Yu -Jie Zhong

Jian-Hong Liao

Tzu-Hao Chiu

Ying-Yann Wu

Samia Kahlal

Jean-Yves Saillard

C W Liu

Hydride-Encapsulated Bimetallic Clusters Supported by 1, 1-Dithiolates

Two mixed-metal hydride clusters, corresponding to x = 3 and 4, were structurally characterized from a set of atomically precise heptanuclear clusters, CuxAg7-x(H){S2P(O i Pr)2}6 (x = 1-6). An interstitial hydride lying at the center of a tricapped tetrahedral cage was located and refined anisotropically by using X-ray data, and its presence acertained by multinuclear NMR spectroscopy and DFT calculations.

Metal hydrides are known for their glamorous structural diversity and bonding features, 1 as well as their potential applications in hydrogen storage 2 and various catalytic processes. 3 The search for new Cu/Ag hydrido nanoclusters (NCs) is emerging and topical. 4,[5][6][7][8][9][10][11][12][13][14][15][16][17][18][START_REF] Silalahi | [END_REF][20] Several high-nuclearity Cu(I)/Ag(I) hydrides, [Cu25H10(SPhCl2)18] 3-, 5 [Cu16H14 -dpmppm)4] 2+ , 6 [Cu18H17(PPh3)10] + , 7 and [Ag10H8(dppa)6] 2+ , 8 have been reported. A few polyhydrido NCs displaying M(I)/M/(0) character also exist, such as the 2e species [Cu25H22(PPh3)12] + , 7 [Cu29Cl4H22(Ph2phen)12] + , 9 [Cu61(S t Bu)26S6Cl6H14] + , 10 and [Ag40(DMBT)24(PPh3)8H12] 2+ . 11 A pitfall in the structural characterization of these NCs is that the position of the hydrides cannot be accurately determined from X -ray diffraction alone. Nevertheless, the molecular formulation can be determined by the spectroscopic evidences of ESI-MS, multinuclear NMR, in combination with DFT calculations. Our group has isolated and structurally characterized a seri es of tetracapped-tetrahedral octanuclear 12 and tricapped-tetrahedral heptanuclear clusters 13 encapsulating a tetrahedrally coordinated hydride, the position of which was fully established by neutron diffraction on the compound [Cu7(H){S2C(aza-15crown [START_REF] Silalahi | [END_REF] and the Cu(I)/Pd(0) hydrides, [Pd@Cu14(H)2{S2CN( n Pr)2}6(C2Ph)6], 20 were structurally characterized by both single-crystal X-ray and neutron diffraction, from which a new synthetic concept has been established, based on the fact that hydride additions assist the growth of nanoscale copper clusters. 4a Although neutron diffraction is the most reliable technique to locate hydrogen atoms, it is not conveniently available. In addition, the successful performance of a neutron diffraction experiment is not an easy task, and the following points must be overcome: (i) Grow single crystals of sufficiently big size as well as high quality; (ii) Avoid high background noise especially for compounds containing a large number of H atoms. In comparison with neutron experiments, in-house X-ray diffraction rapidly provides results for further analysis. Grabowsky's group applied the Hirshfeld atom refinement (HAR) method to refine hydrogen with aspherical scattering factors in X-ray data. 21 However, the HAR method is mostly applied for organic compounds, but rarely for inorganic cluster compounds.

-5)}6]. 13b Later, Cu(I) polyhydrides [Cu28(H)15{S2CN( n Pr)2}12](PF6), 14 [Cu20(H)11{E2P(O i Pr)2}9] (E = S, 15 Se 16 ), [Cu30(H)18{E2P(OR)2}12] (E = S, Se; R = n Pr, i Pr, i Bu), 17 [Cu32(H)20{S2P(O i Pr)2}12], 18 [Cu11(H)2{S2P(O i Pr)2}6(C2Ph)3],
Mixed-metal hydride clusters are emergingly challenging in this field. Not only will one encounter the unavoidable disorder issues on metal sites, but also face the high probability of co-crystallization from their reaction mixtures. In our previous computational investigations on the octanuclear species [M8(H)L6] + (M = Cu, Ag; L = dithio-or diselenolate), the metal framework was found to exhibit a tetracapped-tetrahedral geometry of which each metal site can be either Cu(I) or Ag(I). 12f The related M7(H)L6 can be obtained from further reduction of [M8(H)L6] + to form a tricapped-tetrahedral metal framework. 13 These results inspired us to synthesi ze mixed-metal hydride clusters. Herein we report a n one-pot synthesis and characterizations of CuxAg7-x(H){S2P(O i Pr)2}6, x = 1-6 (abbreviated as CuxAg7-xH). Surprisingly it is our first successful attempt on performing anisotropic refinement of an interstitial hydride by using X-ray diffraction data.

The synthesis was performed with [Ag(CH3CN)4]PF6, [Cu(CH3CN)4]PF6, and NH4[S2P(O i Pr)2] dissolved in THF solution first followed by adding NaBH4 as a hydride provider. The final products were purified by column chromatography to yield Cu xAg7- x(H){S2P(O i Pr)2}6, x = 1-6. Two crystals grown from the mixtures were hand-picked to collect diffraction data. The crystal structure of the first one turned out to have one metal site (Mtop) statistically occupied by 50% Cu and 50% Ag, thus corresponding to a co-crystallized product of composition [Cu3Ag4(H){S2P(O i Pr)2}6]0.5[Cu4Ag3(H){S2P(O i Pr)2}6]0.5 (1). Intriguingly, the composition of the second crystal is Cu4Ag3(H){S2P(O i Pr)2}6 (2). In the crystal structure of 1, the seven metal atoms constitute a tricappedtetrahedral framework, (μ3-Cu)3(MtopAg3) (Mtop = Cu0.5Ag0.5), of ideal C3v symmetry, with the encapsulated hydride lying at the center of the MtopAg3 tetrahedron. The entire metal core is stabilized by six diisopropyl dithiophosphate (dtp) ligands , reducing the whole molecular symmetry to C3. The 3-fold rotational axis passes through Mtop, the central hydride and the middle of the (Agtri)3 triangular face of the inner tetrahedron (Figure 1a). Unlike in 1, the Mtop position in structure 2 is fully occupied by a Cu atom (Figure 1b). It is worthwhile to mention that the encapsulated H-atom revealed in both 1 and 2, is the first reported example of a hydride bonded to both Cu and Ag atoms.

Selected bond lengths are listed in Table S1. The distances between metal atoms in 1 and 2 are almost identical. Small differences exist in Mtop-Mtri (3. 6) Å in 2). The Mtop(Agtri)3 tetrahedron in 1 can be compared to its analog in [Ag7(H){S2CC(CN)2}6] 6-, 3, which metal framework also reveals a tricapped-tetrahedral geometry, but with an elongated inner tetrahedron (Figure 1c). The Mtop-Mtri distances of the latter are much longer than those in 1 and 2 (Table S1). The S…S bite distance in 3, which is approximately 0.35-0.43 Å shorter than in 1 and 2, could be the explanation for this elongated tetrahedron. The H-Mtop distances are 1.65(2) Å in 2 and 1.83(4) Å in 1. The angles of Mtop-H-Mtri and Mtri-H-Mtri in 1(2) are 109(2)-118(2) o (117.6(5)-124.3(5) o ) and 100(2)-105(2) o (96.3(4)-97.7(4) o ). In fact the tetrahedral coordination of the hydride observed in both 1 and 2 is one among several possible four-coordinated hydride configurations such as in square-planar, 15 elongated tetrahedral, 13a-b distorted seesaw, 17 and trigonal pyramidal modes. [START_REF] Silalahi | [END_REF] The hydride atom in both 1 and 2 was located from the Fourier density map. I t was initially refined isotropically without any constraints in data 1. To our great surprise, a further anisotropic refinement of the hydride using the X-ray data of 1 was successful. In general, such anisotropic refinement usually results in non-positive-definite in hydrogen refinement, because H has only one electron. It is highly correlated to the crystal orientation during the entire data collection. However, the prediction to collect informative reflections related to hydride is extremely difficult. One should ensure that the coverage of the data collection is as high as possible so that more dataset can be incorporated in order to increase the redundancy, and the most important thing is that a low-temperature experiment should be performed to reduce the temperature effect. The data analysis demonstrated that those of 1 have higher resolution (d~0.73 Å) and redundancy (4.15) than those of 2 (d~0.78 Å, redundancy 3.17). It can only be assumed that more informative reflections were collected in the data of 1. Sum of the above reasons, the hydride in 1 could be anisotropically refined to convergence with unexpected success. It is of note that the hydrides in [CuHP(NMe2)3]6 were located and refined by using X-ray data collected at -162 ℃. 22 DFT calculations 23 on two simplified C3 models for the clusters present in 1 and 2, namely [Cu3Ag4(H){S2PH2}6] (1') and [Cu4Ag3(H){S2PH2}6] (2') reproduce nicely their structures (Table S1). The building of C3 isomers of 1' and 2' in which the three Mcap atoms are Ag (in the place of Cu), destabilizes the systems by 13.6 and 16.4 kcal/mol, (free energy) respectively. Obviously, Cu prefers capping positions. This is also exemplified by calculations on the tetracapped -tetrahedral octanuclear model [Cu4Ag4(H){S2PH2}6] for which the H@Ag4@Cu4 isomer is found more stable than its H@Cu4@Ag4 relative by 6.0 kcal/mol. Exchanging Cutop with one Agtri in 2' destabilizes the cluster by 2.6 kcal/mol, indicating that among the four inner tetrahedral positions, Cu prefers the Mtop one. The Mcap > Mtop > Mtri site preference for Cu is related for its strong affinity to bind sulphur (Mcap: sulphur only) and its stronger than Ag binding to hydride (Mtop-H stronger than Mtri-H).

