Alexis Gosset

Liam Wilbraham

Štěpánka Nováková Lachmanová

Romana Sokolová

Grégory Dupeyre

Fabien Tuyèras

Philippe Ochsenbein

Christian Perruchot

Henri-Pierre Jacquot De Rouville

Hyacinthe Randriamahazaka
email: hyacinthe.randria@univ-paris-diderot.fr

Henri- Pierre Jacquot De Rouville

Lubomír Pospíšil
email: lubomir.pospisil@jh-inst.cas.cz

Ilaria Ciofini
email: ilaria.ciofini@chimie-paristech.fr

Magdaléna Hromadová
email: magdalena.hromadova@jh-inst.cas.cz

Philippe P Lainé
email: philippe.laine@univ-paris-diderot.fr

Electron Storage System Based on a Two-Way Inversion of Redox Potentials

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Abstract.

Molecular-level multielectron handling towards electrical storage, is a worthwhile approach to solar energy harvesting.

Here, a strategy which uses chemical bonds as electron reservoirs is introduced to demonstrate the new concept of "structronics" (a neologism derived from "structure" and "electronics").

Through this concept, we establish, synthesize, and thoroughly study two multicomponent "superelectrophores": 1,8-dipyridyliumnaphthalene, 2, and its N,Nbridged cyclophane-like analogue, 3. Within both of them, a covalent bond can be formed and subsequently broken electrochemically. These super-electrophores are based on two electrophoric (pyridinium) units that are, on purpose, spatially arranged by a naphthalene scaffold.

A key characteristic of 2 and 3 is that they possess a LUMO that develops through space as the result of the interaction between the closely-positioned electrophoric units. In the context of electron storage, this "super-LUMO" serves as an empty reservoir, which can be filled by a two-electron reduction, giving rise to an elongated C-C bond or "super-HOMO". Because of its weakened nature, this bond can undergo an electrochemically driven cleavage at a significantly more

anodic -yet accessible -potential, thereby restoring the availability of the electron pair (reservoir emptying). In the representative case study of 2, an inversion of potential in both of the two-electron processes of bond formation and bond-cleavage is demonstrated. Overall, the structronic function is characterized by an electrochemical hysteresis and a chemical reversibility. This structronic superelectrophore can be viewed as the three-dimensional counterpart of benchmark methyl viologen.

Introduction

In the context of global warming 1,2 the sun is now recognized as a relevant alternative to fossil fuels. 3,4 Multifarious technological and societal changes are deemed to accompany this advent to make possible the desirable change of primary energy source in optimal conditions. [START_REF]helicene Derivatives: Reversible Formation/Destruction of Chiral Fluorophore and Modulation of Chiroptical Properties by Solvent Polarity[END_REF]6 As is the case in natural photosynthesis, its artificial counterpart 7 as well as man-made devices meant to solar energy transduction (e.g. photovoltaics, DSSCs) 8 largely rely on electrons for their functioning. Likewise, both the handling of multielectron/hole accumulation and the storage of electricity is performed using chemical energy, whether these processes occur within nature or within artificial setups such as batteries and fuel cells.

Here we tackle the fundamental issue of the extent to which a working principle for the storage of electrons can be established at the molecular level, combining chemical reversibility (represented by cyclability in secondary electric batteries) with the thermodynamic irreversibility of the bond formation and cleavage processes of fuel cells. To do so, we assess the possibility of using electrochemistry, exploiting non-catalyzed processes of formation and cleavage of a single chemical bond. This bond is conceived for its dual ability of being formed upon attaching electrons as the only reagents, that is upon reduction, and to yield electrons upon breaking it that is, upon oxidation (in analogy to the charging and discharging of secondary batteries). We refer to this concept as structronics (from "structural" and "electronics"), which is characterized by "using a chemical bond as a reservoir for electron storage". As this concept concerns the ability to go back and forth between binary states (chemical reversibility combined with thermodynamic irreversibility), it is closely related to the notion of electrochemical hysteresis (see Scheme S1 in SI) and proceeds according to a two-way inversion of redox potentials. Thereby storage is distinguished from the mere process of charges (electrons) accumulation, 4c,d which rather refers to thermodynamically reversible processes, or alike. 9,10 Fundamentally, this concept represents a core functionality and, as such, with the exchanged electrons as energy vectors may serve as a means for energy storage.

Indeed, in the context of solar energy conversion, given that light absorption is a one-photon process that results in a one-electron charge-separation process, 4c at least two of those photons are formally "consumed" per structronic functional species for its charging i.e. C-C bond formation, hence the relevance of using such a bond for photoelectrochemical energy storage (two-photon/two-electron gated reversible Carbon-Carbon bond). [START_REF] Waidhas | Electrochemically controlled energy storage in a norbornadiene-based solar fuel with 99% reversibility[END_REF] Last, thanks to hysteretic behavior of the structronic elements, these can serve as a means for the molecular-level storage of information (molecular electronics). 12

Molecular design

For molecular-level charge confinement, the adverse effect to be overcome is the ubiquitous electrostatic repulsion of like charges for both charge accumulation and storage. Detailed explanation of these two terms in connection with the herein reported molecular design of charge reservoirs, is given in Section S3 of the SI.

The redox-active units of reservoirs are initially At this stage, the question is raised as to whether 2- for the electrochemical hysteresis (see also Section S25 and Figure S16 in SI). Beyond the frustration effect on electrostructural relaxation of the reduction product (3 0 ), this last super-electrophore is also anticipated to providing us with the opportunity of getting some insights into the questioned relevance of using a 2-electron/multicenter pseudo-pancake covalent bonding as electron reservoir.

Chart 3. Molecular structures of the reference single-pyridinium electrophoric assembly, 1 + , and of the two-pyridinium super-electrophores, 2 2+ and 3 2+ (hosting the bond that functions as a two-electron reservoir) along with ferrocenyl derivatives 1Fc + and 2Fc 2+ . Syntheses and full characterization of various compounds are reported in SI (Section S28). S1 and Section S4 in SI).

Results and discussion

Analysis of the solid-state structure of 2 2+ confirms that the two pyridinium rings connected at positions 1-and 8-of the naphthalene scaffold are not stacked in a strictly parallel fashion but are askew arranged.

The fact that distortion of molecular backbones is observed in both 2 2+ and 1 + regardless of the charge they hold in their native form, indicates that steric hindrance rather than adverse electrostatic repulsion (case of 2 2+ ) primarily governs the overall structural features, as expected. This finding agrees with previous results derived from the study of congested polyacenes. 17,28 Interestingly, and as expected, the bridging of electrophoric subunits in 3 2+ via their N pyridinio atoms is found to improve the cofacial layout of the pyridiniums (better pre-organization), enforcing the naphthalene scaffold to decreasing its distortion, as reflected by ABCD and  angles, which amount to 2.78°, and 13.49°, respectively (see Figure S1 in SI).

A key parameter, especially for 2 2+ and 3 2+ , is the C()-C(') distance (see Chart 2c and d(C()-C(')) in Figure S1). From single-crystal X-ray crystallography of 2 in the native state (i.e. 2 2+ ), which contains two molecular assemblies per asymmetric unit, a single distance is found to be shorter S1.1 in SI).

The calculated LUMO of model compound 1 + is essentially localized on the pyridinium subunit, with major contributions from the  and  positions (Figure 1). S2 in SI), the calculated bond between the two C() atoms is significantly elongated, thereby confirming its relative weakness. Typically, in the case of 2 0 , it is estimated that the length of this electrochemically-

formed strained bond is 1.647 Å (d(C sp3 -C sp3 ) calcd.
) and is characterized by an associated harmonic stretching force constant which is about 70% that of a usual C-C sigma bond (see Table S2 in SI). In fact, as is the case for the SupLUMO, the spatial expanse of the HOMO naturally encompasses the electrophoric components. The bond-length alternation (BLA) pattern computed for the heterocyclic fragments in going from 2 2+ to 2 0 (Table S1. The observation that the SupLUMO of 2 2+ and the SupHOMO of 2 0 essentially have the same spatial expanse and involve essentially the same part of the super-electrophoric assembly further substantiates the notion of an "electron reservoir" (i.e. that the electrons are removed from the same place (SupHOMO) that they were initially stored (SupLUMO)).

At this point, we recall that multi-[n]-center (n > 2) delocalization of like charges is an efficient means to minimize adverse "on-site" Coulomb repulsion, which can be achieved here via the extension of the bonding SupLUMO beyond the two primarily-concerned C() centers. This is indeed achieved here due to the involvement of the C() p z AOs in (i)

the parent LUMOs of pyridinium units (in-plane -type overlap), and (ii) in the through-space interannular bonding interaction (out-of-plane sigma-type overlap structures. However, contrary to the previous case, after reduction of 1[PF 6 ] in CD 3 CN with Na/Hg, a more complex spectrum is obtained (Figure S5) where protons of the scaffold are more affected than those of pyridinium (see Section S12.2), corresponding to a mixture of compounds among which is the - (i.e. 2-2; Table S3 in SI) dimer ([1] 2 ) that is computed to be the most stable (see Table S4). Once again, no effect ascribable to the presence of radical 1 ⦁ could be evidenced.

In fact, upon chemical reduction, whether considering the cases of 1[PF 6 ] or 2[PF 6 ] 2 , there is sufficient time for chemistry to fully develop, which is not necessarily the case with potentiodynamic electrochemical methods. SupLUMO. This latter is specific to the two-pyridinium "supramolecular" assembly (see Table S6 and Figure S8).

In situ

Indeed, as far as the lowest-lying virtual MO of the superelectrophore is concerned, the two heteroaromatic moieties are essentially through-space coupled, with virtually no electronic interaction mediated through the bonds of the naphthalene scaffold (see Figure 1), insofar as it behaves as a cross-conjugated platform when considering the 1,8connectivity. [START_REF] Schmidt | Charge Transfer Pathways in Three Isomers of Naphthalene-Bridged Organic Mixed Valence Compounds[END_REF] This CT band is therefore a diagnostic feature (in red) and 2 0 (in blue). Spectra are obtained by convolution with Gaussian functions centered at calculated absorption energies with a fixed full width at half maximum of 0.45 eV. Intensities were normalized with respect to the experimental peak at 360 nm.

