hereby synthesized UiO-66-NH2, irrespective of the -CD/ligand ratio. Indeed, it is commonly accepted that the Langmuir theory (based on monolayer adsorption) tends to overestimate the apparent surface area of the MOFs. 34 While the PSD was not significantly altered, being centered at ca. 11.8 and 14.8 Å, calculation of the proportion of micropore volume once again revealed that high degrees of microporosity were maintained, reaching approximately 65%. This result is consistent with the increased dangling (or pending) linkers 24 percentage within the framework, possibly due to competitive equilibria. 35,36 Interestingly, this trend obtained with N2 adsorption measurements was also observed using CO2 as adsorbate (adsorption at 273 K), in line with the overall increase in microporosity in UiO-66-NH2 materials prepared from -CD (ESI, Figure S5). Astonishingly, we observed that the addition over 1 equivalent of -CD with respect to the linker, resulted in a dramatic decrease in the specific surface area from 1451 m 2 •g -1 to 51 m 2 •g -1 (Table 1, entries 8-10; ESI, Figure S6). Overall, our results indicate that the stoichiometric -CD / linker molar ratio appears to be the upper limit of the beneficial effect of -CD as comodulator. Table 1 also revealed the advantage of ground materials over unground materials. Indeed, comparing Entries 7 and 11 clearly showed than the unground material led to lower SSA. Unground CDs are probably less dispersed and less available to play their role of comodulator. Accordingly, grinding -CD and linkers has a beneficial influence on its dispersion in the aqueous solution. Indeed, the MOF synthesis took place in an aqueous dynamic heterogeneous medium wherein the components were not fully soluble. It is also worth noting that, when the synthesis was performed with sorbitol using a 1/1 saccharide / linker molar ratio, the BET SSA value was dramatically reduced to 110 m 2 •g -1 . With -D-methylglucopyranose as constitutive building blocks for CD, the BET SSA value was 625 m 2 •g -1 , a value similar to that obtained with the control (no saccharide, SSA = 611 m 2 •g -1 ). This is in stark contrast with our results on-CD. Thermogravimetric analysis (TGA, Figure 5) was also carried out to evaluate the thermal stability of the UiO-66-NH2 derived from -CD. 28,37 The resulting TGA profiles revealed three primary stages of mass loss. After dehydration of the MOF structure from 100 to 150 °C, we observed the dehydroxylation of the Zr6-nodes between 150 and 350 °C. In this temperature range, the   -OH coordinated hydroxyl groups were partly removed, resulting in the transformation of the SBU from [Zr6O4(OH)4] 12+ to [Zr6O6] 12+ . 38 The third thermal event, observed from 350 to 500 °C, corresponds to the exothermic decomposition of the linker. This temperature range at which decomposition occurred, was lower than that reported, usually around 500-550°C for MOFs synthesized under solvothermal conditions (ZrCl4 in DMF). 39,40 Over 500 °C, we observed the decomposition of the framework with only ZrO2 as the final residue, as confirmed by additional XRD (ESI, Figure S7) and DSC analyses (ESI, Figure S8). Throughout the heating process, TG curves relative to MOFs synthesized from various -CD/linker ratios clearly show that the thermal stability does not significantly depend on the amount of -CD. Indeed, all the TG curves are superimposable. This is in line with the hypothesis that CDs do not create defects within the bulk MOF structure. Additionally, no significant difference was observed in the FTIR spectra regardless the amount of -CD (ESI, Figure S4). Noteworthy, our results did not show any characteristic vibration bands for -CD, which indicates that the latter is not contributing to the MOF structure and is removed during the washing process.

To unequivocally exclude the presence of -CD within the MOF structure or on the peripheral surface of UiO-66-NH2 crystallites, we also performed digestion experiments of UiO-66-NH2 (see the protocol in the ESI). We first undertook aqueous alkaline digestion experiments of UiO-66-NH2 using NaOH. The 1 H NMR spectrum of the degraded UiO-66-NH2 showed signals corresponding to the aromatic protons of the linker (ESI, Figure S9). No trace of -CD was detected, strongly indicating the complete absence of -CD.

Accordingly, the role of -CD can therefore be rationalized as a true co-modulator, facilitating the crystallization process without being incorporated into the metal-organic framework.

In order to assess the defect concentration in the UiO-66-NH2 samples synthesized with the use of -CD co-modulator, the modulator to linker ratio needs to be quantified. To determine the number of acetic acid and H2BDC-NH2 molecules per Zr node, an acidic digestion 41 of the MOF was carried out in the presence of maleic acid as an internal reference. 42 From the relative peak area of the distinctive aromatic BDC-NH2 protons (d 7.12 ppm ; s 7.46 ppm ; d 7.79 ppm) and acetic acid methyl protons (2.52 ppm), the amount of modulator was found to be quite constant for the various samples, in the range from 0.08 to 0.11 per {Zr6}-cluster, whereas the amount of BDC-NH2 linker was found to vary more significantly from 5.95 down to 5.08 per {Zr6}-cluster (ESI, Table S1). It should be noted that in the ideal non-defected structure, there would be 6 BDC and 0 acetate per inorganic nodes. Hence, we can conclude that the synthesized MOFs show very few defects within their structures, irrespective of the amount of -CD. The MOF structure obtained using 1 equiv. -CD with respect the linker can be formulated as Zr6O4(OH)4.85(BDC-NH2)5.52(CH3COO)0.11, indicative of 0.85 out of 12 linkers missing per {Zr6}-cluster. When using -CD (which consists of 7 -D-glucopyranoside units) as co-modulator instead of -CD, under the same experimental conditions, no modification of the XRD patterns (ESI, Figure S10) and FTIR spectra (ESI, Figure S11 and S12) was noticed. Once again, this demonstrates the integrity of the UiO-66-NH2 structure.