Due to the intrinsic characters of highly similar polarities, the products, CuxAg7-x(H){S2P(O i Pr)2}6, x = 1-6, cannot be further separated. While the reactions were performed with various stoichiometric ratios Cu:Ag:L:H -= x:(7-x):6:1, (x = 1-6), the resulting products based on their 1 H NMR spectra are very similar. The hydride resonance displays in a wide distribution (4.9 -6.3 ppm) that integrates to one H relative to 12 methylene and 72 methyl protons of a total of six di -isopropyl dithiophosphate ligands (Figure 2). The resonance centered at 5.0 ppm splits into a doublet of doublets ( 1 J1H-107Ag = 86.7 Hz, 1 J1H-109Ag = 99.7 Hz), which is due to the hydride coupled with different isotopomers, 107 Ag and 109 Ag nuclei. This doublet of doublets peak can be assigned to the Cu6AgH compound. Based on the same principle, the resonance centered at 5.44 ppm with a triplet of triplets pattern can be assigned to Cu5Ag2H. It clearly suggests that the hydride coupled with two pretend, magnetically equivalent silver nuclei, and the distribution of the splitting pattern is the superposition of the spectra of the different isotopomers. It is confirmed by the simulated pattern of MxM'2-xH (M, 107 Ag; M', 109 Ag) spin system (Figure 3) with the assumed coupling constant of 1 J1H-107Ag (65.0 Hz) and 1 J1H-109Ag (74.7 Hz). The resonance in the range of 5.9 -6.4 ppm displays a mixture combination that can be resolved as Cu4Ag3H (quartet of quartets), Ag4Cu3H (quintet of quintets), Cu2Ag5H (sextet of sextets), and CuAg6H (septet of septets), respectively. Their simulated spectra are stacked in Figure 3. The sum of all spectra (red) exactly matches with the experimental one (inset of Figure 2). An observed doublet resonance at 1080.5 ppm ( 1 J1H-109Ag = 74.5 Hz) in 109 Ag NMR spectrum (Figure S1) matches with 1 J1H-109Ag (74.7 Hz) of Cu5Ag2H in the 1 H NMR spectrum. Only Cu5Ag2H, Cu4Ag3H, Cu3Ag4H, and Cu2Ag5H species could be observed. The other compounds were not detected due to lower quantities. The corresponding chemical shifts and coupling constants are listed in Table 1. Cu4Ag3H shows the most intense peak in both 1 H, 2 H (Figure S8), and 109 Ag NMR spectra, which is highly correlated with the crystal data. Variable-temperature 1 H NMR experiment was performed (Figure S2). The hydride resonances are all slightly upfielded by ~ 0.15 ppm at -60℃. Their lineshapes did not change upon temperature decreases to -40℃, then became broad at -60℃. The NMR spectroscopic data suggests that fast intrachanges between capping and inner vertices atoms occur at the NMR time scale. Besides, the coupling constant ( 1 JHAg) of each compound in the CuxAg7-xH mixture was discernible. Presumably the metal exchange tends to be intramolecular rather than intermolecular and such a fluxional behavior has been observed in the Ag7(H)L6 analogs (L = [S2CC(CN)2] 2-, [E2P(OR)2] -E = S, Se). 13a,c ESI-MS spectrum displayed a wide distribution of CuxAg7-xH (x = 0 -7). The most and second intense peaks at m/z 2000.3582 and 2044.3331 correspond to the molecular ion [Cu4Ag3(H){S2P(O i Pr)2}6 + Ag + + 2(H2O)] + (calcd. m/z 2000.4689) and [Cu3Ag4(H){S2P(O i Pr)2}6 + Ag + + 2(H2O)] + (calcd. m/z 2044.4444), respectively (Figure S4). Interestingly, both [Cu7(H){S2P(O i Pr)2}6 + Ag + + 2(H2O)] + and [Ag7(H){S2P(O i Pr)2}6 + Ag + + 2(H2O)] + can be barely seen (Figure S5). It is noted that both CuxAg7x(H){S2P(O i Pr)2}6 (x = 0 and 7) species were not observed in the 1 H NMR spectrum. This can be attributed to rearrangement reactions taking place in the gas phase.

Calculations on models of the non-structurally characterized clusters, i.e. CuxAg7-x(H)(S2PH2) 6 (x =1, 2, 5, 6), indicate that their isomers of lowest energy follow the metal site preference rule discussed above (Figure S7). Their computed 1 H NMR hydride chemical shifts display the best agreement with their experimental counterparts (Table 1).

In conclusion, a set of mixed-metal hydride compounds CuxAg7-x(H){S2P(O i Pr)2}6 (x = 1-6) have been synthesized and characterized by multi -nuclear NMR, ESI mass spectroscopy, DFT simulation and both Cu3Ag4H and Cu4Ag3H by single-crystal Xray diffraction. An unprecedented (μ4-H) hydride encapsulated in a hetero-metallic M4 (Ag3Cu) tetrahedral cage, which can be refined anisotropically in crystal 1, is reported. It is expected that the addition of a supplementary Cu(I)/Ag(I) capping metal such as Cu(I)/Ag(I) would lead to a stable octanuclear species since there is one empty site along the C3 axis.

  0890(3)-3.2531(3) Å in 1; 3.0970(5)-3.2543(5) Å in 2), Mcap-Mtop (2.7368(4)-2.7499(4) Å in 1;

  s c r i p t 2.7152(6)-2.7312(6) Å in 2), and Mcap-Mtri (2.9208(3) -3.0324(4) Å in 1; 2.8950(5) -3.0118(

Figure 1 .

 1 Figure 1. ORTEP drawing (30% probability) of (a) compound 1 (O i Pr groups omitted). (b) The Cu4Ag3H core in 2, and (c) the Ag7H core in 3.

Figure 2 .

 2 Figure 2. 1 H NMR (800.24MHz, CDCl3, r.t.) spectrum of CuxAg7-x(H){S2P(O i Pr)2}6. The hydride chemical shift ranges 4.9 -6.3 ppm (inset). The asterisk at 4.95 ppm is the resonance of the BHT, which is commonly added in THF solvent as a stabilizer.

Figure 3 .

 3 Figure 3. Simulated splitting patterns of a hydride in CuxAg7-xH (x = 1-6) in 1 H NMR spectrum. The sum of all simulation is given on the top (red line).

Table 1 .

 1 The chemical shift and coupling constant of CuxAg7-x(H){S2P(O i Pr)2}6, x = 1-6 in 1 H and 109 Ag NMR spectra.

	Cpd. Cu6AgH Cu5Ag2H Cu4Ag3H Cu3Ag4H	4.996 5.441 6.049 6.078	5.40 5.78 5.82 6.45	86.7 65.0 62.9 57.4 m 99.7 74.7 72.3 66.0 a n u s c r i p t 2.1 --16.9 1080.5 74.5 40.9 1120.9 72.7 25.1 1140.2 66.6 --
	Cu2Ag5H CuAg6H A c c e p t e d 6.116 6.33 52.3 6.132 6.13 44.6	60.1 51.3	11.4 3.6	1137.4 --	60.0 --

1 H NMR (ppm) 1 H NMR (ppm) DFT 1 J( 1 H-107 Ag) (Hz) 1 J( 1 H-109 Ag) (Hz) Ratio (%) 109 Ag NMR (ppm) 1 J( 1 H-109 Ag) (Hz)

Acknowledgements

This work was supported by the Ministry of Science and Technology of Taiwan (109-2113-M-259-008) and the GENCI French national computer resource center (project A0050807367). We thank Academia Sinica High-Field NMR Center (HFNMRC) for technical support; HFNMRC is funded by Academia Sinica Core Facility and Innovative Instrument P roject (AS-CFII-108-112).