Based on the remarkable agreement between the experimental results recorded by UV-vis SEC and computed data obtained from the TD-DFT modeling, we may confidently deduce that: (i) Roughly the same overall behavior is observed in the case of 1 + even if spectral changes upon in-situ processes of reduction and re-oxidation are less pronounced (in particular regarding contributions from charge-transfer excitations) than in the case of the dipyridinium derivative (see Figure S9 in SI). This finding is not surprising: a single electrophoric subunit rather than the "synergistic assembly of two coupled pyridinium subunits" is involved. The observed reversibility of the SEC behavior is consistent with the electrochemical behavior (see below, Section 3.5.2.1), which

shows that, when formed as a minor reduction product at low potential scan rate (i.e. v ≤ 0.225 V s -1 ), the dimer is indeed oxidatively cleaved (Ox-2 process in Figure 5-top) and the starting electrophore, 1 + , is recovered. The presence of isosbestic points (Figure S9) is compatible with the formation of both the radical species (1 ⦁ ) and the dimer SupHOMO/[Z -2] (-5.12 eV) reveals that stabilization energy attached to the second heterogeneous electron transfer (∆E = -1.17 eV) is greater than that attached to the first electron transfer (∆E = -0.81 eV). Bearing in mind that bond formation is usually considered a barrierless process, [START_REF] Caruso | Bond Breaking and Bond Formation: How Electron Correlation is Captured in Many-Body Perturbation Theory and Density-Functional Theory[END_REF] this means that the second one-electron reduction process is expected to be "easier" than the first one. To some extent, this finding can be viewed as an indication of an inversion of potentials for the chemical bond formation, provided that one considers the super-electrophoric multicomponent system as a single electronic entity (this is justified by the through-space interaction of the two pyridiniums, which gives rise to the SupLUMO). This inversion is consistent with observation of a single-step two-electron reduction (see below, Red-1 in Figures 4 and5). Now, the question is the extent to which the oxidative bond cleavage also leads to a similar inversion of oxidation potentials. Experimentally, one should therefore observe a single two-electron oxidation. In fact, this is in line with observation of the cyclic voltammograms of 2 (see below, Figure 5-bottom) and comparison of the derived semiintegrated neo-polarograms (see Figure S15 in SI). Brought together, these findings suggest that, from a mechanistic standpoint, both the bond formation and bond cleavage

processes are a result of potential inversion phenomena. Comparison of the LSV wave heights for electrophores identical to 1 + and 2 2+ and that of Fc probes confirms that the number (n) of electrons involved in the reduction processes (Red-1) is equal to the number of Fc units, that is, n = 1 and 2 for 1Fc + and 2Fc 2+ , respectively. [START_REF]For the Red-1 process of 2 2+ , the measurement of |E p -E p/2[END_REF] CVs of 1Fc + and 2Fc 2+ reveal sizably different behaviors (Figure 4). For 1Fc + , one quasi-reversible redox process (Red-1/Ox-1) at -1.278 V 36 is observed along with an oxidation process (Ox-2) in the return scan at E pa-(Ox-2) = -0.362 V, which is characterized by a much smaller peak current. For 2Fc 2+ , the reduction (Red-1) is an irreversible process that occurs at a more positive potential than for 1Fc + (E pc-(Red-1) = -1.002 V), which is also accompanied in the return scan by an irreversible oxidation process (E pa-(Ox- 2) = -0.095 V) that manifests itself as a well-defined broad peak characterized by a current intensity lower than the reduction peak (Red-1). The irreversible oxidation process (Ox-2) is not present in the first anodic scan (starting from In order to get further insight into the very nature of Ox-2 processes observed for one-and two-electron-reduced electrophores 1 + and 2 2+ , we performed a potentiodynamic study as a function of the potential scan rate (v) (Figure 5). (bottom; red traces) solutions (c ≈ 5 × 10 -4 mol L -1 in MeCN with 0.1 M of TBAPF 6 ) recorded at 0.1 V s -1 (solid line) and at 0.625 V s -1 (dashed line).

Mechanistic Insights into Potential Inversion of

CV traces are normalized against the square root of the potential (applied voltage) scan rate v. 38 From comparison of CVs recorded at 0.1 V s -1 and 0.625 V s -1 , one notices that the main difference stems from the status of the Ox-2 process, which becomes virtually indiscernible at higher scan rate for 1 + (ca. v > 0.225 V s -1 ) while it remains virtually unchanged for 2 2+ (this latter behavior was actually verified for v values of up to 1000 V s -1 using an ultramicroelectrode setup). [START_REF]A General Electrochemical Procedure Implemented by a Universal Algorithm[END_REF] In other words, the Ox-2 processes are clearly different in nature for redox-active species 1 + and 2 2+ and the insensitivity to potential scan rate observed for 2 2+ clearly points towards a fast chemical process, and more precisely a process inner to the superelectrophore (i.e. the formation of the reservoir bond). In this case, it is worthwhile establishing the number (n) of electrons involved in the Ox-2 process of 2 0 . This number is typically derived from semi-integration (i.e. convolution) 40

of the CV of 2 2+ recorded at v = 0.1 V s -1 (see Figure S15 in Section S25.1 of SI), which here shows that n is the same for the reduction (Red-1) and the return scan (Ox-2) processes, i.e. n = 2, as expected. Besides, Section S25.1 also reports the comparison of electrochemical behaviors of 2 and 3.

Conversely to 2 (and 3), the quasi-reversibility of the Red-1 process together with the scan rate dependence of Ox-2 in the case of single-pyridinium electrophore (1 + ) are straightforwardly explained by intervening intermolecular process that requires the formation of pairs of reactive species (singly-reduced electrophores i.e. radicals 1 • ). This is the well-documented dimerization process [START_REF] Teplý | Electron Transfer Triggers Fast Dimer/Monomer Switching of Pyridinium and Quinolinium Cations[END_REF][START_REF] Hromadová | Electrochemical reduction of dodecylpyridinium bromide in aprotic solvents: mechanistic studies[END_REF][START_REF] Brandt | Intense redoxdriven chiroptical switching with a 580 mV hysteresis actuated through reversible dimerization of an azoniahelicene[END_REF][START_REF] Plutnar | Electron Transfer Mechanism of Substituted Benzimidazoles: Dimer Switching, Oscillations, and Search for Singlet Fission Properties[END_REF] (electrochemical-chemical -EC dim -mechanism) here giving most likely the ortho-ortho dimers [1] 2 consistently with computational outcomes (see SI, Section S10.2) as well as observations out of the NMR study (see Section 3.4.1 and Figure S5).

Electrochemistry and CV Simulation.

Fast electron transfer followed by the dimerization of a radical intermediate is a common mechanism often encountered in the reduction of pyridinium-based derivatives.

Dimerization is a bimolecular process, which is characterized not only by a positive shift of the redox potential of the product, as compared to that of parent single-pyridinium "monomers", but is also dependent on the bulk concentration. [START_REF] Mairanovsky | The polarographic behavior of quaternary pyridinium salts[END_REF] An archetypal example is the more than centennial synthetic route to methyl viologen (MV). 46 The theory and application of the reduction-dimerization process was developed by Koutecký,[START_REF] Koutecký | Zwei beispiele schneller, in den depolarisationvorgang an der tropfelektrode eingeschalteter, bimolekularer reaktionen[END_REF] Blunt, [START_REF] Gaudiello | On the mechanism of the electrochemical reduction of Nmethylpyridinium ion[END_REF] Olmstead 49 and Savéant. [START_REF] Andrieux | Electrodimerization: I. One-electron irreversible dimerization. Diagnostic criteria and rate determination procedures for voltammetric studies[END_REF] However, abovedescribed experiments unambiguously established that reduction of 2 2+ (Red-1 in Figures 4 and5) is not a one-but a two-electron process (see Section 3.5.2.1). Features, which distinguish the redox mechanism of 2 2+ from bimolecular dimerization are (i) the two-electron reduction, (ii) the independence of redox potentials on the bulk concentration, (iii) observation of Ox-2 process at higher scan rates, (iv) spectroscopic proof of the bond formation inner to the superelectrophore (Section 3.4). These features confirm that the observed reduction mechanism of 2 2+ is consistent with an inner process, namely the electrochemical -bond formation.

Therefore, the reservoir-bond formation has nothing to do with an ordinary bimolecular dimerization. This agrees with theoretical calculations that pointed to the inner bond formation correlated with characteristic structural and electronic rearrangements (Section 3.3). The heterogeneous transfer of two electrons at apparently the same driving force, i.e. at the same applied potential, implies that the standard redox potentials E 0 1 and E 0 2 for the transfer of each electron are either compressed (up to a coalescence) or even inverted (Section 3.2). Detailed analysis of the currentvoltage dependence obtained by the cyclic voltammetry at higher scan rates indeed confirms the potential inversion phenomenon (as suggested in Sections 3.5.1 and 3.5.2.1). The voltammogram using the glassy carbon electrode (see Section S26-A in SI) and the scan rate of 2 V•s -1 (Figure S21) yields similar semi-integrated "neo-polarograms" or "convolution voltammograms" as those shown in Figure S15. The separate semi-integration of each voltammetric half-cycle yields welldefined waves (Figure S22) reaching a limiting value of the semi-integral of the current, , 51 and thus enabling the 𝑚(𝑡)lim analysis of the wave shapes by the log-plot analysis:

log 𝑚(𝑡) 𝑚(𝑡)lim -𝑚(𝑡) vs. 𝐸
The two-electron reduction yields the log-plot trace featuring an S-shaped curve, having two linear asymptotes with the same slope of 59 mV/decade, as is expected for two, fast, one-electron transfers. This allows not only the estimation of the two standard potentials, E 0 1 and E 0 2 , but also provides the demonstration that these latter are inverted (Figure 6). The difference |E 0 1 -E 0 2 | = 60 mV is obtained both at lower and at higher voltage scan rates (0.06

V•s -1 ≤ v ≤ 2 V•s -1
). This insensitivity of the voltage difference with respect to v was further verified by numerical fitting of a series of voltammograms by application of the finite difference techniques. Furthermore, it is worth noting that the magnitude of the experimentally derived potential inversion (ca. 60 mV) is in line with the computational estimate (ca. 40 mV; see Section S7.3 in SI). An analogous log-plot analysis applied to the oxidation wave (Ox-2) is not that straightforward. This peak, observed in the return scan regardless of potential scan rates, is indeed rather small due to the diffusion of the neutral -bonded reduction product toward the bulk. The lack of a reduction counterpart for the oxidation process Ox-2 of the -bonded form indicates that this inner-sphere -bond cleavage is a fast and irreversible process (see Section S26-B in SI). This is in line with the assumption that the Ox-2 process regenerates the electrophore in its starting (native) form,

2 2+ .
The digital simulation of CVs by finite difference methods considered two possible reaction schemes differing in sequences of electron transfers (E) and the chemical reaction (C, here the -bond formation and cleavage). The overall electrochemical reaction is usually referred to as an EEC process and is represented in Chart 4a where A 2 2+ and B 2 0 stand for 2 2+ and sigma-bonded 2 0 . Alternatively, the structural change (i.e. the bond cleavage) could take place right after the first electron withdrawal, as it is shown in Chart 4b.

The latter case would be termed an ECE mechanism. The equilibrium constants and chemical rates are Since the digital simulation analysis of the oxidation wave allows unambiguously identifying of two redox potentials E' 3 and E' 4 the reaction mechanism most likely involves the EEC scheme with potential inversion (Chart 4a) for both the reduction (Red-1) and the oxidation (Ox-2) steps. The fitting of voltammograms recorded over a wide range of v-values, usually confirms the nature of the reaction scheme. [START_REF] Hall | Intramolecular Electron Transfer in Bipyridinium Disulfides[END_REF] Both reaction mechanisms contain a large number of kinetic parameters and equilibrium constants. This raises issues regarding the reliability of fitting outcomes insofar as several combinations of adjustable parameters may give similar results. It is therefore worthwhile getting an estimate at least of some of the input parameters (see Section 3.5.2.1 and Section S26 in SI).