All the N2 adsorption-desorption isotherms obtained with this series prepared with -CD can be still classified as type I, typical of microporous samples (ESI, Figure S13). Furthermore, we observed a strong dependence of the amount and proportion of the comodulator -CD on the samples' textural properties. Our results evidenced that we were able to optimize porosity by adjusting the -CD/linker ratio to 0.5 (Figure 6), which, in these conditions, is similar to that measured with -CD (ESI, Figure S6). The total pore volume was 0.458 cm 3 •g -1 . The BET and Langmuir surface areas of near defect free UiO-66-NH2 were 1187 and 1287 m 2 •g -1 , respectively (ESI, Table S2). Interestingly such values are similar to those obtained using organic solvents, thus revealing the effect of CDs on the SSA of near defect free UiO-66-NH2. It is also worth remembering that, in the literature, the highest value reported so far for the SSA of UiO-66-NH2 synthesized under aqueous conditions in a batch reactor was only 888 m 2 •g -1 . Above this -CD/linker ratio of 0.5, some degree of porosity was sacrificed, but typical specific surface areas could be retained even for the sample obtained using 1 equiv. of CD (displaying reduced BET and Langmuir SSA values of 801 and 889 m 2 .g -1 , respectively). This demonstrates that N2 uptake and specific surface areas may be optimized by the judicious control of -CD/linker ratios.

In order to get further insight into the role of CDs in aqueous synthesis of UiOs, and to assess their performance as a function of their size, -CD (which consists of 8 -D-glucopyranoside units) was also used as a modulator in the synthesis of UiO-66-NH2. Surprisingly, we observed that the modulator's effect on the textural properties of UiO-66-NH2 was deleterious (ESI, Figure S14), although the XRD patterns (ESI, Figure S15) and FTIR spectra (ESI, Figures S16 andS17) revealed that the UiO topology was maintained. Increasing the proportion of -CD from 0 to 1 equivalent of -CD with respect to the linker provoked a dramatic decrease in the surface area from 611 to 39 m 2 •g -1 for the BET model and from 706 to 59 m 2 •g -1 for the Langmuir model (Figure 7). This trend seems to be indicative of the destabilization effect of -CD on the crystal growing MOF, thereby creating defects and limiting the MOF crystal size. This was further proved by the subsequent N2 uptake. N2 uptake was only 0.035 cm 3 •g -1 for UiO-66-NH2 synthesized using 1 equivalent of -CD with respect to the linker (ESI, Table S3). Variation of the specific surface area of UiO-66-NH2 as a function of the amount of -CD.  calculated using the Langmuir method,  using the BET method.

Field emission scanning electron microscopy (FE-SEM) was used to investigate the crystallite morphology. Figure 8 depicts typical FE-SEM images of three selected UiO-66-NH2 samples prepared without cyclodextrin (control) and, with 0.5 equiv. of -CD or 0.5 equiv. of -CD at magnifications of 25,000 and 50,000. Notably, the particle morphology was found to be similar without or with cyclodextrin as co-modulators. Overall, the SEM pictures showed that the materials are comprised of densely packed particles whose shapes mostly polygonal with straight or gently curved grain boundaries. Nevertheless, it can be noticed that the average size of the polygonal particles slightly differs depending on the sample. Therefore, to the control UiO sample (without cyclodextrin), which produce small monodisperse particles typically ranging from 100 to 200 nm in size (Figure 8 Especially in the latter case, the impact on the grain dimension seems to be even more pronounced, with sizes that could sometimes reach or exceed 450 nm (ESI, Figure S18). Complementary experiments were carried out to better understand the role of CDs. Photographs of aqueous solutions containing the Zr precursor and the linker were compared at various mixing time if CDs was added or not as a co-modulator. They clearly showed a bad dispersion of the materials without CD (especially at the beginning of the synthesis), and a good dispersion of the reactants in the presence of CDs (ESI, Figure S19). We also investigated the interaction of the preformed Zr6O4(OH)4(CH3COO)12 oxyacetate cluster with CDs by NMR. To this end, 2D T-ROESY NMR measurements were performed to determine the inclusion ability of the three CDs towards H2BDC-NH2 and Zr6-clusters in water. While no cross-peak was detected in the 2D T-ROESY NMR spectrum of a stoichiometric mixture of -CD/H2BDC-NH2 and -CD/Zr6-clusters in D2O at 25 °C (ESI, Figure S20 S21), strong correlations were revealed in the spectrum of -CD/H2BDC-NH2 43 (ESI, Figure S22), indicative of the inclusion of H2BDC-NH2 into -CD. -CD is well-known to interact with aromatic compounds to form inclusion complexes. 44 However, no correlation was detected for a -CD/Zr6-clusters stoichiometric mixture (ESI, Figure S23), indicating that the Zr6-node was not included into the -CD cavity. As expected, no correlation was observed on the 2D T-ROESY NMR spectrum of -CD/H2BDC-NH2 either (ESI, Figure S24), the molecular recognition of -CD towards H2BDC-NH2 is poor because the -CD cavity is too wide. Note that no interaction was observed between -CD with the cluster (ESI, Figure 25).

