Mickaele Bonneau

Christophe Lavenn

Jia-Jia Zheng

Alexandre Legrand

Tomofumi Ogawa

Kunihisa Sugimoto

Francois-Xavier Coudert

Regis Reau

Shigeyoshi Sakaki

Ken-Ichi Otake

Susumu Kitagawa

Tunable acetylene sorption by flexible catenated metal-organic frameworks

The safe storage of flammable gases, such as acetylene, is essential for current industrial purposes. However, the narrow pressure and temperature range required for the industrial use of pure acetylene (100 < P < 200 kPa at 298 K) and its explosive behaviour at higher pressures make its storage and release challenging. Flexible metal-organic frameworks (MOFs) that exhibit a gated adsorption/desorption behaviour -in which guest uptake/release occur above threshold pressures, usually accompanied by framework deformations -have

shown promise as storage adsorbents. Herein, the pressures for gas uptake and release of a series of zinc-based mixed-ligand catenated MOFs were controlled by functionalizing the two ligands as well as changing their ratio. This affects the deformation energy of the framework, which in turn controls the gated behaviour. The materials offer good performances for acetylene storage with a usable capacity of ~ 90 v/v (77 % of the overall amount) at 298 K and under a practical pressure range (100-150 kPa).

The development of new technologies for efficient gas storage at ambient pressure can improve safety and provide new possibilities for gas cylinder design. The usable gas capacity, which differs from the maximum quantity adsorbed, under mild pressure and temperature conditions is another concern for researchers. In contrast to gases such as carbon dioxide, methane, oxygen, and nitrogen, which have wider compression pressure tolerances, pure acetylene gas is challenging to handle and safely store due to its explosive behaviour when compressed over 200 kPa at room temperature. Consequently, in the last century, cylinders have been filled with acetylene dissolved at high pressure (P > 1500 kPa) in a solvent such as acetone. [START_REF] Matsuda | Highly controlled acetylene accommodation in a metal-organic microporous material[END_REF][START_REF]Safetygram 13[END_REF][START_REF]Solvents for Acetylene Filling[END_REF] However, such acetylene commercial set-up could contaminate the acetylene gas. When cylinders are filled with pure acetylene gas at 200 kPa (safe pressure), it only allows the release of 50 % of its total capacity at 100 kPa (limit pressure for the gas to flow spontaneously), keeping 50 % unused, which is not economically useful and potentially dangerous. The creation of porous materials capable of delivering large quantities of pure acetylene in a highly narrow pressure range (100-150 kPa) at room temperature is therefore essential. [START_REF] Bulow | Method and vessel for the storage of gas[END_REF] As a potential solution, porous materials, particularly metal-organic frameworks (MOFs) or porous coordination polymers (PCPs), possess inherent voids that allow the storage and delivery of large amounts of gases. [START_REF] Kapelewski | Record High Hydrogen Storage Capacity in the Metal-Organic Framework Ni2(m-dobdc) at Near-Ambient Temperatures[END_REF][START_REF] Li | Porous Metal-Organic Frameworks: Promising Materials for Methane Storage[END_REF][START_REF] Moghadam | Computer-aided discovery of a metal-organic framework with superior oxygen uptake[END_REF][START_REF] Vaidhyanathan | Direct observation and quantification of CO₂ binding within an amine-functionalized nanoporous solid[END_REF] In contrast to rigid materials showing a Langmuir type I isotherm, flexible MOFs, also called soft porous crystals (SPCs), possess both ordered network and structural transformability, resulting in a sigmoidal S-shape isotherm and higher usable storage capacity (USC) when the gate pressures are in the desired working pressure range (Fig. 1). [START_REF] Horike | Soft porous crystals[END_REF][START_REF] Warren | Shape Selectivity by Guest-Driven Restructuring of a Porous Material[END_REF][START_REF] Chang | Flexible Metal-Organic Frameworks: Recent Advances and Potential Applications[END_REF][START_REF] Schneemann | Flexible metal-organic frameworks[END_REF] The flexibility depends not only on the binding capacity and mobility of unit ligands and metal ions but also on other factors, including the deformation of the entire framework as a result of the guest molecules in the pores. [START_REF] Zhang | Optimized Acetylene/Carbon Dioxide Sorption in a Dynamic Porous Crystal[END_REF][14][START_REF] Jiang | Interpenetration control in metal-organic frameworks for functional applications[END_REF] Strategies using ligand functionalisation have been developed to investigate the behaviour and properties of flexible MOFs. [START_REF] Martí-Gastaldo | Side-chain control of porosity closure in single-and multiplepeptide-based porous materials by cooperative folding[END_REF][START_REF] Taylor | Tuning the Adsorption-Induced Phase Change in the Flexible Metal-Organic Framework Co(bdp)[END_REF][START_REF] Horcajada | How Linker's Modification Controls Swelling Properties of Highly Flexible Iron(III) Dicarboxylates MIL-88[END_REF][19][START_REF] Ramsahye | Influence of the Organic Ligand Functionalization on the Breathing of the Porous Iron Terephthalate Metal Organic Framework Type Material upon Hydrocarbon Adsorption[END_REF][START_REF] Kundu | Functionalization-Induced Breathing Control in Metal-Organic Frameworks for Methane Storage with High Deliverable Capacity[END_REF] Still, none of them focused on gas storage and especially sensitive gas, which mandates strict temperature and pressure conditions. [START_REF] Mason | Methane storage in flexible metal-organic frameworks with intrinsic thermal management[END_REF][START_REF] Zhu | Tuning the Gate-Opening Pressure in a Switching pcu Coordination Network, X-pcu-5-Zn, by Pillar-Ligand Substitution[END_REF][24][25][START_REF] Zhang | Pore Surface Tailored SOD-Type Metal-Organic Zeolites[END_REF][START_REF] Henke | Flexibility and Sorption Selectivity in Rigid Metal-Organic Frameworks: The Impact of Ether-Functionalised Linkers[END_REF] Regarding acetylene storage, rigid and flexible MOFs have demonstrated high adsorption capacities at 298 K and 100 kPa. However, the desorption pressure does not match the acetylene storage requirements for safe handling described above. All the MOFs reported for their high acetylene capacities were recorded at a pressure of 100 kPa, which is below the ideal working pressure range. [START_REF] He | Metal-organic frameworks with potential for energyefficient adsorptive separation of light hydrocarbons[END_REF][START_REF] Duan | A novel methoxy-decorated metal-organic framework exhibiting high acetylene and carbon dioxide storage capacities[END_REF][START_REF] Pang | A porous metal-organic framework with ultrahigh acetylene uptake capacity under ambient conditions[END_REF][START_REF] Cai | An amino-decorated NbO-type metal-organic framework for high C2H2 storage and selective CO2 capture[END_REF][START_REF] Wang | High Working Capacity Acetylene Storage at Ambient Temperature Enabled by a Switching Adsorbent Layered Material[END_REF] Speculative values were also calculated for acetylene adsorption at high pressure (P > 3500 kPa), far above the explosion limit for acetylene. [START_REF] Wang | High Working Capacity Acetylene Storage at Ambient Temperature Enabled by a Switching Adsorbent Layered Material[END_REF] Herein, we demonstrate the optimum pressure range for stable acetylene storage while maintaining high deliverable capacity. [START_REF] Sun | Tuning the gate opening pressure of a flexible doubly interpenetrated metal-organic framework through ligand functionalization[END_REF][34][START_REF] Foo | Functional Hybrid Porous Coordination Polymers[END_REF][START_REF] Lescouet | Engineering MIL-53(Al) flexibility by controlling amino tags[END_REF] To the best of our knowledge, acetylene sorption isotherms on MOFs at a pressure range between 100-150 kPa have not been reported so far.

We used a doubly catenated structure, i.e., interpenetrated or two-fold interlocked, as an SPC candidate; [Zn 2 (bdc) 2 (4,4'-bpy)] (bdc = 1,4-benzenedicarboxylic acid; 4,4'-bpy = 4,4´bipyridine) (Zn-CAT), where the mutual dislocation of the two identical framework motifs can introduce acetylene, displaying a sigmoidal sorption isotherm. The gate-opening and -closing pressures (P go and P gc ) depend on the deformation energy to create space, attributed to the interaction between the two interpenetrating frameworks. [START_REF] Chen | A Microporous Metal-Organic Framework for Gas-Chromatographic Separation of Alkanes[END_REF][START_REF] Gu | Structural-Deformation-Energy-Modulation Strategy in a Soft Porous Coordination Polymer with an Interpenetrated Framework[END_REF] Therefore, the isotherm can be controlled by the steric and electronic properties of the organic linker (L) of the framework. At a certain temperature, T, the isotherm is a function of P and L as A T = F(P, L) (Fig. 1). The introduction of different ratios of functionalised bdc-X linkers (X = -NO 2 or -NH 2 ) allows the optimization of the sigmoidal sorption at temperature storage and 100 kPa release of acetylene.

The strategy of deformation energy control, as devised for acetylene, is also applicable to CO 2 , O 2, and CH 4 gases in which lower filling and release pressures are preferable for the safer use of the gas cylinder.

Results and Discussion

Synthesis and characterisation

A series of Zn-CAT derivatives composed of a mixture of bdc and a substituted bdc-X (X = -NO 2 or -NH 2 ) ligands were synthesised with precise control of the bdc/bdc-X ratio (Figs.

2a-c and Supplementary Fig. 1). [START_REF] Chen | A Microporous Metal-Organic Framework for Gas-Chromatographic Separation of Alkanes[END_REF] The final linker ratio of these [Zn 2 (bdc) 2-a (bdc-X) a (bpy)] materials (with a varying from 0 to 2) was confirmed by 1 H NMR and showed almost no deviation from the linker ratio expected from the introduced mixture (Fig. 2c and Supplementary Figs. 234567891011121314). The samples were designated according to Zn-CAT-(X) n , that is, the functional moiety (X = -NH 2 , -NO 2 ) and molar percent (n = 0-100%) for clarity. All materials were thermally stable up to 623 K, as observed by thermogravimetric analysis (TGA) (Supplementary Figs. 151617). The SEM images of the as-synthesised samples were then analysed and exhibited homogeneity of particle sizes in the micrometre range (Supplementary Figs. 1819). Powder Xray patterns (PXRD), infrared analysis (IR), UV-vis and crystal structures of the as-synthesised and activated samples confirmed the purity of all materials as well as their isostructural nature (Supplementary Figs. 20212223242526and Tables 12). Interpenetrated Zn-CAT shows a dynamic transition from the closed (when the structure is completely evacuated from guests) to the open form (when a specific gas pressure or temperature is applied). [START_REF] Xiang | Exceptionally High Acetylene Uptake in a Microporous Metal-Organic Framework with Open Metal Sites[END_REF] Thus, the gas sorption properties of the Zn-CAT-(X) n series were investigated.

