Emily Bloch

Eric Besson

Séverine Queyroy

Richard Llewellyn

Stéphane Gastaldi

Philip L Llewellyn

Investigating Unusual Organic Functional Groups to Engineer the Surface Chemistry of Mesoporous Silica to Tune CO 2 -Surface Interactions

Keywords: Carbon dioxide, functionalized mesoporous silica, microcalorimetry, in silico design, SBA-15

As the search for functionalized materials for CO 2 capture continues, the role of theoretical chemistry is becoming more and more central. In this work, a strategy is proposed where ab initio calculations are compared and validated by adsorption microcalorimetry experiments for a series of, so far unexplored, functionalized SBA-15 silicas with different spacers (aryl, alkyl) and

terminal functions (N 3 , NO 2 ). This validation then permitted to propose the use of a nitro-indole surface functionality. After synthesis of such a material the predictions were confirmed by experiment. This confirms that it is possible to fine tune CO 2 -functional interactions at energies much lower than those observed with amine species.

INTRODUCTION

The separation step in any chemical process can represent over 50% of the operational expenditure and separations, in general, represent 10-15% of the world energy use. Adsorption based separation processes such as Pressure Swing Adsorption 1 can provide an interesting alternative to other techniques in terms of energy use, environmental impact, simplicity and overall cost. Indeed research is active in the use of adsorbents with the search for materials with improved properties in terms of uptake (working capacity) and selectivity for example. [START_REF] Mersmann | The Potential of Energy Saving by Gas-Phase Adsorption Processes[END_REF] Increases in selectivity can be obtained by adjusting the pore size to increase the segregation between molecules of different sizes, known as molecular sieving. [START_REF] Caro | Zeolite Membranes -State of their Development and Perspective[END_REF] One can adapt the shape of the pores in order to allow for optimal packing and thus induce entropic effects to aid selectivity. [START_REF] Krishna | Entropy Effects during Sorption of Alkanes in Zeolites[END_REF] One can equally fine-tune the chemistry of the adsorbent surface in order to induce stronger attractions for a target species thus increasing thermodynamic selectivity. [START_REF] Sircar | Basic Research Needs for Design of Adsorptive Gas Separation Processes[END_REF] Examples of the above effects can be found for the zeolite family of materials. As a first example, the compensation cations in the LTA zeolite structure can be found at the pore entrances and thus by changing the cation, the entrance size can be modified and this can lead to adsorbents which are highly selective to small molecules such as water. [START_REF] Cartigny | Frequency-Response Method in the Study of Methane Diffusion in 3 Cation Exchanged Zeolites-A[END_REF] An example of an entropic separation can be found for para-xylene recovery using BaX zeolites in the Eluxyl process. [START_REF] Ash | Eluxyl -A New Paraxylene Separation Process[END_REF] Shape selectivity can also be observed with the ITQ-12 zeolite which is selective towards propane with respect to propene. [START_REF] Barrett | ITQ-12: A New Microporous Silica Polymorph Potentially Useful for Light Hydrocarbon Separations[END_REF] In terms of chemistry, the use of lithium cations greatly improves the selectivity of zeolite X towards carbon dioxide. [START_REF] Baksh | Lithium Type X Zeolite as a Superior Sorbent for Air Separation[END_REF] In terms of possibilities to vary the chemistry of adsorbents, the family of metal-organic frameworks (MOFs) is unprecedented with possibilities to vary either the metal modes, organic linkers to form different structures. [START_REF] Zhou | Metal-Organic Frameworks (MOFs)[END_REF] It is additionally possible to graft organic functional groups to these nodes and linkers as well as incorporate cations or other species into the porous structure. One example is that of the UiO-66 family in which it is possible to vary the organic functions on the terephthalate linker. [START_REF] Bai | Zr-based Metal-Organic Frameworks: Design, Synthesis, Structure, and Applications[END_REF] Additionally, computational methods have been used as tools to screen for the most optimal organic functional groups for carbon dioxide capture. [START_REF] Wu | Revealing the Structure-Property Relationships of Metal-Organic Frameworks for CO 2 Capture from Flue Gas[END_REF] The most interesting material in terms of chemistry has then be synthesized and experimentally verified for CO 2 capture from nitrogen. [START_REF] Yang | A Water Stable Metal-Organic Framework with Optimal Features for CO 2 Capture[END_REF] The use of computational chemistry is extremely useful in such cases. Indeed, with over five thousand MOF structures and an almost infinite range of possibilities to functionalize the materials with organic moieties, a strategy based on computational screening is highly desirable. Furthermore, the creativity of organic synthesis chemists is huge and the range of functions that have been tested with adsorbent materials has been relatively limited. There is thus an opportunity to develop a strategy to explore a wide range of organic functions that could be of interest for a specific application in gas adsorption and separation. Indeed, computational screening is well adapted as it is far less expensive, polluting and time consuming than the equivalent experimental studies involving synthesis and testing. The major drawback is the choice of the right computational parameters in order to account as best as possible for the effects to be screened for, without compromising on computer calculation rate.

As a starting point, it is of importance to concentrate on the organic functions themselves.

From a synthesis standpoint one can simply anchor the functions on a relatively neutral support to compare solely on the effect of the function. To this end, ordered mesoporous silica is an ideal candidate adsorbent support with SBA-15 being of particular interest owing to its large porosity, its high surface area and its easy functionalisation. [START_REF] Vinu | Recent Advances in Functionalization of Mesoporous Silica[END_REF] As an initial test molecule to screen for, carbon dioxide is topical with a current focus in carbon capture and storage. In addition, from a computational point of view, CO 2 is relatively simple to model. This rather simple CO 2 case study, but with rather complex organic functions, will act as a springboard to highlight the usefulness of the strategy proposed and show the possibilities of further development to more complex problems.

There are a number of screening criteria that one can consider. [START_REF] Wiersum | An Adsorbent Performance Indicator as a First Step Evaluation of Novel Sorbents for Gas Separations: Application to Metal-Organic Frameworks[END_REF][START_REF] Rege | A Simple Parameter for Selecting an Adsorbent for Gas Separation by Pressure Swing Adsorption[END_REF] Screening criteria can be based on uptake, or working capacity. Another criteria can be in terms of kinetic or thermodynamic selectivity. In the present study, we have chosen to screen the material in terms of energy. Indeed the interaction of any species with an adsorbent material will have a direct effect on the uptake rate and the difference in interactions for two different species will influence the selectivity. Finally the strength of interaction will equally affect other process parameters such as regenerability and column heat management. All of these points suggest that the interaction strength is a key point that can be used as a screening parameter.

To measure the interaction strength, one can calculate the isosteric enthalpies of adsorption from isotherms measured at different temperatures. [START_REF] Rouquerol | Adsorption by Powders and Porous Solids, Principles, Methodology and Applications[END_REF] A second approach is to directly measure the enthalpies of adsorption using microcalorimetry. [START_REF] Llewellyn | Gas Adsorption Microcalorimetry and Modelling to Characterise Zeolites and Related Materials[END_REF] These experimental enthalpies of adsorption can be compared to energies calculated using either ab initio or GCMC type calculations, thus allowing a validation of the computational approach.

The aim of this article is thus to highlight the development of a strategy to develop materials with optimal organic functions for a given target molecule in a gas adsorption separation/storage type scenario. One of the criteria used in this work is to avoid overly strong interactions such as the chemisorption type interacts that are observed with amine functions and carbon dioxide where various species can be formed including carbamate, carbamic acid, and bidentate carbonate. [START_REF] Knöfel | Study of Carbon Dioxide Adsorption on Mesoporous Aminopropylsilane-Functionalized Silica and Titania Combining Microcalorimetry and in Situ Infrared Spectroscopy[END_REF] The formation of such strongly bound species can be problematic in terms of regeneration. Therefore in this study, species leading to physisorption type interactions with CO have been privileged. Such interactions are both simpler to model and would lead to materials more suitable for adsorptive based separation processes. In this respect, we have chosen -NO and -N 3 functions which are relatively stable, have permanent moments and whose adsorption properties have not yet been explored.

The strategy proposed in this work thus consists of firstly the synthesis of a series of materials in which organic linker and function have systematically been varied. In a second step adsorption microcalorimetry has been performed on these materials and a rationalization of the results has been carried out. At the same time in silico calculations have been made based on the functions synthesized and these results are compared to experiment. These calculations have suggested the use of further moieties with slightly stronger interactions for CO 2 and one of these materials has been prepared. The confirmation of the increased interaction by microcalorimetry experiments permits such an in silico design strategy to be validated. This leads to conclusions and perspectives concerning the applicability of this approach to far more challenging systems.

EXPERIMENTAL Materials synthesis

The organic-inorganic hybrid nanostructured silicas were prepared following the sol-gel process in the presence of a structure-directing agent and an alkyl(triethoxysilane). The syntheses of the trialkoxysilane precursors as well as their characterizations are fully described in the supporting information (SI). In a typical material synthesis, pluronic P-123 (2 g) in deionized water (65 mL) and 12 M hydrochloric acid solution (10 mL) were stirred for 3 h at 40 °C. Tetraethoxysilane (7.66 g, 36.85 mmol) and a trialkoxysilane precursor (4.09 mmol) were added.

The mixture was stirred for 24 h at 40 °C, then warmed without stirring at 100 °C for 2 days, filtrated, washed twice with water, once with ethanol and finally extracted with a Soxlhet apparatus (ethanol) for one day. The wet powder was filtrated, washed twice with ethanol, acetone and diethylether. After one night at 80 °C under vacuum, a powder was recovered. The characterizations of each silica are in SI. The structure of the organic moieties in the pore of the silicas and their loadings are given in Figure 1. The loading 1/X indicates the organic moieties/TEOS molar ratio determined after the characterization of the hybrid silicas.

Si O Si OH N 3 Si O Si OH NO 2 Si N 3 Si OH n Si Si OH N H NO 2 SBA C3-N3 (1/11) : n = 1 SBA C11-N3 (1/18) : n = 9 SBA Ar-N3 (1/24) SBA Ar-NO2 (1/15) SBA Ar-NO2 (1/18) SBA Ar-NO2 (1/24) SBA indole-NO2 (1/30) Figure 1.
Structure of the silicas functionalized with an organic moiety in the pores.

Microcalorimetry

Adsorption enthalpies were measured experimentally using a Tian-Calvet type microcalorimeter coupled with a home-made manometric gas dosing system. [START_REF] Llewellyn | Gas Adsorption Microcalorimetry and Modelling to Characterise Zeolites and Related Materials[END_REF] This apparatus allows the simultaneous measurement of the adsorption isotherm and the corresponding differential enthalpies. Gas is introduced into the system using a step-by-step method and each dose is allowed to stabilize in a reference volume before being brought into contact with the adsorbent located in the microcalorimeter. The introduction of the adsorbate to the sample is accompanied by an exothermic thermal signal, measured by the thermopiles of the microcalorimeter. The peak in the calorimetric signal is integrated over time to give the total energy released during this adsorption step. At low coverage the error in the signal can be estimated to around ±0.2 kJ mol -1 .