K 1 = k 1 /k -1 and K 2 = k 2 /k -2 .
An example of calculated data that quite accurately fit experimental data is given in Figure S26 (see also Table S11 in SI). Fitting indeed reasonably reproduced the i a /i c ratio (Figure S27), the potential differences (E 0 1 -E 0 2 ) = 71 mV and (E 0 3 -E 0 4 ) = 141 mV as well as fulfilling additional conditions stated in Section S26. Most importantly, not only did this simulation distinguish between E 0 3 and E 0 4 , it also demonstrated that potential inversion is at work for the bond cleavage, as is the case for the bond formation. First of all, it's worth noting that, contrary to 3 that relies on cross clamping to tune electrochemical hysteresis (see Section S25.1 in SI), assemblies reported in this section rely on two other steric constraints that are namely the front strain as well as the back (remote) strain (see Chart 2c). So, interestingly, there exist connections between the applied field of molecular-based electrochromism, [START_REF] Suzuki | A new type of tricolor electrochromic system based on the dynamic redox properties of hexaarylethane derivatives[END_REF] that is, the molecular-level optical switching (output) controlled by an electrochemical input, and the basic research efforts devoted to the exploration of "ultralong C-C bonds and of their limits", [START_REF] Suzuki | Ultralong C-C bonds in hexaphenylethane Derivatives[END_REF][START_REF] Suzuki | Bis(10-methylacridinium)s as a Versatile Platform for Redox-Active Functionalized Dyes and Novel Structures[END_REF] that are, actually, beyond [START_REF] Takeda | Hexaphenylethanes with an Ultralong CC Bond: Expandability of the CC Bond in Highly Strained Tetraarylpyracenes[END_REF][START_REF] Bettinger | Tetraphenyldihydrocyclobutaarenes -what causes the extremely long 1.72 Å C-C single bond?[END_REF][START_REF] Kammermeier | 2 + 2] Cycloaddition Products of Tetradehydrodianthracene : Experimental and Theoretical Proof of Extraordinary Long C -C Single Bonds[END_REF] expectations. [START_REF] Pauling | The Nature of the Chemical Bond and the Structure of Molecules and Crystals[END_REF][START_REF] Zavitsas | The Relation between Bond Lengths and Dissociation Energies of Carbon-Carbon Bonds[END_REF] The above-mentioned connection stems from the intriguing electrochemical phenomenon referred to as "Dynamic

Redox" (DyRex) behavior, [START_REF] Suzuki | Ultralong C-C bonds in hexaphenylethane Derivatives[END_REF][START_REF] Suzuki | Bis(10-methylacridinium)s as a Versatile Platform for Redox-Active Functionalized Dyes and Novel Structures[END_REF][START_REF] Suzuki | Hexaphenylethane Derivatives Exhibiting Novel Electrochromic Behavior[END_REF] also referred to as "intramolecular switching of single bonds" 62 (Chart S1 in SI).

Namely, it was shown that covalent assemblies properly incorporating couples of redox-active units can give rise to the formation of a (C sp3 -C sp3 ) 63 carbon-carbon bond that bridges electrochemically active components following a reductive or an oxidative process. For example, typically integrated redox-active chromophoric units are trityliums, [START_REF] Suzuki | A new type of tricolor electrochromic system based on the dynamic redox properties of hexaarylethane derivatives[END_REF][START_REF] Suzuki | Hexaphenylethane Derivatives Exhibiting Novel Electrochromic Behavior[END_REF]64 . [START_REF] Suzuki | Bis(10-methylacridinium)s as a Versatile Platform for Redox-Active Functionalized Dyes and Novel Structures[END_REF] In fact, we were sharing in these respects the same concerns as Suzuki and coworkers, especially about ortho (C2 and C6) reactive positions of pyridinium. [START_REF] Suzuki | Ultralong C-C bonds in hexaphenylethane Derivatives[END_REF] Originally, we were considering the "naked" pyridinium derivative (2 2+ ) as a non-optimized demonstrator compound, likely to be subsequently improved by ensuring its ortho protection with some bulky substituents. The present work demonstrates that such a protection was, in fact, superfluous thanks to the effects of the synergistic interplay of electrophoric units.

Although specifically implemented as a key-ingredient of the design of structronic devices, this synergy was finally found to be more efficient than initially planned i.e. its impact on reactivity had been underestimated (Chart 2c). Indeed, we found that the "SupLUMO" of 2 2+ was functioning as a genuine attractor that diverts or traps the whole reactivity of classical (i.e. isolated) pyridinium.

For a further thorough comparative discussion of Dyrex systems with super-electrophores described in the present work, see Section S27 in SI.

Summary and Conclusions.

The reactivity of pyridinium components within 2 (and 3) was found to be heavily impacted compared to that of naked pyridinium or to a mono-pyridinium model species like 1. The synergistic interplay of the two electrophoric units allows both (i) emergence [START_REF] Luisi | Emergence in Chemistry: Chemistry as the Embodiment of Emergence[END_REF] Finally, super-electrophore 2 which, at first, can be regarded as the result of a rough design, turned out to be a relevant demonstrator and a proof of the "structronics" concept, in spite of its simplicity. This prototype has the great asset of cost-effectiveness thanks to its low engineering level as well as chemical robustness (see Sections 3.4.1 and S27). These features both pertain to the "Sustainocene" criteria. Soc., 1977, 99, 1916-1924. 20 ( 2006, 110, 13003-13006. 22 It's worth noting that the potentially repelling cationic charges are not located at the short contact site where bond formation is expected, hence reducing accordingly electrostatic strain (demand) within the native assembly. 23

When using the reading grid of electrochemical reactivity in the potential range spanning from -2V to +1V (vs. SCE in acetonitrile), the sole pyridylium groups, that are spatially disjointed, are redox active. They are structurally gathered by the electrochemicallyinactive naphthalene scaffold, hence the notion of supramolecular assemblies used for these super-electrophores within the present context of electrochemistry. This reading grid is very similar to that perfected by V. Balzani 24 for the field of Supramolecular Photochemistry, in the context of photochemistry. 24 (a) Balzani, V.; Moggi L.; Scandola F. in Supramolecular Photochemistry, (Eds.: V. Balzani, D. Reidel) Publishing Co., Dordrecht, The Netherlands, 1987, pp. 1-28; (b) Balzani, V. Supramolecular Photochemistry. Tetrahedron, 1992, 48, 10443-10514. 25 As regards MVs, it is worth noting that close cofacial contact of these electrophores is likely to impact their electronic structure and reactivity. This phenomenon is observed whether these MVs are tightly tethered, in high concentration, organized in self-assembled monolayers or encapsulated in supramolecular host-guest assemblies.

The result is the formation of  complex dimers exhibiting redox behaviors distinguishable from that of isolated MV. For representative references, see (a) Burgess, M.; Chénard, E.; Hernández-Burgos, K.; Nagarjuna, G.; Assary, R. S.; Hui, J.; Moore,

  defined in their closed-shell native redox state, represented by their initial charge [z]. This notation extends to their reactive (i.e. open-shell) radical states, which may be obtained following a one-electron reduction [z -1] or a oneelectron oxidation [z + 1] process. Similarly, species obtained after a two-electron reduction are denoted as [z -2]. Based on energetics, cationic units are preferred as electrophoric components for electrons whereas neutral or anionic units are preferred for holes. In both cases, to formally generate closely lying radicals upon reduction or oxidation, thereby formally preparing them for the formation of the reservoir bond (Chart 1). The scope of the present work is restricted to the simplest case of a prototypical covalent assembly based on monocationic electrophoric components (z = 1+) that are involved in the reductive formation (and the oxidative cleavage) of a twoelectron bond, which functions as a two-electron reservoir. Chart 1. Generic representation of the molecular design strategy of the three-dimensional super-electrophore assembly ("structronics" function), which is disconnected from any mechanistic considerations. Electrophoric units (triangles) are pre-organized by the scaffold (spatial layout and strain tuning). The charge of cationic super-electrophore here amounts to [Z]= [z] +[z] = 2+ in its native redox state.

  The whole aim of this proof of concept paper is to engineer the inter-component (through-space) orbital overlap to yield a LUMO which functions as a "two-electron trap". Therefore, one must switch the point of reductive reactivity from the individual electrophoric components to the superelectrophoric assembly as a whole.Based on the above-defined criteria, we consider the pyridinium moiety as electrophoric component (Chart 2). This choice was made firstly because of the intrinsically uneven distribution of atomic contributions to the LUMO over its heteroaromatic ring, characterized by the large involvement of the carbon atom at the  position (C(); Chart 2a) a post-reduction quinoidal structure.[START_REF] Marder | Impact of Backbone Tether Length and Structure on the Electrochemical Performance of Viologen Redox Active Polymers[END_REF] Furthermore, pyridinium derivatives are remarkably versatile, especially regarding their redox properties, which can be tuned over a range of potentials of c.a. 1.5 V.16 As the scaffold, we selected the naphthalene platform (with 1,8connectivity) that has the great asset of being structurally flexible enough through skeletal distortion, 17,18 which is of interest for adjusting the strain on the reservoir bond (whether via cross clamping, front strain or remote (back) strain; Chart 2c). 19 Chart 2. Molecular design of "structronic" prototypes: organizing electro-structural synergy. (a) Pictorial representation of the weighted atomic contributions (p z atomic orbitals, AOs) to the LUMO of a pyridinium ring with emphasis on the large contribution of the C() position. (b) Representation of the in-plane pi-overlap between p z AOs of the two C() atoms of native MV 2+ , which form the basis of the second interannular bond involved in the overall quinoidal electronic structure adopted by MV 0 (following the stepwise two-electron reduction of MV 2+ ). (c) Representation of the sigma-type, out-of-plane, overlap of p z AOs of the C() atoms within the three-dimensional super-electrophore (i.e. SupLUMO build-up) in the structronic context, leading to the formation of the elongated C sp3 -C sp3 reservoir bond following an apparent single-step two-electron transfer (SupHOMO). Connected at positions 1-and 8-of naphthalene (Chart 2), the two pyridinium units within the super-electrophore are anticipated to have their C() carbon atoms in close vicinity, lying at a distance (d(C()-C(')) smaller 20 than the sum of their van der Waals (vdW) radii (1.70 × 2 = 3.40 Å). 21 This critical short contact serves to maximize orbital interaction (overlap) between the two rings at precisely the C() sites where the LUMO of each pyridinium component has a large contribution (Chart 2a & 2c). In other words, instead of the in-plane π-type delocalization that typically takes place within the archetypal reference molecule MV 2+ (Chart 2b), an out-of-plane sigma-type overlap of the LUMOs of the two pyridinium building blocks is targeted as a result of their three-dimensional pre-organization (Chart 2c

  than 3.4 Å, which is indeed d(C()-C('): 2.993 Å / 3.012 Å, as expected. On the contrary, three couples of atoms are found to be in close contact within 3 2+ , thereby satisfying the vdW criterion of a 3.4 Å maximum separation distance: the d(C()-C('), 2.958 Å, and the two d(C()-C(') atom pairs: 3.146 Å and 3.291 Å. It is thus anticipated that bonding interactions will be different in 2 0 and 3 0 . Even if the above picture of inner structural features of super-electrophores 2 and 3 in their native state ([Z] = 2+) are derived from solid-state characterization, one can reasonably expect that virtually identical (although averaged) distortions are present in solution. Overall, the combination of the steric front strain 19 between pendant pyridylium groups with the intrinsic torsional flexibility of naphthalene scaffold, 17 and the prominent contribution of the C() p z orbitals to the LUMO, come together to favor a single through-space short contact within 2. On the contrary, with supplementary cross clamping i.e. within the cyclophane-like model 3, an enforced inner pseudostacking (characterized by the almost cofacial layout of heteroaromatic rings) is observed, which results in multiple short contacts that could lead to a 2-electron/n-center covalent bonding upon reduction (see below). 3.3. Molecular Modeling. DFT calculations were performed to gain further insight into the molecular geometries and electronic structures (frontier molecular orbitals) of the native and reduced forms of 1, 2 and 3. Geometry optimization of 1, 2 and 3 in their native forms, (C()-C(')), ABCD torsion angles, ,  and  (vide supra and Figure S1 and Table

Figure 1 .

 1 Figure 1. Relevant molecular orbitals for single-, 1, and double-2/3 pyridinium electrophores in native [Z], singly reduced [Z -1] and doubly reduced [Z -2] forms. These frontier MOs were calculated using the PBE0 exchange-correlation functional and are plotted with an iso-contour value of 0.04. The corresponding orbital energies are also given in eV.

  1 in SI), namely the shortening of d(C()-C()) (from 1.377 to 1.347 Å) with concomitant elongation of d(N-C()) (from 1.348 to 1.382 Å) and d(C()-C()) (from 1.348 to 1.506 Å), reflects the emerging quinoidal structure that develops in 2 0 . 16b This loss of aromaticity of the pyridinium rings is indicative of their actual involvement in the reduction. 16b For consistency, we found it more convenient to refer to the HOMO of superelectrophore 2 in its [Z -2] state (i.e. 2 0 ) as the SupHOMO.