First, Zn-CAT-(X) n samples were confirmed to behave as a solid solution; that is, the ligands were homogeneously distributed within the backbone structures (Supplementary Fig. 27). [START_REF] Fukushima | Modular Design of Domain Assembly in Porous Coordination Polymer Crystals via Reactivity-Directed Crystallization Process[END_REF][START_REF] Lescouet | Homogeneity of flexible metal-organic frameworks containing mixed linkers[END_REF] The influence of the ratio of linker mixture on the P go /P gc pressures was determined from single gas-adsorption measurements performed at different temperatures and pressures (Fig. 3 and Supplementary Figs. 282930313233343536). Acetylene physisorption of pristine Zn-CAT at 273 K showed P go = 44 kPa and P gc = 19 kPa with an excess amount of 2.87 mol.mol -1 of acetylene adsorbed (Fig. 3). [START_REF] Xiang | Exceptionally High Acetylene Uptake in a Microporous Metal-Organic Framework with Open Metal Sites[END_REF] When Zn-CAT-(NH 2 ) n is used, a shift in P go from 48 to 83 kPa was observed as n increased from 5 to 60% (Fig. 3a). In contrast, for Zn-CAT-(NO 2 ) n , the gateopening pressure shifted to a lower pressure when n increased from 5 to 100 % (from P go = 31 kPa to P go = 1 kPa) (Fig. 3b). The plot of the gate-opening and -closing pressures as a function of the molar percentage of amino groups in Zn-CAT-(NH 2 ) n suggests that P go increases exponentially with the amount of bdc-NH 2 incorporated (Fig. 3c). At a higher molar percentage (n > 60%) of bdc-NH 2 , P go was not observed below 100 kPa; however, the value can be extrapolated from the exponential fit of the lower molar percentage data points and later confirmed by high-pressure measurements (Fig. 3 and Supplementary Figs. 3738394041424344). This extrapolation is possible due to the exponential trend observed at a lower temperature (195 K), where all P go values can be determined for high amino loading (Supplementary Figs. 2829).

Additionally, the nitro group exhibited a significant impact on the exponential decay of P go as a function of n (Fig. 3d).

In-situ gas sorption studies

The structure of Zn-CAT, Zn-CAT-(NH 2 ) 100 , and Zn-CAT-(NO 2 ) 100 were also characterized by in situ PXRD under various acetylene pressures at 195 K (Supplementary Fig. 45) to further investigate the effect of bdc substitution and phase transformations. and increase in the volume of the cell, respectively, compared to pristine Zn-CAT, as seen from the shift of the 100 peak. These results were confirmed by the crystal structure obtained from Xray diffraction experiments (Supplementary Table 1-2). [START_REF] Gu | Structural-Deformation-Energy-Modulation Strategy in a Soft Porous Coordination Polymer with an Interpenetrated Framework[END_REF] The cell volume of activated Zn-CAT-(NO 2 ) 100 (715.9 Å 3 ) is higher than that of Zn-CAT (703.6 Å 3 ) and Zn-CAT-(NH 2 ) 100 (701.9 Å 3 ), which can be attributed to i) the softer interaction of -NO 2 groups with the metal nodes and ii) the steric hindrance of the -NO 2 moiety, which further separates the aromatic part of the ligand. Therefore, the stabilisation of the structure through π-π interactions is reduced, as observed for pristine MOF. [START_REF] Ghoufi | Physics Behind the Guest-Assisted Structural Transitions of a Porous Metal-Organic Framework Material[END_REF] For Zn-CAT-(NH 2 ) 100 , the higher pressure needed for the gate opening to occur is explained by the initially more contracted structure owing to interactions between -NH 2 moieties and oxygen atoms from the metal paddlewheel. [START_REF] Gu | Structural-Deformation-Energy-Modulation Strategy in a Soft Porous Coordination Polymer with an Interpenetrated Framework[END_REF] Finally, all MOF structures are entirely open in the higher pressure range, and they display similar diffraction patterns (Supplementary Figs. 45d-f). Considering that the open and closed forms of each material are isostructural, the difference in the gate-opening pressure is attributed to the functionalised bdc-X linker.

Theoretical investigations of the gate-opening behaviour

To rationalise the effect of linker on the thermodynamics of the gate-opening process, we applied theoretical thermodynamic models to the adsorption isotherms of Zn-CAT and its derivatives (Supplementary information). [START_REF] Coudert | Thermodynamics of Guest-Induced Structural Transitions in Hybrid Organic-Inorganic Frameworks[END_REF][START_REF] Bousquet | Free energy landscapes for the thermodynamic understanding of adsorption-induced deformations and structural transitions in porous materials[END_REF] Researchers have previously shown that a simple two-phase model provides a good description of gate opening in interpenetrated frameworks.

The gate opening is an adsorption-driven first-order phase transition from a nonporous phase to a microporous phase. Here, the -open phase‖ adsorption is described by a Langmuir-type fit of isotherms for all materials at 273 K. The free energy difference (ΔF) associated with the gateopening process have been calculated based on the adsorption and desorption isotherms (see Supplementary Information for the equations). This value indicates that Zn-CAT exhibits a relatively small free energy difference (ΔF = 5.2 kJ/mol) between the two phases (here, all energies are reported per unit cell). This value is slightly larger, but of a similar order of magnitude, than those of other gate-opening materials, such as [Cu(4,4′-bpy)(dhbc) 2 ]H 2 O, which has a ΔF of approximately 4 kJ/mol). [START_REF] Kundu | Functionalization-Induced Breathing Control in Metal-Organic Frameworks for Methane Storage with High Deliverable Capacity[END_REF]45 The relatively small free energy difference between the two phases explains why the material readily opens upon adsorption of small guests.

However, the introduction of -NO 2 and -NH 2 substituents leads to a decrease in ΔF (see Supplementary Table 3). The impact of -NH 2 was determined to be smaller than that of -NO 2 .

The more pronounced decrease in ΔF of Zn-CAT-(NO 2 ) 100 is attributed to the repulsive interaction of the -NO 2 groups in the closed phase. Owing to the smaller energetic cost for gate opening, the opening is easier and occurs at lower gas pressures. This is particularly noticeable in the Zn-CAT-(NO 2 ) 100 phase, where the free energy difference is almost zero, which implies that the material is truly bistable. Minute amounts of adsorbate trigger the gate-opening transition.

This effect was more complex when -NH 2 groups are present. The introduction of -NH 2 groups in Zn-CAT decreased ΔF but did not lead to a decrease in the gate-opening (and closing) pressure. The effect is smaller than that of the -NO 2 groups because -NH 2 is a less bulky substituent. Therefore, the change in ΔF is not the dominant factor, and the driving force is the evolution of affinity in the open phase, discussed in detail later in the paper.

These results suggest the possibility of purposefully controlling the gate-opening and closing pressure by introducing different ratios of functional groups. Additionally, the dependence of gate-opening and closing on the molar ratio of amino or nitro functional groups is not unique to acetylene. Still, it is also applicable to CO 2 , O 2 , and CH 4 , demonstrating the method's versatility (Supplementary Figs. 30313233343536).

Notably, the total adsorbed capacity does not considerably change for all Zn-CAT-(NH 2 ) n compared to that of pristine Zn-CAT for acetylene gas at 273 K (Fig. 3c). The total adsorption capacity at higher loading in the Zn-CAT-(NH 2 ) n series was recorded at a lower temperature (T = 195 K), demonstrating that the total capacity did not decrease for all Zn-CAT-(NH 2 ) n series (Supplementary Fig. 2829). Conversely, the capacity decreases up to 15% for Zn-CAT-(NO 2 ) 100 (from 2.73 mol.mol -1 to 2.32 mol.mol -1 ). This difference is expected due to the bulkier nitro moiety (VDW surface area bdc-NO 2 = 249.2 Å) compared to the amino moiety (VDW surface area bdc-NH 2 = 223.3 Å).

To elucidate the origin of the counter-intuitive observation that the gate-opening pressure decreases after introducing a -NO 2 46). However, E def differs considerably and is of the same order as that of P go . The larger E def for Zn-CAT-(NH 2 ) 100 indicates that the introduction of the -NH 2 group into Zn-CAT results in a less flexible framework; thus, the gate-opening pressure increases. The less flexible framework occurs due to the hydrogen-bonding interactions of the -NH 2 group with the neighbouring bpy and carboxylate ligands (Supplementary Fig. 47a). The hydrogen-bonding interaction between the -NH 2 group and the bpy ligand weakened after C 2 H 2 adsorption, confirmed by the increase in the NH 2 -bpy distance (Supplementary Fig. 47a-b). For Zn-CAT-(NO 2 ) 100 , the introduced -NO 2 group results in a repulsive interaction with the carboxylate ligand (Supplementary Fig. 47c) in the guest-free structure. Consequently, the framework flexibility is increased to reduce the repulsive interaction, explaining the smaller E def of Zn-CAT-(NO 2 ) 100 than that of pristine Zn-CAT, hence decreasing the gate-opening pressure. Indeed, the -NO 2 groups do not exist in a repulsive position after C 2 H 2 adsorption (Supplementary Fig. 47d). These results indicate that introducing substituents into parent MOFs can tune the flexibility of MOFs and readily change the E def of the framework upon gas adsorption, making it possible to control the gate-opening gas adsorption, as observed from the experimental results.

Optimal conditions

Owing to the insight into the structural dynamics of our system obtained from the combination of various characterisation techniques and theoretical calculations, the amino group efficiently adjusted both P go and P gc for practical use at different temperatures (Fig. 4). Due to the limitation of actual physisorption instruments to 100 kPa pressure, we built a system capable of recording the acetylene physisorption at a higher pressure than 100 kPa, which was not been seen before (Supplementary Fig. 37). The optimal compressed condition must not exceed 200 kPa due to the acetylene explosive behavior above this pressure. The discharge gas should happen above 100 kPa because the gas would not flow automatically below this pressure. Thus, it is possible, with this solid-solution system, to fill at low temperature to achieve high adsorption amount at low gate opening pressure while releasing at room temperature above 100 kPa. The optimal conditions for the safe handling and use of compressed pure acetylene gas were tested for Zn-CAT-(NH 2 ) 60 with a maximum released gas of 66 v/v at 100 kPa and 310 K (which is 67 % of the cylinder content) and Zn-CAT-(NH 2 ) 100 , which allowed the release of 77 % of the cylinder content (~ 90 v/v) at 100 kPa and 298 K (Fig. 4 and Supplementary Fig. 44).

Our system has three main advantages compared to the state-of-the-art materials. First, when comparing the usable storage capacity of acetylene for Zn-CAT-(NH 2 ) 100 with the reported MOFs, our material provides an USC of ~ 90 v/v at 150 kPa and degassing at 100 kPa, both at 298 K, which is the highest among the reported MOFs (Fig. 4c and Supplementary Table 5). Second, Zn-CAT-(NH 2 ) 100 achieves 77% of the amount released at 100 kPa, which is a crucial indicator of safety, as the unused or trapped acetylene is potentially dangerous (Fig. 4c).

Compared to the MOFs reported for their high adsorption capacity, all retained at least 50 % of acetylene in their pores. Third, the recyclability tests did not show any change in the uptake or release capacity, which is in contrast with the conventional zeolite-type adsorbents that cannot be reused without strong heating reactivation (Supplementary Fig. 4849).

For industrial development, we demonstrated that our flexible MOF adsorbents exhibit the following practical advantages for storing acetylene compared to the current solvent compression method (Fig. 4d). High acetylene purity is achieved as no solvents that can degrade or contaminate are present. The functionalized Zn-CAT-(X) n can compress large quantities of acetylene (~ 117 v/v) at less than 200 kPa, allowing the development of smaller gas containers at low pressure with filling controlled via a pressure gauge and not by mass. Compared to the solvent-free compression method, the adsorbed gas can be compressed in larger quantities in MOF pores (37-fold increase, Fig. 4d). Finally, the release pressure condition can be easily manageable with the % ratio of the amino group. Indeed, geographically different locations have different climates and thus different temperature regimes in which to operate the acetylene containers. Therefore, the tailor-designed of Zn-CAT-(X) n solid solution offers tunable uptake/release pressures at a desired temperature for various gases.