Conflicts of interest

There are no conflicts to declare.


Benoit Renault

Jacques Saraydaryan

Olivier Simonin

Modeling a Social Placement Cost to Extend Navigation Among Movable Obstacles (NAMO) Algorithms

Current Navigation Among Movable Obstacles (NAMO) algorithms focus on finding a path for the robot that only optimizes the displacement cost of navigating and moving obstacles out of its way. However, in a human environment, this focus may lead the robot to leave the space in a socially inappropriate state that may hamper human activity (i.e. by blocking access to doors, corridors, rooms or objects of interest). In this paper, we tackle this problem of "Social Placement Choice" by building a social occupation costmap, built using only geometrical information. We present how existing NAMO algorithms can be extended by exploiting this new cost map. Then, we show the effectiveness of this approach with simulations, and provide additional evaluation criteria to assess the social acceptability of plans.

I. INTRODUCTION

In 2005, Stilman et al. [START_REF] Stilman | Navigation among movable obstacles: real-time reasoning in complex environments[END_REF] formulated the field of Navigation Among Movable Obstacles (NAMO). The NAMO problem consists in planning a path from a start to a goal position, while moving obstacles if necessary. It extends the well known Piano Mover's Problem by differentiating static and movable obstacles, and allowing the manipulation of the latter. to find a path or to minimize a displacement cost function (eg. travel distance, time, energy). Service robotics, in particular, would definitely benefit from algorithms capable of dealing with manipulable clutter, doors or objects.

Actually, service robotics also imply robot navigation in human environments. Research in Socially-Aware Navigation (SAN) [START_REF] Kruse | Human-aware robot navigation: A survey[END_REF], [START_REF] Rios-Martinez | From Proxemics Theory to Socially-Aware Navigation: A Survey[END_REF], [START_REF] Charalampous | Recent trends in social aware robot navigation: A survey[END_REF] has shown that minimizing disturbance 1 to humans is paramount in order to raise their acceptability of robots in such environments. However, until now, to the best of our knowledge, this necessity of minimizing disturbance to humans (or any other autonomous agents types) has never been considered in NAMO problems. This may lead to cases such as in Fig. 1, where current algorithms would even deteriorate free space connectivity; forcing other agents to either move the object again or take detours to reach their own goal. This is why we propose to define Social NAMO (S-NAMO) problems and algorithms to avoid such situations.

When modifying the environment configuration, the most obvious way for a robot to minimize disturbance to humans is to avoid invading space that humans need to keep free. Determining these semantic spaces could be done by in situ human behavior observation [START_REF] Limosani | Long-term human affordance maps[END_REF], but the robot may not a Benoit Renault and Olivier Simonin are with INSA Lyon, CITI Lab and INRIA Chroma Team benoit.renault@insa-lyon.fr, olivier.simonin@insa-lyon.fr b Jacques Saraydaryan is with CPE Lyon, CITI Lab and INRIA Chroma Team jacques.saraydaryan@cpe.fr 1 In Socially-Aware Navigation, disturbance is used as a synonym for 'discomfort', the feeling/state of being unsafe [START_REF] Rios-Martinez | From Proxemics Theory to Socially-Aware Navigation: A Survey[END_REF]. have the opportunity to observe long enough (due to timing requirements, privacy concerns, or task complexity). Human activity could otherwise be inferred from the objects layout, through the understanding of their respective affordances and related affordance spaces [START_REF] Lindner | Towards a Formalization of Social Spaces for Socially Aware Robots[END_REF]. However, this would require both an extensive prior knowledge of human environments and good enough sensing capabilities to leverage it.

Our claim is that the significance of space (or rather, the importance of leaving it free) can be at least partially inferred without such strong pre-requisites. Indeed, as the field of interior design shows, the layout of human space is very much deliberate: "Providing adequate space for movement is essential and requires careful consideration of ergonomics, scale, and organizational flow" [START_REF] Mitton | Residential Interior Design: A Guide to Planning Spaces[END_REF]. Free space thus has, in and of itself, a strong semantic meaning we can use of accessibility and circulation.

We build upon this observation to compute a social occupation costmap to improve the social acceptability of NAMO plans. For this, we model the environment as passageways corresponding to the skeleton, then associate to each skeleton cell their corresponding space allowance, and convert it into a social cost that we then propagate in a decreasing wave. We evaluate this model by applying it to a reference NAMO algorithm [START_REF] Stilman | Navigation among movable obstacles: real-time reasoning in complex environments[END_REF] in several experimental scenarios. We provide new quantitative performance criteria to evaluate the social acceptability of plans, and study the computational and displacement cost overhead of our approach. This paper is organized as follows. Section II provides an overview of related work. Section III introduces the Social Placement Choice problem and Section IV details the proposed costmap computation. Section V presents how to use the costmap to extend existing NAMO algorithms, and we evaluate the approach with simulated scenarios in Section VI. We conclude and discuss future work in Section VII.

II. RELATED WORK

In a previous state-of-the-art on NAMO algorithms [START_REF] Benoit | Towards S-NAMO: Socially-aware Navigation Among Movable Obstacles[END_REF], we have shown that existing approaches have never considered the consequences of the robot's choices, beyond its interest to reach its goal. All existing NAMO works only consider the minimization of the robot's displacement cost. Only Stilman et al [START_REF] Stilman | Navigation among movable obstacles: real-time reasoning in complex environments[END_REF] & Kakiuchi et al [START_REF] Kakiuchi | Working with movable obstacles using on-line environment perception reconstruction using active sensing and color range sensor[END_REF] suggested the idea of taking object fragility into account, but have not applied it.

The Social/Human-aware variations of problems most related to NAMO mainly focus on direct human interaction. In Manipulation Planning [START_REF] Pellegrinelli | Motion planning and scheduling for human and industrial-robot collaboration[END_REF], [START_REF] Sisbot | A Human-Aware Manipulation Planner[END_REF] , Assembly Planning [START_REF] Unhelkar | Human-Aware Robotic Assistant for Collaborative Assembly: Integrating Human Motion Prediction With Planning in Time[END_REF] as well as in Combined Task/Motion Planning [START_REF] Alami | On human-aware task and motion planning abilities for a teammate robot[END_REF], the focus of research is collaborative manipulation in close proximity and/or handover manipulations. In the end, to the best of our knowledge, there is no existing work on the social acceptability estimation of object placement choice in the NAMO-related motion planning literature either.

We have found that the Social Placement Choice problem we investigate in this paper shares objectives with realistic indoor scene synthesis [START_REF] Zhang | A Survey of 3d Indoor Scene Synthesis[END_REF], in that it also tries to find implicit or explicit rules to help decide of socially acceptable object placement For example, in [START_REF] Yu | Make it home: automatic optimization of furniture arrangement[END_REF], Yu et al. used a set of rules, automatically learned from human-designed indoor environments (e.g. distance/orientation to walls/specific objects, pathways between doors,. . . ) to arrange furniture in a convincing-enough way for humans. However, there are two major differences with our problem: in realistic indoor scene synthesis, the aim is not to preserve an existing humanmade arrangement of space designed by humans, and since there is no robot, the optimization of displacement cost for an agent actually moving the objects is not addressed. Also, it is assumed that any required semantic data is available, but acquiring such knowledge in situ is no easy task for a robot, as the active work in semantic mapping capabilities shows [START_REF] Charalampous | Recent trends in social aware robot navigation: A survey[END_REF]. This motivates our choice to build a new model requiring only a binary occupancy grid of the fixed obstacles, such a grid already being a pre-requisite of commonly-used navigation frameworks like the ROS Navigation Stack [START_REF] Lu | Layered costmaps for context-sensitive navigation[END_REF].

Finally, in a survey on human-aware robot navigation, Kruse et al. [START_REF] Kruse | Human-aware robot navigation: A survey[END_REF] show that a common social navigation approach is to build a costmap representing zones to avoid or favour for the robot's trajectory. While they recognize it may oversimplify the problem, it allows for simple search, representation, and combination of many types of social considerations. That is why we decide to build our model as a social occupation costmap.