Two separate experiments were carried out at 30 °C using large (0.6 bar) doses up 6 bar and small (0.1 bar) doses up to 1 bar respectively. Around 0.2 g of sample was used and was outgassed at 150 °C for 16 hours under secondary vacuum prior to each experiment. For each injection of gas, equilibrium was assumed to have been reached after 90 minutes. This was confirmed by the return of the calorimetric signal to its baseline (<5 μW). The gas used for the adsorption was obtained from Air Liquide and was of N47 quality (99.997% purity)

Computer simulation

All simulations have been performed with Gaussian 09 program. [START_REF] Frisch | Revision D.01[END_REF] The starting structures were constructed with Agui 10.1. 21 All systems contained one molecule of CO 2 and one organic molecule of generic formula : H3Si-linker-function. For all the systems, a full geometry optimization was performed in vacuo at the MP2/6-311+g(d,p) level, and a harmonic frequency analysis was followed to characterize the located stationary points as true minima. The basis set superposition error (BSSE) was removed from the calculation of the complexation energies using the counterpoise correction protocol implemented into Gaussian. [START_REF] Boys | Calculation of Small Molecular Interactions by Differences of Separate Total Energies -Some Procedures with Reduced Errors[END_REF][START_REF] Simon | How Does Basis Set Superposition Error Change the Potential Surfaces for Hydrogen Bonded Dimers?[END_REF] For each system, several conformations (with different starting position of the CO 2 molecule) were optimized, some of which lead to the same minimum.

RESULTS AND DISCUSSION

Materials characterization

Prior to the evaluation of the effect of organic functionalization on CO 2 adsorption, the prepared hybrid nanostructured silicas were fully characterized by nitrogen physisorption at 77K, thermogravimetric analysis, XRD and [START_REF] Yang | A Water Stable Metal-Organic Framework with Optimal Features for CO 2 Capture[END_REF] C and 29 Si NMR spectroscopies. This characterization allows several materials parameters to be determined which are shown in Table 1. In general terms it would seem that as the loading of the function in the material increases, then the BET area decreases. It would equally seem that the BET area equally decreases when a longer linker unit is used. However, caution must be taken with such conclusions concerning BET area as the organic function density of the materials will change with the loading of organic material and any comparison per mass may not be rational and this point is further discussed below.

To further understand the role of the functionalization on the CO 2 adsorption properties, calorimetry experiments were performed and different parameters were explored.

Effect of ligand loading

An initial point that has been studied is the effect of the concentration of organic functions on the adsorption properties of carbon dioxide at 303K. In this respect, three samples were prepared with different loadings of the 'NO 2 ' function with aryl 'C 6 H 4 ' spacers. The number of ligands per nm 2 almost doubles for each increment from 0.41, to 0.75 and finally 1.6 for the 1/24, 1/18 and 1/15 samples respectively (Table 1). Figure 2 shows the results obtained for these three samples for CO 2 adsorption at 303K. Figure 2a shows the adsorption isotherms given as amount adsorbed per gram of solid. It would thus initially seem that the different loading of organic functions has no effect on uptake. However, as discussed above, one has to bear in mind that the silica and organic function have different densities and that this representation, per mass, does not compare equivalent volumes or equivalent quantities of 'silica' material. As a comparison per equivalent amount of silica is not straightforward, another approach that can be taken is to compare uptake per amount of accessible area using the results obtained from the BET analysis (Table 1). The isotherms replotted in terms of CO 2 uptake per m 2 of available (BET) surface is given in Figure 2b. It is clear from this plot that the more functions per surface, the higher uptake as would be expected, and that the overlap observed in Figure 2a is fortuitous. To summarize, as expected and at a first glance, the amounts adsorbed are proportional to the number of organic functions per nm². The isotherms show that comparing materials solely in terms of uptake is not a simple task.

Comparing the adsorption enthalpies may be a way forward. In this respect, the energies of adsorption obtained with CO 2 at 303 K are shown in Figure 2c. In this comparison, the materials with the lower concentration of functions show similar adsorption energies, and this could be expected if we consider that the CO 2 interacts with each function. However slightly higher Pressure / bar energies are measured with the sample with the highest density of organic functions per nm², which may suggest cooperative effects between neighboring functions.

Effect of linker and terminal function

In choosing a terminal organic functional group to integrate into a material, one can attempt to separate the effect of the linker (or spacer) with that of the function. Indeed, the use of a linker with an aryl group will provide electrons which will affect the polarity of the function.

Furthermore, the length of the linker may equally affect the chemistry of the organic functional group as well as the geometry. With this in mind, three linkers have been used with the same function (-N 3 ). These linkers include a C 3 alkyl chain, a C 11 alkyl chain and an aryl spacer. The results for CO 2 adsorption at 303 K are shown in Figure 3. The amount of CO 2 adsorbed per unit surface, Figure 3a, seems to show an inverse trend with respect to the BET area (Table 1) and there seems to be no trend with respect to the number of organic functions per nm 2 . Indeed, in terms of enthalpy of adsorption, there equally seems to be no significant difference in energies with CO 2 uptake. These results seem to suggest that the linker has little effect on the CO 2 uptake in the particular cases chosen here. However, in other studies it has been shown that the inclusion of aryl groups or the position of the function in a chain can affect gas uptake. [START_REF] Zelenak | Amine-Modified SBA-12 Mesoporous Silica for Carbon Dioxide Capture: Effect of Amine Basicity on Sorption Properties[END_REF] To compare the effect of the organic functional group present (N 3 versus NO 2 ) on the CO 2 adsorption properties, the same linker was used and the results are shown in Figure 4. In, this comparison, there are approximately the same functions per unit surface rendering the comparison slightly easier. There seems to be a greater uptake and slightly higher enthalpy of adsorption for CO 2 with the N 3 moiety with respect to the NO 2 moiety.

However any differences are small.

Effect of passivation

From the above results several questions arise. The first of these pertains to the fact that the silica based materials that have been prepared contain a significant quantity of hydroxyl groups.

It can thus be difficult to separate the effect of the organic functional group on the CO 2 adsorption with respect to the effect of surface hydroxyls. With this in mind, some of the samples were passivated with Me 3 SiCl. Table 1 shows that the passivation led to a decrease in surface area by a factor 2. The comparison of samples before and after passivation is given in Figure 5.

Figure 5 highlights that the passivation has the same effect whatever the function present in the material. Indeed as can be expected, passivation leads to a decrease in surface area which translates to a decrease in CO 2 uptake (SI). Furthermore, the passivation neutralizes interactions with -OH groups and this translates to a decrease in adsorption energy which depends on the organic functional group. Indeed this decrease is more noticeable in the case of the -N 3 function (4 kJ mol -1 , Figure 5b) than with the -NO 2 function (2.5 kJ mol -1 , Figure 5c). Inversely, this suggests that the CO 2 would interact more strongly with the -NO 2 alone than with the -N 3 group alone. Nevertheless, this section highlights that the energies that we measure are most probably the result of a synergetic combination of CO 2 interactions with both the organic functional group and surface hydroxyls, and which could equally be resultant of cooperative effects.

To further aim to understand the effect of linker and terminal function on the CO 2 interactions, CO 2 adsorption was equally performed on the pure silica material which was then equally subjected to a passivation treatment. The energies of adsorption are also shown in Figure 5a and equally evidence a slight decrease in interaction energy (±1 kJ mol -1 ), again related to loss of the possibility of CO 2 interaction with surface -OH groups.

Molecular modelling of the interactions in play

To further understand and predict organic functional groups for a given probe molecule, ab initio calculations were performed. Nevertheless, as we have partially seen above, the adsorption of CO 2 depends on many criteria: the geometry of the material, the loading of the organic functions included in the material, the affinity of these organic functions for the gas, etc…

The part due only to the nature of the organic function and linker can be estimated in silico by computing the complexation energy between the functions and the gas. It is then possible to compare the enthalpies of adsorption at low loading with the calculations of the complexation energy of one molecule of CO 2 and one organic functional group, obtained with quantum mechanical simulations. This would be the first step towards an in silico screening of potential functions for gas physisorption.

In order to compute reliable complexation energies, Second order Møller-Plesset simulations with 6-311+g(d,p) basis set were performed. We expect that this level of theory would take properly into account correlation's effects. BSSE corrected complexations energies for each complex (CO 2 +molecule) are reported in Table 2. For a given complex, several conformations (shown in Figure 6 and in the SI) corresponding to energy minima could be found and explains the multiple entries in the table. If we take into account only the lowest energy minima (statistically more representative), these results indicate that N 3 and NO 2 functions have similar complexation energies. The aryl and alkyl linkers also behave similarly. These results are confirmed experimentally above. Hydrogen bonding with the silanol groups on the material surfaces has also been investigated.

For the C 3 N 3 molecule, the complexation energy is -12.6 kJ mol -1 if one SiOH is added to the molecule (OH-SiH 2 -O-SiH 2 -linker-function). This further interaction represents a gain of around 3 kJ mol -1 . This value compares well with the decrease in enthalpy measured experimentally for the materials obtained before and after passivation (Figure 5). Thus this in silico approach seems to provide an interesting correspondence with experiment.

However, in order to induce stronger interactions with the CO 2 molecule, further linkers and conformations were explored. The effect of the SiOH suggests the use of an indole linker which may induce an enhanced complexation energy due to a secondary interaction via hydrogen bonding between the NH of the indole and CO 2 (see Table 2). To simplify the syntheses, a nitro group was preferred rather than an azido and the corresponding nitro-indole material was prepared from commercially available 7-nitro-indole as depicted in scheme 1. The CO 2 adsorption measurement was carried out as shown in Figure 7a. This result indeed shows a slightly enhanced adsorption energy and this allows a plot of measured adsorption energy at low coverage as a function of predicted complexation energy to be constructed (Figure 7b).

The plot in Figure 7b further suggests that a general correlation can be made between the in silico screening and experimental results which is encouraging for future research in terms of material design for a given probe molecule. This also means that the simplifications implied by our simulation model (only one chain, only one CO 2 , no surface, etc…) have a negligible impact on the selection of the best organic moieties. This sample may show interest with respect to CO 2 capture for N 2 or CH 4 as it can be seen from Figure 7c that the uptakes of these latter gases are quite negligible with respect to CO 2 . This is confirmed as the adsorption energies at low coverage are equally much lower for N 2 (17 kJ mol -1 ) and CH 4 (15 kJ mol -1 ) than for CO 2 .

Finally, IAST calculations 25 of selectivity lead to a value of 35 for a 15% CO 2 / 85% N 2 mixture at 1 bar, representative of flue gas, and 15 for a 50% CO 2 / 50% CH 4 mixture. These encouraging results of enthalpy and selectivity suggest that it can be possible to fine tune materials chemistry with the use of less standard organic functions. This is especially the case when would like to move away from the quite, maybe overly, strong interactions that can be found when using amine functionalization. Further, the overall aim of this study was to develop a strategy of materials design, using a quite simple case study. This can clearly be adapted to other gas and vapor targets more appropriate for such hybrid materials. As such the use of in silico screening with more complex organic functions may be a way forward in studying more challenging problems such as propane/propene separations and the recovery of toxic vapors for example.

Indeed, where subtle differences in gas mixture or where toxic molecules are involved, a computer aided screening strategy is optimal for materials design.

CONCLUSIONS

This work has highlighted that computer simulations can be used to model the interactions of carbon dioxide with organic functional groups that can be incorporated into mesoporous silica.

We suggest that the use of microcalorimetry may be a more pertinent data parameter than uptake in the characterization of materials where only subtle differences are required or that in instances where effects such as available surface area or functional group density may also play roles.