Conclusion

Herein, we demonstrated the design of a functional MOF with practical gate pressures via a modification of the molecular framework. The designed system has a mechanism by which the substituent exchange of the ligand controls the gate pressures of the target guest molecule by modulating the deformation energy of the interpenetrated framework. In the case of acetylene storage, these conditions were met using Zn-CAT-(NH 2 ) 60 at 310 K and Zn-CAT-(NH 2 ) 100 at 298 K, which allowed the release of more than 66 % of acetylene at 100 kPa. This method is versatile, as it uses readily available MOFs and achieves the required performance with only minor modifications. These results open the way for the application of flexible MOFs to the practical storage of targeted gases from the discovery of cooperative adsorption phenomena. Single gas adsorption isotherms. MOF samples were prior activated through thermal activation at 423 K for 12 h before measurement. The adsorption isotherms were measured with BELSORP-CRYO (C 2 H 2 and CO 2 ) and BELSORP-MAX (C 2 H 2, CO 2 , CH 4 and O 2 ) volumetric adsorption equipment from Bel Japan. Targeted relative pressures in the range of 0.01-100 kPa were defined and limits of excess and allowance amount were set to 5 and 10 cm 3 .g -1 , respectively. Equilibrium conditions for each point were: 0.03 % pressure change within 300 s.

The dead volume was determined using helium gas.

Gate opening and closing pressure and total capacity determination

The gate pressure points were determined at the inflection point of the adsorption and desorption isotherms. The total capacity reported are the maximum amount of gas adsorbed by the material.

In-situ PXRD coupled with gas sorption measurements. The measurements were performed on BELSORP-18PLUS (MicrotracBEL, Japan, Corp.) automated volumetric sorption analyzers, equipped with cryostat temperature controllers. The in-situ PXRD/adsorption measurements were carried out using a Rigaku SmartLab with Cu-Kα radiation connected to BELSORP-18PLUS volumetric adsorption equipment. Those apparatuses were synchronized with each other and each PXRD pattern was obtained at each point of the sorption isotherms. The samples were prior activated through external thermal activation at 423 K for 12 h and further 2 h internal activation at 423 K before measurement.

Acetylene high pressure storage test set-up measurements. The experimental set-up used for acetylene storage measurements is shown in Supplementary Fig. 37. The gas manifold consisted of two lines fitted with mass flow controller. One line is used to feed an inert gas, in our case, helium gas is used as reference gas before each experiment and as gas purge after each acetylene measurement. The other line, contains pure acetylene gas. In a typical experiment, 10 g of

  Increasing the gas pressure resulted in a transition from a close to open phase structure, which occurred at different pressures for the three samples. The gate-opening pressures, which correspond to the phase transition observed on the PXRD patterns, were observed at P go = 0.04, 0.21, and 1.16 kPa for Zn-CAT-(NO 2 ) 100 , Zn-CAT, and Zn-CAT-(NH 2 ) 100 , respectively. The difference in the parent structure can explain this difference. Below P go , the PXRD patterns of activated MOFs (closed form) for Zn-CAT-(NH 2 ) 100 and Zn-CAT-(NO 2 ) 100 show a decrease

Fig. 1 |

 1 Fig. 1 | Adsorption S-shape isotherms of flexible MOF depending on pressure and organic linker.A type I adsorption isotherm of rigid MOF is shown in gray for comparison. The functionalization of Zn-CAT with -NO 2 or -NH 2 groups leads to a shift in the gate pressure to the lower pressure (blue circles) or higher pressure range (red circles), respectively. The orange area represents the targeted practical range (100-150 kPa) for safe acetylene uses.

Fig. 2 |Fig. 3 |

 23 Fig. 2 | Presentation of the Zn-CAT-(X) n derivatives as SPC candidate. a) Synthetic scheme of Zn-CAT-(X) n materials. b) (Left) crystal structure of activated MOF-508 along the b axis and (right) crystal structure of activated Zn-CAT along the c axis showing a pillar ligand connecting the 2D layers. c) Molar percentage of the bdc-X linker within the material obtained as a function of the molar percentage of bdc-X introduced in the reaction. The obtained ratio was estimated from the 1 H NMR spectra of digested sample available in the Supplementary Fig. 2-14.

Fig. 4 |

 4 Fig. 4 | Efficiency of Zn-CAT-(X) n for acetylene storage. Obtained (filled symbols) and predicted (empty symbols) gate a) opening and b) closing pressures of Zn-CAT and Zn-CAT-

& 2 .

 2 Crystallographic data for as-synthesized and activated Zn-CAT-(NO 2 ) 100 structures have been deposited in the Cambridge Structural Database with CCDC numbers 2036574 -2036575.respectively.

-CAT-(NH 2 ) 100 and Zn-CAT-(NO 2 ) 100

  group into Zn-CAT but increases after introducing an -NH 2 group, quantum chemical calculations were carried out (see Supplementary Information for computational details). For clarity, only C 2 H 2 adsorption onto pristine Zn-CAT and fully substituted MOFs, that is, Zn, were considered. As shown in Table 4 of the Supplementary Information, the binding energy (BE) of C 2 H 2 with MOFs increased from Zn-CAT-(NH 2 ) 100 (-9.3 kcal/mol) > Zn-CAT (-9.9 kcal/mol) > Zn

-CAT- (NO 2 ) 100

  (-10.6 kcal/mol), where a more negative BE value indicates stronger C 2 H 2 adsorption with increased energy stabilisation. The results suggest that C 2 H 2 adsorption begins to occur at a higher pressure for Zn-CAT-(NH 2 ) 100 but at lower pressure for Zn-CAT-(NO 2 ) 100 , consistent with the experimental results. The BE was decomposed into the interaction energy (E int ) between the MOF and C 2 H 2 molecules and that of the adsorbed C 2 H 2 molecules and the deformation energy (E def ) of the MOF structure by C 2 H 2 adsorption. The E int values were similar for C 2 H 2 adsorption onto Zn

- CAT, Zn-CAT-(NH 2 ) 100 , and Zn-CAT-(NO 2 ) 100

  , which is reasonable because the adsorption positions of C 2 H 2 in these MOFs are similar (Supplementary Fig.

Acknowledgements

This work was supported by Air Liquide via the 2016 Air Liquide Scientific Challenge, a KAKENHI Grant-in-Aid for Specially Promoted Research (JP25000007), Scientific Research (S) (JP18H05262), and Early-Career Scientists (JP19K15584) from the Japan Society of the Promotion of Science (JSPS). Synchrotron XRD measurements were performed at the Japan Synchrotron Radiation Institute (JASRI), Super Photon ring-8 Gev (Spring-8) (Proposal numbers 2018B1820 and 2019A1136). We acknowledge iCeMS Analysis Centre for access to analytical facilities.

We are grateful to CNRS-Kyoto LIA "SMOLAB". In addition, we thank Air Liquide Japan, Dr.

Patrick Ginet and Laurent

Prost as well as the technical staff for advices and experimental assistance. Kitaura, R., Fujimoto, K., Noro, S.-i., Kondo, M. & Kitagawa, S. A Pillared-Layer Coordination Polymer Network Displaying Hysteretic Sorption: [Cu2(pzdc)2(dpyg)]n (pzdc= Pyrazine-2,3-dicarboxylate; dpyg=1,2-Di(4-pyridyl)glycol). Angew. Chem. Int. Ed. 41, 133-135, doi:10.1002/1521-3773(20020104)41:1<133::aid-anie133>3.0.co;2-r (2002).

adsorbent powder was activated under vacuum at 423 K in a separate oven for 12 h. The vessel is filled with helium and plugged in the set-up. Before starting each experiment, the column is evacuated and heated at 423 K for 2 h before the acetylene filling at a flow rate of 50 cm 3 .min -1 .

The acetylene filling process was performed at low temperature (273 K or 243 K). After reaching the maximum loading, the vessel was slowly warmed to 290 K, 298 K or 310 K. The acetylene excess amount was released during the heating process. After filling, the desorption is made by vacuum pump until complete release of the gas.

Data availability X-ray crystallographic data have been deposited at the Cambridge Crystallographic Data Centre (http://www.ccdc.cam.ac.uk/) with reference numbers CCDC 2036574-2036575. A copy of the data can be obtained free of charge via https://www.ccdc.cam.ac.uk/structures/. All other data supporting the findings of this study are available within the article and its Supplementary Information. Data are also available from the corresponding author upon reasonable request.

Author Contributions

Competing Interests statement

The authors declare no competing financial interests. The mixture was kept at 373.15 K for 24 h. After cooling down to room temperature, the resulting precipitated is collected via centrifugation and washed three time with DMF (3 x 40 mL) and three times with EtOH (3 x 40 mL). The material is dried under vacuum at 323.15 K for 12 h to give a white/yellow powder in ~ 90 to 98 % yields (6 to 6.9 g).


Kulangara Sandeep

Kripa Joseph

Jérôme Gautier

Kalaivanan Nagarajan

Meleppatt Sujith

K George Thomas

Thomas W Ebbesen
email: ebbesen@unistra.fr

Manipulating the Self-assembly of Phenyleneethynylenes under Vibrational Strong Coupling

The chemical and physical properties of molecules and materials are known to be modified significantly under vibrational strong coupling (VSC). In order to gain insight into the effects of VSC on - interactions involved in molecular self-assembly, themselves sensitive to vacuum electromagnetic field fluctuations, the aggregation of two structural isomers (linear and V-shaped) of phenyleneethynylene under cooperative coupling was investigated. By coupling the aromatic C=C stretching band, the assembly of one of the molecules results in the formation of spheres as opposed to flakes under normal conditions. As a consequence, the electronic absorption and emission spectra of the self-assembled structures are also modified significantly. The VSC induced changes depend not only on the type of vibration that is coupled but also on the symmetry of the phenyleneethynylene isomer. These results confirm that VSC can be used to drive molecular assemblies to new structural minima and thereby provide a new tool for supramolecular chemistry. TOC