Based on all the above results, we sought to develop a mechanistic insight into the roles of CDs on the MOF growth. As initially envisaged, -CD, -CD and -CD could interact differently with the Zr6-clusters as co-modulators, while other saccharides did not. This strongly suggests that the CD structure plays an important role in modulating the dynamic motion of the frameworks. It is known that the complexation of native CDs with metal ions can affect the supramolecular arrangements and packing, as recently reviewed. 45,46 Besides the connection with the carboxylate groups of acetic acid, the growing peripheral sites of UiO-66-NH2 could be temporarily stabilized by CDs that greatly alter the growing process to generate highly ordered structures. We suggest that, during the MOF crystal growth process, -CD, -CD and -CD regulate the coordination of BDC-NH2 ligands onto the Zr6-clusters to form regular MOF structures (Figure 9). From the dimensions of CDs and Zrclusters, three cases could be distinguished. -CD (external diameter =1.75 nm; internal diameter = 0.75-0.83 nm) 47 hampers the coordination of BDC-NH2 to the Zr6-cluster (Ø ~ 1.1 nm), resulting in a poorly-defined MOF structure. 48 Indeed, y-CD could transiently include the Zr6 clusters in line with the respective sizes. This hypothesis agrees with data from the literature. Indeed, it is reported that -CD could spontaneously interact with a number of different oxo-and metal-clusters that can be described as nanoscale objects of about typically 2.5-3 nm in diameter. [49][50]51,52 Conversely, the smaller size of -CD (external diameter =1.46 nm; internal diameter = 0.47 -0.53 nm) 47 would not disturb the BDC-NH2 bonding formation with the Zr6-clusters to make the 3D infinite framework. The respective dimensions of -CD and Zr6-clusters would not permit inclusion of the cluster within the -CD cavity.-CD, for its part, has an intermediate behavior between -CD and -CD because of its dimensions (external diameter =1.54 nm; internal diameter = 0.60 -0.65 nm) 47 . -CD being wider than -CD, its interaction area with Zr6clusters could be greater. Therefore, a smaller amount of the -CD may already lead to MOF growth inhibition. However, in a welldefined concentration range, the effect of -CD on the MOF structure is determining as the largest crystallites obtained in this study result from the utilization of -CD as co-modulator (Figure 8). Our results could also be rationalized by other concomitant Firstly, CDs could act as dispersing agents. As evidenced by photographs on the suspension of the reactants (ESI, Figure S19), -CD facilitates the dispersion of H2BDC-NH2 (which is not fully solubilized in water) via adsorption on hydrophobic linker aggregates. Indeed, the possibility could not be excluded that -CD interacts with linker aggregates. In the literature, it has already been shown that CDs help dispersing hydrophobic organic particles in aqueous media. 53,54 In our case, the improved dispersion enables the growth of UiO-66-NH2 featuring record high SSA reported so far for aqueous syntheses. Secondly, CDs could form supramolecular inclusion complexes with the linker. As revealed by 2D NMR experiments,-CD also probably participates in the inclusion of H2BDC-NH2 within its cavity, while -CD and -CD are unable to do so. The formation of such supramolecular interaction is clearly not favorable to the MOF growth. Hence, when the concentration of is too high, such inclusion complex between -CD and the linker may displace equilibria to such an extent that the amount of nonincluded linker became sub-stoichiometric with respect to Zr6clusters, resulting in MOF structures featuring lower SSA.

Experimental

Modulated hydrothermal synthesis of UiO-66-NH2 ZrOCl2•8H2O (0.322 g, 1 mmol) was dissolved in a mixture of water/acetic acid (5 mL/5 mL). 2-aminoterephthalic acid (0.181 g, 1 mmol) and the saccharide were ground under ball-milling conditions (15 min, 30 Hz) and then added to the preformed Zr acetate precursor in the previous solution. The reaction mixture was heated at 90 °C for 24 h. The precipitate was then washed successively with DMF (2 × 10 mL), water (2 × 10 mL) and methanol (2 × 10 mL) to remove residual reagents from the MOF pores. The sample was dried under vacuum at 100 °C for 24 h to yield the final product.

Conclusions

In conclusion, we have developed a new procedure to promote the synthesis of UiO-66-NH2 under aqueous conditions. Our strategy relies on the use of both acetic acid and saccharides as co-modulators. By simply introducing appropriate saccharides as co-modulators in well-defined quantity, we can control the growth of UiO-66-NH2 featuring high SSA and N2 sorption properties. While sorbitol leads to amorphous structures, the utilization of -D-methylglucopyranoside as co-modulator has no effect on the SSA and N2 sorption properties. -CD, for its part, proves to be inappropriate as it prevents the growth of Zr-MOF. Conversely, -CD and -CD in well-defined concentration ranges act as very effective co-modulators in combination with acetic acid. While CD-free aqueous syntheses of Zr-MOFs typically result in altered properties when compared to solvothermal approaches, hydrothermal syntheses of Zr-MOFs in the presence of α-CD and β-CD were found to reproduce them. Through modulated synthesis,-CD and -CD bind competitively to the metal clusters during framework formation, resulting in crystalline products with higher porosity and SSA compared to Zr-MOFs obtained under typical aqueous conditions. Additionally, CDs help dispersing the poorly watersoluble H2BDC-NH2 linker and facilitate its coordination to Zr clusters. Investigations are currently on-going to widen the scope of the strategy herein developed to other MOFs.

Fig 1 .