31

 31 Figure S4 in SI). This observation is fully consistent with outcomes of ab-initio calculations that predict a change in the chemical shift, , of 102.9 ppm for C() (from 172.2 for 2 2+ to 69.3 ppm: see Figure S4 in SI). This upheaval accurately reflects the change of hybridization of the C(/') carbon atoms from sp 2 to sp 3 that accompanies the electrochemically gated interannular covalent bond formation. 31 In addition, through variable temperature NMR experiments we probed the stability of the newly formed 5membered ring. Even at 333 K, no signal change or broadening (for instance, from a homolytic rupture of the new C sp3 -C sp3 bond) was observed that could be attributed to intervening paramagnetic species. In fact, we will show below that the singly reduced radical intermediate cannot accumulate because of potential inversion 32 (Section 3.5.2.2). Not only is the radical cation too short-lived compared to characteristic time of NMR (around a few seconds) but also observation that no radical species could be detected upon heating is informative regarding the relative robustness of the reservoir bond.

Figure 2 .

 2 Figure 2. 1 H NMR spectra (400 MHz, CD 3 CN, 298 K) of 2[PF 6 ] 2 (c = 10 -2 mol L -1 ) before (a) and after (b) chemical reduction, showing the diagnostic chemical shift of the H(,'), H(,') and N-Me protons of the electrophoric subunits; See also Section S29 in SI.

  Spectroelectrochemical (SEC) Studies.Spectroelectrochemistry has the twofold advantage over its chemical counterpart (e.g. the use of Na/Hg amalgam or cobaltocene for reduction purposes) that the potential energy of electrons as "reactants" can be continuously varied and their availability controlled over time. Great assets of SEC stem from the potential control of electron availability, that is, the monitoring of a given redox process. In this sense, the probed phenomena are not necessarily the same as those observed by chemical means. The UV-vis-NIR electronic absorption SEC study was performed with 1 millimolar solutions of 1[PF 6 ] and 2[PF 6 ] 2 in 0.1M TBAPF 6 in MeCN (Figure 3-top). Regarding the electronic spectrum of 2[PF 6 ] 2 , three main absorption bands are observed in the near UV region that are characterized by maxima at 221 nm, 253 nm and 357 nm. These absorption features are accurately reproduced by TD-DFT calculations (vide supra)that allowed the assignment of electronic transitions (Figure3-bottom). The first two higher-energy bands (with simulated maxima at 218 and 247/253 nm) are π-π* transitions centered on the naphthalene scaffold and on the pyridinium subunits, respectively (see TableS6and FigureS8in SI). Most importantly, the third, lowest-lying band (i.e. the red-edge feature), calculated at 386 nm, is assigned to a 99 %) charge transfer (CT) character from the electron-rich naphthalene moiety to the virtual

  supporting the existence of the target SupLUMO. The spectrum of the two-electron-reduced species, 2 0 , characterized by only two major absorption bands at 228 and 295 nm, is obtained upon varying the applied potential down to -1.20 V (vs. quasi-reference electrode Ag/AgCl). As is the case for the native state, the TD-DFT simulated spectrum of 2 0 compares very well with the experimentally recorded trace, with two bands of calculated maxima at 224 and 295 nm, respectively. Theory tells us that the latter results from a pure π-π* transition centered on the naphthalene unit, as confirmed by the structured shape (vibrational progression) experimentally observed for the lowest-energy band. The observation that the lowest-energy band of CT nature, evidenced for 2 in its native state, is no longer present in the spectrum of 2 electrophoric assembly switches from being electron-deficient (within 2 2+ ) to electron-rich (within 2 0 ) upon reduction. Lastly, we emphasize the isosbestic points at 215, 241, 279 and 311 nm observed during the course of the reductive SEC experiment, and that the spectrum of the super-electrophore in its native form (2 2+ ) is restored upon reoxidation, showing the same isosbestic points (see Figure S6 in SI)both processes of two-electron reduction and oxidation (see Section 3.5.2.1). It is worth noting that the results of the FTIR vibrational absorption (SEC) study further substantiate outcomes of the UV-vis SEC by evidencing the bond-length alternation (BLA) attached to the quinoidal form of the reduced pyridylium components within 2 0 and by confirming the chemical reversibility of the redox transformation 2 2+ /2 0 (see Section S21 in SI).

Figure 3 .

 3 Figure 3. (Top) Spectroelectrochemical UV-vis experiments of a solution of 2 2+ (red) in MeCN (c = 1×10 -3 mol L -1 with 0.1 M of TBAPF 6 ). The spectrum of 2 0 (blue) is obtained after achieving potential of -1.2 V against Ag/AgCl quasi-reference electrode (reduction process). Asterisks (*)

  [1] 2 to the extent that their calculated spectral signatures are very similar (see Figures S9 and S10 in SI).

  Figure 1 suggests that the process of interannular covalent bond formation, which occurs inside super-electrophores 2 and 3, is fundamentally different from the bimolecular process of dimerization. The first added electron enters the SupLUMO and is delocalized across both pyridinium units (i.e. not localized on a single pyridinium as would be the case for a genuine "inner" sigma dimerization). Only a slight structural change is computed for the first electron transfer, whereas the second electron transfer is clearly accompanied by a pyramidalization of each of the two C() atoms. This unambiguously confirms the change of their hybridization from sp 2 to sp 3 . As regards the prototype two-pyridinium super-electrophore, 2, the comparison of the respective energies (Figure 1) of the SupLUMO/[Z] (-3.14 eV), SupSOMO/[Z -1] (-3.95 eV) and

  Study. Electrophores 1 + and 2 2+ were derivatized with redox probes that undergo a monoelectronic redox process (n = 1), namely ferrocene (Fc) units that are appended via saturated propylene linkers at N pyridinio atoms (1Fc + and 2Fc 2+ in Chart 3). Cyclic voltammograms (CVs) at a stationary electrode and linear sweep voltammograms (LSVs) at a rotating disk electrode (RDE) are shown in Figure 4.

  -0.6 V to +1.0 V) and appears only after the reduction process (Red-1) is complete. Such a sequence-dependence indicates that Ox-2 is an electrochemical-chemical(E-C) process related to the oxidation of a reduction product of Red-1, which therefore also involves an E-C process (see below). Further, the anodic shift of the Red-1 reduction potential (∆E Red-1 = +0.276 V) in going from 1Fc + to 2Fc 2+ is consistent with the presence of an interaction between the two pyridinium units within 2Fc[PF 6 ] 2 and thus within 2[PF 6 ] 2 .

Figure 4 .

 4 Figure 4. Cyclic voltammetry (top) and rotating disk electrode (RDE) experiments (bottom) of 1Fc[PF 6 ] (blue traces) and 2Fc[PF 6 ] 2 (red traces) solutions (c ≈ 5 × 10 -4 mol.L -1 in MeCN with 0.1 M of TBAPF 6 ) both recorded at scan rate v of 0.1 V s -1 and rotation speed r of 2000 rpm.37

Figure 5 .

 5 Figure 5. Cyclic voltammograms of 1[PF 6 ] (top; blue traces) and 2[PF 6 ] 2

Figure 6 .

 6 Figure 6. Log-plot analysis of semi-integrated neo-polarogram (Figure S22) calculated for the reduction step (0.5 mM 2 2+ and TBAPF 6 0.1M in MeCN, glassy carbon electrode, scan rate 2 V s 1 ). Extrapolation of two asymptotes to y-axis = 0 yields two redox potentials for the first E 0 1 and the second E 0 2 electron transfer step. The potential difference |E 0 1 -E 0 2 | is 60 mV.

Chart 4 .

 4 Reaction scheme involving the two-electron reduction and reoxidation with a coupled chemical reaction (EEC mechanism for both steps; top, (a)); The EEC mechanism applies for the reduction, whereas ECE pathway holds for the oxidation (bottom, (b)). The point is that for both (a) and (b) schemes, forward pathway to B 2 0 is different from backward pathwayelectron storage; k 0 n are standard rate constants of heterogeneous electron transfers.

Chart 5 .

 5 data indicates that both the reduction (Red-1) and the oxidation (Ox-2) obey the EEC mechanism given in Chart 4a. Finally, pathways for reduction and oxidation are however different with respect to the involved redox-active species (A/B in Chart 4a), thereby accounting also for the experimentally evidenced electrochemical hysteresis. Chart 5 explicitly summarizes the two-electron reduction/oxidation cycles operative, with a two-way potential inversion for super-electrophore 2 (and also 3). The pictorial representation of the mechanism underpinning the working principle of the structronic electrochemical paradigm thoroughly investigated in the present work. The functional assembly (2) referred to as a structronic prototype is akin to a kind of 3-dimensionnal MV ("3D-MV") but displaying an electrochemical behavior sharply different from that of the 2D, MV benchmark species.

  of the SupLUMO, which acts as a "reactivity sink" towards the target reservoir-bond formation inner to the super-electrophoric assembly and (ii), the protection of the super-electrophore from side reactions, like is intermolecular -dimerization. The lack of reactivity at  sites of these pyridinium moieties within the 3-D structronic construct is reminiscent of that of the wellknown 2-D (roughly planar) covalent MV species (as a oneelectron redox shuttle or mediator). However, depending on the nature of electrophores, the manner by which the two electrons are transferred is sharply different, due to fundamentally different working principles (electrochemical paradigms). In the case of MV, the lack of reactivity of the one-electron-reduced intermediate is basically due to delocalization of this electron over the whole molecular skeleton whereas in the case of 2 and 3, it is rather related to the transient nature of the one-electron-reduced species due to potential inversion as well as its buried nature due to the inner location of pivotal SupLUMO/SupSOMO. This latter situation is due to inversion of electrochemical potentials and, to a lesser extent, the lowered spin density at positions as a result of its withdrawal towards the C() sites, hence the uncompetitive intermolecular dimerization. Noteworthy, the inversion of potentials is demonstrated for both the filling of the SupLUMO reservoir (two-electron reduction with bond formation) and for the emptying of the filled reservoir taking the form of the SupHOMO (two-electron oxidation with bond breaking).

. Solid-State Characterization. The molecular structures of 1[PF 6 ], 2[PF 6 ] 2 and 3[PF 6

	3.1. Synthetic Strategy. For the purpose of the present
	study, besides the synthesis of the precursors of mono-and
	di-pyridylnaphthalene	via	Suzuki-Miyaura	coupling	(see
	Section S26 in SI), we have adopted a strategy that allows,
	in a single step, the pyridinium formation (quaternarization)
	or the Fc-derivatization via a saturated spacer that ensures
	the electronic decoupling of redox-active subunits (i.e. Fc
	and	pyridinium).	This	approach,	that	relies	on	3-
	iodopropylferrocene 26 as a key synthon, is potentially
	implementable with various kinds of compounds having pendant
	pyridyl groups, when it is desired, for instance, to attach
	redox probes to count the electrons involved in a redox
	process while avoiding diffusion effects, 14,16a,27 as is the case
	here.
	3.2
	For consistency reasons, all studies in solution (whether
	experimental or theoretical) were performed with acetonitrile
	(MeCN)	solvent,	using	an	implicit	solvent	model	for
	calculations (see Experimental Section).