Modulation of the self-assembly of organic molecules to form particular supramolecular structures has enormous importance in chemistry and biology. [START_REF] Lehn | Toward Self-Organization and Complex Matter[END_REF][START_REF] Webber | Supramolecular Biomaterials[END_REF][START_REF] Aida | Functional Supramolecular Polymers[END_REF] Typically, the equilibrium resulting from weak non-covalent interactions leads to the formation of a single thermodynamically favored, supramolecular self-assembly at the nano/micrometer scale. [START_REF] Tidhar | Pathway-Dependent Self-Assembly of Perylene Diimide/Peptide Conjugates in Aqueous Medium[END_REF][START_REF] Cravotto | Molecular Self-assembly and Patterning Induced by Sound Waves. The Case of Gelation[END_REF][START_REF] Fernández | Morphological Changes in the Self-assembly of a Radial Oligo-phenylene Ethynylene Amphiphilic System[END_REF] Factors such as the polarity of the solvent, temperature, light, sound, mechanical force, the concentration of monomer, and electric field are known to perturb such weak noncovalent interactions and thereby the equilibria and the structure of the self-assembled product. [START_REF] Cravotto | Molecular Self-assembly and Patterning Induced by Sound Waves. The Case of Gelation[END_REF][START_REF] Fernández | Morphological Changes in the Self-assembly of a Radial Oligo-phenylene Ethynylene Amphiphilic System[END_REF][START_REF] Chen | Self-assembled π-stacks of Functional Dyes in Solution: Structural and Thermodynamic Features[END_REF][START_REF] Babu | Functional π-Gelators and Their Applications[END_REF][START_REF] Ajayaghosh | First Phenylenevinylene Based Organogels: Self-Assembled Nanostructures via Cooperative Hydrogen Bonding and π-Stacking[END_REF][START_REF] Peng | Dynamic Surface-assisted Assembly Behaviours Mediated by External Stimuli[END_REF][START_REF] Zhao | Dynamic Control of Chirality and Self-assembly of Double-stranded Helicates with Light[END_REF][START_REF] Adhikari | Light-induced Unfolding and Refolding of Supramolecular Polymer Nanofibres[END_REF][START_REF] Guo | Long-range Mechanical Force Enables Self-assembly of Epithelial Tubular Patterns[END_REF] Very recently, it was reported that light-matter strong coupling can also induce changes in intermolecular interactions affecting the self-assembly of a polymer and the polymorphism of a metal-organic framework. [START_REF] Hirai | Selective Crystallization via Vibrational Strong Coupling[END_REF][START_REF] Joseph | Supramolecular Assembly of Conjugated Polymers under Vibrational Strong Coupling[END_REF] Light-matter strong coupling has attracted much attention over the past decade as a means to modify a variety of material properties such as chemical reactivity, transport, ferromagnetism, superconductivity, and non-linear optical response. In particular, vibrational strong coupling (VSC) has been found to have a very large impact on the chemical and physical properties of molecules. [START_REF] Garcia-Vidal | Manipulating Matter by Strong Coupling to Vacuum Fields[END_REF][START_REF] Nagarajan | Chemistry under Vibrational Strong Coupling[END_REF] VSC alters the potential energy surfaces which in turn modify the chemical reaction kinetics and yields, and symmetry has been found to play a fundamental role. [START_REF] Pang | On the Role of Symmetry in Vibrational Strong Coupling: The Case of Charge-Transfer Complexation[END_REF][START_REF] Sau | Modifying Woodward-Hoffmann Stereoselectivity Under Vibrational Strong Coupling[END_REF] Modification of the potential energy surfaces, as a result of VSC, can also lead to changes in the intermolecular interactions. [START_REF] Hirai | Selective Crystallization via Vibrational Strong Coupling[END_REF][START_REF] Joseph | Supramolecular Assembly of Conjugated Polymers under Vibrational Strong Coupling[END_REF][START_REF] Haugland | Intermolecular Interactions in Optical Cavities: An Ab Initio QED Study[END_REF] In our earlier work, VSC was shown to transform a conjugated polymer gel into flakes. [START_REF] Joseph | Supramolecular Assembly of Conjugated Polymers under Vibrational Strong Coupling[END_REF] However, the modification of intermolecular interactions, followed by the selfassembly of simple organic molecules under VSC, has not yet been reported. To investigate such possibility, we have studied two structural isomers of phenyleneethynylene (PE) molecules: linear and V-shaped isomers having different polarity. PE type molecules are known to aggregate through - stacking and CH- interactions. [START_REF] Kar | Emergence of Chiroptical Properties in Molecular Assemblies of Phenyleneethynylenes: The Role of Quasidegenerate Excitations[END_REF] Interestingly, VSC has a profound effect on the self-assembly of V-shaped isomer but not on the other, as presented below.

The PE molecules (PE1 and PE2) were synthesized by palladium-catalyzed Heck-Cassar-Sonogashira-Hagihara cross-coupling reactions (Scheme 1). [START_REF] Kar | Emergence of Chiroptical Properties in Molecular Assemblies of Phenyleneethynylenes: The Role of Quasidegenerate Excitations[END_REF][START_REF] Bunz | Poly(aryleneethynylene)s: Syntheses, Properties, Structures, and Applications[END_REF] In this synthesis, diiodobenzene is allowed to react with two equivalents of phenylacetylene in the presence of Pd(PPh3)2Cl2, CuI, and triethylamine at room temperature under an inert atmosphere. [START_REF] Bunz | Poly(aryleneethynylene)s: Syntheses, Properties, Structures, and Applications[END_REF] Details of the synthesis and characterization ( 1 H, [START_REF] Guo | Long-range Mechanical Force Enables Self-assembly of Epithelial Tubular Patterns[END_REF] C NMR, m/z, and elemental analysis) of the PE are given in the Supporting Information (SI).

Scheme 1: Synthesis of PE molecules: (i) Pd(PPh3)2Cl2, CuI, triethylamine, and tetrahydrofuran.

The linear isomer is labeled as PE1 and the V-shaped one as PE2.

VSC experiments were carried out using a microfluidic Fabry-Perot (FP) cavity. The cavity was assembled using two parallel Au mirrors separated with an 8 m Mylar spacer. [START_REF] Nagarajan | Chemistry under Vibrational Strong Coupling[END_REF] The mirrors were prepared by sputtering 10 nm Au onto the surface of CaF2 windows. [START_REF] Nagarajan | Chemistry under Vibrational Strong Coupling[END_REF] To prevent the direct interactions between the PE molecules and the Au mirrors, 100 nm of polyvinyl alcohol (PVA) was deposited on the Au surface by spin-coating to act as an insulating layer. Details of the fabrication of FP cavity are given in the SI (see also url:

https://seafile.unistra.fr/d/7bb78e5a4607424f94b5/). FP cavities will produce many resonant modes separated by fixed energy intervals (the free spectral range or FSR). [START_REF] Nagarajan | Chemistry under Vibrational Strong Coupling[END_REF] To achieve VSC, a cavity mode is tuned to the molecular vibration of choice by simply compressing the Mylar spacer between the Au mirrors by adjusting the screws holding the setup together. In a typical experiment, a large number N of molecules are coupled to a resonant optical cavity mode as illustrated in Figure 1. This results in the formation of two bright vibro-polaritonic states, P+ and P-and N-1 dark states (DS). [START_REF] Nagarajan | Chemistry under Vibrational Strong Coupling[END_REF][START_REF] Scholes | Entropy Reorders Polariton States[END_REF] The P+ and P-states are separated by the Rabi splitting energy (ℏΩ 𝑣𝑅 ). [START_REF] Scholes | Entropy Reorders Polariton States[END_REF] The VSC occurs even in the dark because the coupling involves the zero-point energy of both the optical mode and the vibrational transition.

C=C -PE2 -Toluene

The vibrational spectrum of toluene in a cell (i.e., non-cavity made by assembling of two CaF2 windows coated with PVA separated by a 8 M Mylar spacer without the Au mirror coating)

was recorded and compared with the infrared (IR) spectrum of the PE molecules (Figure 2A,B).

The C=C stretching frequency of the toluene and the PE molecules are very close at ca.1603 cm - 1 , which enables cooperative coupling of toluene and PE. In other words, toluene is strongly coupled and by intermolecular interactions in the solvation shell, it drives the VSC of the resonant PE molecules. [START_REF] Lather | Improving Enzyme Catalytic Efficiency by Co-operative Vibrational Strong Coupling of Water[END_REF][START_REF] Hagenmüller | Cavity-Enhanced Transport of Charge[END_REF] The concentration of the PE molecules was 1 mM in toluene and thus, the absorbance of neat toluene (ca. 9 M) completely dominates at 1603 cm -1 . The cavity modes have an FSR 336 cm -1 as shown by the red curve in Figure 2C. The strong coupling was achieved with the 5 th cavity mode of the FP cavity at this wavenumber (Figure 2D). The Rabi splitting was found to be 24 cm -1 which is larger than the full width at half maximum (FWHM) of the cavity modes (20 cm -1 ) and the vibrational transitions (7 and 9 cm -1 , toluene and PE respectively), thereby meeting the necessary conditions for VSC. As a control experiment, we made a perfectly offresonance cavity, where none of the vibrational transitions of toluene and PE are in resonance with the optical modes of the cavity (Figure 2C). As a result of the aggregation, in both cases, we observed a decrease in the absorbance with time. Interestingly, a small but significant difference in the absorption spectrum of the self-assembled structures is found under VSC. It is well known in the literature that the different types of aggregate packing result in a change in electronic absorption and emission properties in such conjugated molecules. [START_REF] Kar | Emergence of Chiroptical Properties in Molecular Assemblies of Phenyleneethynylenes: The Role of Quasidegenerate Excitations[END_REF][START_REF] Antina | Effect of ms-Substitution on Aggregation Behavior and Spectroscopic Properties of[END_REF] Thus, we also checked the fluorescence of the different samples under on-resonance and off-resonance conditions and compared it with the emission spectrum of the monomer (Figure 3B,D). As can be seen, VSC induces drastic changes in the fluorescence spectrum of PE2, confirming that the self-assembled structure has a different packing under strong coupling. It should be noted that the fluorescence spectrum is not modified by the thin Au film in the present experimental conditions which is why the off-resonance spectrum (Figure 3D) is identical to the emission outside a cavity (Figure S3C). Unlike the fluorescence spectrum, the absorption spectrum does not change much in the long wavelength region (Figure 3). It suggests that the modified emission might come from an excited state dimer or higher order assembly. Indeed, as shown below, the packing formed under on-resonance condition is very different as can be seen from electron microscopy images. It should be noted that PE1 shows no changes under identical VSC conditions as can be seen in the SI (Figure S1).

Next, scanning electron microscopy (SEM) was used to visualize the structural changes induced by VSC (Figure 4). Interestingly under cooperative VSC of PE2, mostly spherical structures are formed (Figure 4A,B) as is clearly visible in the sample whereas in the absence of strong coupling, the aggregation of PE2 resulted in the formation of 2D amorphous flakes (Figure 4 C,D). More example of such SEM images from different runs are given in the SI (Figure S2).

Interestingly, the spherical structures are very large, ranging from 2 to 20 m in diameter, and sparse, probably due to few nucleation events. They consist of an outer wall and a dense core which scatters brightly in the SEM image, and these are separated by a dark zone suggesting much less material present. We verified that there are no chemical modifications in these aggregates by dissolving them and recovering the monomer mass-spectrum. The emission of the dissolved material also recovers to the original monomeric spectrum. As a control experiment, the aggregation experiments in a cell under similar conditions was also performed. The absorption, emission, and the structure of the aggregates are identical to that of off-resonance conditions as shown in the SI (Figure S3). All the above SEM images were taken with dry samples after opening the holder. It was checked that before opening, spherical structures were already present in the solution by using optical microscopy. An example of an optical image is given in the SI (Figure S4). We also analyzed the kinetics of aggregation carefully which involves a latency period after which the spectral changes become very significant (Figure 3) and follow a pseudo-first order E rate. This process under VSC is found to be slightly faster (0.39 min -1 ) than that of the offresonance (0.35 min -1 ) condition but not enough to draw any conclusions (SI, Figure S5).

Obviously, the aggregation process is complex and VSC might not affect significantly the rate limiting step even if it modifies the intermolecular interactions. Nevertheless, the slightly faster rate of aggregation under VSC will favor the formation of isotropic structures such as those observed. In another experiment, the aromatic C-H stretching band at 3027 cm -1 of PE2 was also cooperatively coupled with toluene (see Figure S6 in the SI) but in this case, VSC had no effect on the aggregation, resulting in the formation of amorphous flakes as in the absence of strong coupling. Finally, as shown in SI, cooperative VSC of PE1, the linear isomer, resulted in no change in the spectroscopic data nor in the structural features of the aggregates.

Considering all the above results, it is clear that the modifications induced by VSC are sensitive to both the type of vibration that is coupled and to the structural features of the isomers.

PE1 and PE2 have different symmetry and as a consequence the polarity of PE2 (C2v point group) is higher than the centrosymmetric PE1 (𝐷 ∝ℎ ) which should make it more sensitive to any changes in the polarity or polarizability induced by VSC on the solvent and/or the solute. [START_REF] Haugland | Intermolecular Interactions in Optical Cavities: An Ab Initio QED Study[END_REF]48 After all, symmetry is clearly a key parameter that influences the effects of VSC. [START_REF] Pang | On the Role of Symmetry in Vibrational Strong Coupling: The Case of Charge-Transfer Complexation[END_REF][START_REF] Sau | Modifying Woodward-Hoffmann Stereoselectivity Under Vibrational Strong Coupling[END_REF] The fact that coupling the C-H stretching mode has no observable consequences while the C=C band of PE2 does strongly influence the aggregation process, is also reasonable when one considers the role of - interactions in the self-assembly of such aromatic molecules. Furthermore, - interactions are due to London-type dispersive forces that are sensitive to the electromagnetic fluctuations that pervade the universe and which are involved in the strong coupling process with the cavity mode.