 1 Fig 1. Schematic structural illustration of UiO-66-NH2. a) [Zr6O4(OH)4] 12+ cluster as secondary building unit (SBU), b) 2aminoterephthalate carboxylate (BDC-NH2) as an organic linker, and c) the face-centred-cubic (fcc) crystal structure of UiO-66-NH2. Purple sphere: octahedral cavity; green sphere: tetrahedral cavity; NH2 groups are omitted for clarity, due to disordered configuration

Fig 2 .

 2 Fig 2. Powder XRD patterns of UiO-66-NH2 derived from mixtures containing sorbitol (a) and -D-methylglucopyranoside (b).

a

  Average of at least two experiments. b Specific surface area calculated from the Brunauer-Emmmet-Teller equation in the P/P° range from 0.001 to 0.05; c Specific surface area calculated from the Langmuir equation in the P/P° range from 0.001 to 0.70; d Micropore surface area determined by the t-plot method; e Micropore volume determined by the t-plot method; f Total pore volume estimated at P/P° = 0.95; g Average pore size based on 4 × V total/ BET SSA; h Prepared without grinding -CD and H2BDC-NH2 ligand.

Fig 3 .

 3 Fig 3. (A) N2 sorption isotherms collected at 77 K of NH2-UiO-66 synthesized using various amounts of ground -CD with respect to ground H2BDC-NH2 [Adsorption (filled circles) and desorption (open circles)]. (B) with the corresponding pore size distribution plots determined via NLDT : 0 equiv. -CD/ H2BDC-NH2 (a, grey), 0.05 equiv. (b, pink), 0.10 equiv. (c, orange), 0.25 equiv. (d, blue), 0.5 equiv. (e, red), 0.75 equiv. (f, green), and 1 equiv. (g, violet).

Fig 4 .

 4 Fig 4.Variation of the specific surface area (SSA) of UiO-66-NH2 with the amount of -CD. SSA calculated for UiO-66-NH2 synthesized using sorbitol or-D-methylglucopyranoside as saccharide sources were added for comparison.  calculated using the Langmuir method,  calculated using the BET method.

Fig 5 .

 5 Fig 5. TGA curves of UiO-66-NH2 synthesized under modulated hydrothermal conditions using -CD to ligand ratios of 0 (black), 0.25 (blue), 0.5 (red), 0.75 (green), and 1 (purple).

Fig 6 .

 6 Fig 6. Variation of the specific surface area of UiO-66-NH2 as a function of the amount of -CD.  calculated using the Langmuir method,  calculated using the BET method.

  -a and Figure 8-b), slightly more polydisperse and larger crystallites were observed for UiO-66-NH2 synthesized with -CD (Figure 8-c and Figure 8-d) and -CD (Figure 8-e and Figure 8-f) as co-modulators.

Fig 8 .

 8 Fig 8. Representative FE-SEM images of UiO-66-NH2 synthesized without -CD (a and b), with 0.5 equiv. -CD with respect to the linker (c and d), and with 0.5 molar equiv. of -CD with respect to the linker (e and f). Magnification: 25,000x (left column) and 50,000x (right column).

Fig 9 .

 9 Fig 9. a) Structural representation of a molecular oxo-acetato {Zr6}-cluster highlighting the longest (1.12 nm) and shortest (0.79 nm) intracluster O⋅⋅⋅O distances, as defined by the red and blue arrows. From reference 48. b) Structural representations of native cyclodextrins (-CD, -CD and -CD) with their average internal and external diameters. From reference 47. c) Possible interactions of -CD, -CD and -CD with the {Zr6}-cluster. Color scheme: Zr white, O red, C black, H grey.

Acknowledgements

The authors acknowledge the Chevreul Institute (FR 2638), Ministère de l'Enseignement Supérieur et de la Recherche, Region Hauts de France and FEDER program for supporting and partially funding this work. Guillaume Hoyez is grateful to the University of Artois for the funding of his PhD grant. We thank also Dr. Ahmed Addad (UMET, University of Lille) and Prof. Christian Mathieu (UCCS, University of Artois) for FE-SEM analyses, Dr. Jérémy Ternel (UCCS, University of Artois) for NMR analyses and Dr. Nicolas Kania (UCCS, University of Artois) for TGA measurements.

Notes and references

Conflicts of interest

There are no conflicts to declare.


Véronique Merrien-Soukatchoff

Muriel Gasc-Barbier

Role of positive temperature variations on rock slopes outcrops (a review)


Super-resolution capacity of variance-based stochastic fluorescence microscopy

Simon Labouesse [START_REF] Sheppard | Three-dimensional image formation in confocal microscopy[END_REF], Jérôme Idier [START_REF] Klar | Subdiffraction resolution in far-field fluorescence microscopy[END_REF], Marc Allain (3), Guillaume Giroussens (3), Thomas Mangeat [START_REF] Sheppard | Three-dimensional image formation in confocal microscopy[END_REF] and Anne Sentenac (3)*

(1) LITC Core Facility, Centre de Biologie Intégrative, Université de Toulouse, CNRS, UPS, 31062, Toulouse, France.

(2) Nantes Université, École Centrale Nantes, LS2N, CNRS, Nantes, France. and (3) Aix Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel, Marseille, France.