Microcalorimetry can be used to both characterize the effect of function, linker and surface and compares well with the in silico modeling. Interestingly the computer modeling can further be used to follow the effect of surface hydroxyls in the proximity of the organic functions studied.

This experimental validation of the in silico modeling thus opens an approach to screen many other organic functions in order to find materials with optimal features for gas capture.

As a perspective, this screening may permit to explore more complex organic functions and linkers to lead to composite interactions where several anchor points can be used to interact with more intricate target molecules. Indeed, this exploration can be expanded to cover little or unknown organic functions to be attached that can be synthesized and incorporated using one pot materials synthesis as described above. This computer aided approach can be used to treat this problem from the materials perspective where one could screen one given organic function's potentiality to interact with a range of target molecules. Overall, this work is the first step in a strategy to use in silico screening for the optimization of materials design and synthesis.

Figure 2 .

 2 Figure 2. CO 2 adsorption at 303K on the series of 'Ar-NO 2 ' samples with different loadings of

Figure 3 .

 3 Figure 3. CO 2 adsorption on materials with -N 3 functions and various linkers : alkyl C 3 , alkyl

Figure 4 .

 4 Figure 4. CO 2 adsorption on materials with the same linker and different functions: -N 3 and -

Figure 5 .

 5 Figure 5. CO 2 adsorption enthalpies on (a) silica, (b) aryl-N 3 and (c) aryl-NO 2 materials before

Figure 6 .

 6 Figure 6. Two conformations of CO 2 -C 3 N 3 complex

Figure 7 .

 7 Figure 7. (a) Adsorption of CO 2 at 303 K on mesoporous silica functionalized with 7-nitroindole

Table 1 .

 1 Characteristics of some of the materials prepared

	Material	Linker	Function	A BET (m².g -1 )	Function	Function per nm²
					per -SiO 2
	SBA C3-N3 (1/11)	C 3 H 6	N 3	580	1/11	1.4
	SBA C11-N3 (1/18)	C 11 H 22	N 3	470	1/18	1
	SBA Ar-N3 (1/24)	C 3 H 6 O-	N 3	793 (371) a	1/24	0.45
		C 6 H 4
	SBA Ar-NO2 (1/15)	C 3 H 6 O-	NO 2	350	1/15	1.6
		C 6 H 4
	SBA Ar-NO2 (1/18)	C 3 H 6 O-	NO 2	632 (270) a	1/18	0.75
		C 6 H 4
	SBA Ar-NO2 (1/24)	C 3 H 6 O-	NO 2	900	1/24	0.41
		C 6 H 4

a values in parentheses correspond to passivated materials.

Table 2 .

 2 BSSE corrected complexations energies between CO 2 and different molecules.

	Molecule	Conformation	E complexation (kJ mol -1 )
	C 3 N 3	a	-9.6
	C 3 N 3	b	-5.4
	C 3 NO 2	a	-9.6
	C 3 NO 2	b	-9.6
	Ar N 3	a	-7.2
	Ar N 3	b	-9.6
	Ar NO 2	a	-8.4
	Ar NO 2	b	-8.4
	Indole-N 3		-13.4
	Indole-NO 2		-12.6

ASSOCIATED CONTENT

Supporting Information. Details of energy minima conformations, synthesis and characterization (NMR, SAXS, N 2 adsorption and TGA) are given in the supporting information.

AUTHOR INFORMATION *philip.llewellyn@univ-amu.fr

Author Contributions

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


François Vibert

Emily Bloch

Michèle P Bertrand

Séverine Queyroy

Stéphane Gastaldi
email: stephane.gastaldi@univ-amu.fr

Eric Besson
email: eric.besson@univ-amu.fr.

Evidence for the Contribution of Degenerate Hydrogen Atom Transport to the Persistence of Sulfanyl Radicals Anchored to Nanostructured Hybrid Materials

Keywords: TEM, SAXS, Nitrogen adsorption/desorption analysis, ATG), simulations, EPR decay curves]

Nanostructured functionalized silicas were used as a platform to compare the behaviour of anchored arylsulfanyl radical depending on the nature of the precursor (diazene/thiol). The radicals generated from thiols exhibit higher half-lifetimes than the radicals issued from diazenes. The ability of thiols to maintain the sulfanyl radical density via degenerate hydrogen atom transfer is likely to account for this sharp difference.

Introduction

The continuous interest in persistent and stable radicals 1 arises from their broad spectrum of applications, ranging from spectroscopy (EPR, 2 DNP 3 ) to the design of smart devices. 4 The persistence and the stability of these families of radicals can originate either from steric effects, as exemplified by triarylmethyl radicals, or from electronic effects, as illustrated with the phenalenyl radical, but most often it results from a subtle interplay of both effects.

5 Furthermore, it is known that the persistence of a radical can be improved by sorption onto materials 6 such as zeolites, 7 or by grafting onto a silica. 8 In such situations, the increase of radical lifetimes has been rationalized in term of limited diffusion and/or interactions with the surface. Progress in the synthesis of mesoporous silicas via the sol-gel process in the presence of structure-directing agents enables the preparation of organic-inorganic hybrid nanostructured materials. The organic moiety can be selectively located and regularly distributed on the pore surface of the silica with a sharp control of the loading. 9 The multiscale control of the structure of mesoporous silicas was used to investigate the parameters responsible for the increase of the lifetime of transient radicals anchored to the material.

In previous studies, we have shown that the structure of the hybrid material played a key role in the improvement of the lifetimes of radicals anchored in the framework 10 or on the pore 11 of mesoporous silicas. The presence of an organic monolayer on the pore surface was shown to be of crucial importance to disfavour the coupling reaction, i.e. the formation of disulfides, which is the main contributor to the decay of the EPR signal in the case of arylsulfanyl radicals. 11 In these materials, the arylsulfanyl half-lifetime (t 1/2 ) is considerably improved, up to a 10 9 factor compared to the reference value of 200 μs estimated by flash photolysis for phenylsulfanyl radical in solution. 12 However, in systems as complex as hybrid materials, no simple rationale can be drawn to account for the increase of the radical lifetime because of the numerous parameters that are involved in the phenomenon. We found it important to evaluate the incidence of the nature of the organic functional groups in the neighbourhood on the radical persistence. In a first approach, the arylsulfanyl radicals were generated in the pore of the silica from the homolysis of a S-H bond. Thus, the created radicals were surrounded by hydrogen atom donors. In order to decrease the reducing nature of the pore surface, a diazene was designed as an alternative precursor for the very same arylsulfanyl radicals. The comparative results are discussed in the following.

Results and discussion

A simple photolysis can trigger the fragmentation of the diazene group. The loss of dinitrogen leads to a pair of alkyl radicals. 13 The β-sulfur carbon-centered radicals are well known to undergo a fast β-fragmentation 14 releasing the sulfur-centered radicals (Scheme 1). The dissymmetric diazene precursor 1 was prepared in two steps via the substitution of mesylate 5 with thiol 4 followed by the olefin hydrosilylation (Scheme 2). The dissymmetric diazene 5 was prepared from aminoester 9, via its conversion in bis-sulfonamide 8. The extrusion of sulfur dioxide led to diazene 7 which was subsequently converted into mesylate 4.

The triethoxysilane 1 was used to prepare mesostructured silicas SBA n -1 [START_REF]The resulting materials were named SBA n -1 in which 1 specifies the nature of the organic precursor and n indicates the TEOS/radical precursor molar ratio determined after the characterization of the hybrid silicas[END_REF] by direct synthesis, 9 that is, the cocondensation of 1 and tetraethyl orthosilicate (TEOS) in the presence of P123 (PEO 20 PPO 70 PEO 20 ) as structure directing agent. This process enabled the preparation of SBA-15 type of mesoporous silicas as illustrated by their surface areas, their

Please do not adjust margins Please do not adjust margins pore diameters and pore volumes (Table 1). In both series SBA n -1 and SBA n -2, as usually observed, the surface areas increased while the concentration in organic functions decreased. Concerning the pore diameter, the difference between the two materials was more pronounced. While pore diameters were quite regular for SBA n -2, they notably increased for SBA n -1 for the lower loadings what might be related to the steric hindrance of the radical precursors. The hybrid silica SBA 47 -1 was passivated upon reaction with ClSiMe 3 . The characterization of all materials demonstrated that the diazene moiety was stable to the sol-gel process conditions.

Scheme 1 Radical generation

Scheme 2 Diazene precursor synthesis

The behaviour of the arylsulfanyl radicals generated from the photolysis of these precursors was investigated by EPR. In a typical procedure, the functionalized material was degassed (10 -5 mbar) in a 4 mm quartz-glass tube and arylsulfanyl radicals were generated via the light-induced decomposition of the diazene inside the spectrometer cavity (xenon lamp (200-800 nm)). An anisotropic signal, resulting from the immobilization of the radicals, was recorded at room temperature after irradiation whatever the functionalized silica. The EPR parameters of these signals were in agreement with the formation of arylsulfanyl radicals 10,11,16 for which the g tensor is axisymmetrical and the spin density is localized on the sulfur atom. 17 A representative spectrum is shown in Fig. 1a. For both SBA n -1 and SBA n -2, g iso 18 ranging from 2.008 to 2.010 were calculated from graphically determined g⊥ and g //. These values were consistent with the formation of an arylsulfanyl radical. The half-lifetimes of the anchored arylsulfanyl radicals were measured by monitoring the decay of the signal intensity after turning off the irradiation. They were determined from the double integrated EPR signal. The decay curves were fitted with exponential or multi-exponential models which did not allow the mechanism involved to be determined. Several competitive processes are likely to participate to the signal decay. The half-lifetimes were graphically estimated from the fitted decay curves. The radical half-lifetimes recorded for the arylsulfanyl radicals generated from the various hybrid materials are collected in Table 2. Half-lifetimes increased from 3.2 h for the lowest loading (1/73, entry g) to more than 63 h for the highest loading (1/21, entry e). As the radical is identical and the loadings are similar, the reasons for the differences observed between the two types of precursors must be sought in the modification of the close environment of the radical. Once generated, the sulfanyl radical can evolve through: (i) the formation of a S-S bond via the coupling of two sulfanyl radicals, (ii) the formation of a S-H bond via a hydrogen atom transfer (HAT) between the sulfur-centered radical and a nearby group, (iii) other termination reactions involving the material surface. In our materials, the probability to generate close radical centres is low. Therefore, the probability of dimerisation is also low. We have already shown that the coupling occurs without activation barrier when the two radicals are correctly positioned, but is hindered otherwise.