Finally, it should be recalled that the cooperative coupling is a collective phenomenon that leads to the formation of collective states that potentially extend over the volume of the optical mode. [START_REF] Garcia-Vidal | Manipulating Matter by Strong Coupling to Vacuum Fields[END_REF][START_REF] Nagarajan | Chemistry under Vibrational Strong Coupling[END_REF] This might also favor the self-assembly process.

In conclusion, the effect of VSC on modification of the self-assembly of aromatic molecules reveal that strong coupling can perturb the - interactions and thereby drive the system to a different structural minimum in the potential energy landscape. This confirms that VSC is a new tool to control self-assembly of molecules and more broadly to manipulate supramolecular chemistry.

ASSOCIATED CONTENT

Supporting Information

The Supporting Information is available free of charge at https://pubs.acs.org/doi/XXXX Synthesis and characterization of PE1 and PE2, fabrication of the FP cavity, cavity tuning, self-assembly of PE1, SEM images of the PE2 aggregates, cell experiments, optical microscopy images, kinetics of self-assembly, and the self-assembly of PE2 under cooperative coupling of C-H bonds.

Figure 1 :

 1 Figure 1: Schematic representation of the generation of polaritonic states, P+, P-, DS (left panel)

Figure 2 :

 2 Figure 2: (A,B) Fourier transform infrared (FTIR) spectrum of toluene (A, blue trace) taken in a

Figure 3 :

 3 Figure 3: (A-D) Spectroscopic investigations of the self-assembly of PE2 molecules (1 mM in

Figure 4 :

 4 Figure 4: SEM images of the structures formed after the self-assembly of PE2 under VSC (A,B)

Acknowledgements

We acknowledge support from the International Center for Frontier Research in Chemistry (icFRC, Strasbourg), the ANR Equipex Union (ANR-10-EQPX-52-01), and CSC (ANR-10-LABX-0026 CSC) within the Investissement d'Avenir program ANR-10-IDEX-0002-02. T.W.E. acknowledges the funding support of the ERC (project no. 788482 MOLUSC), QuantERA project RouTe, and the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 811284 (UHMob).

AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

BODIPY Dyes in


Kuidong Wang
email: kuidong.wang@unistra.fr

Ai-Yin Liu

Hui-Hsin Hsiao

Cyriaque Genet

Thomas Ebbesen
email: ebbesen@unistra.fr

Large Optical Nonlinearity of Dielectric Nanocavity-Assisted Mie Resonances Strongly Coupled to an Epsilon-near-zero Mode

Keywords: strong coupling, dielectric nanocavity, Mie resonances, epsilon-near-zero mode, third-order optical nonlinearity

Strong coupling provides a powerful way to modify the nonlinear optical properties of materials.

The coupling strength of the state-of-the-art strongly coupled systems are restricted by weak field confinement of the cavity, which limits the enhancement of the optical nonlinearity. Here, we investigate a strong coupling between Mie resonant modes of high-index dielectric nanocavities and an epsilon-near-zero mode of an ultrathin indium tin oxide film and obtain an anti-crossing splitting of 220 meV. Static nonlinear optical measurements reveal a large enhancement in the intensity-independent effective optical nonlinear coefficients, reaching more than three orders of magnitude at the coupled resonance. In addition, we observe a transient response of ~300 fs for the coupled system. The ultrafast and large optical nonlinear coefficients presented here offer a new route towards strong coupling-assisted high-speed photonics.

INTRODUCTION

Strong coupling provides a new platform for manipulating the nonlinear optical properties of materials. In such systems, an optical mode is coupled resonantly to a given optical transition (optical mode, exciton or vibration), they both exchange energy faster than each dissipative and incoherent processes. [START_REF] Garcia-Vidal | Manipulating matter by strong coupling to vacuum fields[END_REF][START_REF] Törmä | Strong coupling between surface plasmon polaritons and emitters: a review[END_REF][START_REF] Ebbesen | Hybrid Light-Matter States in a Molecular and Material Science Perspective[END_REF][START_REF] Christ | Waveguide-Plasmon Polaritons: Strong Coupling of Photonic and Electronic Resonances in a Metallic Photonic Crystal Slab[END_REF] Strong coupling can efficiently enhance many nonlinear optical phenomena of the coupled material, such as second-harmonic generation, [START_REF] Chervy | High-Efficiency Second-Harmonic Generation from Hybrid Light-Matter States[END_REF][START_REF] Chervy | Room Temperature Chiral Coupling of Valley Excitons with Spin-Momentum Locked Surface Plasmons[END_REF][START_REF] Liu | Nonlinear Optics at Excited States of Exciton Polaritons in Two-Dimensional Atomic Crystals[END_REF] third-harmonic generation, [START_REF] Barachati | Tunable Third-Harmonic Generation from Polaritons in the Ultrastrong Coupling Regime[END_REF][START_REF] Liu | Dispersion of Third-Harmonic Generation in Organic Cavity Polaritons[END_REF] four-wave mixing [START_REF] Bruno | Negative Refraction in Time-Varying Strongly Coupled Plasmonic-Antenna--Epsilon-Near-Zero Systems[END_REF] as well as ultrafast nonlinearities. [START_REF] Xiang | Manipulating optical nonlinearities of molecular polaritons by delocalization[END_REF][START_REF] Xiang | Nonlinear infrared polaritonic interaction between cavities mediated by molecular vibrations at ultrafast time scale[END_REF] Apart from these specific nonlinear properties, the intrinsic third-order nonlinear response, which can be described by a nonlinear refractive index 𝑛 2 and a nonlinear absorption coefficient 𝛽, has also been studied very recently in the case of strong coupling, [START_REF] Wang | Large optical nonlinearity enhancement under electronic strong coupling[END_REF] where strongly coupled J-aggregated molecules yielded two-orders-of-magnitude enhancement factors on both 𝑛 2 and 𝛽. These strong-coupling-enhanced nonlinear coefficients facilitate the high efficiency of various third-order nonlinear optical phenomena. However, the relative weak field confinement of the Fabry-Perot cavity used in previous work restricts the coupling strength, thereby limits the nonlinear enhancement. One possible option to enhance the local field is to use a plasmonic cavity, considering that a plasmonic structure can confine light into a subwavelength volume and thereby significantly boost the coupling strength. [START_REF] Törmä | Strong coupling between surface plasmon polaritons and emitters: a review[END_REF] In a remarkable study, Boyd and coauthors showed that indeed very large thirdorder nonlinearities can be obtained when strongly couple the plasmonic antennas to an ultrathin indium tin oxide (ITO) film. [START_REF] Alam | Large optical nonlinearity of nanoantennas coupled to an epsilon-near-zero material[END_REF] Plasmonic nanostructures mainly enhance the electric near field, while affecting only weakly the magnetic near field. [START_REF] Kuznetsov | Optically resonant dielectric nanostructures[END_REF] This diminishes the technological potential of such systems in the context of nonlinear optics, including light frequency conversion and optical modulation. [START_REF] West | Searching for better plasmonic materials[END_REF] 3 High-index dielectric nanostructures appear promising as alternative candidates to provide optical modes required for strong coupling. Indeed, dielectric resonant nanocavities have various advantages, such as low optical loss, high radiation efficiency and reasonable bandwidth.

Moreover, dielectric nanostructures with Mie-type resonances can sustain both electric dipolar (ED) and magnetic dipolar (MD) modes with comparable strength. [START_REF] Kuznetsov | Optically resonant dielectric nanostructures[END_REF] In this case, the magnetic dipole stems from the circular displacement currents within each dielectric resonator induced by the incoming light. This makes it possible to enhance both the electric and magnetic near-fields at the nanoscale. [START_REF] Kuznetsov | Optically resonant dielectric nanostructures[END_REF][START_REF] Bakker | Magnetic and Electric Hotspots with Silicon Nanodimers[END_REF] Considering their large breakdown threshold to laser power, dielectric nanostructures are expected to be an outstanding platform for various nonlinear optical phenomena, including harmonic generations [START_REF] Liu | Resonantly Enhanced Second-Harmonic Generation Using III-V Semiconductor All-Dielectric Metasurfaces[END_REF][START_REF] Koshelev | Subwavelength dielectric resonators for nonlinear nanophotonics[END_REF][START_REF] Shcherbakov | Enhanced Third-Harmonic Generation in Silicon Nanoparticles Driven by Magnetic Response[END_REF][START_REF] Yang | Nonlinear Fano-Resonant Dielectric Metasurfaces[END_REF][START_REF] Liu | An all-dielectric metasurface as a broadband optical frequency mixer[END_REF][START_REF] Liu | Enhanced high-harmonic generation from an all-dielectric metasurface[END_REF] and ultrafast optical switching. [START_REF] Shcherbakov | Ultrafast All-Optical Switching with Magnetic Resonances in Nonlinear Dielectric Nanostructures[END_REF][START_REF] Shcherbakov | Ultrafast all-optical tuning of direct-gap semiconductor metasurfaces[END_REF][START_REF] Grinblat | Efficient ultrafast all-optical modulation in a nonlinear crystalline gallium phosphide nanodisk at the anapole excitation[END_REF] Taking advantage of the remarkable nonlinear optical properties of the high-index dielectric nanostructures and epsilon-near-zero (ENZ) media, [START_REF] Reshef | Nonlinear optical effects in epsilon-near-zero media[END_REF][START_REF] Wang | High Contrast, Femtosecond Light Polarization Manipulation in Epsilon-near-Zero Material Coupled to a Plasmonic Nanoantenna Array[END_REF][START_REF] Bohn | All-optical switching of an epsilon-near-zero plasmon resonance in indium tin oxide[END_REF] here, we first present evidence of strong coupling between spectrally overlapping ED and MD modes of a dielectric nanocavity and an ENZ mode from an ultrathin ITO layer, reaching an anti-crossing splitting as large as 220 meV. Then, by making use of the Z-scan technique, [START_REF] Sheik-Bahae | Sensitive measurement of optical nonlinearities using a single beam[END_REF] we observe large values of effective optical nonlinear coefficients of the system at the coupled resonance, that is more than three-orders-ofmagnitude larger than that of bare ITO and flat silicon films. These large optical nonlinearities result from the modified properties of the coupled system, which are related to the strong local field in these structures. In addition, the coupled system exhibits femtosecond response time of ~300 fs that stems from the thermalization of pump-induced hot electrons in both constituents. The large, ultrafast nonlinear response of this system is a very promising path for nonlinear photonics.

RESULTS AND DISCUSSION

Linear optical measurements. The coupled system in our study consists of an amorphous silicon disk array, which was fabricated on top of a 33-nm ultrathin ITO layer, as shown in Figure 1a. The silicon disks, with an average height of 200 nm, were fabricated on a sputtered silicon film with a standard electron-beam lithography (EBL) and a reactive ion etching (RIE) techniques (Figure 1b). The dimensions of the unit cell were 1 µm × 1 µm in order to ensure strong far-field signal in transmittance measurements. The details of the sample's preparation are presented in the Methods part. In order to systematically explore the strong coupling between the ED/MD resonant modes and the ENZ mode, the diameter of the silicon disk was intentionally designed to support localized ED/MD modes at the wavelengths corresponding to the ENZ region of the ITO (Figure 1c). The linear permittivity of the ITO layer was measured by a spectroscopic ellipsometry (M-2000, J.A. Woollam Co. Inc.), and the corresponding zero permittivity (real part) occurs at 1400 nm, as presented in Figure 1c.