III. SOCIAL PLACEMENT CHOICE PROBLEM

Following the quote of [START_REF] Mitton | Residential Interior Design: A Guide to Planning Spaces[END_REF] in Sec. I, "Socially-acceptable object placements" can be defined as object configurations that best preserve the adequacy of space for movement in regards to ergonomics, scale and organizational flow. We reformulate this notion into the S-NAMO problem of Social Placement Choice hereafter (illustrated in Fig. 1).

Definition 1: SPC problem In S-NAMO, the Social Placement Choice problem is to determine for a robot R the plan P that brings it from initial configuration q tinit R to a goal configuration q t end R while minimizing both the resulting displacement cost2 DC(P ) and total social occupation cost SOC(P ) caused by the occupation of obstacles M i it moves.

Let us define notations and structures of the problem:

• a mobile robot R • a set of fixed undeformable obstacles that cannot be moved by the robot F =< F 1 , ..., F m > • a set of movable undeformable obstacles that can be moved by the robot

M =< M 1 , ..., M n >, F ∩ M = ∅ • a physical representation of the world W , with R, F, M ⊂ W .
For the sake of simplicity, elements E i ∈ W are considered undeformable polygons in a 2D plane, allowing us to define a world state at time t as a set of element configurations

W t =< q t E1 , ..., q t Ep > with q t Ei = (x t Ei ∈ R, y t Ei ∈ R, θ t Ei ∈ [0, 2π]). • a discrete representation G(W t ) of W t as a 2D binary
occupancy grid without loss of generality. Cells are addressed by tuples of coordinates (x, y) ∈ N 2 . • another 2D grid G SC built from G(W t ) (thus sharing the same dimensions, position, orientation and resolution), containing in each cell a social occupation cost SC(x, y) ∈ R + . We write Cells(E i , t) the cells occupied by an element E i at time t. The social occupation cost SOC(M i , t) is written as the sum SC(x, y) of each cell occupied by M i at t:

SOC(M i , t) = ∀(x,y)∈Cells(Mi,t) SC(x, y) (1)
The navigation plan P followed by the robot is made of path components. A path component is a sequence of robot configurations q i R where the robot moves without manipulating the object (transit path) p transit , or where the robot moves with the object (transfer path) p transf er .

The displacement cost DC(P ) of the plan P is expressed as the sum of the estimated displacement cost DC(p) for each path component p:

DC(P ) = ∀p∈P DC(p) (2)
The displacement cost DC(p) is the sum between each successive robot pose q i R ∈ p of an estimation function f (q i , q i+1 ) of distance, time or energy, that depends on hypotheses as to the robot capabilities and the physical characteristics of obstacles:

DC f (p) = |p|-1 i=1 f (q i , q i+1 ) (3)
A. Heuristic hypotheses Following our argumentation in Section I, let us assume that all free space in human environments can be considered as passageways that are designed for a certain number of people to walk abreast (which holds especially true in public buildings [START_REF] Interior | Arrete du 25 juin 1980[END_REF]). Since we only assume to have the binary occupancy grid of static obstacles, our focus can only be the preservation of the environment's original topology and space allowances. Thus, we propose the following two heuristic hypotheses to ensure this: 1) AVOID-MIDDLE: The original free space allowance of a passageway must be preserved as much as possible, so it must be the least divided (see Fig. 2): thus, the cells in the "middle of the space" must have the highest local costs; 2) AVOID-NARROW: The narrower a passageway, the more risk there is of blocking it with an object: thus, narrow spaces must have higher costs than wide spaces.

B. Computation steps overview

The above heuristic hypotheses are translated into a four step algorithm to build the social occupation costmap from the binary occupancy grid of fixed obstacles ) 1) Skeletonize: Find the set of free cells SK(G(F )) that represents the middle of all passageways3 (Fig. 3.A.); 2) Compute the minimal distance for each cell in SK(G(F )) to the fixed obstacles ∈ F , i.e. measure the original space allowance of passageways (Fig. 3.A.); 3) Associate a social cost to each cell in SK(G(F )) by relating the previously determined minimal distance from obstacles to appropriate human-sized constants using f conv (Fig.

C. Skeleton & Space Allowance

Determining the skeleton of a binary occupancy grid and the minimal distance to obstacle cells are well-known problems in the literature, and can respectively be done using skeletonization [START_REF] Saha | Skeletonization: Theory, Methods and Applications[END_REF] and distance transform [START_REF] Fabbri | 2d Euclidean Distance Transform Algorithms: A Comparative Survey[END_REF] algorithms.

Given the tremendous amount of choice of skeletonization algorithms in the literature, we focused our choice on three of the most popular: Zhang-Suen [START_REF] Zhang | A Fast Parallel Algorithm for Thinning Digital Patterns[END_REF], Guo-Hall [START_REF] Guo | Parallel thinning with two-subiteration algorithms[END_REF] and Medial-Axis [START_REF] Blum | A Transformation for Extracting New Descriptors of Shape[END_REF]. We ended up choosing the Guo-Hall algorithm, because, it has a very good contour noise immunity, and does not produce end-of-edge artifacts like Zhang-Suen. Just like the Medial-Axis method, it guarantees the preservation of connected components, which is paramount to properly represent passageways. An example of such skeletonization is available on Fig. 3A.

To express the original space allowance of passageways (i.e. their "broadness"/"narrowness"), we must measure the minimum euclidean distance from fixed obstacles (in contrast to taxicab or chessboard distance), since passageways do not necessarily follow perfect diagonals or straight lines in the grid. We do that using the Euclidean Distance Transform (Fig. 3 A.). Since the minimum distance to obstacles is only half of the space allowance between the two nearest obstacles, we double it to get the space allowance d allow (x, y).

D. From Space Allowance to Social Occupation Cost

To transform the previously computed space allowance in each cell ∈ SK(G(F )) into a meaningful social cost that respects the AVOID-NARROW rule, we must use realworld measurements that define what is narrow or wide for humans. Basically, we want the cost to be maximal for any space allowance that is below a reference human diameter d h , then decrease as the space allowance increases.

To find a convincing d h value, we turn to anthropometrics, and use the DINED [START_REF] Johan | DINED -anthropometric database[END_REF] database. We derive the value of d h from standing breadth over the elbows, which is the widest measurement in a standing posture with arms at rest. According to the most recent 2004 data for Dutch adults between the age of 20 to 60, we can reasonably cover about 95% of the population by setting d h = 0.55m. This value is coherent with the average adult space clearance for ambulatory human movement of 0.559m provided by [START_REF] Mitton | Residential Interior Design: A Guide to Planning Spaces[END_REF].

Then, to determine a relevant decrease profile, we turn to a commonly used law-defined measurement in french construction [START_REF] Interior | Arrete du 25 juin 1980[END_REF]: "UP" or "Unités de Passage" (litt. "Passage Units"). UPs describe the minimal space allowance between obstacles/walls for the passageway to allow proper emergency evacuation for n persons: 1UP = 0.90m (meant to comply with standard wheelchair width), 2UP = 1.40m, and for three and beyond nUP = n * 0.60m. An indoor passageway is rarely more than 5UP = 3m large because the same law recommends to increase the number of passageways rather than create ones that are more than 5UPs large: therefore, we choose to keep the social cost constant beyond 5UP. For the sake of keeping social occupation cost values comparable, we reduce the interval of acceptable values to [0, 1]. The cost values associated with each UP between 1 to 5 can be adjusted depending on the relative importance users give to preserving space allowance. Fig. 4 shows a set of values we have found to yield a reasonably convincing social occupation cost representation in our experiments of Section VI. Intermediate values are determined by simple linear interpolation.

0 d h 1UP 0.9 2UP 1.4 3UP 1.8 4UP 2.4

E. Propagation

As explained in the introduction of Section IV, we need a propagation procedure that guarantees a decreasing cost from the skeleton values, in order to guarantee the benefit of getting the obstacle away from the "middle of space".

For this, we use a variant of the Wavefront Propagation algorithm [START_REF] Lavalle | Planning Algorithms[END_REF] where we start from the skeleton's set of cells SK. We then iteratively mark all unlabeled 4-neighbors cells with the λ-decayed minimum value of their own neighbor cells from the previous iteration, with λ ∈]0, 1[, as described in equation 4. Since the choice of λ value depends on the grid resolution, we will discuss it in Section VI.

f prop : W → R[0, 1] f prop (x, y) = f conv (x, y) if (x, y) ∈ SK λ • min(N eigh(x, y)) if (x, y) / ∈ SK (4)
The resulting costmap is illustrated in 3D in Figure 3.C. As expected, the middle of space is populated by ridges created by local social occupation cost maxima, and the narrowest spaces are associated with the highest costs, encouraging placement choices outside the corridor and away from the room's center, especially in front of the opening.