] 2 were determined by single-crystal X-Ray diffraction (see Figure

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	Financial supports by the Grant Agency of the Czech Republic
	5,6b Of course, there is still a long way to go to reach truly workable devices that could enter the marketplace. Indeed, it is now a matter of designing electroactive molecular-based materials with a high density of "structronic" elements so as to obtain workable storage devices from the standpoint of charge-density. This could possibly be obtained by grafting "structronic" assemblies at the surface of conductive electrodes (e.g. graphene) or by embedding these functional elements within a matrix of conducting polymers or inside nanoporous conductive covalent organic frameworks (COFs), using electricity of photovoltaic origin. Actually, possible utilizations of super-electrophores span from the field of solution organic molecular chemistry (e.g. as new generation super-electron-donors, SED, 71 which are adjustable and electrochemically rechargeable to deliver couples of redox equivalents) to the field of electrochemical energy storage devices, thus functioning as building blocks of organic 72 or redox-flow 73 batteries. From the standpoint of homogeneous artificial photosynthesis and, above all, of advanced Photochemical (18-04682S), and the Academy of Sciences of the Czech Republic (RVO: 61388955, 61388963), the Ministry of Education of the Czech Republic (Barrande project 7AMB15FR027), the French Ministries of Europe and Foreign Affairs (MAE) and of Advanced Education, Research and Innovation (MESRI) (PHC Barrande, 2015 project N° 34012SC), the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 648558) and the French National Research Agency, ANR ("E-StorIc" project : ANR-14-CE05-0002), are gratefully acknowledged. Notes and references The ice-core record: climate sensitivity and future greenhouse warming. Nature, 1990, 347, 139-145. 2 Crutzen, P. J. Geology of mankind. Nature, 2002, 415, 23. 3 (a) Molecular devices featuring sequential photoinduced charge separations for the storage of multiple redox equivalents. Coord. Chem. Rev. 2011, 255, 2578-2593. 5 Faunce, T. Towards a global solar fuels Structural Changes in N-Aryl-Expanded Pyridiniums: project-Induced Establishing Their Unusual,
	Molecular Devices (PMDs), 24 it's worth bearing in mind that
	what	can	be	done	electrochemically	can	be	achieved
	photochemically, provided that suitable photosensitizers are
	involved.	Thus,	the	"structronic"	(+	photosensitizers)
	functional component(s) can be associated with various
	catalysts (whether molecular or nanoparticulate) and with
	electron donors 74 (whether sacrificial or not) to give

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Acknowledgment

molecular-level principles able to manage multi-photon/multielectron i.e. photosynthetic transduction processes.

ASSOCIATED CONTENT

AUTHOR INFORMATION

Corresponding authors

hyacinthe.randria@univ-paris-diderot.fr; Ilaria.ciofini@chimie-paristech.fr; magdalena.hromadova@jh-inst.cas.cz; lubomir.pospisil@jh-inst.cas.cz philippe.laine@univ-paris-diderot.fr

Conflicts of interest.

The authors declare no competing financial interest.


D Provitolo
email: damienne.provitolo@geoazur.unice.fr

A Model to Assess the Role of Spatial Urban Configurations on Crowd Evacuation Dynamics During Terrorist Attacks

Introduction: Modern Terrorist Attacks

Terrorism has always existed in history and has taken different forms: anarchist attacks, political attacks, or attacks by colonial states, such as the terrible Jallianwala Bagh massacre, which took place on April 13, 1919 in Amritsar, Punjab, when a crowd of non-violent protesters were fired upon by troops of the British Indian Army.

In this article we focus specifically on behavior of populations facing new forms of terrorism, such as the attacks perpetrated by sects, groups or "lone wolves" that are becoming more and more prevalent. Three types of acts can be distinguished:

• shooting and bombing attacks, for example in bus and subway stations (Madrid, Spain, 2004;Paris, France, 1995, 2015;Nigeria, 2014;Belgium, 2016;Kashmir, 2019), at airports (Brussels airport, 2016;Atatürk airport, 2016), in urban areas with simultaneous terrorist acts (Mumbai, India, 2008, with 12 coordinated shooting and bombing attacks lasting four days). • attacks using CBRN (chemical, biological, radiological, and nuclear weapons) such as the sarin gas attack on the Tokyo subway committed by the Aum sect (March 20, 1995) [START_REF] Dauphiné | Risques et catastrophes-Observer, spatialiser, comprendre, gérer[END_REF]). • in response to tightened police controls aiming to reduce bombing attacks, new types of ad hoc weapons are now being employed, such as suicide planes on the World Trade Center (2001), lorries or cars that are simply driven directly through a crowd massed for a cultural event or strolling peacefully. These attacks are most often led by "lone wolves", who are difficult to detect beforehand and equally difficult to locate because they are "nested" among civilians. On July 14, 2016, 86 people who attended the Bastille Day fireworks on the "Promenade des Anglais" in Nice, France, were killed; 458 more were wounded.

These attacks, perpetrated in crowds, in frequented places, or on public transport at rush hour, give rise to diversified human reactions. One can observe behaviors of sideration (paralysing panic), of flight panic, of agitation in all directions, of reasoned flight, of sheltering, or mutually offered assistance [START_REF] Crocq | Les paniques collectives[END_REF], etc.

Except for sideration, sheltering and escape behaviors, whether taken under the influence of panic (flight panic) or of reason (reasoned escape), produce trajectories of displacement. When an attack happens, the topography of the area is also very important. The dynamics of human reactions and the associated displacements are indeed guided by the space and the alternatives that it offers, as displayed in Figure 3.1, especially in terms of evacuation, flight, and accessibility to refuge areas, such as safe places and squares. There are few data available to identify the dynamics of the displacements associated with these behaviors. Of course, it is difficult to artificially reproduce a disaster, which would otherwise allow us to observe the diversity of human reactions that could occur, to follow the spatio-temporal dynamics, and to analyze the impact of spatial configurations on these dynamics. To overcome these limits, it is possible to develop mathematical models from which hypotheses are tested by means of a human behavior computer simulation model. The Com2SiCa research team 1 proposed the Panic-Control-Reflex (PCR) model [START_REF] Provitolo | Les comportements humains en situation de catastrophe: de l'observation à la modélisation conceptuelle et mathématique[END_REF] as shown in Figure 3.2, which is a model that simulates the possible human behavior that can occur during sudden onset and unpredictable disasters, such as terrorist attacks. This model is formed by a system of ordinary differential equations to describe behavioral dynamics over time [START_REF] Verdière | Mathematical Modeling of Human Behaviors During Catastrophic Events[END_REF], 2015[START_REF] Cantin | Mathematical Modeling of Human Behaviors During Catastrophic Events: Stability and Bifurcations[END_REF].

Neuroscience research explains that during a disaster, humans are rarely stuck in one type of behavior. The population switches between different behavioral states, some of which are the result of instinctive, others of reasoned, reaction. As the brain switches from one behavioral state to another, in the context of terrorist 1 https://geoazur.oca.eu/en/research-geoazur/2158-com2sica-how-to-comprehend-and-simulate-humanbehaviors-in-areas-facing-natural-disasters attacks and therefore in a situation of sudden and unforeseen threat, the entire impacted population first adopts a behavioral reaction, Reflex r(t), under the influence of surprise and the suddenness of the event, before transiting to the Panic reflex behavior p(t), or Controlled behavior c(t) [START_REF] Provitolo | Les comportements humains en situation de catastrophe: de l'observation à la modélisation conceptuelle et mathématique[END_REF]. Reflex r(t) and Panic p(t) behavior are managed by the reptilian brain: they are instinctive, automatic reactions, allowing one to react extremely quickly to the threat, either by being stunned and paralyzed r(t), or by fleeing as quickly as possible due to the panic fear p(t). Controlled behavior c(t) is managed by the pre-frontal cortex, which concerns reasoned and self-control reactions. These can take different forms during a catastrophe, for example, in the form of evacuation, escape, containment, sheltering, search for help, mutual aid, or, on the contrary, looting, etc. Despite their diversity, the PCR model aggregates all of these controlled behaviors.

In this article we propose an extension of the PCR model in order to take into account the influence of spatial configuration in the mathematical modeling of the dynamics of human reactions in the face of traumatic situations, such as terrorist attacks. We call this extended model the Coupled Weighted PCR (CWPCR).

In Section 3.2, we will present the Weighted Panic Control Reflex model in its graphic and mathematical form, as published by the Com2SiCa research team. This model is improved from the PCR model in order to take into account the role of spatial configurations on behavioral dynamics. In Section 3.3 we introduce the CWPCR model, which incorporates the pressure and counterpressure of the crowd in each place via a bottleneck effect, which takes into account the narrowness and the length of the streets and the size of places. We will consider an oriented network with three nodes representing three places or public squares of different sizes, linked by narrow streets. In this network, we compute the equilibrium points that show the motion of the crowd after a terrorist attack. Section 3.4 is therefore devoted to the analysis of the impact of the size of the intermediate place of this network on the evacuation of the population in the face of a terrorist attack, by means of numerical simulations. The numerical results highlight that, depending on their respective size, intermediate places modulate the dynamics and the speed of flow of the crowd. In this sense, they become strategic places, both for the planners who must think about the organization of the area to host public events and festivals, and also for the terrorists who can use these strategic places to multiply the effect of their harmful actions by trapping the escape movements between two areas of action. Finally, in Section 3.5, a brief conclusion will be drawn.

Weighted PCR System: A Model for Analyzing the Dynamics of Human Behavior During a Disaster

Network of Squares and Streets Modelling Spatial Urban Configurations

Just before a dramatic event like a terror attack in a city, the crowd is generally spread across several places, public squares and streets. In the aftermath of the initial shock, people rush through the streets to reach what they think will be more secure places, as one can see in In this chapter, we define a city by a mathematical graph, where the places and public squares are called vertices or nodes and are denoted 𝑁 1 , 𝑁 2 , ⋯ , 𝑁 𝑝 , and the streets, escalators, doors, and stairs are the oriented edges (𝑁 𝑖 → 𝑁 𝑗 ) linking these nodes. They are oriented, because the flow from one place toward another is not symmetric during terrorist attacks. Our aim is to model the motion of the crowd through such edges. To achieve this, we must introduce some "geographical" particularities of the city, like the size of places, the narrowness of streets, and the number of people initially present in each place. This is why we need to upgrade the standard PCR model, into the Weighted PCR (WPCR) model, by introducing new data, with Weight standing for the relative sizes of crowd, places, and streets.

FIGURE 3.3

Possible paths of rushing people in a city network. (IGN -BD Orthophoto, 2017, 50 cm resolution).

First, on each node 𝑁 𝑘 , 𝑘 = 1, 𝑝 we call 𝑟 𝑘 (𝑡), 𝑐 𝑘 (𝑡), 𝑝 𝑘 (𝑡), 𝑞 𝑘 (𝑡), 𝑏 𝑘 (𝑡) the number of people being in reflex, control, panic, daily usual behavior, and return normal life and 𝑉 𝑘 (𝑡) the total number of people present at this node

𝑉 𝑘 (𝑡) = 𝑟 𝑘 (𝑡) + 𝑐 𝑘 (𝑡) + 𝑝 𝑘 (𝑡) + 𝑞 𝑘 (𝑡) + 𝑏 𝑘 (𝑡) (3.1)
Second, in order to more precisely model the characteristics of the city, we introduce 𝑊 𝑘 , the maximum capacity of the number of people who can be present in each node 𝑁 𝑘 (i.e. due to the size of the corresponding place). Each maximum capacity is a constant. Each time one must have

𝑉 𝑘 (𝑡) ≤ 𝑊 𝑘 (3.2)

Equations of the WPCR System on Each Node

The WPCR simulates the human behavior, in one specific place, during a catastrophic event.

As explained in the introduction, it is based on neuroscience studies. During the event, the switches from one behavioral state to another are caused by transitional dynamics, as schematized in Figure 3.2 due to:

i. Causal relationships (B1, B2, C1, C2). Once the population is in the reflex behavior state, a part of it can evolve toward controlled behaviors at the rate B1, while another part transitions towards panic behaviors at the rate B2. Likewise, a part of the panicked population may switch to controlled behavior at the rate C1. According to the evolution of the situation, individuals who have adopted a controlled behavior may switch back to panic behavior at the rate C2. This process can be iterated many times.

ii. Processes of imitation and contagion, which are well known in crowd psychology and have been termed "emotional contagion" [START_REF] Hatfield | Emotional Contagion[END_REF]. The imitation processes are modelled identically to epidemiological propagation [START_REF] Provitolo | Un exemple d'effets de dominos : la panique dans les catastrophes urbaines[END_REF]).