Improving the resolution of fluorescence microscopy beyond the diffraction limit can be achieved by acquiring and processing multiple images of the sample under different illumination conditions. One of the simplest techniques, Random Illumination Microscopy (RIM), forms the super-resolved image from the variance of images obtained with random speckled illuminations. However, the validity of this process has not been fully theorized. In this work, we characterize mathematically the sample information contained in the variance of diffraction-limited speckled images as a function of the statistical properties of the illuminations. We show that an unambiguous two-fold resolution gain is obtained when the speckle correlation length coincides with the width of the observation point spread function. Last, we analyze the difference between the variance-based techniques using random speckled illuminations (as in RIM) and those obtained using random fluorophore activation (as in Super-resolution Optical Fluctuation Imaging, SOFI).

The light intensity recorded by the camera of a fluorescence microscope cannot exhibit spatial frequencies above 2/λ where λ is the wavelength of the emitted light. This low-pass filtering, due to the loss of the evanescent waves at the detector plane, cannot be circumvented. Therefore the challenge of super-resolution imaging is to recover spatial frequencies of the sample fluorescence density beyond 2/λ from data that are frequency limited to 2/λ. A widespread solution consists in processing multiple images obtained by changing the illumination, like translating focused spots [START_REF] Sheppard | Three-dimensional image formation in confocal microscopy[END_REF][START_REF] Klar | Subdiffraction resolution in far-field fluorescence microscopy[END_REF][START_REF] Sheppard | Super-resolution in confocal imaging[END_REF][START_REF] Müller | Image scanning microscopy[END_REF] or rotating and translating periodic light patterns [START_REF] Heintzmann | Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating[END_REF][START_REF] Gustafsson | Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy[END_REF]. The data processing of most techniques using structured illuminations requires the knowledge of the illumination patterns, either explicitely as in Structured Illumination Microscopy [START_REF] Heintzmann | Laterally modulated excitation microscopy: improvement of resolution by using a diffraction grating[END_REF][START_REF] Gustafsson | Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy[END_REF] or implicitely as in confocal or Image Scanning microscopy [START_REF] Sheppard | Three-dimensional image formation in confocal microscopy[END_REF]. In this context, Random Illumination Microscopy [START_REF] Mangeat | Super-resolved live-cell imaging using random illumination microscopy[END_REF][START_REF] Dertinger | Fast, background-free, 3d super-resolution optical fluctuation imaging (sofi)[END_REF][START_REF] Ventalon | Quasi-confocal fluorescence sectioning with dynamic speckle illumination[END_REF] stands out as an exception as it does not require the knowledge of the illumination patterns: the super-resolved image is formed from the variance of multiple diffraction-limited images recorded under different random speckled illuminations. While attractive because of its simplicity and significant image improvement [START_REF] Mangeat | Super-resolved live-cell imaging using random illumination microscopy[END_REF], RIM variance-based processing lacks a rigorous analysis of its resolution potential, the non-linearity of the variance being a significant obstacle to its derivation. In this work, we study the sample information that can be extracted from the variance of speckled images as a function of the statistical properties of the random illumination and we derive the condition under which the variance can provide a resolution gain.

To model the data provided by a fluorescence microscope under an inhomogeneous illumination, we introduce the point spread function of the microscope h and the illumination intensity function E. Importantly, these two functions are defined at a macroscopic scale inside the sample, through the averaging over regions large enough to contain thousands of atoms (typically of the order of a thousand nm 3 ), to wash out the microscopic fluctuations. In this context, we define the macroscopic fluorescence density ρ such that V ρ(r)E(r) is the energy (detected by the camera) of the fluorescent light emitted by a macroscopic volume V centered about r. Hereafter we neglect the Poisson noise. The fluorescence density depends on the fluorophore concentration and the molecular brightness.

With these definitions, the microscope image can be written as,

I(r) = ρ(r ′ )E(r ′ )h(r -r ′ )dr ′ (1)
where r indicates a position in the image domain that is conjugated to a point in the object domain. This model can be applied to two-or three-dimensional imaging configurations. In the Fourier space, Eq. (1) reads,

Ĩ(ν) = h(ν) ρ(ν -ν ′ ) Ẽ(ν ′ )dν ′ (2)
where f (ν) = f (r)e -i2πν•r dr stands for the Fourier transform of f . If E is a constant, as in a standard fluorescence microscope, the recorded image depends only on the sample spatial frequencies belonging to the support of the Optical Transfer Function (OTF) h, noted W h , which is at best a disk of radius 2/λ (in the 2D case) or exhibits a torus-like shape in the 3D case [START_REF] Sentenac | Unified description of threedimensional optical diffraction microscopy: from transmission microscopy to optical coherence tomography: tutorial[END_REF].

On the other hand, if E is not a constant, and noting W E the support of Ẽ, the recorded image depends on the sample spatial frequencies in the domain

W hE = {ν -µ | ν ∈ W h , µ ∈ W E }, (3)
which is no more limited by 2/λ and corresponds to the Fourier support of hE.

Now, sensitivity to spatial frequencies of the sample outside W h is a necessary but not a sufficient condition for being able to form a super-resolved image. One also needs a technique for extracting the high spatial frequencies of the sample from the diffraction-limited images I n obtained for various illumination intensities E n . For example, in an ideal confocal microscope, the n th illumination is a focal spot centered about r n and the measured data is

I n (r n ) = ρ(r ′ )E n (r ′ )h(r n -r ′ )dr ′ .
If one neglects the Stokes shift and forms the illumination through the same optical path as the collection, one gets (using the reciprocity theorem [START_REF] Sentenac | Unified description of threedimensional optical diffraction microscopy: from transmission microscopy to optical coherence tomography: tutorial[END_REF]) E n (r) = h(r nr) and

I n (r n ) = ρ(r ′ )h 2 (r n -r ′ )dr ′ (4)
which permits to recover the fluorescence density over

W hE = W h 2 with W h 2 = {ν -ν ′ | ν ∈ W h , ν ′ ∈ W h }.
Note however that the super resolution capacity of an ideal confocal microscope requires the use of an infinitely small pinhole which is impossible in practice. Generally, when the different illumination intensities are known, the sample frequencies in W hE can be obtained explicitely from a linear combination of the diffraction-limited images (see [START_REF] Gustafsson | Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy[END_REF] for exemple).