V. EXTENSION OF NAMO ALGORITHMS

A. NAMO Algorithms structure

From our previous state-of-the-art work [START_REF] Benoit | Towards S-NAMO: Socially-aware Navigation Among Movable Obstacles[END_REF], we draw the conclusion that NAMO algorithms share a common structure. Basically, as long as no satisfying 4 plan to reach the goal has been/can be found, these algorithms iteratively or recursively loop through known movable obstacles, each using a different way to choose the order in which to do so.

Once an obstacle is selected for manipulation search, a reachable non-colliding pose to leave it at needs to be found so that it creates an opening for the robot. While APPROACH 1 finds the relocation configuration as the robot explores its action space, APPROACH 2 first samples configurations, regularly or randomly, within a set of constraints (Alg.1). Usual constraints for pose sampling are: distance from initial obstacle pose, and a discrete set of accepted rotations. Each NAMO approach explores the action space with different algorithms, generally based on graph-search like Dijkstra, A* or RRT (see NAMO algorithms comparison table in [START_REF] Benoit | Towards S-NAMO: Socially-aware Navigation Among Movable Obstacles[END_REF]). Computation of transit paths to and from obstacles and action space exhaustion exit conditions are not shown for the sake of brevity.

B. Social occupation cost integration

Minimizing both the displacement cost of existing NAMO algorithms and our social occupation cost is a typical multiobjective optimization problem. Such a problem is generally solved by comparing the cost of all available solutions. In our case, this would mean exploring the entire robot action space to find all reachable non-colliding poses for the obstacle and computing the associated social occupation costs. Exploring such a high-dimensional problem is however intractable, even in small spaces like in Fig. 1.

That is why we propose to compute a heuristic compromise cost CC(x, y) for the set of potentially reachable cells to move the selected obstacle at. We do that with the following process, illustrated in Fig. 5: 1) Create an occupancy grid taking into account all obstacles except the robot R and selected obstacle, M i . 2) Inflate the grid by the inscribed circle radius of M i .

This operation allows to find a subset of potentially acceptable cells for M i . 3) Compute a goal-less Dijkstra search from M i 's center cell. The set of potentially reachable cells for moving M i is thus obtained, with an underestimate of the distance d Mi (x, y) to reach them 5 . 4) Compute the compromise cost CC for each potentially reachable cell with a weighted sum of the following normalized costs: occupation cost (SC(x, y)), distance d Mi (x, y) and a heuristic euclidean distance from cell to NAMO goal d goal (x, y):

CC(x, y) = w 1 * d Mi + w 2 * d goal + w 3 * SC w 1 + w 2 + w 3 (5)
where

VI. EXPERIMENTAL EVALUATION

A. Algorithm choice

We choose to implement our method upon the founding algorithm of Stilman et al. [START_REF] Stilman | Navigation among movable obstacles: real-time reasoning in complex environments[END_REF] since all following works on NAMO have been inspired by it, and because it focuses more on the method to choose which obstacles to evaluate and their order than the actual manipulation search method. Indeed, in [START_REF] Stilman | Navigation among movable obstacles: real-time reasoning in complex environments[END_REF], this search is mentioned to be a bounded Breadth-First Search over the robot action space (APPROACH 1 in Alg. 1), and we implemented it as is for the baseline NAMO to compare against. However, to take full advantage of our computations from Section V, we changed this subroutine for an A* search that focuses on the best compromise cell for the S-NAMO variant (APPROACH 2) 6 .

B. Simulator & simulation parameters

We implemented our approach in a custom open-source simulator 7 that only considers object geometry for interaction (no kinematics nor dynamics) akin to the one used in [START_REF] Stilman | Navigation among movable obstacles: real-time reasoning in complex environments[END_REF], and provide an integration layer with ROS for visualization in Rviz. Fig. 1 and 6 illustrate the simulator. It is also presented in the video provided with the paper.

All grids are set to a resolution of 0.1m/cell and the robot action space allows for backward/forward unit translations of 0.1m, and unit rotations of 30 • . We assume that the robot can be reduced to a circle (of radius 0.2m) to accelerate collision checks. Full geometry and movability-related world knowledge is provided to the robot, as required by the baseline algorithm [START_REF] Stilman | Navigation among movable obstacles: real-time reasoning in complex environments[END_REF]. Such knowledge could be provided by an external motion capture system and markers on movable obstacles like the baseline authors did in [START_REF] Stilman | Planning and executing navigation among movable obstacles[END_REF]. Displacements with an obstacle are penalized by a factor of 2 to reflect the extra effort (additional semantic data like obstacle weigth could result in a finer approach). All experiments are run with λ = 0.97 (so that social costs decrease fast enough given the chosen resolution), w 1 = 10, w 2 = 2 and w 3 = 15, and robot action space exploration stops when a solution is at a cell with a compromise cost within the best 1%. The choice of weights here is manually tuned over the given scenarios, but could be optimized through machine-learning over a wider range of real-world scenarios.

C. Comparison criteria

We now need criteria for measuring both the displacement cost efficiency and social acceptability of solutions. For displacement cost efficiency, as in all of NAMO literature, we use the total euclidean distance for transfer and transit paths, respectively L transit and L transf er . As for social acceptability, we define the following criteria:

• Total Social Occupation Cost: Derived from our model, it is the sum of SC for all movable obstacles. It represents the global space accessibility disturbance for humans.

ST (W t ) = ∀Mi∈M S (Mi,t) (6)
• Number of connected components: In the grid representation of W t inflated by half the reference human diameter d h defined in Section IV, it is the number of interconnected sets of cells C i h (W t ) that make up the global set of accessible cells for humans. It allows to measure the topology variation.

N cc(W

t ) = |{C i h (W t )}| (7)
• Space fragmentation percentage: In the same grid as before, it relates the size of the biggest free space component C max h (W t ) to the total size of free space. It allows to observe the variation in space allowance.

f rag(W t ) = 1 - |C max h (W t )| |C acc h (W t )| (8)
A plan improves social acceptability by lowering any of these criteria between W tinit to W t end . Finally, we also provide the total planning time T planning to represent the computational cost of each plan.

D. Experimentations

Ltransit We evaluated our approach with four scenarios of increasing complexity, called [A] "corridor-to-room" (Fig. 1), [B] "crossing", [C] "after-the-feast" and [D] "CITI-lab" (Fig. 6). In scenarios [A] and [C], the robot is only required to reach one goal, however in [B] and [D], it has to sequentially reach several. Table I compares results obtained in these 4 scenarios between the baseline NAMO algorithm and its S-NAMO counterpart. We comment these results hereafter.

Scenario [A] shows the worst-case scenario that can happen with current NAMO approaches: unintentional connectivity loss. At the cost of little extra computation and reasonable further displacement of the obstacle (the total distance is almost the same), our approach results in complete defragmentation. The low initial fragmentation is due to lack of knowledge beyond the corridor.

Scenario [B] allows to highlight an interesting phenomenon: NAMO algorithms, when faced with a movable obstacle blocking a tight intersection branch they need to pass will naturally tend to block another branch because of the sole focus on displacement cost. As the robot needs to get from one room to the next, it repeatedly blocks another corridor and systematically induces an additional computation cost, that after 4 iterations is already 6 times greater than our S-NAMO approach. After the 4th iteration the chosen obstacle placement even blocks all branches for a human being of diameter d h .

Scenario [C] shows that, of the three criteria we propose to evaluate social acceptability, only the one based on our social cost computation ST allows us to differentiate the quality of solutions that produce the same relative change in number of connected components and space fragmentation.

Finally, Scenario [D] shows that our approach is capable of growing up to scale in big realistic environments. Indeed, despite the environment size and complex topology, Fig. 6 corridors around the middle obstacle and the tight passages between desks in the surrounding the offices are properly detected as less appropriate than the wider areas like office corners, the entry hall on the left and the big intersection on the right. The results in Table I show that despite this increase in complexity compared to the other scenarios, our approach still succeeds in achieving more socially acceptable solutions than the baseline approach, for almost the same computational cost 8 .

While the S-NAMO solutions presented here have better social acceptability than their baseline NAMO counterparts, some of them could still be improved. For example, in case [C], if the moved obstacle were a stool, it would have been better to keep it closer and aligned to the tables than the wall. This could only be solved better if this extra semantic information were available. A potential approach would be to apply conditional cost reductions and augmentations around points of interest, but this would be the object of future work.