In order to explore the linear optical properties of the strongly coupled ENZ mode with the Mie resonances, it is necessary to briefly discuss the optical behavior of each constituent. The ENZ mode is a confined optical mode and can only be supported by an ultrathin layer of ENZ materials. [START_REF] Vassant | Berreman mode and epsilon near zero mode[END_REF][START_REF] Vassant | Epsilon-Near-Zero Mode for Active Optoelectronic Devices[END_REF][START_REF] Campione | Theory of epsilon-near-zero modes in ultrathin films[END_REF] Its dispersion relation can be derived by solving Maxwell's equations of a threelayer system, as follows [START_REF] Campione | Theory of epsilon-near-zero modes in ultrathin films[END_REF] 1

+ 𝜀 0 𝑘 𝑧2 𝜀 2 𝑘 𝑧0 = 𝑖tan(𝑘 𝑧1 𝑑) ( 𝜀 1 𝑘 𝑧2 𝜀 2 𝑘 𝑧1 + 𝜀 0 𝑘 𝑧1 𝜀 1 𝑘 𝑧0 ) (1)
where

𝑘 𝑧𝑖 2 = 𝜀 𝑖 𝜔 2
𝑐 2 -𝑘 || [START_REF] Törmä | Strong coupling between surface plasmon polaritons and emitters: a review[END_REF] and 𝜀 𝑖 are the longitudinal wave number and the relative permittivity for free space (𝑖 =0), ENZ material (𝑖 =1) and glass substrate (𝑖 =2), respectively. 𝑘 || is the transverse wave number, 𝜔 is the angular frequency, 𝑑 is the film thickness of the ENZ material, and 𝑐 is the speed of light in free space. When the film thickness is much thinner than the skin depth of the ENZ material, such as the 33-nm ITO used in our study, the dispersion of the ENZ mode will be close to the ENZ frequency and stay nearly constant at small 𝑘 || region, as shown in Figure 2a.

Since the dispersion of the ENZ mode lies on the right side of the air light line, a hybrid structure that can support a wide range of wavevector components is required to efficiently excite the ENZ mode. [START_REF] Jun | Epsilon-Near-Zero Strong Coupling in Metamaterial-Semiconductor Hybrid Structures[END_REF] In addition, the electric field intensity distribution of the ENZ mode can be calculated by using field-continuity conditions (inset of Figure 2a). Obviously, the field is concentrated inside the film, which enables high coupling efficiency when the ENZ material interacts with adjacent nanostructures.

Regarding the high-index dielectric nanostructure, a disk shape was used here because it can bring the lowest-order electric and magnetic dipolar resonances into spectral overlap within the ENZ region by choosing a suitable height-to-diameter aspect ratio. [START_REF] Staude | Tailoring Directional Scattering through Magnetic and Electric Resonances in Subwavelength Silicon Nanodisks[END_REF] Therefore, it is sufficient to change the diameter of the disk while keeping the height and unit cell dimensions fixed in our simulations (COMSOL Multiphysics). The calculated transmittance spectra of the silicon disk array on a glass substrate (no ITO layer) are presented in Figure 2b. The overlapping ED and MD modes redshift as the diameter of the silicon disk increases, and eventually spectrally split when the disk diameter is larger than ~570 nm. This spectral separation takes place because the electric dipole resonance exhibits a much stronger redshift with respect to the nanodisk diameter as compared to the spectral shift of the magnetic dipole resonance based on the magnetic wall boundary condition of the cylindrical dielectric cavity. [START_REF] Long | The resonant cylindrical dielectric cavity antenna[END_REF] The result agrees well with the diameterdependent transmittance spectra of the silicon disks that discussed in the literature. [START_REF] Staude | Tailoring Directional Scattering through Magnetic and Electric Resonances in Subwavelength Silicon Nanodisks[END_REF] The parameters of the silicon disks are intentionally selected here so that both the ED and MD modes can appear at the ENZ region of the ITO. The linear optical response of the silicon disk-ITO strongly coupled systems were recorded at normal incidence (Model V-670, JASCO). As shown in Figure 2c, the measured transmittance spectra of the coupled systems at various disk diameters exhibit pronounced splittings at 1400 nm, as well as an anti-crossing feature. This feature is the signature of the strong coupling between the ENZ mode and nanodisk resonant modes. [START_REF] Törmä | Strong coupling between surface plasmon polaritons and emitters: a review[END_REF][START_REF] Jun | Epsilon-Near-Zero Strong Coupling in Metamaterial-Semiconductor Hybrid Structures[END_REF] When the diameter of the silicon disk increases to 580 nm, the splitting of the ED and MD components at both coupled resonances occur, which are consistent with the separation of two modes at large disk diameters in pure silicon disk as given in Figure 2b (details are shown in Supporting Information S1). Interestingly, there are also resonances at ~1000 nm, that possibly come from the electric quadrupole (EQ) and collective resonance as discussed in Supporting Information S2. Similar characteristics are also displayed in the simulated spectra of the coupled systems (Figure 2d). In contrast to the wavelength position of the resonances, the linewidths and amplitudes of the measured spectra are different from that of the simulation.

These differences mainly come from the imperfect disk shape and height unavoidable through the fabrication process, because the simulated resonances are sensitive to the shape and height of the silicon disk when the disk diameter is fixed. Another possible reason is the different between refractive index of each material used in our simulations and actual values in experiment. The strong coupling can also be proved by comparing the scattering cross section (SCS) spectrum of a single silicon disk with and without ITO layers (Figure 2e), here only localized resonances exist and there is no collective mode from grating effects. Considering a single Si disk/substrate system with 505-nm diameter disk, the overlapping ED/MD modes appear at 1290 nm, which are close to ENZ wavelength. When an ultrathin ITO layer is inserted between the single silicon disk and the substrate, two new coupled resonances (one red branch and one blue branch with respect to the ENZ wavelength) are induced due to strong coupling. The coupled resonances in the SCS spectrum resemble that in measured transmittance spectrum (blue curve in Figure S1). Multipolar decomposition analysis indicates that the electric and magnetic dipoles dominate both the coupled resonances (Figure 2f). Although the electric quadrupole and magnetic quadrupole also play a role in the coupled resonances, their contribution can be neglected in our discussion due to their small amplitudes. This suggests that both the ED and MD modes are strongly coupled to the ENZ mode.

Nonlinear optical measurements. The remarkable optical nonlinear response of the ENZ material and high-index dielectric nanostructures lead us to examine the third-order optical nonlinearity of the strongly coupled system via Z-scan measurements. The details of our Z-scan setup was presented in Supplementary Information of our previous work. [START_REF] Wang | Large optical nonlinearity enhancement under electronic strong coupling[END_REF] . Usually, the thirdorder optical nonlinear process can be shown by a linear relation between the peak and valley contrast of the normalized transmittance (∆𝑇 p-v ) and the peak intensity of the incident light at the focus. [START_REF] Sheik-Bahae | Sensitive measurement of optical nonlinearities using a single beam[END_REF] In our strongly coupled system, ∆𝑇 p-v increases nearly linearly with the peak intensity at a representative wavelength (Supporting Information S3). This indicates that the transmittance variation in Z-scan measurements can be used to characterize the nonlinear refractive index and nonlinear absorption coefficient because of its third-order nonlinear origin. Here, the effective nonlinear coefficients, 𝛽 eff and 𝑛 2eff are used to describe the third-order optical nonlinearity in silicon disk/ITO strongly coupled system. This way, we can estimate 𝛽 eff and 𝑛 2eff from the traces of the open-and closed-aperture Z-scan, respectively (Supporting Information S4). Since both 𝛽 eff and 𝑛 2eff are strongly dependent on wavelength (Figures 3a and3b), the peak intensity required In order to explore the nonlinear response under optical strong coupling conditions, we choose a silicon disk/ITO coupled sample with the largest anti-crossing splitting (220 meV), which corresponds to a disk diameter of 505 nm (blue curve in Figure S1). As shown with the log plots in Figures 3a and3b, both 𝛽 eff and 𝑛 2eff display very large absolute values at the coupled resonances. By contrast, they exhibit relatively small values near ITO's ENZ wavelength. The trends of the dispersive nonlinear coefficients follow the linear transmittance spectrum of the coupled system, and the spectral positions of the maximum nonlinear coefficients match that of the coupled hybrid resonances (blue curve in Figure S1). More precisely, near the wavelength of one of the coupled resonances (1240 nm), 𝛽 eff and 𝑛 2eff reach the maximum absolute values of 9.41 × 10 4 cm/GW and 3.51 cm 2 /GW, which are more than three orders of magnitude larger than both the 𝛽 of a 200-nm thick silicon film (17.8 cm/GW) and the maximal 𝑛 2 of the bare 33nm ITO at its ENZ wavelength (1.71 × 10 -3 cm 2 /GW). At the wavelength of another coupled resonance (~1550 nm), the effective nonlinearity coefficients are also enhanced by ~2 orders of magnitude compared with those of the bare ITO and flat silicon films. Because the optical nonlinearity of the ITO layer at ENZ wavelength is very small compared with that of the coupled system, it is important to explore the role of the ED/MD modes of the silicon disk on this optical nonlinearity enhancement. We investigated the open-and closed-aperture Z-scan measurements of a reference sample where a silicon-disk array was directly fabricated on a float glass substrate (no ITO layer). The Si disk/substrate sample has overlapping ED/MD resonances (~1240 nm, Fig. S6a) that are quite close to one of the coupled resonances in the Si disk/ITO/substrate system used in our optical nonlinearity measurements (blue curve in Figure S1). The values of 𝛽 and 𝑛 2 of the Si-disk/substrate at the ED/MD resonant wavelength are measured to be 9.46 × 10 3 cm/GW and 0.42 cm 2 /GW, respectively, which are still much smaller than that of the Si disk/ITO/substrate system at the coupled resonance (see Figure S6). This confirms that the strong-coupling-assisted new mode plays a major role on the large nonlinearity reported here. The Z-scan results of the bare ITO and the flat silicon films, as well as the Si-disk array/substrate (without ITO) are discussed in Supporting Information S6. In addition, the optical nonlinear coefficients at the coupled resonances measured here are also more than two orders of magnitude larger than that of reported plasmonic metamaterial, [START_REF] Neira | Eliminating material constraints for nonlinearity with plasmonic metamaterials[END_REF] and more than one order of magnitude larger than that of the nonlinear ITO in the ENZ region [START_REF] Alam | Large optical nonlinearity of indium tin oxide in its epsilonnear-zero region[END_REF] and than that recently reported J-aggregated molecule in the strong coupling regime. [START_REF] Wang | Large optical nonlinearity enhancement under electronic strong coupling[END_REF] The signs of the measured values of 𝛽 eff and 𝑛 2eff with respect to the optical wavelength presented in Figures 3a and3b are important, the negative and positive values of 𝛽 eff correspond respectively to the saturable absorption and reverse saturable absorption, while the positive and negative values of 𝑛 2eff denote the positive and negative "Kerr lens", respectively.