Please do not adjust margins Please do not adjust margins Inside the hybrid materials, the studied arylsulfanyl radicals are anchored and have no flexibility nearby the radical center. Therefore, steric hindrance, which increases with the loading, disfavors the disulfide formation. The coupling process is slowed down at the highest loadings, 11 which is consistent with the slight increase of t 1/2 registered for SBA 27 -1 compared to SBA 47 -1 and SBA 190 -1 (Table 2, entries a-c). In the case of SBA n -2, degenerate HAT between a thiol group and a sulfanyl radical occurs readily. Whereas, movements of the whole chain are needed to position properly the two reacting sulfur atoms for dimerisation, degenerate HAT occurs at larger S...S distances and only requires that a S-H bond and the sulfur atom of a neighbouring sulfanyl radical are aligned, which is facilitated by the free rotation around the C-S bond. In order to visualize how easily could HAT proceed at the surface of the different materials, ab initio simulations of a very simple model system composed of one chain of CH 3 -CH 2 -CH 2 -O-Ph-SH and one chain of CH 3 -CH 2 -CH 2 -O-Ph-S  were performed. The energy landscape of such a system shows a very large number of transition states (TS) for the degenerate HAT, according to the geometry of the chains approach. This approach can be characterized by the C-S...S-C dihedral angle (φ). Therefore, an energy scan has been performed at the B3LYP-6-311+G(d,p) level of theory, searching for transition states and varying φ. All the transitions states found have S, H and S atoms aligned with a distance between the two sulfur atoms equal to 3.2 Å. This scan shows (Fig. 2) that TS for the degenerate HAT can be found when the distance between the terminal groups of the chains (i.e. the carbons that should be linked to the silica) d T is comprised between 7.5 Å (when the torsion is close to 0°) and 19.5 Å (when the torsion is close to 180°). When the surface functional density grows larger than 1.8 functions per nm 2 , d T becomes smaller than 7.5 Å and the steric hindrance keeps the sulfur atoms at a distance larger than 3.2 Å, preventing HAT. When the surface functional density diminishes below 0.25 functions per nm 2 , d T becomes larger than 19.5 Å and the sulfur atoms become too far away to perform HAT. Reaction paths for six of these TS have been followed by integrating the intrinsic reaction coordinate (IRC). The energy barrier for degenerate HAT have been evaluated from the stationary points of these IRC scans. It is small and does not vary significantly with the terminal groups distance or the dihedral angle: ΔG ≠ varies from 5.8 to 9.0 kcal mol -1

, while ΔH ≠ varies from 4.4 to 5.0 kcal mol -1 (c.f. Supplementary Information). The energies for the simplest system (Ph-SH + Ph-S  ) have been evaluated using the G3(MP2)RAD procedure [START_REF] Henry | [END_REF] in order to validate that the B3LYP/6-311+G(d,p) level of theory gives reasonable energy values. For this system, the B3LYP/6-311+G(d,p) activation parameters are ΔG ≠ = 8.2 kcal mol -1 and ΔH ≠ = 5.5 kcal mol -1

, while the G3(MP2)RAD . According to these results, among the loadings tested experimentally, the probability that a HAT could happen is small for SBA 73 -2 (as the number of chains is small and the distance between chains is large), but becomes larger when the loading rises (SBA 27 -2 and SBA 21 -2) as the conditions for HAT are geometrically favourable and the number of chains (therefore the number of possible contacts between chains) grows. This degenerate transfer would propagate all along the pore surface until a termination reaction becomes faster than the next available HAT. 20 Quantum mechanical tunneling (QMT) increases the efficiency of HAT. [START_REF]For general reviews[END_REF] Calculations showed that the degenerate HAT is at least two times faster than classical thermodynamics would have predicted (see ESI). This process can be related to the proton transport in acid functionalized mesoporous material.

22

In the case of SBA n -1, the termination reaction leading to a disulfide can still take place but the absence of thiol groups in the close vicinity precludes degenerate HAT. However, due to the structural features of the precursor, other reactions can be considered that might contribute to the decay of the sulfanyl radical signal. Hydrogen atom transfer from the methylene or a methyl group of a neighbouring diazene can be envisaged, but this transfer is enthalpically disfavoured (BDE ArS-H = 77.8 kcal mol -1 / C sp3 -H = 92.1 or 100.5 kcal mol -1

). [START_REF] Luo | Comprehensive Handbook of Chemical Bond Energies[END_REF] The generation of such a carboncentered radical should trigger a β-fragmentation releasing a diazenyl radical and consequently another t-butyl radical. Thus, the fate of t-butyl radical must also be considered. The latter is likely to diffuse rapidly in the silica pore and dimerize or disproportionate. Moreover, it could reduce the lifetime of the arylsulfanyl radical. First, via a recombination process that would lead to the corresponding t-butylsulfide. In addition, as it is a good hydrogen atom donor since its C-H BDE was estimated 36.7 kcal mol -1 , [START_REF] Denisov | Handbook of Free Radical Initiators[END_REF] it could deliver a hydrogen atom to an arylsulfanyl radical. This would lead to a thiol mainly surrounded by diazene precursors which in turn would generate another sulfanyl radical under irradiation. It must be underlined that no t-butyl radical was ever detected in any of the reported EPR experiments. This observation argues in favour of a fast evolution of the t-butyl radicals that would occur before the decay of arylsulfanyl radical starts being monitored. For the above mentioned reasons, these processes should not contribute significantly to the observed phenomenon, the average half-lifetime (1h) remains considerably increased compared to the 200 s half-lifetime reported for phenylsulfanyl radical in solution.

12 In summary, the most impressive change in half-lifetime for a given type of arylsulfanyl radical according to the nature of the precursor is intimately connected to the availability of a fast HAT. In the case of diazene precursors, the absence of fast HAT increases the probability of termination reactions. Independently, the thermolysis of dissymmetric diazenes SBA n -1 localized on the pore was investigated. The influence of the localization (wall/pore) on the arylsylfanyl radical lifetime was evaluated by comparison with the data obtained from symmetric diazenes embedded in the framework of the nanostructured silica. 10 The thermogravimetric analysis (TGA) showed that the thermal decomposition of diazene moieties occurred at 473 K. The decay of the corresponding arylsufanyl radicals was investigated at this temperature by EPR. Due to the temperature, the EPR signal was isotropic with a 2.009 g-factor (Fig. 1b), i.e. a value close to the values recorded at 293 K (Table 3). The half-lifetimes of the anchored arylsulfanyl radicals were measured by monitoring the decay of the signal of highest intensity. They were in the same order of magnitude than those registered when the symmetric diazene precursor was located in the wall of the silica. 10 The registered values (approximately 12 min) were still much higher than the reference value reported by Thyrion in solution at room temperature (200 μs). 12 At such a temperature, neither the effect of the monolayer structure nor passivation did improve the radical lifetime. For the sake of comparison, the same experiment was performed with SBA n -2. The heating of SBA n -2 at 473 K led to the generation of a weak EPR signal probably due to a weak efficiency of the thermal homolysis of the S-H bond. An additional irradiation did not enable to improve the signal intensity. This difference in behaviour showed the diazene precursor is the most appropriate precursor to observe arylsulfanyl radicals at high temperature.

Conclusions

Progress in the synthesis of nanostructured silicas enables the identification of the phenomenons responsible for the transformation of a transient arylsulfanyl radical into a persistent one. The comparison of the behaviour of arylsulfanyl radicals generated from two different precursors highlights the importance of the radical surroundings. At room temperature, the persistence of arylsulfanyl radicals derived from diazene precursors (1h/200 s in solution) results from the intrinsic effect of confinement which slows down the formation of disulfide likely to be the main source of decay of their EPR signal. Potential minor causes of signal decay, that must be sought in the structure of the diazene moiety in the precursor, are discussed even though their involvement could not be ascertained. The contrasting behaviour of thiol derived radicals, which exhibit a spectacular increase in half-lifetime (up to 63 h) is rationalized on the basis of fast degenerate HAT from thiol groups in the close vicinity which competes with the termination processes. From this experimental comparative study, supported by calculation, we concluded that in the case of aromatic thiol, the amazing half-lifetimes recorded result from the synergistic effects of the structure of the organic monolayer and the fast degenerate HAT. This HAT process is a dynamic phenomenon which contrasts with the steric and/or Please do not adjust margins Please do not adjust margins electronic effects that are commonly used to improve radical lifetimes.

At 473 K the situation is reversed, the occurrence of HAT no longer improves the lifetime of the sulfanyl radicals generated from thiol precursors, diazene precursors were more efficient as regard to the radical persistence.

Experimental

Methyl

2-[N-(tert-butyl)aminosulfonamido]-2methylpropanoate (8).

To a solution of methyl αaminoisobutanoate hydrochloride (9) (4.0 g, 26.0 mmol, 1 equiv) in acetonitrile (45 mL) was added sulfuryl chloride (3.51 g, 26.0 mmol, 1 equiv) at room temperature. The mixture was stirred for 7 hours at 80 °C. Sulfuryl chloride (3.51 g, 26.0 mmol, 1 equiv) was again added and the solution was stirred overnight at 80 °C. After cooling and evaporation of the solvent, the residue was triturated with Et 2 O (3 times). Et 2 O phase was dried over MgSO 4 . After concentration, the corresponding sulfamoyl chloride was obtained (4.82 g, 22.4 mmol, 86%) and used without further purification. 1 H NMR (400 MHz, CDCl 3 ) δ: 6.90 (br s, 1H, NH), 3.84 (s, 3H, OCH 3 ), 1.73 (s, 6H, 2xCH 3 ). 13 C NMR (100 MHz, CDCl 3 ) δ: 173.5 (CO), 61.6 (OCH 3 ), 53.7 (C), 24.6 (2xCH 3 ). To a 0 °C solution of freshly prepared sulfamoyl chloride (4.82 g, 22.4 mmol, 1 equiv) in CH 2 Cl 2 (45 mL) was added t-butylamine (4.92 g, 67.2 mmol, 3 equiv). The mixture was stirred at room temperature. The reaction was monitored by TLC. After completion (2 h), the mixture was diluted with water and extracted three times with EtOAc. Organic extracts were washed successively with HCl (1 N), water, brine and then dried over MgSO 4 . After concentration, sulfonamide 8 (5.03 g, 19.9 mmol, 89%) was obtained and used without further purification. To a solution of 8 (2.0 g, 7.9 mmol, 1 equiv) in pentane (10 mL) was slowly added NaOH (652 mg, 16.3 mmol) in NaOCl (29 mL, 0.56 M). The mixture was stirred for 2 hours at room temperature. The mixture was diluted with water and extracted three times with pentane. Organic extracts were washed with brine and then dried over MgSO 4 . After concentration, diazene 7 (670 mg, 3.6 mmol, 46%) was obtained and used without further purification.

1 H NMR (400 MHz, CDCl 3 ) δ: 3.69 (s, 3H, OCH 3 ), 1.42 (s, 6H, 2xCH 3 ), 1.17 (s, 6H, 3xCH 3 ).

2-[(E)-2-[1-(Methansulfonyloxy)-2-methylpropan-2-yl]diazen-1-yl]-2-methyl propane (5).

To alcohol 6 (1.19 g, 7.5 mmol, 1 equiv) and triethylamine (1.08 g, 10.7 mmol, 1.4 equiv) in CH 2 Cl 2 (90 mL) under argon was slowly added methanesulfonyl chloride (1.02 g, 8.9 mmol, 1.2 equiv) at 0 °C. The reaction mixture was allowed to warm slowly to room temperature. After 1 h (TLC monitoring), the mixture was diluted with water, washed with 1M HCl and brine and then dried over MgSO 4 .

After concentration, the corresponding mesylate 5 was obtained (1.75 g, 7.4 mmol, 99%) and used without further purification.

1 H NMR (300 MHz, CDCl 3 ) δ: 4.31 (s, 2H, CH 2 ), 2.96, (s, 3H, O 2 SCH 3 ), 1.16 (s, 6H, CH 3 ), 1.13 (s, 9H, CH 3 ).

(E)-2-(1-(4-(Allyloxy)phenylthio)-2-methylpropan-2-yl)-1-tertbutyldiazene (3).