On the other hand, when the illuminations are unknown speckles as in Random Illumination Microscopy, there is no simple way for recovering the sample frequencies from the diffraction-limited images. While a link between the (numerically intractable) full covariance of the images and the sample was derived in [START_REF] Idier | On the superresolution capacity of imagers using unknown speckle illuminations[END_REF], there is currently no rigorous result concerning the sample information that can be extracted from the sole variance.

Let us consider an experiment in which multiple images of a sample are recorded under different speckled illuminations E n that are considered as different realizations of the same random process E. The variance of these diffraction-limited images, 2 , where E stands for the ensemble average over the random illuminations, can be expressed as a function of the speckle autocovariance [START_REF] Idier | On the superresolution capacity of imagers using unknown speckle illuminations[END_REF],

V ρ (r) = E[I 2 (r)] -E[I(r)]
C(r, r ′ ) = E[E(r)E(r ′ )] - E[E(r)]E[E(r ′ ), as
V ρ (r) = dr 1 dr 2 (5) h(r -r 1 )ρ(r 1 )C(r 1 , r 2 )ρ(r 2 )h(r -r 2 ).
In the following, we assume that E is a second-order stationnary random process so that C(r 1 , r 2 ) = C(r 1r 2 ) and the Fourier support of C is equal to the Fourier support of E, W E [START_REF] Priestley | Spectral analysis and time series: probability and mathematical statistics[END_REF].

Since the variance depends on the square of raw images that are frequency limited to W h , its Fourier support corresponds to W h 2 .

On the other hand, each raw image is sensitive to the spatial frequencies of the sample in W hE , so we expect the variance to be also sensitive to the spatial frequencies of the sample in W hE (we show on a specific example in appendix B, that this is indeed the case). These preliminary remarks bring out the main question posed by variance-based super-resolved microscopy approaches: what sample information in W hE can we extract from an image that is frequency limited to W h 2 ? We first consider RIM configuration in which the illumination is performed through the same objective as the observation and the Stokes shift can be neglected. In this case, the speckle autocovariance C is equal to the point spread function h of the microscope [START_REF] Mangeat | Super-resolved live-cell imaging using random illumination microscopy[END_REF][START_REF] Goodman | Speckle phenomena in optics: theory and applications[END_REF] and W E is similar to W h . Thus, the domain of sample spatial frequencies acting on RIM raw images, W hE , matches the Fourier support of the variance, W h 2 . In the following, we demonstrate that, indeed, if h = C, there is a bijection between the RIM variance and the sample frequencies within W h 2 .

We start by noting that the Fourier transform of C, C, is positive as C is an autocovariance function. We define

h E such that hE = C. One easily shows that h E (r 1 - x)h E (r 2 -x)dx = C(r 1 -r 2 )
. Using this decomposition of the speckle covariance, V RIM ρ can be cast as,

V RIM ρ (r) = B ρ,ρ (r) (6)
where

B U,V (r) = M U (r, x)M V (r, x)dx (7)
with

M V (r, x) = h(r -r 1 )V (r 1 )h E (r 1 -x)dr 1 (8)
and U and V are integrable real functions. To pursue the demonstration, we need to point out several properties of B U,V and M V . First, it is easily seen that B U,V is symmetric with respect to (U, V ), B U,V = B V,U and bilinear,

B U+U ′ ,V +V ′ = B U,V + B U ′ ,V ′ + B U,V ′ + B U ′ ,V .
Second, the Fourier transform of M V with respect to (r, x),

MV (ν, µ) = h(ν) Ṽ (µ + ν) C(µ), (9)
is bounded, so M V is an analytic function. Third, we show in appendix A that, if

h = C, B U,V (r)V (r)dr = |M V (r, x)| 2 U (r)drdx. ( 10
)
We now consider two fluorescence densities, ρ 1 (r) ≥ 0 and ρ 2 (r) ≥ 0 which have the same RIM variance, B ρ1,ρ1 (r) = B ρ2,ρ2 (r). Using the bilinearity and symetry of B U,V , we can show that B ρ1,ρ1 -B ρ2,ρ2 = B ρ1+ρ2,ρ1-ρ2 = 0. Thus, one obtains