In contrast to the static optical nonlinear measurements recorded by the Z-scan technique, the time-resolved dynamics of the strongly coupled system was also exploited by modulating the amplitude of the transmitted probe light via a control beam. The schematic of the measurements is presented in Figure 3c, and the details of the pump-probe setup can be found in the Methods part. As shown in Figure 3d, with the illumination of the pump pulse, the transmittance of the probe light first reduces drastically within 100 fs near zero delay, which is restricted by the pulse width of the laser. The transmittance then recovers back to the original state within an ~200-fs exponential decay. The negative value of the normalized transmittance change corresponds to a light-induced absorption process for the coupled system, and the femtosecond Kerr-type nonlinearity results from the laser-induced thermalization of the hot electrons inside both the silicon disk and ITO film. [START_REF] Shcherbakov | Ultrafast All-Optical Switching with Magnetic Resonances in Nonlinear Dielectric Nanostructures[END_REF][START_REF] Alam | Large optical nonlinearity of indium tin oxide in its epsilonnear-zero region[END_REF][START_REF] Grinblat | Sub-20 fs All-Optical Switching in a Single Au-Clad Si Nanodisk[END_REF] The 300-fs switching time achieved here, which corresponds to a calculated bandwidth of 1.46 THz for Gaussian pulses with a time-bandwidth product of 0.44, [START_REF] Li | Ultrafast All-Optical Graphene Modulator[END_REF] is even shorter than the optical modulation in plasmonic antenna/ITO coupled system [START_REF] Wang | High Contrast, Femtosecond Light Polarization Manipulation in Epsilon-near-Zero Material Coupled to a Plasmonic Nanoantenna Array[END_REF] that we recently reported, making our silicon disk/ITO coupled system ideal for ultrafast optical processing and switching applications.

As we stressed above, we attribute the large nonlinear coefficients at the coupled resonances to the strong electromagnetic field enhancement of the silicon disk/ITO strongly coupled system. In order to correlate the optical nonlinear coefficients and the local field intensity (𝐼 loc ), we analyze the electric field distribution at the coupled resonances by describing the electric field inside both the dielectric nanodisk and the ITO layer as a linear superposition of electric and magnetic components. [START_REF] Smirnova | Multipolar Third-Harmonic Generation Driven by Optically Induced Magnetic Resonances[END_REF][START_REF] Kruk | Nonlinear Optical Magnetism Revealed by Second-Harmonic Generation in Nanoantennas[END_REF] Here we define an enhancement factor for the local electric field intensity as 𝜂 𝐼 = 𝐼 loc /𝐼 0 , where 𝐼 0 is the intensity of the incident light in vacuum. As we already explained in our former work, [START_REF] Wang | Large optical nonlinearity enhancement under electronic strong coupling[END_REF][START_REF] Wang | Plasmon-enhanced optical nonlinearity for femtosecond all-optical switching[END_REF] the enhancement of the effective third-order susceptibility (𝜒 eff [START_REF] Ebbesen | Hybrid Light-Matter States in a Molecular and Material Science Perspective[END_REF] )

and 𝜂 𝐼 can be directly related by a mean field approximation model within different nanocavity Here the polarization direction of the normal incident light is parallel to the x axis.

14 systems. This approach leads to an enhancement factor of 𝜒 eff [START_REF] Ebbesen | Hybrid Light-Matter States in a Molecular and Material Science Perspective[END_REF] that follows 𝜂 𝜒(3)eff = 𝜂 𝐼 2 .

Considering the local electric field distribution (E loc ) at one of the coupled resonances (1240 nm) as presented in Figure 4, 𝜂 𝐼 is estimated to be ~34 by averaging an unit volume of both silicon disk and ITO layer based on the simulated electric field profile. This gives rise to a three-orders-ofmagnitude enhancement on 𝜒 eff [START_REF] Ebbesen | Hybrid Light-Matter States in a Molecular and Material Science Perspective[END_REF] at the resonant condition. A comparison with the electric field distribution at off-resonant wavelength (1120 nm) can also be found in Figure S7 in Supporting Information. Regarding to the optical Kerr effect induced by a single light beam, the intensityindependent effective complex nonlinear refractive index 𝑛 ̃2eff follows [START_REF] Boyd | Nonlinear optics[END_REF][START_REF] Boyd | The third-order nonlinear optical susceptibility of gold[END_REF]

𝑛 ̃2eff = 3 4𝑛 0eff 𝑛 0eff ′ 𝜀 0 𝑐 𝜒 eff (3) (2)
where

𝑛 ̃2eff = 𝑛 2eff + 𝑖𝑛 2eff ′′ = 𝑛 2eff + 𝑖 𝑐 2𝜔
𝛽 eff , 𝑛 0eff and 𝑛 0eff ′ are the complex and the real part of the effective linear refractive index, respectively. 𝑛 2eff ′′ is the imaginary part of the effective nonlinear refractive index. The relation clearly suggests that the effective nonlinear refractive index can be boosted in the same proportion as the enhancement of 𝜒 eff [START_REF] Ebbesen | Hybrid Light-Matter States in a Molecular and Material Science Perspective[END_REF] . Therefore, by simulating the electric field intensity enhancement at various wavelengths (See Figure S8), the dispersive 𝜂 𝜒(3)eff and the wavelength-dependent enhancement of 𝑛 ̃2eff (𝜂 |𝑛 ̃2eff| ) can be estimated. Both are plotted as black curves in Figures 3a and3b. As expected, the simulated enhancement of the effective nonlinear refractive index well follows the dispersive trend of the effective optical nonlinear coefficients of the strongly coupled system. It should be noted that equation ( 2) can be applied here because in our system the coupled resonant wavelengths are relatively far from the ENZ wavelength (1400 nm). [START_REF] Caspani | Enhanced Nonlinear Refractive Index in $\ensuremath{\epsilon}$-Near-Zero Materials[END_REF][START_REF] Del Coso | Relation between nonlinear refractive index and third-order susceptibility in absorbing media[END_REF] When considering an ENZ-effect dominated system, for instance a thick ITO film, the free carrier nonlinearity should also be considered, as discussed in detail in the literature. [START_REF] Guo | Ultrafast switching of tunable infrared plasmons in indium tin oxide nanorod arrays with large absolute amplitude[END_REF][START_REF] Kinsey | Epsilon-near-zero Al-doped ZnO for ultrafast switching at telecom wavelengths[END_REF][START_REF] Clerici | Controlling hybrid nonlinearities in transparent conducting oxides via two-colour excitation[END_REF][START_REF] Secondo | Absorptive loss and band non-parabolicity as a physical origin of large nonlinearity in epsilon-near-zero materials[END_REF][START_REF] Wang | Extended Drude Model for Intraband-Transition-Induced Optical Nonlinearity[END_REF][START_REF] Khurgin | Fast and Slow Nonlinearities in Epsilon-Near-Zero Materials[END_REF] Although the large optical nonlinear coefficients of the silicon disk/ITO strongly coupled system are estimated from the data acquired under strong laser irradiance, the enhancement is independent on peak power of the incident light (Figure S3d), i.e. it is an intrinsic optical property of the coupled system.

CONCLUSION

We have observed the signature of optical strong coupling between the ED/MD modes of the silicon nanodisk and the ENZ mode of an ultrathin ITO layer. Through wavelength-dependent Z-scan measurements performed on this coupled system, we obtain large values on both 𝑛 2eff

(3.51 cm 2 /GW ) and 𝛽 eff (9.41 × 10 4 cm/GW) at the coupled resonant wavelength. These large nonlinearities are ascribed to a significant enhancement of the electric field in the coupled system.

In addition, the coupled system demonstrates an ultrafast modulation of its optical response of ~300 fs.

Since the optical nonlinearity coefficients of the coupled system are strongly related to the local electric field enhancement, designing a specific dielectric nanocavity with higher field confinement could be a direct way to further improve the values of 𝑛 2eff and 𝛽 eff for the system.

Because the excitons of semiconductors also show remarkable optical nonlinearities, [START_REF] Ohara | Excitonic enhancement of optical nonlinearities in perovskite CH 3 NH 3 PbCl 3 single crystals[END_REF] one can expect large optical nonlinearity by coupling the excitons to the ED/MD modes of the dielectric nanocavity. The study of ultrafast, large third-order optical nonlinearity in a dielectric nanodisk/ITO coupled system here offers a new route to explore various third-order optical nonlinear phenomena via strong coupling for high speed active photonic and optoelectronic devices.

METHODS

Sample preparation. The sample used in our study is a silicon nano-disk array which was fabricated on top of a commercially 33-nm ITO layer (sheet resistance 50 Ω/square) via standard EBL and RIE techniques. Between the ITO film and the substrate, there is an ~25-nm SiO2 passivation layer in order to ensure electric insulation. Before fabrication, an ~200-nm thick amorphous silicon film was sputtered on top of the ITO layer (0.21 nm/s, PLASSYS MP300). For the EBL process, two layers of the positive PMMA e-beam resists (system 600K/950K of AR-P series, AllresistGmbH) were used in order to improve the accuracy of the exposed pattern. The 600-K and 950-K resists were spin coated sequentially with r.p.m. of 4000 and 2000, respectively, on top of the silicon film. The film thickness of the 600-K and 950-K resists are 220 and 100 nm, respectively. After baking the spin coated resists at 180°C for 90 s, the designed areas were exposed to a 30-keV electron beam (AURIGA, Zeiss). The exposed part of resist was then developed in a conventional solution of methyl isobutyl ketone-isopropyl alcohol (MIBK-IPA)

(1:3) for 30 s and then rinsed with an IPA solution for 30 s to complete the development process.

Next, the Cr mask was obtained by evaporating a 30-nm chrome (Cr) film (0.05 nm/s, PLASSYS ME300) and removing the unexposed resist with acetone (lift-off process). The Cr disks can then be transferred to the silicon film by using Ar and CHF3 plasma gases under the pressure of 2×10 -5 Torr in a RIE chamber (PlasmaLab 80 Plus, Oxford Inst.). Finally, the Cr mask was removed by immersing the sample in a Chromium etchant solution in ~2 minutes. The silicon disks, which possess average diameters from 450 nm to 600 nm, were designed so that they support localized ED and MD modes at wavelengths that can strongly couple to the ENZ mode of ITO at 1400 nm.

The choice of the unit cell dimensions is also crucial because the wavelength of collective resonances from metasurface rely on the periodicity of the structure. Here we choose 1 µm×1 µm unit cell dimensions in order to minimize the influence of the collective resonances in the ENZ region of the ITO.

Time-resolved dynamics measurements. The temporal response of the coupled system was performed with a degenerate optical pump-probe setup. The 1000-Hz, 1240-nm light applied here was generated from an optical parametric amplifier (OPA, TOPAS) which is seeded by a Ti:Sapphire laser system (Spitfire, Spectra-Physics). The output beam was split into two portions as the pump and probe beams. The two beams were set to be nearly parallel (~3° intersection angle)

and illuminating the sample surface normally. The transient transmitted intensity of the probe beam was acquired by a biased InGaAs photodetector (DET10N2, Thorlabs, Inc.). The time delay between the pump and the probe pulses was set by routing the probe beam through a motorized stage with retroreflectors.

ASSOCIATED CONTENT Supporting Information

The Supporting Information is available free of charge at…

Figure 1 .

 1 Figure 1. Schematic of the coupled system and permittivity of the ITO. (a) Illustration of the coupled system between silicon disks and ITO film. (b) SEM image of the silicon disks on top of an ITO layer. (c) Measured real (black curve) and imaginary (gray curve) parts of the permittivity of the ITO film used in this work. The ENZ region and ENZ wavelength are indicated by a blue shadow and a red dot, respectively.

Figure 2 .