A suspension of sodium (177 mg, 7.7 mmol, 1 equiv) in MeOH (15 mL) under argon was stirred until the reaction was completed. The thiophenol 4 (1.28 g, 7.7 mmol, 1 equiv) dissolved in MeOH (2 mL) was added at 0 °C and the mixture was stirred for 15 min at room temperature. The Please do not adjust margins Please do not adjust margins solvent was evaporated and the residue diluted with Et 2 O before being filtrated. The residue was washed with Et 2 O and dried under vacuum to give the expected thiolate as a pale yellow solid (1.39 g, 7.4 mmol, 96%) that was used without further purification. A suspension of mesylate 5 (1.75 g, 7.4 mmol, 1 equiv) and the freshly prepared sodium salt (1.39 g, 7.4 mmol, 1 equiv) in DMF (6 mL) under argon was stirred for one night at 50 °C. The reaction was monitored by TLC. After completion, the mixture was diluted with water and extracted four times with EtOAc. Organic extracts were washed with water, brine, dried over MgSO 4 and concentrated. The residue was purified using a silica gel column (1-2% Et 2 O in pentane) to give 3 as a yellow oil (1.1 g, 3.6 mmol, 49%). Experimental procedures for the synthesis and characterisation of materials: Thermogravimetric (TGA) measurements were carried out with a TGA Q500 apparatus (TA Instruments) under dynamic air atmosphere (sample flow rate 40 ml/min). SAXS experiments were performed on SAXSess-MC2 (Anton-Paar, GmbH, Austria) with a sealed copper tube as X-ray source (wavelength is 0.15417 nm (Cu Kα)) and CCD camera as detection system. The N 2 adsorption/desorption isotherms were obtained at 77 K on a Micrometrics ASAP2010. The specific surface area was determined according to the Brunauer, Emmett and Teller (BET) method and the pore size distribution was calculated from the desorption isotherms using the Barrett Joyner Halenda (BJH) method. [START_REF] Rouquerol | Adsorption by Powders and Porous Solids: Principles, Methodology and Applications[END_REF] Prior to adsorption, the samples were outgassed at 353 K overnight under a vacuum pressure of 2×10 ). BET Surface Area: 451 m²/g. BJH Desorption Average Pore Diameter: 5.1 nm. TGA: Mass Loss = 7.1%. SAXS: No signal. SBA 47 -1: Following the previous procedure, tetraethoxysilane (2.12 mL, 9.5 mmol, 19 equiv) and organic precursor 1 (235 mg, 0.5 mmol, 1 equiv). 13

Fig. 1

 1 Fig. 1 EPR spectra for (a) the photolysis at 293 K and for (b) the thermolysis at 473 K of SBA47-1 For diazene-based substrates SBA n -1, half-lifetimes ranged from 52 min to 1.3 h. The difference observed between the highest (1/27, entry a) and the lowest loading (1/190, entry b) was low. Thiol-based substrates SBA n -2, behaved differently.Half-lifetimes increased from 3.2 h for the lowest loading (1/73, entry g) to more than 63 h for the highest loading (1/21, entry e). As the radical is identical and the loadings are similar, the reasons for the differences observed between the two types of precursors must be sought in the modification of the close environment of the radical. Once generated, the sulfanyl radical can evolve through: (i) the formation of a S-S bond via the coupling of two sulfanyl radicals, (ii) the formation of a S-H bond via a hydrogen atom transfer (HAT) between the sulfur-centered radical and a nearby group, (iii) other termination reactions involving the material surface. In our materials, the probability to generate close radical centres is low. Therefore, the probability of dimerisation is also low. We have already shown that the coupling occurs without activation barrier when the two radicals are correctly positioned, but is hindered otherwise.

Fig. 2

 2 Fig.2Energy scan of TS for HAT. The functional density was estimated from the end-of-chains distance by supposing that functions are spread on the surface in a regular square pattern.

13 C

 13 NMR (100 MHz, CDCl 3 ) δ: 175.6 (CO), 59.0 (C), 54.6 (C), 52.8 (OCH 3 ), 29.9 (3xCH 3 ), 25.6 (2xCH 3 ). HRMS (ESI): m/z: calcd for [M+H] + C 9 H 21 N 2 O 4 S: 253.1217, found: 253.1218. (E)-Methyl 2-(2-tert-butyldiazenyl)-2-methylpropanoate (7).

13 C

 13 NMR (75 MHz, CDCl 3 ) δ: 76.7 (CH 2 ), 67.5 (C-N), 67.0 (C-N), 37.1 (CH 3 ), 26.6 (CH 3 ), 21.7 (CH 3 ). HRMS (ESI): m/z: calcd for [M+H] + C 9 H 21 N 2 O 3 S: 237.1267, found: 237.1268.

1H

  NMR (400 MHz, CDCl 3 ) δ: 7.33 (d, J = 8.8 Hz, 2H, ArH), 6.82 (d, J = 8.8 Hz, 2H, ArH), 6.04 (ddt, J = 17.2, 10.5, 5.3 Hz, 1H, CH=CH 2 ), 5.40 (dq, J = 17.3, 1.5 Hz, 1H, CH=CH 2 ), 5.28 (dq, J = 10.5, 1.5 Hz, 1H, CH=CH 2 ), 4.50 (dt, J = 5.3, 1.

13 C

 13 CPMAS NMR (101.6 MHz) δ: 157.5, 132.2, 130.5, 127.2, 114.5, 74.9, 69.0, 46.5, 25.4, 22.6, 15.1 (P123), 7.1. 29 Si CPMAS NMR (79.5 MHz) δ : -56.

Table 1 .

 1

		Silica	Surface area a	Pore diameter b	Pore volume c
	a	SBA27-1	451	5.1	0.77
	b	SBA47-1	744	6.8	0.97
	c	SBA190-1	640	9.5	1.16
	e	SBA21-2	370	6.2	0.58
	f	SBA37-2	479	4.4	0.76
	g	SBA73-2	628	5.6	0.96
	a m 2 /g. b nm. c cm 3 /g.

Table 2

 2 Half-lifetime of the arylsulfanyl radicals generated at 293 K by photolysis of SBAn

-1 and SBAn-2

		Silica	g//	g⊥	giso a	t1/2 b	density c
	a	SBA27-1	2.015	2.006	2.009	1.3 h	0.67
	b	SBA47-1	2.015	2.006	2.009	52 min	0.24
	c	SBA190-1	2.014	2.006	2.009	52 min	0.08
	e	SBA21-2	2.014	2.008	2.010	> 63 h	1.14
	f	SBA37-2	2.014	2.006	2.009	9.5 h	0.50
	g	SBA73-2	2.015	2.005	2.008	3.2 h	0.19

a giso = (2g⊥ + g//)/3. b Determined from the double integrated EPR signal at 293 K. c Number of functions per nm

2

Table 3

 3 Half-lifetime of the aryl sulfanyl radicals generated at 473 K by thermolysis on

	SBAn-1
		Silica	g	t1/2 (min) a
	a	SBA27-1	2,009	13
	b	SBA47-1	2,009	12
	c	SBA47-1 *	2,009	3

a Determined from the double integrated EPR signal at 473K.

  -3 mbar. All solid-state Cross Polarization Magic Angle

	Spinning (CPMAS) NMR spectra were obtained on a Bruker
	Avance-400 MHz NMR spectrometer operating at	13 C and	29 Si
	resonance frequencies of 101.6 MHz and 79.5 MHz,
	respectively.	13 C and	29 Si CPMAS experiments were performed
	with a commercial Bruker double-bearing probe. About 100
	mg of samples were placed in zirconium dioxide rotors of 4
	mm outer diameter and spun at a Magic Angle Spinning rate of
	10 kHz. The CP technique	26 was applied with a ramped	1 H-
	pulse starting at 100% power and decreasing down to 50%
	during the contact time in order to circumvent Hartmann-
	Hahn mismatches.	27,28 The contact times were 2 ms for	13 C
	CPMAS and 5 ms for	29 Si CPMAS. To improve the resolution, a
	dipolar decoupling GT8 pulse sequence	29 was applied during
	the acquisition time. In order to obtain a good signal-to-noise
	ratio, 6144 scans were accumulated using a delay of 2 s in	13 C
	CPMAS experiment, and 4096 scans with a delay of 5 s in	29 Si
	CPMAS experiment. The	13 C and	29 Si chemical shifts were
	referenced to tetramethylsilane. Commercially available
	tetraethylorthosilicate (TEOS) was distilled before used. SBA n -2 was prepared according to a literature procedure. 11
	Synthesis of Materials SBA n -1:
	SBA 27 -1: In a typical procedure, pluronic P-123 (2 g) in
	deionized water (14 mL) and 2 M hydrochloric acid solution (60
	mL) was stirred for 3 h at 40 °C. Tetraethoxysilane (2.0 mL, 9.0
	mmol, 9 equiv) and organic precursor 1 (471 mg, 1.0 mmol, 1
	equiv) were added. The mixture was stirred for 24 h at 40 °C,
	then warmed without stirring at 100 °C for 2 days, filtrated,
	washed twice with water, once with ethanol and finally
	extracted with a Soxlhet apparatus (ethanol) for one day. The
	wet powder was filtrated, washed twice with ethanol, acetone
	and diethylether. After one night at 80 °C under vacuum, a
	pale yellow powder was recovered. The molar composition of
	the isolated mixture was as follows: (1-x) M TEOS: x M 1: 0.017
	M P123 Polymer: 188 M H 2 O: 5.8 M HCl, where x denotes the
	number of moles of precursor 1.

  C CPMAS NMR (101.6 MHz) δ : 156.9, 132.4, 127.9, 114.4, 74.9, 69.4, 25.1, 22.5, 15.2 (P123), 7.7. : 9.5 nm. TGA: Mass Loss = 1.2%. SAXS: d(100) = 11.2 nm, a = 12.9 nm. General procedure for mesoporous silicas passivation. To non-passivated silica (1 g) in suspension in toluene (75 mL) were added triethylamine (5.5 mL) and trimethylsilylchloride (4.1 mL). The medium was heated for one night at 70 °C and then for one hour at 100 °C before being filtrated and washed once with toluene and once with ethanol. The recovered powder was stirred in ethanol during 4 h and then filtrated, washed twice with ethanol and twice with diethylether. After one night under vacuum at 80 °C, an orange/brown powder was recovered. Experimental Procedures for EPR Analysis: EPR experiments were performed on an ELEXSYS Bruker instrument and the Bruker BVT 3000 set-up was used to control the temperature. The photolysis instrument (ORIEL version 66901 with an energy supplier version 68911) is equipped with a 300X UXL306 arc Xe lamp (200-800 nm) with an optical fiber (1 m, version 77620). In a 4 mm quartz-glass tube, 35 mg of functionalized silica were degassed with three freeze-pumpthaw cycles with a 10 -5 mbar vacuum pump. EPR spectra for direct observation of sulfur centered radical experiments were recorded with the following parameters: modulation amplitude = 1 G, receiver gain = 90 dB, modulation frequency = 100 kHz, power = 20 mW, sweep width = 200 G, conversion time = 24 ms, sweep time = 25 s, number of scans = 2.