The imitation is valid in both directions and is modelled by the function F(r,c) (see Equation (3.5)) for emotional contagion between reflex and controlled behavior (using the damping coefficients α1 and α2), by the function G(r,p) (see Equation (3.6)) for emotional contagion between reflex and panic behavior (δ1 and δ2), and by the function H(c,p) (see Equation (3.7)) for emotional contagion between controlled and panicked behavior (μ1 and μ2).

iii. Domino effects, which illustrate a succession of events (s1 and s2) correspond, for example, to a new attack in an urban area or to a closed door during an evacuation. In the PCR model, the parameters s1 and s2 are either constant or built in a periodic form.

The (WPCR) model is defined on each node k by

𝑋 ̇= 𝛷(𝑡, 𝑋) (3.3)
with 𝑋 ̇= 𝑑𝑋

𝑑𝑡

, 𝑋 = (𝑟, 𝑐, 𝑝, 𝑞, 𝑏) 𝑇 ∈ ℝ 5 and 𝛷 given by 𝛷(𝑡, 𝑋) = (𝛷 𝑖 (𝑡, 𝑋)) 𝑇 , 𝑖 = 1, . . . ,5,

where the functions 𝛷 𝑖 are defined by

{ 𝛷 1 (𝑡, 𝑋 𝑘 ) = 𝑟̇𝑘(𝑡) = 𝛾(𝑡)𝑞 𝑘 (𝑡)(𝑊 𝑘 -𝑟 𝑘 (𝑡)) -(𝐵 1 + 𝐵 2 )𝑟 𝑘 (𝑡) + 𝑠 1 (𝑡)𝑐 𝑘 (𝑡) + 𝑠 2 (𝑡)𝑝 𝑘 (𝑡) +𝐹(𝑟 𝑘 (𝑡), 𝑐 𝑘 (𝑡))𝑟 𝑘 (𝑡)𝑐 𝑘 (𝑡) + 𝐺(𝑟 𝑘 (𝑡), 𝑝 𝑘 (𝑡))𝑟 𝑘 (𝑡)𝑝 𝑘 (𝑡) 𝛷 2 (𝑡, 𝑋 𝑘 ) = 𝑐̇𝑘(𝑡) = -𝜑(𝑡)𝑐 𝑘 (𝑡)(𝑊 𝑘 -𝑏 𝑘 (𝑡)) + 𝐵 1 𝑟 𝑘 (𝑡) + 𝐶 1 𝑝 𝑘 (𝑡) -𝐶 2 𝑐 𝑘 (𝑡) -𝑠 1 (𝑡)𝑐 𝑘 (𝑡)
-𝐹(𝑟 𝑘 (𝑡), 𝑐 𝑘 (𝑡))𝑟 𝑘 (𝑡)𝑐 𝑘 (𝑡) + 𝐻(𝑐 𝑘 (𝑡), 𝑝 𝑘 (𝑡))𝑐 𝑘 (𝑡)𝑝 𝑘 (𝑡) 𝛷 3 (𝑡, 𝑋 𝑘 ) = 𝑝̇𝑘(𝑡) = 𝐵 2 𝑟 𝑘 (𝑡) -𝐶 1 𝑝 𝑘 (𝑡) + 𝐶 2 𝑐 𝑘 (𝑡) -𝑠 2 (𝑡)𝑝 𝑘 (𝑡) -𝐺(𝑟 𝑘 (𝑡), 𝑝 𝑘 (𝑡))𝑟 𝑘 (𝑡)𝑝 𝑘 (𝑡) -𝐻(𝑐 𝑘 (𝑡), 𝑝 𝑘 (𝑡))𝑐 𝑘 (𝑡)𝑝 𝑘 (𝑡) 𝛷 4 (𝑡, 𝑋 𝑘 ) = 𝑞̇𝑘(𝑡) = -𝛾(𝑡)𝑞 𝑘 (𝑡)(𝑊 𝑘 -𝑟 𝑘 (𝑡)) 𝛷 5 (𝑡, 𝑋 𝑘 ) = 𝑏 ̇𝑘(𝑡) = 𝜑(𝑡)𝑐 𝑘 (𝑡)(𝑊 𝑘 -𝑏 𝑘 (𝑡))

(3.4) The parameters involved in Equation (3.4) are real positive coefficients: 𝐵 𝑖 ≥ 0, 𝐶 𝑖 ≥ 0, 𝑖 = 1, 2; 𝛼 𝑖 ≥ 0, 𝛿 𝑖 ≥ 0, 𝜇 𝑖 ≥ 0, 𝑖 = 1, 2; 𝑠 𝑖 ≥ 0, 𝑖 = 1, 2.

The imitation functions 𝐹, 𝐺 and 𝐻 are real valued functions defined on ℝ× ℝ by

𝐹(𝑟(𝑡), 𝑐(𝑡)) = -𝛼 1 𝑓 1 ( 𝑟(𝑡) 𝑐(𝑡)+𝜀 ) + 𝛼 2 𝑓 2 ( 𝑐(𝑡) 𝑟(𝑡)+𝜀 ) (3.5) 𝐺(𝑟(𝑡), 𝑝(𝑡)) = -𝛿 1 𝑔 1 ( 𝑟(𝑡) 𝑝(𝑡)+𝜀 ) + 𝛿 2 𝑔 2 ( 𝑝(𝑡) 𝑟(𝑡)+𝜀 ) (3.6) 𝐻(𝑐(𝑡), 𝑝(𝑡)) = 𝜇 1 ℎ 1 ( 𝑐(𝑡) 𝑝(𝑡)+𝜀 ) -𝜇 2 ℎ 2 ( 𝑝(𝑡) 𝑐(𝑡)+𝜀 ), (3.7)
where 𝜀 is a positive number and 𝑓 𝑖 , 𝑔 𝑖 , ℎ 𝑖 for 𝑖 = 1,2 are real valued functions defined on ℝ. They have a decreasing shape indicating that the behavior imitation is symmetric. Moreover they are normalized,

0 ≤ 𝑓 𝑖 (𝑢) ≤ 1, 0 ≤ 𝑔 𝑖 (𝑢) ≤ 1, 0 ≤ ℎ 𝑖 (𝑢) ≤ 1, ∀𝑢 ∈ 𝑅, 𝑖 = 1,2 (3.8)
Because this model does not take the mortality rate into account, the population remains constant. Therefore Equation (3.4) is considered when time is proceeding from an initial time 𝑡 0 ≥ 0, with initial condition 𝑟 𝑘 (𝑡 0 ) + 𝑐 𝑘 (𝑡 0 ) + 𝑝 𝑘 (𝑡 0 ) + 𝑞 𝑘 (𝑡 0 ) + 𝑏 𝑘 (𝑡 0 ) = 𝑉 𝑘 (𝑡 0 ) = = 𝑟 𝑘,0 + 𝑐 𝑘,0 + 𝑝 𝑘,0 + 𝑞 𝑘,0 + 𝑏 𝑘,0 = 𝑉 𝑘,0 ≤ 𝑊 𝑘 (3.9) that satisfies the following properties

𝑟(𝑡 0 ) > 0, 𝑐(𝑡 0 ) > 0, 𝑝(𝑡 0 ) > 0, 𝑞(𝑡 0 ) > 0, 𝑏(𝑡 0 ) > 0 (3.10)
We suppose that the characteristic parameters 𝐵 1 , 𝐵 2 , 𝐶 1 , 𝐶 2 , of each node have the same value, because they depend on cultural and psychological factors specific to each individual rather than to spatial configurations and crowd context. It is why we use function 𝛷 𝑖 instead of function 𝛷 𝑘,𝑖 .

Transitional Dynamics

Both forcing functions, 𝛾 and 𝜑 respectively, model the beginning of the disaster and the return to a quiescent daily behavior. Their shape can be adapted to various scenarios. When 𝑡 is sufficiently large, they satisfy 𝛾(𝑡) = 𝜑(𝑡) = 1. In catastrophic situations it is considered that 𝛾 is a stiff function, ranging from 0 to 1 in a very brief interval of time [START_REF] Verdière | Mathematical Modeling of Human Behaviors During Catastrophic Events[END_REF], 2015[START_REF] Cantin | Mathematical Modeling of Human Behaviors During Catastrophic Events: Stability and Bifurcations[END_REF]) because if we consider a bomb attack, the entire crowd that is near the explosion passes from daily to reflex behavior in an instant, and it takes a very long time for people to return to their normal state.

Therefore, one can suppose that a terror attack is shaped by two characteristic times: 𝑡 𝑠 (for start) and 𝑡 𝑒 (for end) with 𝑡 0 < 𝑡 𝑠 < 𝑡 𝑒 for which (3.15)

3.3 Coupled WPCR system on a network

Flows and bottleneck coupling

As previously defined, nodes are linked by edges. We now aim to model the motion of the crowd through such edges (i.e. streets, stairs, escalators, doors, etc.).

We also seek to identify the obstacles that slow the escape of the crowd in the aftermath of the initial shock by analyzing the topology of the network of streets and places in a city. This information can potentially be used to improve the design of a city to facilitate the escape of a crowd towards more secure places.

For the sake of simplicity, as we show in Figure 3.5, we consider first a simplified oriented network with only three nodes (𝑁 1 ; 𝑊 1 ) ; (𝑁 2 ; 𝑊 2 ) ; (𝑁 3 ; 𝑊 3 ) and two edges (𝑁 1 → 𝑁 2 ) ; (𝑁 2 → 𝑁 3 ). Such a simplified network can be straightforwardly complexified by adding as many nodes and edges as necessary, without any difficulty. However, it is better to first focus our attention on the nature of the obstacles in this simplified network.

FIGURE 3.5

Three node network: people in every state are rushing from node 1 towards node 2, and from node 2 towards node 3, keeping the same behavioral class. We suppose that during the short interval of time when people are traveling inside one edge, they remain in the same behavioral class as displayed in Figure 3.5. In this figure, the "geographical" edges (𝑁 1 → 𝑁 2 ) ; (𝑁 2 → 𝑁 3 ) (i.e. the streets linking node 1 to node 2, and node 2 to node 3) are split in three "behavioral edges" meaning that on the same street people in reflex, panic, or controlled behavior are escaping from node 1 to node 2 (and from node 2 to node 3), therefore people in each particular behavior in node 1 are meeting people in the same behavior in node 2 (and in the same manner from node 2 to node 3).

To continue focusing on the special coupling that we are introducing here, we suppose that imitation mechanisms are not activated (i.e. 𝐹 ≡ 𝐻 ≡ 𝐺 ≡ 0, which is equivalent to 𝛼 𝑖 = 𝛿 𝑖 = 𝜇 𝑖 = 0 for 𝑖 = 1,2), and, furthermore, there is no domino effect (𝑠 𝑖 = 0, 𝑖 = 1, 2) and we also suppose that we are in the interval 𝐼 𝑡𝑟𝑎𝑛𝑠 = [𝑡 𝑠 , 𝑡 𝑒 ] where only transitional dynamics are considered. Of course, it is easy to relax such limitations, which are not dependent upon the coupling, by not eliminating the corresponding terms in the equations.

In each node 𝑁 𝑘 , 𝑘 = 1, 2, 3 such transitional dynamics are the solution of the system

{ 𝑟k(𝑡) = - (𝐵 1 + 𝐵 2 )𝑟 𝑘 (𝑡) 𝑐k(𝑡) = 𝐵 1 𝑟 𝑘 (𝑡) + 𝐶 1 𝑝 𝑘 (𝑡) -𝐶 2 𝑐 𝑘 (𝑡) 𝑝̇𝑘(𝑡) = 𝐵 2 𝑟 𝑘 (𝑡) -𝐶 1 𝑝 𝑘 (𝑡) + 𝐶 2 𝑐 𝑘 (𝑡) (3.16)
which is the reduction of System (3.4) in the transition interval (and a particular case of Equation (3.15)).