VII. CONCLUSION

This paper first introduced a social placement cost model based on two heuristic hypotheses as to the significance of free space in human environments: avoiding narrow spaces, and not leaving objects in the middle of space. This model relies solely on the analysis of the environment's binary occupancy grid of fixed obstacles. Then, we showed how to extend NAMO to S-NAMO algorithms that result in plans with better social acceptability. Experimental results on scenarios of increasing complexity show the scaling capability of our method. To the best of our knowledge, this is the first approach to propose an answer to social concerns in NAMO problems.

Our future works will focus on incorporating more complex semantic information to further improve social acceptability. We will also apply our model to other NAMO algorithms (e.g. [START_REF] Levihn | Locally optimal navigation among movable obstacles in unknown environments[END_REF], [START_REF] Levihn | Planning with movable obstacles in continuous environments with uncertain dynamics[END_REF]) in order to manage partial and uncertain knowledge of the environment, thus bringing us closer to real-world conditions and experiments. Finally, we plan to study the implications of other autonomous agents actual presence such as humans.

Fig. 1 :

 1 Fig. 1: Basic pathological case (see notations Sec. III). (a) The robot's goal is in the adjacent room but unreachable due to a movable obstacle. (b) Our social occupation costmap. (c) Classic NAMO solution. (d) Our S-NAMO approach's solution.

Fig. 2 :

 2 Fig.2: Illustrations of the AVOID-MIDDLE and AVOID-NARROW heuristic hypotheses. In the first figure, we can see that leaving an obstacle in a narrow space is more likely to end up reducing connectivity. In the last two figures, we can see that leaving an obstacle furthest from the middle of space least divides it, maximizing the number of people that can access it at once.

  3.B); 4) Propagate the social cost values from the skeleton cells in SK(G(F )) in a decreasing wave defined by f prop (Fig.3.C.).

Fig. 3 :

 3 Fig. 3: Illustration of the 4 step computation: skeletonization & distance transform (A), conversion of space allowance into Social Cost (B) and decreasing wave propagation (C)

Fig. 4 :

 4 Fig. 4: Social Occupation Cost as a function of the Original Space Allowance: SC(x, y) = fconv(d allow (x, y))

Algorithm 1 : 2 if APPROACH 1 then 3 4 else if APPROACH 2 then 5 Sample obstacle poses within constraints; 6 forall sampled poses do 7 Explore robot action space with pose focus; 8 Relax constraints one step; 9 return

 123456789 NAMO manip. search macro algorithm Data: World state W t and most relevant obstacle M i Result: A transfer path p transf er 1 while no satisfying transfer path p transf er found do Explore robot action space without focus; transfer path p transf er ;

  SC , d Mi , and d goal are respectively normalized versions of SC, d Mi , and d goal in [0, 1]. Finally, we use this knowledge in the manipulation search subprocedure of NAMO algorithms by: • bounding the action space exploration by stopping when a solution leaves M i 's center in one of the cells with the top n percent best CC cost, • sampling a goal obstacle pose among the cells with the top n percent best CC cost to focus the search.

Fig. 5 :

 5 Fig. 5: Compromise cost (CC) computation steps

Fig. 6 :

 6 Fig. 6: Experimental scenarios & NAMO vs. S-NAMO results. [B] is 4 rooms joined by corridors, each with a goal (executed counterclockwise from the right). Path to 1st goal is the only one shown for the sake of readability. [C] is a room filled with unmovable tables (black squares) and miscellaneous movable obstacles "after a party". [D] is a grid representing our lab's 2nd floor obtained from a 3D scan. First goal is in 2nd room from the left at the top, second one is in the first room from the right at the bottom.

TABLE I :

 I Performance criteria comparison table. Displacement and computation costs for [B] and [D] are cumulated over goals. ST and N cc have no units. T planning is given as average with standard deviation over 50 iterations.

typically energy, time, or distance

That is, the thinnest discrete version (one cell wide) of the environment's shape equidistant to the obstacles boundaries.

"satisfying" meaning displacement cost optimal or not, depending on the algorithm

The goal cells are filtered out of this set afterwards so that the robot can't put the obstacle on it.

The accompanying video shows how both algorithms unfold: https://team.inria.fr/chroma/files/2020/07/iros_ 2020_Renault.mp4

Provided with all simulation code and data, available at: https://gitlab.inria.fr/brenault/s-namo-sim

The high amount of detected connected components is due to the chosen reference human diameter d h : in the offices, space is considered fragmented between some desks because of this. This does however not make the comparison any less relevant.


R Salles

B Abécassis

E Derat

D Brouri

A Bernard

Q Zhang

A Proust

C Desmarets

G Izzet

Hierarchical Self-Assembly of polyoxometalate-based organo Palladium(II) Metallomacrocycles via Electrostatic Interactions

Introduction

Individual molecular building blocks that can self assemble into different complex molecular architectures with proper molecular organization are currently drawing a vigorous attention. [START_REF] Robinson | Dimensional Control and Morphological Transformations of Supramolecular Polymeric Nanofibers Based on Cofacially-Stacked Planar Amphiphilic Platinum(II) Complexes[END_REF][START_REF] Lescop | Coordination-Driven Syntheses of Compact Supramolecular Metallacycles toward Extended Metallo-organic Stacked Supramolecular Assemblies[END_REF][START_REF] Chan | Synthesis of Luminescent Platinum(II) 2,6-Bis-(N-dodecylbenzimidazol-2 '-yl)pyridine Foldamers and Their Supramolecular Assembly and Metallogel Formation[END_REF][START_REF] Jiang | Hierarchical Self-Assembly of a Porphyrin into Chiral Macroscopic Flowers with Superhydrophobic and Enantioselective Property[END_REF][START_REF] Rest | Strategies to create hierarchical self-assembled structures via cooperative non-covalent interactions[END_REF][START_REF] Mariani | Elucidating Electrostatic Self-Assembly: Molecular Parameters as Key to Thermodynamics and Nanoparticle Shape[END_REF][START_REF] Kim | Development of Toroidal Nanostructures by Self-Assembly: Rational Designs and Applications[END_REF][START_REF] Yu | Supramolecular Amphiphiles Based on Host-Guest Molecular Recognition Motifs[END_REF] Indeed the path that goes from discrete molecules to nanostructured macro or mesoscopic assemblies is a cutting-edge challenge for present and future chemistry. [START_REF] Sun | Soft Materials with Diverse Suprastructures via the Self-Assembly of Metal-Organic Complexes[END_REF][START_REF] Yan | Hierarchical assemblies of coordination supramolecules[END_REF][START_REF] Yamada | Designer coordination polymers: dimensional crossover architectures and proton conduction[END_REF] The way has already been partially paved by research on coordination-driven self-assembly [START_REF] Chakrabarty | Supramolecular Coordination: Self-Assembly of Finite Two-and Three-Dimensional Ensembles[END_REF] although supramolecular architectures regulated by redox-active units have been considerably less investigated. [START_REF] Herasymchuk | Coordination-driven assembly of a supramolecular square and oxidation to a tetra-ligand radical species[END_REF][START_REF] Bivaud | A Metal-Directed Self-Assembled Electroactive Cage with Bis(pyrrolo)tetrathiafulvalene (BPTTF) Side Walls[END_REF][START_REF] Tominaga | Finite, spherical coordination networks that self-organize from 36 small components[END_REF] Polyoxometalates (POMs) are anionic oxo-clusters that display unique redox properties and are thoroughly investigated for their promising technological applications. [START_REF] Ji | Polyoxometalate-functionalized nanocarbon materials for energy conversion, energy storage and sensor systems[END_REF][START_REF] Miras | Engineering polyoxometalates with emergent properties[END_REF][START_REF] Li | Ionic Complexes of Metal Oxide Clusters for Versatile Self Assemblies[END_REF] They can be functionalized into organic-inorganic hybrids though the covalent grafting of organometallic fragment onto the polyoxometalate framework. [START_REF] Izzet | Tailor-made Covalent Organic-Inorganic Polyoxometalate Hybrids: Versatile Platforms for the Elaboration of Functional Molecular Architectures[END_REF][START_REF] Proust | Functionalization and post-functionalization: a step towards polyoxometalate-based materials[END_REF] Thanks to their nanoscale size, their tunable redox properties, and their sensitivity to various types of interactions, hybrid POMs are promising building blocks for the elaboration of complex nanostructured molecular architectures. For instance, hybrid POMs displaying remote coordination sites can self-assemble in metallomacrocycles upon metal coordination where positive and negative charges can further be associated. We previously developed such an approach by connecting ditopic Keggin types and Dawson type hybrids donors to linear metal acceptors. [START_REF] Piot | Control of the hierarchical self-assembly of polyoxometalate-based metallomacrocycles by redox trigger and solvent composition[END_REF][START_REF] Izzet | Hierarchical Self-Assembly of Polyoxometalate-Based Hybrids Driven by Metal Coordination and Electrostatic Interactions: From Discrete Supramolecular Species to Dense Monodisperse Nanoparticles[END_REF] However this system afforded mixtures of triangle and square shape secondary building units leading to defects in the stacking of the metallomacrocycles, owing to the mismatch between these two species. This drove us in investigating the self-assembly of similar hybrid ditopic donors with 90° degrees metal acceptor, since a unique type of square-shape metallomacrocycle (containing 2 hybrid POMs and 2 metal acceptors) should be formed using this strategy. We herein report the self-assembly of bis silylated Keggin-based building units in the presence of [Pd(eda)(MeCN) 2 ](BF 4 ) 2 , a well-known metal acceptor displaying two orthogonal accessible coordination sites imposed by the ethylenediamine (eda) ligand. We show that in a highly dissociating solvent these hybrid POMs self-assemble into a unique discrete supramolecular architecture that can further self-aggregate into different hierarchical self-assemblies according to the solvent composition.