					29 Si
	CPMAS NMR (79.5 MHz) δ : -57.2 (T	2 ), -65.2 (T	3 ), -92.1 (Q	2 ), -
	101.4 (Q	3 ), -110.2 (Q 4 ). BET Surface Area: 744 m²/g. BJH
	Desorption Average Pore Diameter: 6.8 nm. TGA: Mass Loss =
	4.4%. SAXS: d = 11.9 nm.
	SBA 190 -1: Following the previous procedure, tetraethoxysilane
	(2.17 mL, 9.75 mmol, 39 equiv) and organic precursor 1 (118
	mg, 0.25 mmol, 1 equiv).	13 C CPMAS NMR (101.6 MHz) δ:
	132.1, 114.4, 73.1, 24.3, 21.9, 15.1 (P123), 6.5.	29 Si CPMAS
	NMR (79.5 MHz) δ: -66.7 (T 3 ), -91.4 (Q	2 ), -101.3 (Q 3 ), -110.5
	(Q

4

). BET Surface Area: 640 m²/g. BJH Desorption Average Pore Please do not adjust margins Please do not adjust margins Diameter

J. Name., 2013, 00, 1-3 | 3

J. Name., 2013, 00, 1-3 | 5

J. Name., 2013, 00, 1-3 | 7

Acknowledgements

The authors acknowledge the Agence Nationale de la Recherche for funding (ANR-12-JS07-005).

Notes and references


A Radgui 1i

C Demonceaux

E Mouaddib

M Rziza

D Aboutajdine

Optical flow estimation from multichannel spherical image decomposition

Keywords: Omnidirectional images, optical flow, spherical wavelets

The problem of optical flow estimation is largely discussed in computer vision domain for perspective images. It was also proven that, in terms of optical flow analysis from these images, we have di culty distinguishing between some motion fields obtained with smooth camera motion. The omnidirectional cameras developed recently provided images with large filed of view.

These images will contain global information about motion and allows to remove the ambiguity present in the case of planar projection in terms of optical flow analysis. Nevertheless, these images contain significant radial distortions that is necessary to take into account when treating these images to estimate the motion. In this paper, we shall describe new way to compute e cient optical flow for several camera motions given synthetic and real omnidirectional images. Our formulation of optical flow estimation problem will be given in the spherical domain. The omnidirectional images will be mapped on the sphere and used in multichannel image decomposition

INTRODUCTION

Optical flow estimation is one of the basic problems in computer vision.

It is the projection of the 3D velocity on the image and can be extracted from the image brightness variation. It was used by robotics researchers in many tasks such as : object detection and tracking, motion detection, ego-motion estimation, robot navigation or visual odometry (see for example [START_REF] Kim | An omnidirectional vision-based moving obstacle detection in mobile robot[END_REF] [2] [START_REF] Wang | An intelligent surveillance system based on an omnidirectional vision sensor[END_REF] [4] [START_REF] Winters | Omni-directional vision for robot navigation[END_REF] [START_REF] Gluckman | Ego-motion and omnidirectional cameras[END_REF] and references therein).

Given an image sequence I(x, y, t), the optical flow estimation consists in measuring the velocity (v x , v y ) of pixel (x,y) between t and t+1. Assuming the brightness constant across successive frames, this velocity can be computed by the well-known brightness constancy constraint equation (BCCE) eq.( 1): @I @x .v x + @I @y .v y + @I @t = 0,

No unique solution of eq.( 1) is possible, since on each location and each time,

we have to solve a single equation to determine two scalar unknowns(v x , v y ).

This was called the "aperture problem". Equation ( 1) only provides the normal velocity component. To overcome this problem more than one equation is needed to obtain the two velocity components at each position (x, y).

For traditional cameras, a number of di↵erent approaches to recovering optical flow have been proposed by adding other hypotheses of the flow. Horn

and Schunck [START_REF] Horn | Determining optical flow[END_REF] are the first to introduce a smoothness constraint in order to solve the aperture problem. Later a number of approaches are proposed in di↵erent frameworks. These methods can be grouped into correlation [START_REF] Anandan | A computational framework and an algorithm for the measurement of visual motion[END_REF],

energy [START_REF] Adelson | Spatiotemporal energy models for the perception of motion[END_REF] [10], phase [START_REF] Fleet | Computation of component image velocity from local phase information[END_REF] [12] [START_REF] Tsao | A neural scheme for optical flow computation based on gabor filters and generalized gradient method[END_REF], di↵erential [START_REF] Horn | Determining optical flow[END_REF] [14] [START_REF] Nagel | On a constraint equation for the estimation of displacement rates in image sequences[END_REF] and multichannel decomposition approaches [16][17][18] [START_REF] Bruno | Robust motion estimation using spatial gabor-like filters[END_REF]. It should be noted that the di↵erential method proposed by Lucas and Kanade [START_REF] Kanade | An iterative image registration technique with an application to stereo vision[END_REF], even if it is an old method, is the most commonly used in robotics applications since it is a simple method and often gives acceptable results. A review of a number of the most popular optical flow techniques can be found in [START_REF] Barron | Performance of optical flow techniques[END_REF] where the di↵erent approaches were compared. In this review, Fleet et al. [START_REF] Barron | Performance of optical flow techniques[END_REF] found that a phase-based approach [START_REF] Fleet | Computation of component image velocity from local phase information[END_REF] performed the best numerically than energy [START_REF] Adelson | Spatiotemporal energy models for the perception of motion[END_REF] [10], region-matching [START_REF] Anandan | A computational framework and an algorithm for the measurement of visual motion[END_REF] and di↵erential approaches [START_REF] Horn | Determining optical flow[END_REF] [14] [START_REF] Nagel | On a constraint equation for the estimation of displacement rates in image sequences[END_REF]. Later, it was proven that the multichannel decomposition of the optical flow equation gives robust estimation [START_REF] Weber | Robust computation of optical-flow in a multiscale di↵erential framework[END_REF] [18] [START_REF] Bruno | Robust motion estimation using spatial gabor-like filters[END_REF]. The multichannel image decomposition methods give a good compromise between robustness and time computation.

Recently, a new evaluation methodology and benchmarks for optical flow estimation methods in perspective images is proposed in [START_REF] Baker | A database and evaluation methodology for optical flow[END_REF]. It replace the quantitative evaluation of optical flow given by Barron et al. [START_REF] Barron | Performance of optical flow techniques[END_REF]. It gives set of databases and comparison of several well-known methods.

Referring to all these works, the problem of optical flow estimation was largely discussed in computer vision domain for perspective images. It was also proved that estimation methods from these images have di culty distin-guishing between small pure translations and small pure rotations. Recently, the developed omnidirectional cameras with large field of view have been able

to overcome the limited field of view introduced by planar projections [START_REF] Gluckman | Ego-motion and omnidirectional cameras[END_REF].

The omnidirectional images with hemispherical field of view contain global information about motion with the presence of the focus of expansion (FOE)

and/or the focus of contraction (FOC) in the images. Consequently, the optical flow filed analysis from omnidirectional images is more e cient also with smooth camera motion.

The catadioptric images are obtained using a combination of lenses and mirrors. These images are distorted due to the non-linear projection of the scene points in the image. Thus, the traditional treatments developed for perspective images are not appropriate for the deformed catadioptric images. Different works tried to adapt the existing methods by proposing an adaptation of the concepts definition of neighborhood, gradient function, Gaussian function, and so forth... We may classify the proposed works into two di↵erent approaches:

• Optical flow computation directly on the catadioptric image: A method to estimate optical flow on the catadioptric images was proposed on [START_REF] Demonceaux | Optical flow estimation in omnidirectional images using wavelet approach[END_REF] using wavelet approach based on a brightness change constraint equation. This method does not take into account the radial distorsion present in omnidirectional images. In our previous work [START_REF] Radgui | An adapted lucas-kanade's method for optical flow estimation in catadioptric images[END_REF], we have suggested a new di↵erential method adapted to omnidirectional images.

This method is based on motion model designed for paracatadioptric images and an adapted neighborhood that takes into account the shape of the mirror. This method [START_REF] Radgui | An adapted lucas-kanade's method for optical flow estimation in catadioptric images[END_REF] uses a conventional gradient operator to compute image derivatives necessary for flow computation. This is why it was not appropriate for omnidirectional images.

• Optical flow computation using image processing on the sphere: The works of Daniilidis et al. [START_REF] Daniilidis | Image processing in catadioptric planes: Spatiotemporal derivatives and optical flow computation[END_REF], Tosic et al. [START_REF] Tosic | Multiresolution Motion Estimation for Omnidirectional Images[END_REF], Mochizuki et al. [START_REF] Mochizuki | Featureless visual navigation using optical flow of omnidirectional image sequence[END_REF] and Bagnato et al. [START_REF] Bagnato | Optical flow and depth from motion for omnidirectional images using a tv-l1 variational framework on graphs[END_REF] belong to this category. Mochizuki et al. [START_REF] Mochizuki | Featureless visual navigation using optical flow of omnidirectional image sequence[END_REF] proposed a formulation of Horn-Schunck method [START_REF] Horn | Determining optical flow[END_REF] on the spherical images. They computed optical flow on the sphere that yields monocular disparity on the antipodal point on the equator. They used this estimation to compute the navigation direction.

A recent work by Bagnato et al. [START_REF] Bagnato | Optical flow and depth from motion for omnidirectional images using a tv-l1 variational framework on graphs[END_REF] also deals with the problem of computing optical flow using a formulation on the sphere. The authors have extended an existing and recent TV-L1 variational method [START_REF] Zach | A duality based approach for realtime tvl 1 optical flow[END_REF] and have proposed a new formulation of this method as a graph-based framework on the sphere. They have defined spherical di↵erential operators and proved that it is more accurate than planar di↵erential operators applied to omnidirectional images to compute optical flow especially near the poles in spherical images.

Most authors in Omnidirectional vision use the sphere as an appropriate domain for omnidirectional images. The spherical image processing will take into account the specific geometry of omnidirectional cameras [START_REF] Gluckman | Ego-motion and omnidirectional cameras[END_REF] [24] [29] [START_REF] Tosic | Multiresolution Motion Estimation for Omnidirectional Images[END_REF]. Our formulation of optical flow estimation is given on the sphere and the omnidirectional images are mapped onto a unit sphere and treated in the spherical domain. Then we propose to adapt approach based on multichanel decomposition of optical flow equation [START_REF] Bruno | Robust motion estimation using spatial gabor-like filters[END_REF]. This method proved its e ciency in the case of perspective images and it was indicated that it gives the best results for optical flow estimation [START_REF] Bernard | Discrete wavelet analysis: A new framework for fast optic flow computation[END_REF] [19] [START_REF] Weber | Robust computation of optical-flow in a multiscale di↵erential framework[END_REF]. For this reason, we propose to use a spherical wavelet and decompose the optical flow equation on the sphere to estimate the optical flow field. The reminder of the paper is organized as follows. In section 2, we shall present the optical flow constraint equation on the sphere and the stereographic projection necessary for all transformations between the catadioptric image and the unit sphere. Section 3 describes our approach to compute optical flow using spherical wavelet to constrain the optical flow equation on the sphere. In section 4, the experimental results are given and comparative measurements are discussed for synthetic and real images. We shall eventually give our conclusion in section 5 and provide guidelines for future works.