Note: in the WPCR model, the terms 𝑟k(𝑡), 𝑐k(𝑡), 𝑝k(𝑡), 𝑞k(𝑡), 𝑏 ̇𝑘(𝑡) can be considered as flows, because a flow is a quantity of something divided by a unit of time. There are two kinds of flow. In Equations (3.4), (3.15), (3.16) flows are "behavioral", as they represent the quantities of people changing their behavior per unit of time. Now we consider also "motion" flows, which are the quantities of people in each behavior class, moving from one node to another node, per unit of time. Of course, both kinds of flow are combined to produce a global equivalent in the following equations.

Many studies on pedestrian flows have been published [START_REF] Daamen | First-Order Pedestrian Traffic Flow[END_REF][START_REF] Dias | Investigating collective escape behaviors in complex situations[END_REF][START_REF] Kretz | Experimental study of pedestrian flow through a bottleneck[END_REF][START_REF] Liao | Experimental study on pedestrian flow through wide bottleneck. The Conference on Pedestrian and Evacuation Dynamics 2014 (PED2014)[END_REF][START_REF] Seyfried | The Fundamental Diagram of Pedestrian Movement Revisited[END_REF][START_REF] Zhang | Empirical study of a unidirectional dense crowd during a real mass event[END_REF][START_REF] Zhou | Pedestrian Evacuation Time Model for Urban Metro Hubs Based on Multiple Video Sequences Data[END_REF]. We consider, in particular, the survey of [START_REF] Daamen | First-Order Pedestrian Traffic Flow[END_REF], in which the graphs of six different experiments showing the relationship between the flow of pedestrians and their densities are displayed. All these graphs show clearly a nonlinear relationship of a logistic type between density and flow.

Moreover, Daamen and co-authors developed a first-order traffic flow theory to describe two-dimensional pedestrian flow operations in the case of an oversaturated bottleneck in front of which a large, high-density region has formed. Such a mathematical model also highlights the logistic relationship for any bottleneck width as one can see in Figure 3.6.

FIGURE 3.6

Relationship between flow and density of pedestrians going through an oversaturated bottleneck. (Adapted from [START_REF] Daamen | First-Order Pedestrian Traffic Flow[END_REF].

We now introduce our hypothesis for the coupled WPCR (CWPCR), based on this type of logistic relationship. Again, for the sake of simplicity, we consider only unidirectional motion of the crowd (i.e. motion on an oriented graph), as the one displayed in Figure 3.5, because it is supposed that a terrorist attack occurs in node 1 and that people try to escape from this node towards nodes 2 and 3.

When the crowd is moving from one node to another, its speed and the corresponding motion flow depends on three factors. The first factor reflects the narrowness and the length of the street. More people can go from one place to the next if the street is large, rather than in the case of a narrow street. This topological characteristic will be modelled by "roughness" coefficients 𝜂 1,2 , 𝜂 2,3 . In fact, in the considered coupling, we suppose that people cannot change their behavior when they move from one node to another (e.g. controlled people remain controlled, panicked people remain panicked, and so on), and thus we use three such roughness coefficients 𝜂 𝑐,1,2 , 𝜂 𝑝,1,2 , 𝜂 𝑟,1,2 , 𝜂 𝑐,2,3 , 𝜂 𝑝,2,3 , 𝜂 𝑟,2,3 to refer to the population in reflex, panic, or controlled situations, respectively. Of course, they can have the same value. The second factor is proportional to the number of people present in the node from which the crowd is escaping. This is equivalent to the pressure of the crowd in these places. The third factor reflects the counterpressure due to the maximal capacity of the place toward which the people are fleeing, conjugated with the number of persons already present there. The combination of pressure and counterpressure gives a bottleneck effect.

We propose to model this bottleneck effect by a nonlinearity of a logistic type to keep the same philosophy as [START_REF] Daamen | First-Order Pedestrian Traffic Flow[END_REF].

Thus, the bottleneck coupling corresponding to the one of Figure 3.5, is given by the system

{ 𝑟1(𝑡) = - (𝐵 1 + 𝐵 2 )𝑟 1 (𝑡) -𝜂 𝑟,1,2 𝑟 1 (𝑡)(𝑊 2 -𝑐 2 (𝑡) -𝑝 2 (𝑡) -𝑟 2 (𝑡)) 𝑐1(𝑡) = 𝐵 1 𝑟 1 (𝑡) + 𝐶 1 𝑝 1 (𝑡) -𝐶 2 𝑐 1 (𝑡) -𝜂 𝑐,1,2 𝑐 1 (𝑡)(𝑊 2 -𝑐 2 (𝑡) -𝑝 2 (𝑡) -𝑟 2 (𝑡)) 𝑝1(𝑡) = 𝐵 2 𝑟 1 (𝑡) -𝐶 1 𝑝 1 (𝑡) + 𝐶 2 𝑐 1 (𝑡) -𝜂 𝑝,1,2 𝑝 1 (𝑡)(𝑊 2 -𝑐 2 (𝑡) -𝑝 2 (𝑡) -𝑟 2 (𝑡)) 𝑟2(𝑡) = - (𝐵 1 + 𝐵 2 )𝑟 2 (𝑡) + 𝜂 𝑟,1,2 𝑟 1 (𝑡)(𝑊 2 -𝑐 2 (𝑡) -𝑝 2 (𝑡) -𝑟 2 (𝑡)) -𝜂 𝑟,2,3 𝑟 2 (𝑡)(𝑊 3 -𝑐 3 (𝑡) -𝑝 3 (𝑡) -𝑟 3 (𝑡)) 𝑐2(𝑡) = 𝐵 1 𝑟 2 (𝑡) + 𝐶 1 𝑝 2 (𝑡) -𝐶 2 𝑐 2 (𝑡) + 𝜂 𝑐,1,2 𝑐 1 (𝑡)(𝑊 2 -𝑐 2 (𝑡) -𝑝 2 (𝑡) -𝑟 2 (𝑡)) -𝜂 𝑐,2,3 𝑐 2 (𝑡)(𝑊 3 -𝑐 3 (𝑡) -𝑝 3 (𝑡) -𝑟 3 (𝑡)) 𝑝2(𝑡) = 𝐵 2 𝑟 2 (𝑡) -𝐶 1 𝑝 2 (𝑡) + 𝐶 2 𝑐 2 (𝑡) + 𝜂 𝑝,1,2 𝑝 1 (𝑡)(𝑊 2 (𝑡) -𝑐 2 (𝑡) -𝑝 2 (𝑡) -𝑟 2 (𝑡)) -𝜂 𝑝,2,3 𝑝 2 (𝑊 3 -𝑐 3 (𝑡) -𝑝 3 (𝑡) -𝑟 3 (𝑡)) 𝑟3(𝑡) = - (𝐵 1 + 𝐵 2 )𝑟 3 (𝑡) + 𝜂 𝑟,2,3 𝑟 2 (𝑡)(𝑊 3 -𝑐 3 (𝑡) -𝑝 3 (𝑡) -𝑟 3 (𝑡)) 𝑐̇3(𝑡) = 𝐵 1 𝑟 3 (𝑡) + 𝐶 1 𝑝 3 (𝑡) -𝐶 2 𝑐 3 (𝑡) + 𝜂 𝑐,2,3 𝑐 2 (𝑡)(𝑊 3 -𝑐 3 (𝑡) -𝑝 3 (𝑡) -𝑟 3 (𝑡)) 𝑝̇3(𝑡) = 𝐵 2 𝑟 3 (𝑡) -𝐶 1 𝑝 3 (𝑡) + 𝐶 2 𝑐 3 (𝑡) + 𝜂 𝑝,2,3 𝑝 2 (𝑡)(𝑊 3 -𝑐 3 (𝑡) -𝑝 3 (𝑡) -𝑟 3 (𝑡))
(3.17) with initial conditions satisfying 𝑟 1,0 + 𝑐 1,0 + 𝑝 1,0 = 𝑉 1,0 ≤ 𝑊 1 ; 𝑟 2,0 + 𝑐 2,0 + 𝑝 2,0 = 𝑉 2,0 ≤ 𝑊 2 ; 𝑟 3,0 + 𝑐 3,0 + 𝑝 3,0 = 𝑉 3,0 ≤ 𝑊 3

(3.18)

In this system the bottleneck coupling, concerning, for example, the controlled population that is moving from node 1 to node 2 is given by the term:

𝜂 𝑐,1,2 𝑐 1 (𝑡)(𝑊 2 -𝑐 2 (𝑡) -𝑝 2 (𝑡) -𝑟 2 (𝑡)) (3.19)
in the second equation of (3.17)

𝑐1(𝑡) = 𝐵 1 𝑟 1 (𝑡) + 𝐶 1 𝑝 1 (𝑡) -𝐶 2 𝑐 1 (𝑡) -𝜂 𝑐,1,2 𝑐 1 (𝑡)(𝑊 2 -𝑐 2 (𝑡) -𝑝 2 (𝑡) -𝑟 2 (𝑡))
In this bottleneck coupling (3.19), 𝜂 𝑐,1,2 is the parameter that models the topological characteristic of the street linking node 1 to node 2. The second factor 𝑐 1 (𝑡) of Equation (3.19) reflects the pressure of controlled people in node 1 willing to escape towards node 2 and also the proportionality of people escaping with respect to people staying in node 1. Finally, the factor (𝑊 2 -𝑐 2 (𝑡) -𝑝 2 (𝑡) -𝑟 2 (𝑡)) shows the counterpressure which is maximum (i.e. the term vanishes) when 𝑊 2 = 𝑐 2 (𝑡) + 𝑝 2 (𝑡) + 𝑟 2 (𝑡) because, in this case, there is no more room for people coming from node 1. This bottleneck coupling is nonlinear, as shown on Figure 3.7.

FIGURE 3.7

Graph of the bottleneck coupling function.

As we consider only transitional dynamics where 𝑞 𝑘 (𝑡) = 𝑏 𝑘 (𝑡) = 0, Equation (3.19) can be written

𝜂 𝑐,1,2 𝑐 1 (𝑡)(𝑊 2 -𝑉 2 (𝑡)) (3.19)

Fixed points of the three-node system

The fixed-point research allows us to identify the point of equilibrium towards which the system tends during the transitional period. This equilibrium point highlights the primordial role of the size of both nodes 2 and 3 in the context of evacuation dynamics. It is important to note that the mathematically calculated equilibrium point does not necessarily correspond to the equilibrium situation sought by crisis management personnel. The population stranded in the initial place (node 1) remains very vulnerable to the terrorist threat.

To stay closer to reality, we further assume that people in reflex situation are stunned and paralyzed. They cannot rush from one node to another as indicated in Figure 3.8. Therefore, the transitions between 𝑟 1 , 𝑟 2 , 𝑟 3 are forbidden. People in this stunned state can only change their behavior (from 𝑟 𝑘 to 𝑝 𝑘 or k c ). achieve this goal Equation (3.17) is simply modified to vanish parameters 𝜂 𝑟,𝑖,𝑗 , 𝑖 = 1, 2, 𝑗 = 𝑖 + 1.

The fixed point (𝑟 1 * , 𝑐 1 * , 𝑝 1 * , 𝑟 2 * , 𝑐 2 * , 𝑝 2 * , 𝑟 3 * , 𝑐 3 * , 𝑝 3 * ) of the system toward which the solution of Equation (3.17) converges, is straightforwardly computed.