Design and synthesis of the POM-based elementary building unit

We previously showed that the metal driven self-assembly of Keggin type hybrids led to more stable assembly than Dawson-type analogues probably owing to a lower charge and hence a lower entropy contribution of the resulting self-assembly. [START_REF] Izzet | Metal-Directed Self-Assembly of a Polyoxometalate-Based Molecular Triangle: Using Powerful Analytical Tools to Probe the Chemical Structure of Complex Supramolecular Assemblies[END_REF][START_REF] Piot | Charge Effect on the Formation of Polyoxometalate-Based Supramolecular Polygons Driven by Metal Coordination[END_REF] Furthermore Keggin type POMs display a pseudo axial symmetry (not present in the Dawson analogues) that should lead to a the formation of a unique pseudo D 2h supramolecular assembly upon metal coordination in which the aromatic rings of the organic linker should be chemically equivalent and therefore, give a single set of NMR signals. Therefore, we focused on the Keggin-type POM hybrid [PW 11 O 39 {Si 2 OC 26 H 16 N 2 }] 3-named K Si [pyr]. This hybrid contains two pyridine functions connected to a bis-siloxane organic tether grafted on the lacunary site of the monovacant POM [PW 11 O 39 ] 7-(Figure 1). A Sonogashira cross-coupling reaction, as previously developed in the group (ref ?), was used to attach the pyridine function to the parent hybrid platform containing two iodo-aryl groups. The hybrid K Si [pyr] was isolated as a tetrabutylammonium (TBA) salt and characterized by [START_REF] Robinson | Dimensional Control and Morphological Transformations of Supramolecular Polymeric Nanofibers Based on Cofacially-Stacked Planar Amphiphilic Platinum(II) Complexes[END_REF] H NMR, mass spectrometry, and chemical analysis (Figures 1 &S1).

Synthesis of the discrete supramolecular assembly

The addition of [Pd(eda)(MeCN) 2 ](BF 4 ) 2 to a solution of the POM-based building block K Si [pyr] (5 mM) in DMSO-d 6 was followed by 1 H NMR spectroscopy. The monitoring of the reaction shows the progressive formation of a unique set of new signals and the concomitant disappearance of those of the starting POM-based building block. The reaction is complete after the addition of 1.2 equivalents of [Pd(eda)(MeCN) 2 ](BF 4 ) 2 (Figure 2 & S2), which suggests that the resulting species presents a 1:1 stoichiometry between the POM and the Pd unit while a minor excess of Pd species, observed at ca. 5.5 ppm is necessary for the reaction of K Si [pyr] to be complete. Interestingly, chemical shifts of the new species at intermediate [Pd(eda)(MeCN) 2 ](BF 4 ) 2 concentration are the same as those of the final species which was not the case in the previous POM-Pd and POM-Co systems using linear metal acceptors. In these last systems we observed the prior formation of a molecular dimer with a POM:metal (2:1) stoichiometry at intermediate concentration of the metal linker, which evolved into the metallomacrocycles in the presence of 1 equiv. of metal linker. [START_REF] Piot | Control of the hierarchical self-assembly of polyoxometalate-based metallomacrocycles by redox trigger and solvent composition[END_REF][START_REF] Piot | Charge Effect on the Formation of Polyoxometalate-Based Supramolecular Polygons Driven by Metal Coordination[END_REF] We concluded that the formation of the cyclic assembly in such system was not noncooperative, probably owing to a high entropic cost due to the global gathering of the associated counter-ions. In the present case the targeted assembly spontaneously forms without the intermediate dimer at POM:Pd 2:1 stoichiometry.

We also performed DOSY NMR to evaluate the diffusion coefficients of the starting hybrid POM and after the addition of 1.2 equiv. of [Pd(eda)(MeCN) 2 ](BF 4 ) 2 . In both cases the set of aromatic signals of the hybrid POM decays similarly according to the pulsed-field gradient strengths with a unique diffusion coefficient : D 1 = 1.2 x 10 -9 m 2 .s -1 for the starting hybrid and D 2 = 7.7 x 10 -10 m 2 .s -1 for the supramolecular species (Figure 2 and S3-S4). Furthermore, in the last case, the major peak at 5.6 ppm, corresponding to the eda ligand, displays a diffusion coefficient close to D 2 , suggesting the participation of the Pd(eda) moieties in the supramolecular assembly (note that the slight discrepancy in the diffusion coefficient of the aromatic signals and those of the ethylenediamine moieties probably arises from some exchange of the Pd(eda) moieties in the supramolecular assembly and in the minor unbounded forms at the time scale of the DOSY). By contrast the minor peaks at 5.55 and 5.50 ppm displays considerably higher diffusion coefficient, that can not be properly evaluated with a unique diffusion coefficient (Figure S5), that we attribute to unbounded Pd(eda) species in equilibrium with the supramolecular assembly. We previously evaluated the ratio in the diffusion coefficient between a given assembly and the starting POM hybrid platform on a series of hybrid POM assemblies of different nuclearities (dimer, trimer and tetramer). We found that the diffusion coefficient ratio between a dimer and the starting hybrid POM was about 1.7, a value that is rather close to D 1 /D 2 = 1.6, which suggests the formation of a dimeric macrocyclic species.

Structural characterizaion of the metallomacrocycle: molecular modeling & SAXS analysis

To gain insight into the structure of the cyclic oligomer, optimization of geometries of the square shape supramolecular species was performed by energy minimization with a DFTB method owing to the large size of the system. More specifically, we build the putative K Si [pyr] 2 .[Pd(eda)] 2 system and its geometries was optimized using the XTB program developed by Grimme et al. [START_REF] Grimme | A Robust and Accurate Tight-Binding Quantum Chemical Method for Structures, Vibrational Frequencies, and Noncovalent Interactions of Large Molecular Systems Parametrized for All spd-Block Elements (Z=1-86)[END_REF] using the last generation of semi-empirical tight-binding method, namely GFN2-xTB. [START_REF] Bannwarth | GFN2-xTB-An Accurate and Broadly Parametrized Self-Consistent Tight-Binding Quantum Chemical Method with Multipole Electrostatics and Density-Dependent Dispersion Contributions[END_REF] To ensure that the geometry found is a stable conformer, simulated annealing was performed after a first geometry optimization, and then the most stable conformer was re-optimized. This computational structure can be seen in Figures 3 andS6. One can notice its relative flatness since the dihedral angle between the four units (defined by the two Pd atoms and the two oxygens of bridging the silicon atoms) is 1.97° (Table S1). The two POM units are separated by a distance of 14.70 Å while the distance between the two Pd atoms is 24.03 Å. The two angles Pd-POM-Pd are found to be 116.91° and 117.27°, closed to the ideal 120°. We also performed SAXS to complete the analysis of the supramolecular system. This technique is indeed well suited to characterize the structure of molecular assemblies and aggregates with sizes that range from 1 to 100 nm that we and others have successfully used to characterize the formation of POM-based self-assemblies. [START_REF] Nyman | Small-angle X-ray scattering to determine solution speciation of metal-oxo clusters[END_REF][START_REF] Li | A General Approach to Access Morphologies of Polyoxometalates in Solution by Using SAXS: An Ab Initio Modeling Protocol[END_REF][START_REF] Wu | Complexation of Polyoxometalates with Cyclodextrins[END_REF] A significant SAXS signal is visible for the molecular building block K Si [pyr] in DMSO-d 6 solution, consistent with well-dispersed nanometric objects. As previously observed with hybrid POMs, electrostatic repulsion between the charged species results in a small decrease in intensity at small wave vectors. The SAXS diagram of the supramolecular assemblies displays oscillations in intensity that are characteristic of a distance larger than the POM dimension (Figure 4). We also observed that in DMSO-d 6 the POM-Pd supramolecular systems evolved under the beam preventing scan accumulation. If we ignore the decrease in intensity caused by the electrostatic interactions, we can notice that the maximum intensity increases by a factor of ca. 2 between the POM-Pd supramolecular assembly and the parent K Si [pyr] building unit, in agreement with the formation of a molecular dimer. To further consolidate this, the theoretical SAXS pattern of the optimized structure of K Si [pyr] 2 .[Pd(eda)] 2 was computed using the program CRYSOL [START_REF] Svergun | CRYSOL -A program to evaluate x-ray solution scattering of biological macromolecules from atomic coordinates[END_REF] from the atomic coordinates of the optimized structure. We observe that the intensity at small q values and the first oscillation are very well reproduced, which fully support the macrocyclic structure (Figure 4.A.).