B ρ1+ρ2,ρ1-ρ2 (r)[ρ 1 (r) -ρ 2 (r)]dr = 0 (11)
which, using Eq. ( 10), implies,

|M ρ1-ρ2 (r, x)| 2 [ρ 1 (r) + ρ 2 (r)]drdx = 0. ( 12
)
We now assume that ρ 1 + ρ 2 stays strictly positive in a non-empty open set Ω. In this case, Eq. ( 12) is satisfied if and only if M ρ1-ρ2 (r, x) = 0 for r ∈ Ω and for all x. Since M ρ1-ρ2 is analytic, M ρ1-ρ2 (r, x) = 0 for all x and for r ∈ Ω implies that M ρ1-ρ2 (r, x) = 0 for all x and all r, thus Mρ1-ρ2 (ν, µ) = 0 for all ν and µ. From Eq. ( 9), the nullity of Mρ1-ρ2 is obtained only if ρ1 (η) -ρ2 (η) = 0 for η ∈ W h 2 . Hence, if ρ 1 and ρ 2 have the same RIM variance, they have the same spatial frequencies in W h 2 . Conversely, if two samples ρ 1 and ρ 2 have the same frequency content in W h 2 , they generate the same variance image. We have thus demonstrated that there is a one-to-one correspondence between the spatial frequencies of the variance of diffraction-limited speckled images and the spatial frequencies of the sample fluorescence density in the super-resolved Fourier domain W h 2 provided the speckle autocovariance function C be similar to the observation point spread function h. Now, what happens if the speckle autocovariance is different from the point spread function? If the speckle correlation length is larger than the width of the point spread function, i.e. W E ⊂ W h , it is always possible to filter the recorded images so that h = C. In this case, the variance of the modified images gives access to the sample spatial frequencies in W E 2 at least.

On the other hand, if the speckle correlation length is smaller than the width of the point spread function, W h ⊂ W E , we show in appendix B, that, while the variance is sensitive to the sample frequencies in W hE , it does not necessarily allow their recovery, even on the restricted domain W h 2 . This remark applies in particular to configurations where the observation point spread function and the fluorescence density vary slowly over the speckle grain size. This is the case in fluctuation imaging techniques using near-field speckles and far-field detection [START_REF] Choi | Wide-field super-resolution optical fluctuation imaging through dynamic near-field speckle illumination[END_REF] or optical speckles and acoustic detection [START_REF] Chaigne | Super-resolution photoacoustic fluctuation imaging with multiple speckle illumination[END_REF]. In these techniques, hereafter called speckle-SOFI, the expression of the variance simplifies to,

V speckle-SOFI (r) ≈ C 0 h 2 (r -r ′ )ρ 2 (r ′ )dr ′ . (13)
where C 0 = C(r)dr. We observe that the variance is now linearly linked to the square of the sought parameter (optical absorption or fluorescence density) that is filtered over W h 2 . Now, knowing the Fourier transform of ρ 2 in W h 2 does not mean that ρ can be retrieved over W h 2 . Actually, when the speckled illumination is assumed to be spatially uncorrelated, it is possible to find samples with different fluorescence density spectra in W h 2 that have the same variance image, see appendix B. At this point, it is interesting to differentiate fluctuation imaging using quasi-uncorrelated speckled illuminations from Super-resolution Optical Fluctuation Imaging (SOFI). In SOFI, the intensity fluctuations observed in the recorded images come from the random activation of the fluorophores and not from the illumination (which is kept homogeneous and equal to E 0 during the whole experiment). To account for this phenomenon, one needs to explicit further the characteristics of the fluorescence density ρ which is related to the fluorophore concentration and the molecular brightness. We define the fluorophore concentration g at the macroscopic scale such that, V g(r) is the number of fluorophores contained in a macroscopic volume V centered about r. Next, we introduce the mean molecular brightness b which accounts for the fluorophores' quantum yield and for the environment-dependent ability of the incident/emitted photons to reach the fluorophore/detector. If all the fluorophores are activated in V , the mean brightness b is defined such that V g(r)b(r)E 0 is the energy measured by the camera of the photons emitted from V . In other terms, if all the fluorophores are activated, the fluorescence density is the product of the fluorophore concentration times the mean brightness, ρ = g × b.

In SOFI, only a few fluorophores of V are activated during the image recording and they change at each novel image. Let us assume that they follow a Poisson point process of intensity proportional to the total number of fluorophores in V . Then, the number of activated fluorophores in V observed when recording one image becomes a Poisson variable of parameter V g(r)p(r) where p is the mean percentage of activation. Under this assumption, we show in appendix D that the variance of SOFI images reads,

V SOFI (r) = E 2 0 h 2 (r -r ′ )b 2 (r ′ )g(r ′ )p(r ′ )dr ′ . ( 14
)
While RIM is able to recover the fluorescence density ρ = g × b over W h 2 , SOFI has a similar super-resolution capacity, but the latter applies to a distinct density g × b 2 × p = ρ × b × p.

It is worth noting that, if the mean brightness b is homogeneous, RIM and SOFI are able to restore the fluorophore concentration g over W h 2 , (provided the mean activation percentage p in SOFI is also homogeneous). On the contrary, even if b is homogeneous, fluctuation imaging using quasi-uncorrelated speckled illuminations (speckle-SOFI) can only restore the square of the fluorophore concentration, g 2 , over W h 2 . Thus, SOFI and speckle-SOFI yield a priori different results and their umbrella denomination as 'fluctuation imaging' can be misleading.

In conclusion, we have shown that the variance of images obtained under random speckled illuminations gives the same fluorescence density as an ideal confocal microscope, provided the speckle correlation length matches the width of the observation point spread function (which is the case for RIM). Our demonstration provides a solid theoretical ground to the two-fold resolution gain, the optical sectioning and the linearity to fluorescence observed in RIM [START_REF] Mangeat | Super-resolved live-cell imaging using random illumination microscopy[END_REF]. Also, we have pointed out the difference between the variance of SOFI-images obtained using the random activation of fluorophores and the variance of images obtained with quasi-uncorrelated speckled illuminations (speckle-SOFI). If the fluorophores brightness is homogeneous, SOFI variance is linearly linked to the Noting F the low pass filter operator that removes all the frequencies outside W h 2 , we find,

F [f 2 ](x) = F [g 2 ](x) = 75 + 13 √ 2 cos(2πkx).
Appendix D: Modeling SOFI at the macroscopic scale

Generally, SOFI data are modeled with a discrete sum that depends on the fluorophore positions [START_REF] Dertinger | Fast, background-free, 3d super-resolution optical fluctuation imaging (sofi)[END_REF]. However, it is clearly impossible to recover the fluorophores positions from SOFI (second-order) image, except if a constraint of sparsity is assumed. It may thus be interesting to relate SOFI images to sample characteristics that are defined at a macroscopic scale, such as the fluorophore concentration.