{ 𝑟 1 * = 0 𝑐 1 * = 0 𝑝 1 * = 0 𝑟 2 * = 0 𝑐 2 * = 0 𝑝 2 * = 0 𝑟 3 * = 0 𝑐 3 * = 𝐶 1 (𝑉 1,0 +𝑉 2,0 +𝑉 3,0 ) 𝐶 1 +𝐶 2 𝑝 3 * = 𝐶 2 (𝑉 1,0 +𝑉 2,0 +𝑉 3,0 ) 𝐶 1 +𝐶 2 if 𝑉 1,0 + 𝑉 2,0 + 𝑉 3,0 ≤ 𝑊 3 , (3.21
{ 𝑟 1 * = 0 𝑐 1 * = 0 𝑝 1 * = 0 𝑟 2 * = 0 𝑐 2 * = 𝐶 1 (𝑉 1,0 +𝑉 2,0 +𝑉 3,0 -𝑊 3 ) 𝐶 1 +𝐶 2 𝑝 2 * = 𝐶 2 (𝑉 1,0 +𝑉 +𝑉 3,0 -𝑊 3 ) 𝐶 1 +𝐶 2 𝑟 3 * = 0 𝑐 3 * = 𝐶 1 𝑊 3 𝐶 1 +𝐶 2 𝑝 3 * = 𝐶 2 𝑊 3 𝐶 1 +𝐶 2 if 𝑊 3 ≤ 𝑉 1,0 + 𝑉 2,0 + 𝑉 3,0 ≤ 𝑊 2 + 𝑊 3 , (3.22) { 𝑟 1 * = 0 𝑐 1 * = 𝐶 1 (𝑉 1,0 +𝑉 2,0 +𝑉 3,0 -𝑊 2 -𝑊 3 ) 𝐶 1 +𝐶 2 𝑝 1 * = 𝐶 2 (𝑉 1,0 +𝑉 2,0 +𝑉 3,0 -𝑊 2 -𝑊 3 ) 𝐶 1 +𝐶 2 𝑟 2 * = 0 𝑐 2 * = 𝐶 1 𝑊 2 𝐶 1 +𝐶 2 𝑝 2 * = 𝐶 2 𝑊 2 𝐶 1 +𝐶 2 𝑟 3 * = 0 𝑐 3 * = 𝐶 1 𝑊 3 𝐶 1 +𝐶 2 𝑝 3 * = 𝐶 2 𝑊 3 𝐶 1 +𝐶 2 if 𝑊 2 + 𝑊 3 ≤ 𝑉 1,0 + 𝑉 2,0 + 𝑉 3,0 , (3.23)
The values of this fixed point mean that:

Situation 1: if the number of people initially staying in the three nodes is less than the capacity of node 3 (i.e. 𝑉 1,0 + 𝑉 2,0 + 𝑉 3,0 ≤ 𝑊 3 ), after a while, both nodes 1 and 2 become empty and all the crowd has sought refuge in node 3.

Alternatively, if this number is greater than the capacity (i.e. 𝑉 1,0 + 𝑉 2,0 + 𝑉 3,0 > 𝑊 3 ), then two situations can occur: (when defined), instead of parameters 𝐵 1 and 𝐵 2 , becomes important for the pace at which the "reservoir" of people in reflex behavior is emptied.

Of course, across the world there are different cultures, which lead to different behaviors. These behaviors can be modelled varying parameters.

For example, if populations are not made aware of major risks and not prepared for them, it causes a panic reaction (B1 < B2), this behavior is then regulated by the ratio 𝐶 1 𝐶 2 . The higher this ratio, the more the population remains or transits in the controlled state.

In simulated situation, we choose a set of parameters highlight a weak risk culture, while favoring the return to a controlled behavior: the values, 𝐵 1 = 0.15, 𝐵 2 = 0.45, mean that there is a weak risk culture and, 𝐶 1 = 0.21, 𝐶 2 = 0.07, mean that the panic in the crowd context is compensated by controlled reactions for a part of the population who keep selfcontrol, notably because there is no new threat or sudden attack.

Therefore, in situation 1 when 𝑉 1,0 = 18 000, 𝑉 2,0 = 0, 𝑉 3,0 = 0, 𝑊 2 = 1000, 𝑊 3 = 18 500 and 𝜂 𝑐,1,2 = 𝜂 𝑐,2,3 = 𝜂 𝑝,1,2 = 𝜂 𝑝,2,3 = 0.004, one obtains the following convergence towards the fixed point: This value means that, initially, all the people are in node 1 and both nodes 2 and 3 are empty, but after a certain period of time, both nodes 1 and 2 are empty and everyone has reached node 3. One can see the flow of people through the two edges (𝑁 1 → 𝑁 2 ) ; (𝑁 2 → 𝑁 3 ), in Figure 3.9.

FIGURE 3.9

Situation 1: Convergence towards the fixed point (0, 0, 0, 0, 0, 0, 0, 13 500, 4 500). In this figure the change of behavioral states is symbolized by dotted arrows, and the motion between nodes by plain arrows is as displayed in Figure 3.8.

Influence of the spatial configuration on the pace of evacuation

We now seek to identify the obstacles that slow the escape of the crowd in the aftermath of the initial shock by analyzing the topology of the network of streets and places in a city. This information can potentially be used to improve the design of a city to facilitate the escape of a crowd towards more secure places.

Scaling the parameters

Fixing the value of all the parameters in PCR, WPCR or CWPCR is a very complicated task, which has not yet been done on an experimental basis. In the framework of the Com2SiCa project, an experimental protocol is under scrutiny, in order to achieve these results in the near future. However, it is important to note that only relative values between parameters are important, because there is a relationship between the unit of time and the unit used for the parameters. In other words, considering all the parameters of the adimensional Equation (3.17), (i.e. 𝐵 𝑖 ≥ 0, 𝐶 𝑖 ≥ 0, 𝜂 𝑐,𝑖,𝑗 , 𝜂 𝑝,𝑖,𝑗 , 𝜂 𝑟,𝑖,𝑗 , 𝑖 = 1, 2, 𝑗 = 𝑖 + 1) integrated with respect to the variable time 𝑡, it is nearly equivalent to consider such parameters multiplied by the same constant 𝜅 and integrated using the time variable 𝜏 = 𝑡 𝜅 (e.g. t can be considered in seconds, minutes, or hours). There is not, strictly speaking, equivalence because such parameters are linearly used in the WPCR model, however in the CWPCR model the coupling is nonlinear and a slight distortion intervenes during a transient short period for some variables.

Influence of the intermediate place capacity on the evacuation dynamics

Intermediate places play a central role in the fluidity or, to the contrary, the congestion of movement between a dangerous place and a place of shelter. This can be shown in the following numerical experiments: with the same parameter values, except for the size of node 2, we analyse the speed at which the people are emptying the place of the terrorist (node 1).

As previously described (Section 3.3.2), we consider the values of the parameters 𝐵 1 = 0.15, 𝐵 2 = 0.45, 𝐶 1 = 0.21, 𝐶 2 = 0.07 and we choose 𝜂 𝑐,1,2 = 𝜂 𝑐,2,3 = 𝜂 𝑝,1,2 = 𝜂 𝑝,2,3 = 0.004.

We consider the following values of 𝑊 2 : 50, 100, 200, 1000 for which we display the dynamics in Figure 3.10.

FIGURE 3.10

Situation 1: Convergence towards the fixed point (0, 0, 0, 0, 0, 0, 0, 13 500, 4 500) for the values of parameters 𝐵 1 = 0.15, 𝐵 2 = 0.45, 𝐶 1 = 0.21, 𝐶 2 = 0.07 , 𝜂 𝑐,1,2 = 𝜂 𝑐,2,3 = 𝜂 𝑝,1,2 = 𝜂 𝑝,2,3 = 0.004, 𝑉 1,0 = 18 000, 𝑉 2,0 = 0, 𝑉 3,0 = 0, 𝑊 3 = 18 500, and the values of 𝑊 2 = 50 (red curves), 𝑊 2 = 100 (green curves), 𝑊 2 = 200 (blue curves), 𝑊 2 = 1000 (black curves).

In the case 𝑊 2 = 1000 (black curves) the flight of the entire population from place 1 to place 2 (which can be a small square) and then to place 3 is very fast; it lasts less than 10 minutes (Figures 3.10c,f,i), because place 3 can accommodate the entire population. There is a massive influx of panicked people (Figure 3.10d), which is greater than the controlled one (Figure 3.10e), into place 2, which empties very quickly as the majority of controlled people reach the safe shelter (Figure 3.10h). However, it can be noted that a significant number of the panicked population remains in the refuge place, and this number is only slowly decreased (see bump in Figure 3.10g). This is explained by the fact that the flight dynamics are not hindered by obstacles or bottlenecks, and the fleeing populations have not enough time to change their behavioural state.

On the other hand, if 𝑊 2 = 50 (red curves), the evacuation of the total population from place 1 to place 2 and then place 3 is much slower (about 35 minutes instead of less than 10 minutes, (Figures 3.10c, f, i)). The panicked population has time to calm down in place 1, because there is no new attack (it has been assumed that there is no domino effect, (𝑠 𝑖 = 0, 𝑖 = 1, 2)). It is thus a population that is mainly in a state of reasoned behavior that arrives in place 3 (Figures 3.10g,h).

It can be highlighted from this first analysis that the faster the speed of change of location (hence decreasing the vulnerability of populations), the faster this speed leads to significant flows of panic in both places 2 and 3. There is a paradox here: the fast self-safety movement of populations leads to situations of collective panic that are more difficult to manage. This fact must be taken into account by emergency services and emergency physicians.

Although the deaths that occur are not included in this CWPCR model version, one can imagine that the escape of panicked populations would give rise to more victims.

As has already been said, the simulation results shed light on the importance of the size of the intermediate places and their role in the fluidity or, to the contrary, on the congestion of movements between a dangerous place and a place of shelter.

Depending on their respective size, intermediate places will modulate the dynamics and the speed of flow of the crowds. In this sense, they become strategic places both for the planners, who must think about the organization of the area to host public events, and also for the terrorists who can use these strategic places to multiply the effect of their harmful actions by trapping the escape movements between two areas of action.

Conclusion

In this article we have developed a new model of weighted human behavior coupled with street and place networks, in the context of an urban terrorist attack, thus improving the PCR model with bottleneck coupling and by taking into account the capacity of each place and the number of people stranded in these places. The simulation results in a simple network with three nodes (places or public squares) and two edges (streets) that demonstrates the key role of the capacity of an intermediate place in the dynamics of evacuation from dangerous to safe places.

  Points of the Three-Node System 13 3.4 Influence of the Spatial Configuration on the Pace of Evacuation 18 3.4.1 Scaling the Parameters 18 3.4.2 Influence of the Intermediate Place Capacity on the Evacuation Dynamics 18

FIGURE 3. 1

 1 FIGURE 3.1 Street and places in Mediterranean city of Italy (© D.Provitolo, March, 2019.)

FIGURE 3. 2

 2 FIGURE 3.2 Graphic representation of the Panic-Control-Reflex behavior model (PCR) in the exceptional situation of disaster. (Source: Provitolo et al. 2015).

Figure

  Figure 3.3.

{

  𝛾(𝑡) = 1 , ∀𝑡 ≥ 𝑡 𝑠 𝜑(𝑡) = 0 , ∀𝑡 < 𝑡 𝑒 (3.11)As an example, for 𝐼 𝑡𝑟𝑎𝑛𝑠 = [3.2,43.2], these functions can be defined by𝜑(𝑡) =

FIGURE 3. 4

 4 FIGURE 3.4Forcing functions 𝛾(𝑡), and 𝜑(𝑡), 𝐼 𝑡𝑟𝑎𝑛𝑠= [3.2,43.2]

(

  FIGURE 3.8Three node network: people in the reflex behavior state are stranded in their original node.

Acknowledgements

This work has been supported by the French government, through i) the National Research Agency (ANR) under the Societal Challenge 9 "Freedom and security of Europe, its citizens and residents" with the reference number ANR-17-CE39-0008, co-financed by French Defence Procurement Agency (DGA) and The General Secretariat for Defence and National Security (SGDSN), and ii) the UCA JEDI Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-15-IDEX-01.