Optical flow constraint equation on the sphere

Geyer and Daniilidis have introduced unifying theory for all central catadioptric sensors [START_REF] Geyer | A unifying theory for central panoramic systems and practical implications[END_REF]. They proved that central catadioptric projection is equivalent to a central projection to a virtual sphere followed by projection from the sphere to the retina. This second projection depends on the shape of the mirror. Fig. [START_REF] Kim | An omnidirectional vision-based moving obstacle detection in mobile robot[END_REF] shows the equivalence between any catadioptric projection and the two-step mapping via the sphere.

The parameter ⇠ defines the shape of the mirror. In our case, we consider parabolic mirror where ⇠ = 1. However the method can easily be adapted to the general case. Let P s (✓, ') = P s (X s , Y s , Z s ) be the point on the sphere, having a radius equal to one, where:

8 > > > < > > > : X s =sin ✓ cos ' Y s =sin ✓ sin ' Z s = cos✓ (2)
The stereographic projection of P s from the sphere to the catadioptric plane can be expressed on cartesian coordinates as:

8 < : x= Xs 1 Zs y= Ys 1 Zs (3)
Using eq.( 2)and eq.( 3), we obtain the image point P i(x, y) expressed on spherical coordinates as: 8 < :

x=cot ✓ 2 cos ' y=cot ✓ 2 sin ' (4)
Note that the similar mapping model can easily be derived for all omnidirectional systems [30] [31]. In this paper, we look for a recovering of optical flow vectors on the sphere. For this reason, the omnidirectional images are mapped on the sphere and the formulation of Brightness Change Constraint

Equation is rewritten on spherical coordinates. If I s (✓, ', t) denotes the temporal spherical image on the unit sphere S 2 and rI s = ( @Is @✓ , 1 sin ✓ @Is @' ) T denotes the spatial gradient vector on the sphere, the total derivative [START_REF] Daniilidis | Image processing in catadioptric planes: Spatiotemporal derivatives and optical flow computation[END_REF] gives:

1 sin ✓ @I s @' v ' + @I s @✓ v ✓ + @I s @t = 0 (5)
This equation is equivalent to eq.( 1) and represents the optical flow constraint equation on the unit sphere. The intensity of point on the sphere can be obtained by any interpolation of pixel intensities on catadioptric image. In our case, we used bicubic interpolation. (v ✓ , v ' ) are the components of the flow vector in the tangential coordinates system. The partial derivatives of the spherical images can be obtained using the stereographic projection model (eq.( 4)):

2 4 @Is @✓ @Is @' 3 5 = 2 4 @x @✓ @y @✓ @x @' @y @' 3 5 2 4 @I @x @I @y 3 5 = 2 4 cos ' 2 sin 2 ✓ 2 sin ' 2 sin 2 ✓ 2 cot ✓ 2 sin ' cot ✓ 2 cos ' 3 5 2 4 @I @x @I @y 3 5 (6)
As in perspective case, we have a single equation to solve eq.( 5) to determine two unknowns: v ✓ and v ' ). An additional assumption must be given to solve the aperture problem on the sphere. Our proposed solution, detailed in the following section, consists in using a decomposition of spherical optical flow equation in spherical wavelets base assuming the constancy of optical flow over the support of the wavelets.

Optical flow from multichannel spherical image decomposition

We based our new approach on the work of Bruno et al. [START_REF] Bruno | Robust motion estimation using spatial gabor-like filters[END_REF] and Bernard in [START_REF] Bernard | Discrete wavelet analysis: A new framework for fast optic flow computation[END_REF]. It was proven in these references that multichannel decomposition of the optical flow equation gives robust estimation and a good compromise between robustness and time computation. Due to temporal aliasing, this method works well especially in multiscale framework [START_REF] Weber | Robust computation of optical-flow in a multiscale di↵erential framework[END_REF] [18] [START_REF] Bruno | Robust motion estimation using spatial gabor-like filters[END_REF]. This method consists in:

• Filtering an image sequence by a set of spatio-temporal filters or convolving an image sequence in wavelet base;

• Applying the optical flow constraint equation on each channel of the filter or of the wavelet.

The 2D Gabor filter used in [START_REF] Bruno | Robust motion estimation using spatial gabor-like filters[END_REF] and the 2D discrete wavelets used in [START_REF] Bernard | Discrete wavelet analysis: A new framework for fast optic flow computation[END_REF] are not appropriate to the distorted images given by catadioptric camera and should be adapted. Our proposed solution consists in using spherical wavelets base [START_REF] Antoine | Wavelets on the sphere : Implementation and approximations[END_REF] to constrain the optical flow equation on the sphere. We chose the commonly used wavelet, Morlet wavelets [START_REF] Demanet | Gabor wavelets on the sphere[END_REF], as a special case of Gabor wavelets. We first convolve the spherical optical flow equation with a set of spherical Morlet wavelets defined in di↵erent directions. The number of directions will then determine the size of the over-determined system to solve and get the optical flow vectors on the sphere. This convolution di↵ers from the classical 2D convolution usually used for filtering images. In this section, we shall define a spherical wavelets and convolution on the sphere.

Both definition will be used to constrain eq.( 5).

Spherical wavelets

Our decomposition is based on directional spherical wavelets base. This choice of wavelets is motivated by the trust that wavelet bases are a welldesigned tool for our proposed approach. Wavelet bases have a natural multiscale structure. As a local frequency analysis tool, wavelets analysis favorably compares to filtering because it is far less computation intensive and still provides complete information on the signal. We based our spherical image decomposition on spherical wavelets proposed by Antoine et al. in [START_REF] Antoine | Wavelets on the sphere : Implementation and approximations[END_REF]. They defined an interesting family of spherical wavelets which can be applied to any image represented in the spherical domain. Natural candidates in our purpose are directional wavelets on the sphere [START_REF] Demanet | Gabor wavelets on the sphere[END_REF].

In [START_REF] Antoine | Wavelets on the sphere : Implementation and approximations[END_REF] it was shown that any Euclidian wavelets can be mapped to sphere using an inverse stereographic projection. We chose the spherical Morlet wavelets M , represented in eq.( 7) to decompose spherical images. k0 is the norm of the wave vector of the wavelet in the Euclidian limit (accounting for the number of oscillations). ' 0 is the angle of the wavelet and a is the scale of the wavelet. Note that this function is admissible for k 0 large enough (>= 6) [START_REF] Antoine | Wavelets on the sphere : Implementation and approximations[END_REF]. Fig. [START_REF] Yoshizaki | Free space detection from catadioptric omnidirectional images for visual navigation using optical flow[END_REF] shows the spherical Morlet wavelet defined in two di↵erent directions on half-spheres.

M (✓, ') = 1 2 p e ik 0 tan(✓/2)/ cos(' ' 0 ) e 2 tan(✓/2) 2 /a 2 (1 + tan(✓/2) 2 /a 2 ) (7) with : = 4a 2 ((a 2 1) cos ✓ + (a 2 + 1)) 2

Wavelet Transform on the sphere

Wavelet transformation is di↵erent from the Euclidean one. We shall present the definition of a convolution on the sphere. This convolution will be used to compute the spherical wavelet transform of images to calculate the optical flow vectors.

Let ⌘ = (✓, ') be the point on the sphere S 2 .

is the spherical wavelet defined on the sphere. If the rigid rotation g is an element of the group of rotations SO(3) acting on the sphere S 2 , is rigidly transported on the sphere according to the operator R g as:

R g (⌘) = (g 1 ⌘)
The continuous wavelet transform of spherical image I s (✓, ') can be expressed as a convolution on the sphere with the spherical wavelet as:

W I s (⌘) =< I s , R g >
The convolution on the sphere is defined by Driscoll and Healy [START_REF] Driscoll | Computing fourier transforms and convolutions on the 2-sphere[END_REF] as:

hI s , R g i = Z S 2 Is(⌘) (g 1 ⌘)d⌘ (8)
Where : d⌘ = sin ✓d✓d'. This convolution is performed in the Fourier domain using a simple multiplication like in R n . In practice, we will use the open source YAWTB Toolbox that contain an implementation of several directional wavelets on the sphere based on the work of Antoine et al. [START_REF] Antoine | Wavelets on the sphere : Implementation and approximations[END_REF].

Optical flow estimation

In order to estimate the velocity vector at each point on the sphere using Eq.( 5), we need to add some constraints to the spherical optical flow equation to solve the aperture problem. This assumption is based on the constancy of optical flow over the support of the spherical wavelets.

Let us consider the projection of the optical flow equation (eq.( 5)) onto a wavelets base i (✓, ') (with i= 1... N, and N is the number of wavelet directions). This allows to write :

< M (✓, '), 1 sin ✓ @I s @' v ' + @I s @✓ v ✓ + @I s @t >= 0 (9)
Where <> denotes the spherical convolution operator defined bellow in section(3.2). The multichannel decomposition approach is based on the assumption that (v ✓ , v ' ) are constant over support of the wavelets i (✓, ') for all i = 1...N . Note that the support of the wavelet must not be large to maintain the assumption of constant optical flow. Eq.( 9) become:

< M (✓, '), 1 sin ✓ @I s @' > v ' + < M (✓, '), @I s @✓ > v ✓ + < M (✓, '), @I s @t >= 0 (10)
Taking N directions for the directional wavelets, it leads a system of 2 ⇤ N equations since spherical Morlet wavelets are complex. 0

B B B B B B B B B @ real < 1 (✓, '), 1 sin ✓ @Is @' > real < 1 (✓, '), @Is @✓ > imag < 1 (✓, '), 1 sin ✓ @Is @' > imag < 1 (✓, '), @Is @✓ > .... ..... real < N (✓, '), 1 sin ✓ @Is @' > real < N (✓, '), @Is @✓ > imag < N (✓, '), 1 sin ✓ @Is @' > imag < N (✓, '), @Is @✓ > 1 C C C C C C C C C A | {z } 0 @ v ' v ✓ 1 A = 0 B B B B B B B B B @
real < 1 (✓, '), @Is @t > imag < 1 (✓, '), @Is @t > ......

real < N (✓, '), @Is @t > imag < N (✓, '), @Is @t > 1 C C C C C C C C C A | {z } (11)
To solve this over-determined system for the two unknowns v ✓ and v ' , the least squares method is used. Therefore, the solution of ( 11) is given by minimizing:

! V = arg min v ✓ ,v' N X i=1 [< i (✓, '), 1 sin ✓ @I s @' > v ' + < i (✓, '), @I s @✓ > v ✓ + < i (✓, '), @I s @t >] 2 (12)
If we denote the over-determined system by : A.

2 4 v ' v ✓ 3 5 = B ( 13
)
The solution of the system is given as:

2 4 v ' v ✓ 3 5 = (A 0 ⇤ A) 1 ⇤ A 0 ⇤ B (14)
In practice, the linear system is solved with least square method. The estimated solution is taken into account according to two fixed thresholds. We compare the conditioning of a matrix (A 0 ⇤ A) with a first threshold. In addition, we test if the estimated solution V = (v ' , v ✓ )is acceptable or not by comparing the error ||AV B|| with a second fixed threshold. These two comparisons inform us on the stability of the linear system and allow to verify if our hypothesis of optical flow constancy is true or not and thus decide to take into account or not the estimated solution.