In structured illumination microscopy, one assumes that all the fluorophores are activated. The fluorescence density ρ is written as the product of the fluorophore concentration g with a mean brightness b. The intensity recorded by the camera is modeled as,

I(r) = h(r -r ′ )E(r ′ )g(r ′ )b(r ′ )dr, ( D1
)
where E is the inhomogeneous illumination intensity and h the microscope point spread function.

In SOFI, the illumination E 0 is homogeneous but the fluorophores oscillate between an activated and nonactivated state. Thus, only a subset of the fluorophores present in the sample contributes to the image intensity at a given time t. The activated fluorophores in the (macroscopic) volume V centered about r can be seen as points popping up at random and independently of one another. This process is conveniently modeled with a Poisson point process of intensity proportional to the number of fluorophores in V . Under this assumption, the number of activated fluorophores in V at time t, written as V q(r, t), where q is the activated fluorophore concentration, is a Poisson variable of parameter V g(r)p(r) with p the mean percentage of activation. With this definition, the intensity of the image recorded at t can be written as, I(r, t) = E 0 h(rr ′ )q(r ′ , t)b(r ′ )dr ′ .

(D2)

It is thus a filtered Poisson variable whose time variance reads [16, Chap. 5],

V SOFI (r) = E 2 0 h 2 (rr ′ )b 2 (r ′ )g(r ′ )p(r ′ )dr ′ .

fluorophore concentration while speckle-SOFI variance is quadratically linked to the fluorophore concentration.

Appendix A: Proof of Identity Introducing Eqs. [START_REF] Mangeat | Super-resolved live-cell imaging using random illumination microscopy[END_REF][START_REF] Dertinger | Fast, background-free, 3d super-resolution optical fluctuation imaging (sofi)[END_REF] in the left-hand term of Eq. [START_REF] Sentenac | Unified description of threedimensional optical diffraction microscopy: from transmission microscopy to optical coherence tomography: tutorial[END_REF] yields, A = B U,V (r)V (r)dr where B U,V (r) = dr 1 dr 2 dx h(rr 1 )U (r 1 )h E (r 1x)h(rr 2 )V (r 2 )h E (r 2x).

Integrating over x and using

(where we have assumed that C = h) yields,

Finally, using the expression of M V in Eq. ( 8) and the symmetry of h and h E , we get

Appendix B: What happens when W h ⊂ WE ?

We have shown that provided h = C (which can be obtained with an appropriate filtering of the raw images as soon as W E ⊂ W h ), there is a one to one correspondance between the spatial frequencies of the variance of the speckled diffraction-limited images and that of the sample in W h 2 . We now study a case where h = C and W h ⊂ W E . To simplify the discussion, we assume that (W h , W E ) are centered plain disks with frequency cut-offs ν h and ν E respectively, with ν h < ν E . We further assume that the point spread function h is symmetric so that h is a real positive symmetric function, like C. We consider a sample whose Fourier spectrum is restricted to the null frequency and a high frequency ±k, ρ(r) = A+B cos(2πk•r+ϕ) with (A, B) real positive such that ρ is real positive. The variance of the raw images, given by Eq. ( 5), obtained with such sample reads,

We observe that, as long as k < ν h + ν E , β(k) = 0 and the variance depends on the high spatial frequency of the sample, B. This result confirms the sensitivity of the variance to sample spatial frequencies in W hE . However,

In this case, the variance is sensitive to the amplitudes of the null and high frequencies of the sample, (A, B), but it has lost the information about the phase of the high frequency, ϕ. Worse, this exemple shows that a uniform sample defined by ρ 1 (r) = (A 2 + β(k)B 2 ) 1 2 /α will have the same variance as the inhomogeneous sample defined by ρ(r) = A+B cos(2πk•r+ϕ). Thus, when ν h < ν E , the identifiability of the sample spatial frequencies from the variance is lost, even for frequencies belonging to W h 2 . This assertion is particularly counter-intuitive as it shows that decreasing the size of the speckle grains below the width of the observation point spread function is a priori detrimental to the sample reconstruction.

Appendix C: Imaging with spatially quasi-uncorrelated speckles, speckle-SOFI When the point spread function and fluorescence density are slowly varying over the width of the speckle autocovariance function, the variance of the diffractionlimited images is linearly linked to the square of ρ filtered over W h 2 . In this section, we provide an example of two positive functions with different Fourier contents in the super-resolved domain W h 2 which, when squared, have exactly the same Fourier content in W h 2 .

We consider g the sum of a constant and a onedimensional cosine along the x-axis with a frequency k laying in W h 2 but not in W h , and f the sum of a constant and two cosines with period k and 2k. Note that 2k lays outside W h 2 . We adapt the constant and the cosine amplitudes so that f and g are positive and f 2 and g 2 are equal in W h 2 . A possible solution is,