 2 Figure 2. Linear optical responses of the coupled systems. (a) The dispersion relation of the ENZ mode (red curve) in air/33-nm ITO/glass system calculated from Eq. (1). The black dashed line is the air light line 𝜔 = 𝑐𝑘 || and the gray dashed line denotes the ENZ frequency. Inset of (a), normalized electric field intensity distribution of the 33-nm ITO/substrate system. (b) Simulated transmittance spectra of silicon disks (without ITO) with respect to disk diameter. ED and MD stand for electric dipole and magnetic dipole, respectively. (c) and (d) are the measured and the simulated transmittance spectra of silicon disk-ITO coupled systems as a function of disk diameter, respectively. The white dashed line represents the ENZ mode and the black curves follow the peaks of the coupled resonances. The measured transmittance spectra at different silicon diameters are shown in Figure S1 in Supporting Information. Here the step size in the simulations of the disk diameter is 15 nm. (e) Calculated scattering cross section of a single silicon disk (505-nm disk diameter) with (red curve) and without ITO layer (gray curve). (f) Multipolar decomposition of the 505-nm Si disk/ITO strongly coupled system with respect to wavelength. Gray areas correspond to the two strongly coupled hybrid resonances.

  for measuring a transmittance change at wavelengths far from resonances will easily cause saturation for the wavelengths close to the resonances. To avoid this issue, lower peak intensities were used when measuring optical nonlinearity near the coupled resonances. The corresponding

Figure 3 .

 3 Figure 3. Nonlinear optical responses of the strongly coupled system. Magnitude of (a) 𝛽 𝑒𝑓𝑓 and (b) 𝑛 2𝑒𝑓𝑓 for the strongly coupled system with 505-nm Si disk on an ITO film at various wavelengths. The positive and negative signs of the value are denoted as closed and open symbols, respectively. The black curves are the calculated dispersive enhancement factor of the effective complex nonlinear refractive index for the coupled system. The dashed gray lines denote ENZ wavelength. (c) Schematic of the pump-probe experimental setup. (d) Normalized ultrafast transmittance change of the silicon disk/ITO coupled system under an 1240-nm light pulse illumination. The red curve is the fitting curve for the decay process. The decay time is 200 fs. The pump fluence applied here is 110 µJ/cm 2 .

Figure 4 .

 4 Figure 4. Electric field distribution of the Si disk/ITO system simulated at the coupled resonant wavelength of 1240 nm. (a), (b) and (c) represent the electric field distributions at different crosssectional views of the silicon disk/ITO coupled system when the disk diameter is fixed to 505 nm.

( 1 )

 1 Linear transmittance spectra of the strongly coupled systems; (2) Periodicity-dependent transmittance spectra of the strongly coupled system; (3) Intensity-dependent open-and closed-aperture Z-scan traces for the strongly coupled system; (4) Estimation of optical nonlinear coefficients from Z-scan traces; (5) Parameters of the laser for Z-scan measurements; (6) Nonlinear coefficients of the reference samples; (7) Electric field intensity at various wavelengths.

ACKNOWLEDGMENTS

The authors thank Hicham Majjad for the help of reactive ion etching fabrication, Yifan Yao for the use of the atomic force microscopy and Yuanyuan Zhang for the help of permittivity measurements of ITO with the spectroscopic ellipsometry in East China Normal University. We also thank fruitful discussions with Minghao Li and Gian Lorenzo Paravicini-Bagliani.

Funding

This work of the Interdisciplinary Thematic Institute QMat, as part of the ITI 2021-2028 program of the University of Strasbourg, CNRS and Inserm, was supported by IdEx Unistra (ANR-10-IDEX-0002), by SFRI STRAT'US project (ANR-20-SFRI-0012), by the Labex NIE (ANR-11-LABX-0058 NIE), CSC (ANR-10-LABX-0026 CSC), the EQUIPEX UNION (ANR-10-EQPX-52-01) and PlasHybrid (ANR-18-CE30-0014-02) projects, and by the University of Strasbourg Institute for Advanced Study (USIAS) (ANR-10-IDEX-0002-02) under the framework of the French Investments for the Future Program. The support of the International Center for Frontier Research in Chemistry (icFRC, Strasbourg), the ERC (project no 788482 MOLUSC) and QuantERA (project RouTe) together with the Ministry of Science and Technology of Taiwan

AUTHOR INFORMATION Corresponding authors

Kuidong Wang: kuidong.wang@unistra.fr Thomas Ebbesen: ebbesen@unistra.fr

Notes

The authors declare no competing financial interest.


Omar Iken

Maxime Folschette

Tony Ribeiro

Automatic Modeling of Dynamical Interactions Within Marine Ecosystems

Keywords: logical modeling, dynamic systems, heuristics, interaction graph

Marine ecology models are used to study and anticipate population variations of plankton and microalgae species. These variations can have an impact on ecological niches, the economy or the climate. Our objective is the automation of the creation of such models. Learning From Interpretation Transition (LFIT) is a framework that aims at learning the dynamics of a system by observing its state transitions. LFIT provides explainable predictions in the form of logical rules. In this paper, we introduce a method that allows to extract an influence graph from a LFIT model. We also propose an heuristic to improve the model against noise in the data.

Introduction

Marine ecosystems represent the majority of aquatic systems on Earth. These ecosystems have an important impact, such as regulating climate change, providing food and maintaining biodiversity, etc. The understanding of these systems is therefore particularly interesting, but their complexity makes it difficult to create new models. A key component of these ecosystems is the phytoplankton, which are microscopic organisms present at the surface of all aquatic ecosystems. They are the basis of aquatic food as they are at the lowest level of the oceanic food chain, in addition to being responsible for the production of a large part of the planet's oxygen. Since 1992, the SRN network has gathered samplings of sea water in order to measure phytoplankton population variations along with environmental factors [START_REF]SRN -Regional Observation and Monitoring program for Phytoplankton and Hydrology in the eastern English Channel: SRN dataset[END_REF].

Learning From Interpretation Transition (LFIT) [START_REF] Ribeiro | Learning dynamics with synchronous, asynchronous and general semantics[END_REF] aims to automate the construction of models of dynamic systems from their state transitions. LFIT produces an explainable model that describes the whole system dynamics and can also predict its probable future variations.

In this work, we use LFIT to learn causal models of marine ecosystems from the dataset of the SRN network. The method we propose in this paper can help biologists to identify which factors and species are in interaction. These models are also useful to predict future population changes in plankton species, and thus may help to predict the evolution of climate change.

State Of The Art

LFIT The LFIT framework [START_REF] Ribeiro | Learning dynamics with synchronous, asynchronous and general semantics[END_REF] is based on the notion of dynamical transition, that is, an atomical transition in a discrete time series. Given a set of such transitions, LFIT builds a model of the system dynamics under the form of a set of rules as follows:

v val0 0 head ← v val1 1 ∧ • • • ∧ v valm m body
Such a rule is said to match a given state if, for each i ∈ {1, 2, . . . , m}, the variable v i has the value val i in the current state. Intuitively, the rule above means that the variable v 0 can take the value val 0 in the next state if the rule matches the current state. We distinguish two types of rules on a given matched state: (1) likeliness rules, where the conclusion is observed in at least one transition from this state, and (2) unlikeliness rules, where the conclusion is never observed in any transition from this state. Each rule of both type is weighted by the number of states it matches in the observations.

DATASET

In this work we focus on the SRN dataset [START_REF]SRN -Regional Observation and Monitoring program for Phytoplankton and Hydrology in the eastern English Channel: SRN dataset[END_REF]. This dataset includes long-term time series of marine phytoplankton and physical-chemical measurements. Water samples were collected along the eastern English Channel coast every 15 to 30 days from 1992 to 2020, at different depths and locations. Each sample in the dataset is characterized by its sampling location and depth, sampling date, name of the measured environmental factor or phytoplankton species, and value of measurement. There are several hydrological factors like temperature, oxygen or salinity, associated to different units, while for each phytoplankton species, the number of individuals is counted. Here we focus on one location of interest: the coastal station 1 of Boulogne-sur-Mer, and focus only on 11 hydrological factors and 12 phytoplankton species only sampled at sea surface level, as it was done in [START_REF] Karasiewicz | Realized niche analysis of phytoplankton communities involving hab: Phaeocystis spp. as a case study[END_REF].

Preprocessing

Data sampling being irregular, we re-sample the measurements for the different factors and species on a monthly basis to minimize missing data. Furthermore, since LFIT only works with abstract discrete values for state transitions, we also need to discretize the measurements. This step has a significant impact on the results provided by the LFIT algorithms. Here, hydrological factors are discretized in two values: below average and above average; phytoplankton species in three values: low, below average and above average, where low encodes a population too low to be counted.

After discretization, we have a time series of 308 data point from which are extracted the 307 atomic transitions to be processed by LFIT. These transitions are divided into 80% (253 transitions) for the learning set and 20% (54 transitions) for the test set. The fitted model has 1683 likeliness rules and 1981 unlikeliness rules, and its accuracy is about 67.0% on the test set.

Rules Improvement

Training data being incomplete and noisy, the output of LFIT is not perfect, i.e., rules may be improved and some could be removed. Indeed, LFIT only learns rules that are true on every single transition they match in the training set. This is why we propose a heuristic to simplify rules, in order to improve their quality and reduce their quantity, thus improving improve model accuracy and readability.

For that, we simply generate all subsets of each rule body, and then count how many transitions validate/falsify the new rule, thus giving us two weights: a correct weight, and a wrong weight. For a likeliness rule, we want to maximize the correct weight and minimize the wrong weight, and conversely for an unlikeliness rule. We can then compute the Pareto frontier (of correct/wrong weights) to find the best subsets of a given body, as shown in Figure 1.

From the Pareto front, we extract what we consider the best rules based on the ratio of correct and wrong weights. We chose an arbitrary ratio of 2 for likeliness rules and 1 /2 for unlikeliness rules.

By eliminating the redundant rules (subsumed by another rule) from the improved model, we obtain a set of 1609 likeliness rules and 1405 unlikeliness rules. With this, we manage to obtain a new accuracy of 71.6% on the test set.

Influence Graph

From the detailed interactions in the form of rules, a more abstract view of the system can be extracted in the form of an influence graph containing the mutual influences between variables. This allows to identify whether an environmental factor or a species is a strict activator or inhibitor of another. The point is to identify which variable need to be controlled in order to impose a desired behavior on a targeted species. We propose to extract such model from LFIT output rules by comparing the value of the head of each rule with the value of each feature of the body of the same rule. Figure 2 (left) shows an example of the process on two rules of the same variable a. For instance, a positive influence is considered when a given body value makes the head evolve the same way (such as a 0 in the example which produces a 0 ) and conversely for a negative influence. This produces a score that allows to compute the influence between each pair of variables. Figure 2 (right) shows the resulting influence graph restricted to one phytoplankton species (Led) once all corresponding rules have been processed.

a 0 ← a 0 ∧ b 1 ∧ c 2 =⇒      a + -→ a b ? -→ a c - -→ a a 0 ← a 1 =⇒ a - -→ a

Conclusion

In this paper, we presented new methods to both improve a LFIT model and extract an underlying influence graph that can be used by biologists to get new insights of the studied data. The accuracy of the method could be improved with a stricter pre-processing of the data, a fine-tuning of the rule improvement from the Pareto frontier, and exploring other ways of building the influence graph. Yet, it consists in a first encouraging step towards the automation of marine ecosystem models.

Fig. 1 :

 1 Fig. 1: Distribution and Pareto frontier of weighted combinations for a likeliness rule (left) and an unlikeliness rule (right). Correct weight is in abscissa and wrong weight is in ordinate. The size of circles corresponds to the number of conditions in the combination, green circles being the "best" alternatives to consider.

Fig. 2 :

 2 Fig. 2: Left: an example of influence extraction on variables with 3 expression levels (0, 1 and 2). Right: an example of the final influence graph of the phytoplankton species Led, where negative influences are in red, positive influences in green, and color intensity corresponds to the certainty.

Acknowledgments The authors would like to thank Sébastien Lefebvre and Stéphane Karasiewicz for providing the data and for their help.