The least square estimation described above allows to determine a set of optical flow vectors defined on the tangential coordinate system (v ✓ , v ' , 0). To compare our optical flow field with that obtained using Bruno et al. method [START_REF] Bruno | Robust motion estimation using spatial gabor-like filters[END_REF], this motion field must be projected on the catadioptric image using this transformation:

2 4 v x v y 3 5 = 2 4 @x @✓ @x @' @y @✓ @y @' 3 5 2 4 v ✓ v ' 3 5 (15)
Using eq.( 4), the velocity vector (v x , v y ) in image space can be found by the following transformation of angular velocity (v ✓ , v ' ):

2 4 v x v y 3 5 = 2 4 cos ' 2 sin 2 ✓ 2 cot ✓ 2 sin ' sin ' 2 sin 2 ✓ 2 cot ✓ 2 cos ' 3 5 2 4 v ✓ v ' 3 5 (16)

Experiments and Results

To show the improvement given by our proposed method, we will compare it to four previously published methods.

We will compare our method first to a similar method developed for perspective camera given by Bruno et al. [START_REF] Bruno | Robust motion estimation using spatial gabor-like filters[END_REF]. This method is based on decomposition of optical flow equation using Gabor filter bank. The parameters of Gabor filter are fixed to preserve the stability of the linear system and to give the smallest error estimation. We have fixed experimentally the optimal parameters of Gabor filter according to the size of the movement applied to the virtual cameras. Figure [START_REF] Wang | An intelligent surveillance system based on an omnidirectional vision sensor[END_REF] shows angular errors obtained for many values of scale , central frequency f 0 and the number of filters N . According to the motion in the synthetic sequences, the optimal Gabor filter banks parameters are sets as follows:

2 [3.0, 6.0], f 0 < 0.14 N = 6.

In [START_REF] Bruno | Robust motion estimation using spatial gabor-like filters[END_REF], the optical flow estimation is performed in multiresolution framework.

We use image decomposition with Gabor filter bank defined in three level of multiresolution estimation where G 2 k f 0 ,2 k ,N corresponds to the Gabor filter bank in level k. Equivalently the spherical wavelets support is arranged to have the same size as that fixed in [START_REF] Bruno | Robust motion estimation using spatial gabor-like filters[END_REF]. For this reason, we fixed a scale of wavelets in a = 0.05 and a = 0.03 respectively for large and small motion.

For all sequences, we took six orientations of the spherical Morlet wavelets and 2D Gabor filter.

We have compared our proposed method also to that of Kanade et al. [START_REF] Kanade | An iterative image registration technique with an application to stereo vision[END_REF].

This method is still one of the most popular versions of two-frame di↵erential methods for motion estimation and is often used in robotic applications.

Lucas and Kanade compute the optical flow on a point (x, y), considering that the motion is constant in a fixed neighborhood of this point.

The third method, which we will compare our proposed method with, is our previously published method dedicated to omnidirectional images [START_REF] Radgui | An adapted lucas-kanade's method for optical flow estimation in catadioptric images[END_REF]. This method is an adaptation of Lucas and Kanade's method which uses a new constrain taking into account the distortions existing in the omnidirectional images.

Finally, we will conduct comparison with the results obtained using a method developed by Danilidis et al. [START_REF] Daniilidis | Image processing in catadioptric planes: Spatiotemporal derivatives and optical flow computation[END_REF]. This method can be also considered as an adaptation of Lucas and Kanade's method. It uses the same assumption that the motion is constant in a fixed neighborhood. The main di↵erence between this two adapted methods ( [START_REF] Radgui | An adapted lucas-kanade's method for optical flow estimation in catadioptric images[END_REF] and [START_REF] Daniilidis | Image processing in catadioptric planes: Spatiotemporal derivatives and optical flow computation[END_REF]) is that the first one uses a new definition of neighborhood and new motion model to solve the aperture problem on the image space, whereas the second one is based on the spherical representation of omnidirectional images and solves the same spherical optical flow equation that we use in this paper.

As presented in section 3.3, to test the stability of our linear estimation we fixed the conditioning threshold on 0.1 and the adequacy threshold of estimated solution in 0.1. This allows to do not take into account the aberrant estimation of optical flow. We have considered three kinds of motions:

translation, rotation and combined motion to show the e↵ectiveness of our contribution. These motions are applied to virtual and real cameras moving on a plane perpendicular to their optical axis.

Synthetic sequences

We have tested our algorithm on di↵erent synthetic sequences for which 2D motion fields are known. These sequences have been generated with POVRAY. A virtual parabolic camera observes four textured planes separated by approximatively 100cm. POVRAY generated images of 500 500

pixels. An example of synthetic images is shown in (Fig.( 4)). We have compared our proposed method to that proposed in [START_REF] Bruno | Robust motion estimation using spatial gabor-like filters[END_REF] for perspective images and that proposed in [START_REF] Radgui | An adapted lucas-kanade's method for optical flow estimation in catadioptric images[END_REF] for catadioptric images. To assess the optical flow estimation methods, we use the mean of angular errors [START_REF] Barron | Performance of optical flow techniques[END_REF] between the known velocity vectors,V r = (u r , v r ) and an estimated vectors V e = (u e , v e ). Note that in all these experimentations we do not consider the image points around the center of images. In these locations both methods generate significant errors. In the rest of the image space the flow vectors prove that the adapted approach on the sphere compute a succeeded velocity vectors even with large camera motion and also in di↵erent positions on the distorted omnidirectional images.

This means that the fixed support of 2D Gabor filter is not appropriate to omnidirectional images.

The mean angular errors obtained from di↵erent sequences using the five implemented methods are indicated in Tab. [START_REF] Kim | An omnidirectional vision-based moving obstacle detection in mobile robot[END_REF]. These results show that the classical approaches ( [START_REF] Bruno | Robust motion estimation using spatial gabor-like filters[END_REF] and [START_REF] Kanade | An iterative image registration technique with an application to stereo vision[END_REF]), even if these methods work well for perspectives are not appropriate to omnidirectional images. Only the case of very small rotational motion of virtual camera gives an acceptable result using these methods.

The angular errors obtained using spherical methods ( [START_REF] Daniilidis | Image processing in catadioptric planes: Spatiotemporal derivatives and optical flow computation[END_REF] and our proposed method in this paper) are inferior to that obtained using adapted method on the image space [START_REF] Radgui | An adapted lucas-kanade's method for optical flow estimation in catadioptric images[END_REF]. These results give the argument that methods dedicated to omnidirectional images are more robust when using a spherical model to represent images. Finally, we can see that angular errors are inferior using our spherical approach especially with large camera motions. This result proves that multichannel spherical images decomposition method proposed in this paper gives good estimation of local motion compared to di↵erential methods [START_REF] Daniilidis | Image processing in catadioptric planes: Spatiotemporal derivatives and optical flow computation[END_REF] [23].

Real sequences

We have also used sequences of real omnidirectional images. The sequences are obtained using a camera which is mounted on a mobile robot and moves on a plane perpendicular to its optical axis. The camera is a SONY DF W SX910. We obtain an image of 1280 ⇤ 960 pixels. The catadioptric sensor mounted in the mobile robot is illustrated in Fig. [START_REF] Adelson | Spatiotemporal energy models for the perception of motion[END_REF] with the image that it gives. The camera is calibrated using the Omnidirectional Calibration Toolbox [START_REF] Mei | Single view point omnidirectional camera calibration from planar grids[END_REF]. fields obtained using a fixed omnidirectional camera and moving one object of the scene. In all these cases, the results show that our proposed method allows an estimation of accurate motion field in various scenes and di↵erent kinds of camera motions.

Conclusion and future work

In this paper we have described a new method to estimate optical flow vectors in omnidirectional images. This method uses the sphere as the space of image processing. We have used spherical wavelets to constrain the optical flow equation on the sphere. The simulation results in real and synthetic images shows that the proposed method allows an accurate estimation of optical flow for a wide range of camera motions.

As future works, we shall first try to estimate optical flow using multichannel decomposition in multiresolution frameworks. We trust that due to temporal aliasing, it will give more accurate optical flow vectors. On the other hand, we have observed that a number of other recent works in omnidirectional vision takes a traditional method like that of Lucas and Kanade [START_REF] Kanade | An iterative image registration technique with an application to stereo vision[END_REF] for optical flow estimation, even if it is not adapted, to recover the egomotion [START_REF] Gluckman | Ego-motion and omnidirectional cameras[END_REF] [35] [START_REF] Vassallo | A general approach for egomotion estimation with omnidirectional images[END_REF] or the epipoles [37] [38]. We believe that using our adapted optical flow field proposed in this paper, instead of the classical one, will improve the estimation of egomotion and epipoles in omnidirectional images.

Figure 1 :

 1 Figure 1: Equivalence between the catadioptric projection and the two-step mapping via the sphere.

Figure 2 :

 2 Figure 2: Spherical Morlet wavelet at scale a = 0.04 and ' 0 = 0 (left) and ' 0 = ⇧/2 (Right)

Figure 3 :

 3 Figure 3: Top: Angular error in function of scale ( N = 6; f 0 = 0.014). Middle: Angular error in function of central frequency f 0 ( N = 6; = 5). Bottom: Angular error in function of number of filters N (f 0 = 0.014; = 5). These experiments are applied to the sequence obtained with translation T = [ 1cm, 0, 0] of virtual camera.

Fig.( 5 )

 5 Fig.[START_REF] Winters | Omni-directional vision for robot navigation[END_REF] shows the angular error map obtained for di↵erent camera motions.

Figure 4 :

 4 Figure 4: Left: Synthetic image composed by four textures plans and on right spherical equivalent image

Fig.( 10 )

 10 shows the optical flow fields for a rotation around Z-axis in real scene. Fig.(11) illustrates a combined motion of camera and shows the optical flow fields. Finally, Fig.(12) represents the optical flow

Figure 5 :

 5 Figure 5: Angular Error Maps obtained using Bruno et al. method (left) and adapted method (right) for di↵erent kinds of motion: Top: Pure translation in X and Y-direction. Middle: Pure rotation of 1 around Z-Axis. Bottom: combined motion

Figure 6 :

 6 Figure 6: Optical flow fields from pure translation of camera. In the top: T = [5cm, 3cm, 0]. In the middle: T = [ 1cm, 0, 0]. In the bottom: T = [0, 3cm, 0]. The fields on the left correspond to the Bruno et al. method and those on the right to the adapted method.

Figure 7 :

 7 Figure 7: Optical flow fields from pure rotation of camera. At the top: R = [0, 0, 2 ]. In the middle: R = [0, 0, 1 ]. At the bottom: R = [0, 0, 0.5 ]. The fields on the left correspond to the Bruno et al. method and those on the right to the adapted method.

Figure 8 :

 8 Figure 8: Optical flow fields from combined motion of the camera. At the top: R = [0, 0, 2 ] and T = [2cm, 1cm, 0]. In the middle: R = [0, 0, 1 ] and T = [2cm, 0, 0]. At the bottom: R = [0, 0, 1 ] and T = [0.5cm, 0.2cm, 0]. The fields on the left correspond to the Bruno et al. method and those on the right to the adapted method.

Figure 9 :Figure 10 :

 910 Figure 9: Omnidirectional camera mounted on mobile robot and the real image that it gives

Figure 11 :Figure 12 :

 1112 Figure 11: original images and optical flow fields obtained using combined motion of a translation along Y-axis and rotation around Z-axis from the Bruno et al. method (left) and the adapted method (right)

Classical