Aggregation induced by the addition of solvents : CD 3 CN & D 2 O

We have previously shown that in the presence of a less dissociating solvent than DMSO (such as acetonitrile) or a protic solvent, POM-based metallomacrocycles that contain positive and negative charges can aggregate through electrostatic interactions between the charged sub-units. (include reference)

Here, in the presence of CD 3 CN or D 2 O, the 1 H NMR spectrum of the POM-Pd supramolecular system drastically broadens. Note that the final concentration should not exceed c POM = 0.5 mM in order to avoid partial precipitation of the solution. The two mixtures (DMSOd 6 /CD 3 CN : 1/4; DMSO-d 6 /D 2 O : 1/1, C POM = 0.5 mM) were analyzed by SAXS (Figure 4) and electron microscopy (Figure 5) to characterize the overall structure of these new supramolecular architectures. In the SAXS patterns, we observe in both DMSO-d 6 /CD 3 CN and DMSO-d 6 /D 2 O mixtures the presence of a new peak, at large q values (q = 0.35-0.5 Å -1 ), which are not present in the SAXS pattern of the discrete species in DMSO-d 6 . These peaks correspond to specific POM-POM distances within the aggregates and indicate that they both display some internal nanostructuration. While in DMSO-d 6 /CD 3 CN the peak can be deconvoluted by a single component centered at q = 0.436 Å -1 (corresponding to a distance d = 1.44 nm), 2 components are necessary to accurately fit the peak in the case of the DMSO-D 2 O solvent (q 1 = 0.376 Å -1 , q 2 = 0.490 Å -1 corresponding to distances d 1 = 1.67 nm and d 2 = 1.28 nm respectively, Figure S7). The larger q value in the presence of water reveals that the POM-POM distances are shorter than in the presence of acetonitrile. At lower wavevectors, the signal intensity is considerably higher than that of the starting buildingblock K Si [pyr] at the same POM concentration (0.5 mM in DMSO-d 6 ). In DMSO-d 6 /D 2 O (1/1) solution, at small q values the intensity keep increasing for q → 0 with a slope between q -1 and q -2 suggesting that the aggregates are anisotropic and contain (at least) one preferential orientation. Among the various models tested to fit the data, the SAXS patterns were best reproduced by using a model of almost infinite flexible cylinders of 5.4 nm in radius (50% polydispersity) and a Kuhn length (i.e., average segment length of the flexible cylinder) of 15 nm (Figure 4). Note that the exact evaluation of the length of the cylinder (above hundreds of nanometers) is not possible with q > 2  10 -3 Å -1 . These results were confirmed by TEM analysis (Figure 5), which shows that the aggregates appear to be ramified flexible cylinders with a cherry-branch structures with a section of 8-10 nm (slightly less than the values obtained by SAXS). The difference in the evaluation of the section between SAXS and TEM may arise from the SAXS model that does not account for the branched nature of the aggregates in solution.

In the case of DMSO-d 6 /CD 3 CN mixture, the slope at low q is lower but the Guinier regime is never reached. TEM micrographs of the aggregates obtained by deposition of the same sample on a grid after further dilution in acetonitrile shows the formation of isotropic polydisperse assemblies (Figure 5) with a diameter comprise between 20 nm and 45 nm (Figure S8). Guided by these results, we tried to fit the SAXS pattern with various models such homogenous spheres with constant electron density or a vesicle model with a monolayer of POMS separating an interior and an exterior part filled with solvent. Both modes do not reproduce well the SAXS intensity. The faint oscillation at around q=0.08 Å -1 together with the slowly increasing intensity at lower q suggest an inhomogeneous electron density inside a close shape. A multishell model indeed reproduces well the oscillation (Figure SXX) but further studies are required to get a more precise structure of the these assemblies. We previously observed that different types of hierarchical self-assemblies can be selectively formed according to the solvent composition as a consequence of different aggregation modes of the metallomacrocycles. In the previous study the size of the aggregates (isotropic and pseudo-1D) were much smaller than those of the current study, probably as a consequence of stronger electrostatic interactions in the current POM-Pd metallomacrocycles. The difference in the aggregation mode between DMSO/CD 3 CN and DMSO/D 2 O mixtures probably arise from the solvation energy of the charged species. Indeed, in the elongated branched structures, the surface to volume ratio is considerably improved. Consequently, in this aggregation mode the number of POM and Pd(eda) complexes pointing toward the solvent is much higher than in the isotropic aggregates. The cherry-branched nanostructures would then preferentially be formed in the presence of water due to improved solvation energy between the charged species and TBA with water, probably at the expense of a lower compacity compared to the isotropic particles in DMSO/MeCN mixtures.

Conclusion

We elaborated a new POM-based metallomacrocycle using a ditopic hybrid POM donor and 90° metal acceptor. Using this synthetic strategy, a single discrete supramolecular species was obtained, by contrast to previous approach relying on the use of linear metal acceptors that led to the formation of mixtures of different metallomacrocycles. We observed that this new supramolecular species is able to further self assemble into different types of nanostructured hierarchical species according to the solvent composition. These complex molecular assemblies display bigger structures than the previously reported systems owing to the smaller size of the metallomacrocycle, which leads to improved electrostatic interactions. The large cherry branched structures displaying large size and close POM-POM distances hold great promise as semiconducting materials for the elaboration of neuromorphic networks.

Figure 1 .

 1 Figure 1. Structure of K Si [pyr] and its resulting organopalladium(II) metallomacrocycle.

Figure 2 :

 2 Figure 2: 1 H NMR (600 MHz, DMSO-d 6 ) with corresponding DOSY 1 H NMR of K Si [pyr] (5mM, bue) and its resulting supramolecular assembly in the presence of 1.2 equiv. [Pd(eda)(MeCN) 2 ](BF 4 ) 2 (red).

Figure 3 .

 3 Figure 3. Optimized structure of K Si [pyr] 2 .[Pd(eda)] 2 .

Figure 4 .

 4 Figure 4. Left) Experimental SAXS pattern of a solution of the molecular building-block K Si [pyr] (0.5 mM in DMSO-d 6 , black) and the resulting supramlecular assembly in the presence of 1.2 equiv. [Pd(eda)(MeCN) 2 ](BF 4 ) 2 (blue) and computed SAXS pattern (CRYSOL) of the optimized structure of K Si [pyr] 2 .[Pd(eda)] 2 (red). Right) Experimental SAXS pattern of a solution of the POM-Pd supramolecular assembly in (c POM = 0.5 mM) in DMSO-d 6 /CD 3 CN : 1/4 (blue); DMSO-

Figure 5 .

 5 Figure 5. TEM micrograph of the POM-Pd system in DMSO/CD 3 CN (left) and DMSO/D 2 O (right) mixtures.

Ackowledgements:

We acknowledge SOLEIL for provision of synchrotron radiation facilities and we would like to thank Thomas Bizien for assistance in using beamline SWING.

SAXS methods:

The small angle X-ray scattering data were acquired on the SWING beamline at syncrhotron SOLEIL (Saint Aubin, France). The nanocrystal dispersions were loaded in glass capillaries with an outside diameter between 1 and 1.5 mm and glass walls of 0,01 mm (WJM-Glas, Germany). The energy of the X-ray beam was 12 keV. The scattered X-ray were detected using an EIGER 4M detector situated at 6.231 or 0.517 m distance from the sample. 10 images were taken with a typical acquisition time of 250 ms and averaged to increase the signal to noise ratio. The SAXS intensity was normalized using beamline specific procedures and radially averaged using FOXTROT 3.4.1.